[Secor]
>> That's something that I don't like about the Sims notation  down
>> arrows used in conjunction with sharps, and up arrows with flats.
[Keenan]
>I think Manuel exempts sharps and flats from this criticism.
Yes indeed, for example, Eb/ is always the nearest tone to 6/5
as E\ is always nearest to 5/4.
Manuel
This subject began on the main tuning list and is in reply to Dave
Keenan's message #35580 of 11 March 2002.
 In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> > Dave,
> >
> > This is just to let you know that I have taken some time off from
the
> > Tuning List to work on the sagittal notation per our last
> > conversation. I've eliminated a number of possibilities that
didn't
> > work to my satisfaction and am now much closer to a final
solution.
>
> Thanks for letting me know. I'm glad someone's still working on it.
> Johny Reinhard took the wind right out of my sails.
I followed that conversation and, although I have strong convictions
about what was discussed, I just didn't have the time to get involved
in it. My thoughts on this are:
1) Johnny is already very familiar with cents, so that is what works
for him. For the rest of us it would take a bit of training to be
able to do the same, and then might we need a calculator to determine
the intervals? When you are writing chords, where do all the cents
numbers go, and how can you read something like that with any
fluency? (But that is for instruments of fixed pitch, which do not
require cents, which brings us to the next point.)
2) Tablatures were mentioned in connection with instruments of fixed
pitch, where cents would be inappropriate. I hate tablatures with a
vengeance! Each instrument might have a different notation, and this
makes analysis of a score very difficult. We need a notation that
enables us to understand the pitches and intervals, regardless of
what sort of instrument is used.
3) Gene mentioned that we are most comfortable with whatever is the
most familiar, but a multiEDO/JI notation such as we are trying to
achieve is going to have something new to learn, no matter what.
It's best to make it as simple and logical as possible, and in my
opinion this is best accomplished by symbols that correspond simply
and directly to tones in each system, maintaining commonality across
those systems as much as possible.
Whether the maximum symbols per note should be one or two is
something that still needs to be resolved, but I think that we should
develop both approaches and see what results. Once that is done,
then we can evaluate each, pro and con. As long as microtonality is
such a niche market, I think that this is one extravagance that we
can afford.
>
> > It depends on the context. [whether 8:13 is more of a major or
minor 6th]
>
> True. So what is it in the most common contexts?
>
> > I hear 8:11 primarily as an augmented fourth (rather than a
perfect fourth),
>
> Me too, although I call it a super fourth, leaving "augmented" for
> 5:7.
>
> > and when the 8:11:13 is
> > sounded, the 13 will sound more like an A than an Aflat. (I
noticed
> > that Paul Erlich said something to this effect in his reply.)
>
> Yes.
>
> > Anyway, this is beside the point, in light of what I have to say
> > below.
>
> OK.
>
> > As it turns out, neither variation of the approach that I am now
> > taking pairs the 7 and 13 factors together, and the one that
looks
> > most promising at the moment requires all defining commas to be
no
> > more than half of an apotome,
>
> This seems like a sensible criterion to me too, perhaps for other
> reasons.
>
> > so it *must* use 1024:1053 instead of
> > 26:27.
>
> OK.
>
> > > If you're still planning to assume that certain very small
commas
> > > vanish (so certain combinations of the primecommas never
occur) then
> > > they had better be very small (like 1 cent or less) to keep the
JI folks happy.
> >
> > I was wondering whether you had a particular comma in mind when
you
> > said this, because a very useful one that I found is around 0.4
cents.
>
> No I didn't. So what are the smallest ETs in which this comma
> consistently fails to vanish?
The comma is 4095:4096 (~0.423 cents). (Has anyone previously found
it and given it a name?) Multiplying 81/80 by 64/63 gives 36/35
(~48.770 cents), which comes very close to 1053/1024 (~48.348 cents),
the ratio which is to define the 13 factor. This very small comma
vanishes in ET's 12, 17, 22, 24, 31, 34, 36, 39, 41, 43, 46, 53, 94,
96, 130, 140, 152, 171, 181, 183, 193, 207, 217, 224, 270, 311, 364,
388, 400, 494, 525, 581, 612, and 742. It does not vanish in 19, 27,
50, 58, 72, 149, 159, or 198, and this seems to be due to
inconsistencies in those ET's. For those systems under 100 in which
it does not vanish, I don't think that ratios of 13 will be used to
define their notation, so this should not be a problem.
>
> > > Except that as Manuel pointed out, it's considered bad form to
combine
> > > an up accidental with a down. In that case you'd need a binary
> > > sequence like 1, 2, 4, 8. At least we do have that with the
larger
> > > commas (and hence lower numbered ETs).
> >
> > That's something that I don't like about the Sims notation 
down
> > arrows used in conjunction with sharps, and up arrows with flats.
>
> I think Manuel exempts sharps and flats from this criticism.
>
> If you had a ruler with only inch marks, what could you find quicker
> (a) two and a half, less an eighth, or
> (b) two and three eights?
Or given these choices,
(a) one and a half, less an eighth, or
(b) one and three eighths, or
(c) three, less onefourth?
The particular intervals that were mentioned (farther above) in
connection with this issue are ratios of 11 and 13 that I like to
think of as semi and sesquisharps/flats. Making these the large
units of notation (according to the number of stems in the symbol)
rather than full sharps and flats makes the reading of these ratios
much simpler and solves the problem of having to decide whether to
notate a halfsharp/flat or a sharp/flat less a half. In effect, by
halving the units I have turned the halves into wholes, which makes
available choice (c) above, which, in addition to being at least as
simple as choice (a), does not combine an up with a down. (As to how
I make the distinction between ratios such as 11/9, 39/32, and 16/13,
you will have to wait a while to see; at present I have two different
methods.)
Anyway, when in doubt I think it will come down to trying it both
ways in order to see what looks better. I mentioned that my latest
approach has two variations, and if I can't determine a clear choice
of one over the other, then I will be presenting both so that we can
decide which is better.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> I followed that conversation and, although I have strong convictions
> about what was discussed, I just didn't have the time to get
involved
> in it. My thoughts on this are:
>
> 1) Johnny is already very familiar with cents, so that is what works
> for him. For the rest of us it would take a bit of training to be
> able to do the same, and then might we need a calculator to
determine
> the intervals? When you are writing chords, where do all the cents
> numbers go, and how can you read something like that with any
> fluency? (But that is for instruments of fixed pitch, which do not
> require cents, which brings us to the next point.)
>
> 2) Tablatures were mentioned in connection with instruments of fixed
> pitch, where cents would be inappropriate. I hate tablatures with a
> vengeance! Each instrument might have a different notation, and
this
> makes analysis of a score very difficult. We need a notation that
> enables us to understand the pitches and intervals, regardless of
> what sort of instrument is used.
These were proposed as notations for performers, not composers or
analysers. As such I see no great problem with the above, or
scordatura.
> 3) Gene mentioned that we are most comfortable with whatever is the
> most familiar, but a multiEDO/JI notation such as we are trying to
> achieve is going to have something new to learn, no matter what.
> It's best to make it as simple and logical as possible, and in my
> opinion this is best accomplished by symbols that correspond simply
> and directly to tones in each system, maintaining commonality across
> those systems as much as possible.
I see the commonality thing as the main justification for the kind of
notation you and I favour. Otherwise, in any given linear temperament
(and an ET is often treated as such) there is a native notation based
on the naturals being some MOS of around 5 to 12 notes, which makes
analysis easier for that temperament, but only that temperament.
> Whether the maximum symbols per note should be one or two is
> something that still needs to be resolved, but I think that we
should
> develop both approaches and see what results. Once that is done,
> then we can evaluate each, pro and con. As long as microtonality is
> such a niche market, I think that this is one extravagance that we
> can afford.
Sure.
> > No I didn't. So what are the smallest ETs in which this comma
> > consistently fails to vanish?
>
> The comma is 4095:4096 (~0.423 cents). (Has anyone previously found
> it and given it a name?)
I don't know of a name, but I expect it is listed in John Chalmer's
lists of superparticulars at a given primelimit. Anyone know a
URL? I was astonished to learn that such lists seem to be finite.
> Multiplying 81/80 by 64/63 gives 36/35
> (~48.770 cents), which comes very close to 1053/1024 (~48.348
cents),
> the ratio which is to define the 13 factor. This very small comma
> vanishes in ET's 12, 17, 22, 24, 31, 34, 36, 39, 41, 43, 46, 53, 94,
> 96, 130, 140, 152, 171, 181, 183, 193, 207, 217, 224, 270, 311, 364,
> 388, 400, 494, 525, 581, 612, and 742.
> It does not vanish in 19, 27,
> 50, 58, 72, 149, 159, or 198, and this seems to be due to
> inconsistencies in those ET's.
No. Every one of these (except I didn't check 198) is
{1,3,5,7,9,13}consistent.
> For those systems under 100 in which
> it does not vanish, I don't think that ratios of 13 will be used to
> define their notation, so this should not be a problem.
Well Gene and I are already using the 13comma (1024:1053) to notate
27tET, and it looks like it would be pretty useful in 50tET too.
There are many others under 100tET where 4095:4096 doesn't vanish.
37tET is another such, where I was planning to use the 13comma for
notation. 37tET is {1,3,5,7,13} consistent.
> > If you had a ruler with only inch marks, what could you find
quicker
> > (a) two and a half, less an eighth, or
> > (b) two and three eights?
So what's your answer? Mine is (a).
> Or given these choices,
> (a) one and a half, less an eighth, or
> (b) one and three eighths, or
> (c) three, less onefourth?
My answer is (c), but I don't get it. It isn't the same measurement as
the other two. In the analogy, I'm considering the inches to be the
naturals (since they are marked), the halves to be the sharps and
flats and the eighths (or fourths) to be some comma.
> The particular intervals that were mentioned (farther above) in
> connection with this issue are ratios of 11 and 13 that I like to
> think of as semi and sesquisharps/flats. Making these the large
> units of notation (according to the number of stems in the symbol)
> rather than full sharps and flats makes the reading of these ratios
> much simpler and solves the problem of having to decide whether to
> notate a halfsharp/flat or a sharp/flat less a half. In effect, by
> halving the units I have turned the halves into wholes, which makes
> available choice (c) above, which, in addition to being at least as
> simple as choice (a), does not combine an up with a down.
Now I get the analogy.
> (As to
how
> I make the distinction between ratios such as 11/9, 39/32, and
16/13,
> you will have to wait a while to see; at present I have two
different
> methods.)
>
> Anyway, when in doubt I think it will come down to trying it both
> ways in order to see what looks better. I mentioned that my latest
> approach has two variations, and if I can't determine a clear choice
> of one over the other, then I will be presenting both so that we can
> decide which is better.
Looking forward to it.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> I don't know of a name, but I expect it is listed in John Chalmer's
> lists of superparticulars at a given primelimit. Anyone know a
> URL? I was astonished to learn that such lists seem to be finite.
I proved that a while back using Baker's theorem on this list.
 In tuningmath@y..., "genewardsmith" <genewardsmith@j...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I don't know of a name, but I expect it is listed in John
Chalmer's
> > lists of superparticulars at a given primelimit. Anyone know a
> > URL? I was astonished to learn that such lists seem to be finite.
>
> I proved that a while back using Baker's theorem on this list.
Good work. Did you (or can you) prove that the smallest on each of John's
lists is the smallest there is?
Here's John Chalmers list of superparticulars
/tuningmath/message/1687
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> Good work. Did you (or can you) prove that the smallest on each of John's
> lists is the smallest there is?
Baker's theorem would give an effective bound, but not a good one.
Proving that the seemingly smallest one is in fact the smallest one
looks to me, as a number theorist, to be a difficult problem in number theory, though certainly not hard in the 3limit. Maybe I should try showing 81/80 is the smallest in the 5limit.
 In tuningmath@y..., "genewardsmith" <genewardsmith@j...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>
> > Good work. Did you (or can you) prove that the smallest on each of
John's
> > lists is the smallest there is?
>
> Baker's theorem would give an effective bound, but not a good one.
> Proving that the seemingly smallest one is in fact the smallest one
> looks to me, as a number theorist, to be a difficult problem in
number theory, though certainly not hard in the 3limit. Maybe I
should try showing 81/80 is the smallest in the 5limit.
>
If it looks hard, forget it. I'm sure you've got better things to do
with your time.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> If it looks hard, forget it. I'm sure you've got better things to do
> with your time.
If I *could* prove it, it might be enough for a paper.
 In tuningmath@y..., manuel.op.de.coul@e... wrote:
> [Secor]
> >> That's something that I don't like about the Sims notation 
down
> >> arrows used in conjunction with sharps, and up arrows with flats.
>
> [Keenan]
> >I think Manuel exempts sharps and flats from this criticism.
>
> Yes indeed, for example, Eb/ is always the nearest tone to 6/5
> as E\ is always nearest to 5/4.
>
> Manuel
My objection is to alterations used in conjunction with sharps and
flats that alter in the opposite direction of the sharp or flat by
something approaching half of a sharp or flat. For example, 3/7 or
4/7 is much clearer than 1 minus 4/7 or 1 minus 3/7, but I would not
object to 1 minus 2/7 (instead of 5/7).
I have been dealing with this issue in evaluating ways to notate
ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded that
the alterations for these should not be in the opposite direction
from an associated sharp or flat. In other words, relative to C, I
would prefer to see these as varieties of Esemiflat rather than E
flat with varieties of semisharps. But (for other intervals)
something no larger than 2 Didymus commas (~43 cents or ~3/8 apotome)
altering in the opposite direction would be okay with me.
George
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > The comma is 4095:4096 (~0.423 cents). ...
>
> > It does not vanish in 19, 27,
> > 50, 58, 72, 149, 159, or 198, and this seems to be due to
> > inconsistencies in those ET's.
>
> No. Every one of these (except I didn't check 198) is
> {1,3,5,7,9,13}consistent.
>
> > For those systems under 100 in which
> > it does not vanish, I don't think that ratios of 13 will be used
to
> > define their notation, so this should not be a problem.
>
> Well Gene and I are already using the 13comma (1024:1053) to
notate
> 27tET, and it looks like it would be pretty useful in 50tET too.
> There are many others under 100tET where 4095:4096 doesn't vanish.
> 37tET is another such, where I was planning to use the 13comma
for
> notation. 37tET is {1,3,5,7,13} consistent.
I spent some time wrestling with 27ET last night, and it proved to
be a formidable opponent that severely limited my options. There is
one approach that allows me to do it justice (using 13  what else
is there?) that also takes the following into account:
With this it looks as if I am going to be stopping at the 17 limit,
with intervals measurable in degrees of 183ET. Once I have made a
final decision regarding the symbols, I hope to have something to
show you in about a week or so.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> My objection is to alterations used in conjunction with sharps and
> flats that alter in the opposite direction of the sharp or flat by
> something approaching half of a sharp or flat. For example, 3/7 or
> 4/7 is much clearer than 1 minus 4/7 or 1 minus 3/7, but I would not
> object to 1 minus 2/7 (instead of 5/7).
Aha! I'm glad you've clarified that. I assume then that you would just
barely allow 1 minus 1/3, since that's closer to 2/7 than 3/7.
> I have been dealing with this issue in evaluating ways to notate
> ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded
that
> the alterations for these should not be in the opposite direction
> from an associated sharp or flat. In other words, relative to C, I
> would prefer to see these as varieties of Esemiflat rather than E
> flat with varieties of semisharps. But (for other intervals)
> something no larger than 2 Didymus commas (~43 cents or ~3/8
apotome)
> altering in the opposite direction would be okay with me.
I totally agree, when the 242:243 or 507:512 vanishes (as in many
ETs), but I don't see how it is possible when notating strict ratios.
If two of the above come out as E] and E}, then the other two must be
Eb{ and Eb[, where ] and } represent an increase, and { and [ a
decrease, by the 11 and 13 comma respectively. Also which ever have no
sharp or flat from F,C or G _will_ have a sharp or flat from A, E or B
and vice versa.
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> I spent some time wrestling with 27ET last night, and it proved to
> be a formidable opponent that severely limited my options. There is
> one approach that allows me to do it justice (using 13  what else
> is there?)
...
The only other option I could see was to notate it as every fourth
note of 108ET (= 9*12ET), using a trinary notation where the 5comma
is one step, the 7comma is 3 steps, and the apotome is 9 steps, but
that would be to deny that it has a (just barely) usable fifth of its
own.
> With this it looks as if I am going to be stopping at the 17 limit,
This might be made to work for ETs, but not JI. The 16:19:24 minor
triad has a following.
> with intervals measurable in degrees of 183ET.
I don't understand how this can work.
> Once I have made a
> final decision regarding the symbols, I hope to have something to
> show you in about a week or so.
I'm more interested in the sematics than the symbols at this stage. I
wouldn't spend too much time on the symbols yet. I expect serious
problems with the semantics.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > I have been dealing with this issue in evaluating ways to notate
> > ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded
that
> > the alterations for these should not be in the opposite direction
> > from an associated sharp or flat. In other words, relative to C,
I
> > would prefer to see these as varieties of Esemiflat rather than
E
> > flat with varieties of semisharps. But (for other intervals)
> > something no larger than 2 Didymus commas (~43 cents or ~3/8
> > apotome) altering in the opposite direction would be okay with me.
>
> I totally agree, when the 242:243 or 507:512 vanishes (as in many
> ETs), but I don't see how it is possible when notating strict
ratios.
> If two of the above come out as E] and E}, then the other two must
be
> Eb{ and Eb[, where ] and } represent an increase, and { and [ a
> decrease, by the 11 and 13 comma respectively. Also which ever have
no
> sharp or flat from F,C or G _will_ have a sharp or flat from A, E
or B
> and vice versa.
In the 17limit approach that I outline below, you will see how I
handle this. I am also outlining a 23limit approach; I went for the
19 limit and got 23 as a bonus when I found that I could approximate
it using a very small comma. The two approaches could be combined,
in which case you could have the 1113 semiflat varieties along with
the 19 or 23 limit, but the symbols may get a bit complicated  more
about that below.
> > I spent some time wrestling with 27ET last night, and it proved
to
> > be a formidable opponent that severely limited my options. There
is
> > one approach that allows me to do it justice (using 13  what
else
> > is there?) ...
>
> The only other option I could see was to notate it as every fourth
> note of 108ET (= 9*12ET), using a trinary notation where the 5
comma
> is one step, the 7comma is 3 steps, and the apotome is 9 steps,
but
> that would be to deny that it has a (just barely) usable fifth of
its
> own.
I thought more about this and now realize that the problem with 27ET
is not as formidable as it seemed. If we use the 80:81 comma for a
single degree and the 1024:1053 comma for two degrees of alteration,
we will do just fine, even if the *symbol* for 1024:1053 happens to
be a combination of the 80:81 and 63:64 symbols (by conflating
4095:4096). For the 27ET notation we can simply define the
combination of the two symbols as the 13 comma alteration, and there
would be no inconsistency in usage, since the 63:64 symbol is *never
used by itself* in the 27ET notation. The same could be said about
50ET. Are there any troublesome divisions above 100 that we should
be concerned about in this regard?
> > With this it looks as if I am going to be stopping at the 17
limit,
>
> This might be made to work for ETs, but not JI. The 16:19:24 minor
> triad has a following.
Yes, I can appreciate that, and there are other uses of ratios of 19
that I value. However, the 19 comma gets us down to less than 3.4
cents deviation (32/27 from 19/16), which is about the same as the
minimax deviation for the Miracle tuning, which is not bad. However,
I anticipate that you believe that the JI purists would still want to
have this distinction, so we should go for 19.
>
> > with intervals measurable in degrees of 183ET.
>
> I don't understand how this can work.
See below.
> > Once I have made a
> > final decision regarding the symbols, I hope to have something to
> > show you in about a week or so.
>
> I'm more interested in the sematics than the symbols at this stage.
I
> wouldn't spend too much time on the symbols yet. I expect serious
> problems with the semantics.
I don't know what problems you are anticipating, but let me outline
what I have, both in the way of semantics and symbols. As I said, I
have both a 17limit and 23limit approach, although I expect that
you will be interested in only the latter. Since there is not much
difference between the two, it is not any trouble to give both.
I have found that the semantics and symbols are so closely connected
that I could not address one without the other, given the limitation
of no more than one symbol in conjunction with a sharp or flat
(possible in my 23limit approach) or of no more than one symbol of
any sort (accomplished in my 17limit approach). For the sake of
simplicity, I will outline only how the modifications to natural
notes are accomplished by single symbols, leaving off the problem of
how these are to be combined with sharps and flats for another time.
Both the 17limit and 23limit approaches use 6 sizes of
alterations. In the sagittal notation these are paired into left and
right flags that are affixed to a vertical stem, to the top for
upward alteration and to the bottom for downward alteration. These
pairs of flags consist of straight lines, convex curved lines, and
concave curved lines. With this arrangement there is a limitation
that two left or two right flags cannot be used simultaneously.
*17LIMIT APPROACH*
The 17limit arrangement is:
Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (3 degrees
of 183)
Straight right flag (sR): 54:55, ~31.8 cents (5deg183)
Convex left flag (xL): 715:729 (3^6:5*11*13), ~33.6 cents (5deg183)
Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (4deg183)
Concave left flag (vL): 4131:4096 (3^5*17:2^12, the 17asflat
comma), ~14.7 cents (2deg183)
Concave right flag (vR): 2187:2176 (3^7:2^7*17, the 17assharp
comma), ~8.7 cents (1deg183)
I was a bit hesitant to use two different 17commas, but I saw that
16:17 could be used as either an augmented prime or a minor second
(e.g., in a scale with a tonic triad of 14:17:21). Rather than
choose between the two, I found that it is handy to have both
intervals, especially when going to the 19 limit.
With the above used in combination, the following useful intervals
are available:
sL+sR: 32:33 (the 11assemisharp comma), ~53.3 cents (8deg183)
sL+xR: 35:36, ~48.8 cents, which approximates
~sL+xR: 1024:1053 (the 13assemisharp comma), ~48.3 cents (7deg183)
sR+xL: 26:27 (the 13assemiflat comma), ~65.3 cents (10deg183)
xL+xR: 5005:5184 (5*7*11*13:2^6*3^4), ~60.8 cents, which approximates
~xL+xR: 704:729 (the 11assemiflat comma), ~60.4 cents (9deg183)
vL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates
~vL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents
(6deg183)
That is how I make the distinction between all the different ratios
of 11 and 13. Note that ratios of 11 have like left and right flags,
while ratios of 13 have dissimilar flags, making them easy to tell
apart. With the singlesymbol sagittal notation it is necessary to
have alterations exceeding half an apotome, since there is no way to
notate a (twoflag) sagittal sharp/flat *less* a (twoflag)
~semiflat/~semisharp alteration.
The following combinations are probably not as useful, but are
available anyway:
xL+vR: 1555840:1594323 (2^7*5*11*13*17:3^3*13), ~42.3 cents, which
also approximates
~xL+vR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents
(6deg183)
vL+vR: 524277:531441 (2^19:3^12, the Pythagorean comma), ~23.5 cents
(3deg183)
There are also two other combinations that I didn't see any use for:
vL+sR, ~46.5 cents (7deg183)
sL+vR, ~30.2 cents (4deg183)
Why 183? Since I'm going only to the 17 limit here, it's one in
which the 19 comma (512:513) vanishes, and it represents the building
blocks of the notation rather nicely as approximate multiples of 7
cents. The apotome in 183 is 17 degrees.
*23LIMIT APPROACH*
And here is the 23limit arrangement, which correlates well with 217
ET (apotome of 21 degrees):
Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (4 degrees
of 217)
Straight right flag (sR): 54:55, ~31.8 cents (6deg217)
Convex left flag (vL): 4131:4096 (3^5*17:2^12, the 17asflat comma),
~14.7 cents (3deg217)
Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (5deg217)
Concave left flag (vL): 2187:2176 (3^7:2^7*17, the 17assharp
comma), ~8.7 cents (2deg217)
Concave right flag (vR): 512:513 (the 19asflat comma), ~3.4 cents
(1deg217)
The difference between this and the 17limit approach is that I have
removed the 715:729 alteration and added the 512:513 alteration,
while reassigning the 17commas to different flags. No combination
of flags will now exceed half of an apotome.
With the above used in combination, the following useful intervals
are available:
sL+sR: 32:33 (the 11assemisharp comma), ~53.3 cents (10deg217)
sL+xR: 35:36, ~48.8 cents, which approximates
~sL+xR: 1024:1053 (the 13assemisharp comma), ~48.3 cents (9deg217)
vL+sR: 4352:4455 (2^8*17:3^4*5*11), ~40.5 cents, which approximates
~vL+sR: 16384:16767 (2^14:3^6*23, the 23 comma), ~40.0 cents
(8deg217)
xL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates
~xL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents
(8deg217)
The vL+sR approximation of the 23 comma deviates by 3519:3520 (~0.492
cents).
All of the above provide a continuous range of intervals in 217ET,
which I selected because it is consistent to the 21limit and
represents the building blocks of the notation as approximate
multiples of 5.5 cents.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > >
> > > I have been dealing with this issue in evaluating ways to notate
> > > ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded
> that
> > > the alterations for these should not be in the opposite
direction
> > > from an associated sharp or flat. In other words, relative to
C,
> I
> > > would prefer to see these as varieties of Esemiflat rather than
> E
> > > flat with varieties of semisharps. But (for other intervals)
> > > something no larger than 2 Didymus commas (~43 cents or ~3/8
> > > apotome) altering in the opposite direction would be okay with
me.
> >
> > I totally agree, when the 242:243 or 507:512 vanishes (as in many
> > ETs), but I don't see how it is possible when notating strict
> ratios.
> > If two of the above come out as E] and E}, then the other two must
> be
> > Eb{ and Eb[, where ] and } represent an increase, and { and [ a
> > decrease, by the 11 and 13 comma respectively. Also which ever
have
> no
> > sharp or flat from F,C or G _will_ have a sharp or flat from A, E
> or B
> > and vice versa.
>
> In the 17limit approach that I outline below, you will see how I
> handle this.
OK. So you have gone outside of onecommaperprime and
one(sub)symbolperprime. But you have given fair reasons for doing
so in the case of 11 and 13.
> I am also outlining a 23limit approach; I went for
the
> 19 limit and got 23 as a bonus when I found that I could approximate
> it using a very small comma. The two approaches could be combined,
> in which case you could have the 1113 semiflat varieties along with
> the 19 or 23 limit, but the symbols may get a bit complicated 
more
> about that below.
You don't actually give more below about combining these approaches.
But I had fun working it out for myself. I'll give my solution later.
> I thought more about this and now realize that the problem with
27ET
> is not as formidable as it seemed. If we use the 80:81 comma for a
> single degree and the 1024:1053 comma for two degrees of alteration,
> we will do just fine, even if the *symbol* for 1024:1053 happens to
> be a combination of the 80:81 and 63:64 symbols (by conflating
> 4095:4096). For the 27ET notation we can simply define the
> combination of the two symbols as the 13 comma alteration, and there
> would be no inconsistency in usage, since the 63:64 symbol is *never
> used by itself* in the 27ET notation. The same could be said about
> 50ET.
You're absolutely right.
> Are there any troublesome divisions above 100 that we should
> be concerned about in this regard?
Not that I can find on a cursory examination.
> I anticipate that you believe that the JI purists would still want
to
> have this distinction, so we should go for 19.
Correct. I'll skip the 183ET based one.
> I
> > wouldn't spend too much time on the symbols yet. I expect serious
> > problems with the semantics.
>
> I don't know what problems you are anticipating, ...
Well none have materialised yet. :)
> I have found that the semantics and symbols are so closely connected
> that I could not address one without the other,
Yes. I see that now.
> Both the 17limit and 23limit approaches use 6 sizes of
> alterations. In the sagittal notation these are paired into left
and
> right flags that are affixed to a vertical stem, to the top for
> upward alteration and to the bottom for downward alteration. These
> pairs of flags consist of straight lines, convex curved lines, and
> concave curved lines. With this arrangement there is a limitation
> that two left or two right flags cannot be used simultaneously.
Given these constraints I think your solution is brilliant.
> *23LIMIT APPROACH*
>
> And here is the 23limit arrangement, which correlates well with
217
> ET (apotome of 21 degrees):
I don't think you can call this a 23limit notation, since 217ET is
not 23limit consistent. But it is certainly 19primelimit.
> Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (4 degrees
> of 217)
> Straight right flag (sR): 54:55, ~31.8 cents (6deg217)
> Convex left flag (vL): 4131:4096 (3^5*17:2^12, the 17asflat
comma),
> ~14.7 cents (3deg217)
> Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (5deg217)
> Concave left flag (vL): 2187:2176 (3^7:2^7*17, the 17assharp
> comma), ~8.7 cents (2deg217)
> Concave right flag (vR): 512:513 (the 19asflat comma), ~3.4 cents
> (1deg217)
>
> The difference between this and the 17limit approach is that I have
> removed the 715:729 alteration and added the 512:513 alteration,
> while reassigning the 17commas to different flags. No combination
> of flags will now exceed half of an apotome.
>
> With the above used in combination, the following useful intervals
> are available:
>
> sL+sR: 32:33 (the 11assemisharp comma), ~53.3 cents (10deg217)
> sL+xR: 35:36, ~48.8 cents, which approximates
> ~sL+xR: 1024:1053 (the 13assemisharp comma), ~48.3 cents
(9deg217)
> vL+sR: 4352:4455 (2^8*17:3^4*5*11), ~40.5 cents, which approximates
> ~vL+sR: 16384:16767 (2^14:3^6*23, the 23 comma), ~40.0 cents
> (8deg217)
> xL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates
> ~xL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0 cents
> (8deg217)
>
> The vL+sR approximation of the 23 comma deviates by 3519:3520
(~0.492
> cents).
>
> All of the above provide a continuous range of intervals in 217ET,
> which I selected because it is consistent to the 21limit and
> represents the building blocks of the notation as approximate
> multiples of 5.5 cents.
Once I understood your constraints, I spent hours looking at the
problem. I see that you can push it as far as 29limit in 282ET if
you want both sets of 11 and 13 commas, and 31limit in 311ET if you
can live with only the smaller 11 and 13 commas. But to make these
work you have to violate what is probably an implicit constraint, that
the 5 and 7 commas must correspond to single flags. Neither of them
can map to a single flag in either 282ET or 311ET and so the mapping
of commas to arrows is just way too obscure.
217ET is definitely the highest ET you can use with the above
additional constraint.
I notice that leftright confusability has gone out the window. But
maybe that's ok, if we accept that this is not a notation for
sightreading by performers. However, it is possible to improve the
situation by making the leftright confusable pairs of symbols either
map to the same number of steps of 217ET or only differ by one step,
so a mistake will not be so disastrous. At the same time as we do this
we can reinstate your larger 11 and 13 commas, so you have both sizes
of these available. The 13 commas will have similar flags on left and
right, while the 11 commas will have dissimilar flags. It seems better
that the 11 commas should be confused with each other than the 13
commas, since the 11 commas are closer together in size.
To do this you make the following changes to your 217ETbased scheme.
1. Swap the flags for the 17 comma and 19 comma (vL and vR)
2. Swap the flags for the 7 comma and 11assemisharp/5 comma. (xR and
sR)
3. Make the xL flag the 11assemiflat/7 comma instead of the
17asflat comma.
So we have
Steps Flags Commas

1 vL 19comma 512:513
2 vR 17comma 2187:2176
3 vL+vR 17asflatcomma 4131:4096
4 sL 5comma 80:81
5 sR 7comma 63:64
6 xL or xR
7 vL+sR or xL+vR
8 vL+xR
9 sL+sR 13assemisharp comma 1024:1053
10 sL+xR 11assemisharp comma 32:33
11 xL+sR 11assemiflat comma 704:729
12 xL+xR 13assemiflat comma 26:27
21 apotome
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> So we have
> Steps Flags Commas
> 
> 1 vL 19comma 512:513
> 2 vR 17comma 2187:2176
Oops! That should have been:
Steps Flags Commas

1 vR 19comma 512:513
2 vL 17comma 2187:2176
3 vL+vR 17asflatcomma 4131:4096
4 sL 5comma 80:81
5 sR 7comma 63:64
6 xL or xR
7 vL+sR or xL+vR
8 vL+xR
9 sL+sR 13assemisharp comma 1024:1053
10 sL+xR 11assemisharp comma 32:33
11 xL+sR 11assemiflat comma 704:729
12 xL+xR 13assemiflat comma 26:27
> 21 apotome
If we have more than one comma per prime, we lose the very desireable
property of uniqueness, and automatic translation becomes harder.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > > >
>
> OK. So you have gone outside of onecommaperprime and
> one(sub)symbolperprime. But you have given fair reasons for
doing
> so in the case of 11 and 13.
Gene had a comment about this, however:
 In tuningmath@y..., "genewardsmith" <genewardsmith@j...> wrote:
> If we have more than one comma per prime, we lose the very
desireable
> property of uniqueness, and automatic translation becomes harder.
I assume that JItoET, ETtoJI, or ETtoET translation by computer
software is what is being referred to here. I think that the process
has a similarity to conversion of computer image files: In general,
conversion of a highresolution image to one of lower resolution will
be more successful than the reverse; likewise, translation of music
from JI or a largenumber ET (finer resolution) to a lowernumber ET
(coarser resolution) will be more successful, or at least more
straightforward. (Of course, there will exceptions to this if the
music is simple enough, but I am stating a general principle.) The
only direction where the lack of primal uniqueness is likely to pose
a problem is in going from coarser to finer resolution, i.e., the
direction which we would not expect to be very successful in the
first place.
Something we'll have to keep in mind is how much primal uniqueness
should be traded off against human comprehension of the symbols. I
think that the deciding factor should be in favor of the human, not
the machine  software can be written to handle all sorts of
complicated situations; but where dual commas have been introduced,
it was for the purpose of *clarifying* the melodic function of the
intervals involved to the human reader of the notation. Whoever
might get involved in writing the algorithm for any sort of
translation would need to be aware of these things and would have to
consider providing appropriate menu entries that would govern the
translation.
For what it's worth, each comma (or more accurately, diesis) in the
pairs of 11and13defining intervals is the apotome's complement of
the other, and it should be a simple enough matter to follow the
principle of using the one that doesn't alter in the opposite
direction in combination with a sharp or flat. Likewise, the two 17
commas are the Pythagorean comma's complement of each other, and the
proper one can be clearly determined by whether it is associated with
a sharp or flat (or natural). And only 2176:2187 is used in
combination with other defining commas in the notation, so that
restricts the situations in which the two 17 commas pose a problem in
translation.
> > I am also outlining a 23limit approach; I went for the
> > 19 limit and got 23 as a bonus when I found that I could
approximate
> > it using a very small comma. The two approaches could be
combined,
> > in which case you could have the 1113 semiflat varieties along
with
> > the 19 or 23 limit, but the symbols may get a bit complicated 
more
> > about that below.
>
> You don't actually give more below about combining these
approaches.
I meant more about the symbols, not about how to combine the
approaches, which I didn't get to until the weekend.
> But I had fun working it out for myself. I'll give my solution
later.
>
> > I thought more about this and now realize that the problem with
27ET
> > is not as formidable as it seemed. ... The same could be said
about
> > 50ET.
>
> You're absolutely right.
I'm glad you agree. It looks like we've gotten somewhere with the
ET's.
> > Are there any troublesome divisions above 100 that we should
> > be concerned about in this regard?
>
> Not that I can find on a cursory examination.
I had never given much thought to notating divisions above 100, but I
would like to see how well the JI notation will work with these.
Which ones between 94 and 217 would you consider the most important
to be covered by this notation (listed in order of importance)?
> > I anticipate that you believe that the JI purists would still
want to
> > have this distinction, so we should go for 19.
>
> Correct. I'll skip the 183ET based one.
Agreed!
> > > I wouldn't spend too much time on the symbols yet. I expect
serious
> > > problems with the semantics.
> >
> > I don't know what problems you are anticipating, ...
>
> Well none have materialised yet. :)
I imagined that it would have been the gap between the sum of the 19
and 17 commas (~12.1 cents) and the 5 comma (~21.5 cents) in a binary
sequence, but even that is no problem in 181, 193, and 217 (to give a
few examples). And if 217 seems suitable, then we should stick with
it. (Over the weekend I happened to notice that it's 7 times 31 
in effect a division built on meantone quartercommas!)
> > *23LIMIT APPROACH*
> >
> > And here is the 23limit arrangement, which correlates well with
> > 217ET (apotome of 21 degrees):
>
> I don't think you can call this a 23limit notation, since 217ET
is
> not 23limit consistent. But it is certainly 19primelimit.
I thought that, in the event somebody *absolutely must* have 23, one
could allow a little bit of slack if the following were taken into
account:
 In tuning@y..., "gdsecor" <gdsecor@y...> wrote [main tuning list,
#33699]:
> But for a small fraction of a cent 46EDO misses 17limit
consistency
> for a single pair of intervals (15/13 & 26/15)! I don't think this
> precludes using it for 17limit harmony (no EDO of lower number can
> compete with it), and I know from experience that 15limit
harmonies
> can be successfully employed in 31EDO without any disorientation
> whatsoever (with 13 being implied, much more successfully, in my
> opinion, than 7 is in 12EDO), even if you have a couple of pairs
of
> intervals that go over the boundary of consistency by a couple of
> cents or so. (The same can be said for 19/13 and 26/19 in 72
EDO.)
> I would compare this to briefly driving a car very slightly onto
the
> shoulder, but not far enough off the road to lose control.
After all, 23 is inconsistent only in combination with two other odd
numbers in the 27 limit. But, as you say, we do have the 19prime
limit consistency, and, in addition, 21, 25, and 27 are all
consistent with that.
> > Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (4
degrees of 217)
> > Straight right flag (sR): 54:55, ~31.8 cents (6deg217)
> > Convex left flag (vL): 4131:4096 (3^5*17:2^12, the 17asflat
comma), ~14.7 cents deg217)
> > Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (5deg217)
> > Concave left flag (vL): 2187:2176 (3^7:2^7*17, the 17assharp
comma), ~8.7 cents deg217)
> > Concave right flag (vR): 512:513 (the 19asflat comma), ~3.4
cents (1deg217)
> >
> > With the above used in combination, the following useful
intervals
> > are available:
> >
> > sL+sR: 32:33 (the 11assemisharp comma), ~53.3 cents (10deg217)
> > sL+xR: 35:36, ~48.8 cents, which approximates
> > ~sL+xR: 1024:1053 (the 13assemisharp comma), ~48.3 cents
(9deg217)
> > vL+sR: 4352:4455 (2^8*17:3^4*5*11), ~40.5 cents, which
approximates
> > ~vL+sR: 16384:16767 (2^14:3^6*23, the 23 comma), ~40.0 cents
(8deg217)
> > xL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates
> > ~xL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0
cents (8deg217)
> >
> > The vL+sR approximation of the 23 comma deviates by 3519:3520
(~0.492 cents).
> >
> > All of the above provide a continuous range of intervals in 217
ET,
> > which I selected because it is consistent to the 21limit and
> > represents the building blocks of the notation as approximate
> > multiples of 5.5 cents.
>
> Once I understood your constraints, I spent hours looking at the
> problem. I see that you can push it as far as 29limit in 282ET if
> you want both sets of 11 and 13 commas, and 31limit in 311ET if
you
> can live with only the smaller 11 and 13 commas. But to make these
> work you have to violate what is probably an implicit constraint,
that
> the 5 and 7 commas must correspond to single flags.
That's correct (referring to the constraint).
> Neither of them
> can map to a single flag in either 282ET or 311ET and so the
mapping
> of commas to arrows is just way too obscure.
I don't know exactly what you mean by this. As single flags these
would just have to indicate more degrees, and in order to fill in the
gaps I would have to come up with a 4th kind of flag and another
comma to go with it (but to what purpose?). In any case, your
conclusion stands:
> 217ET is definitely the highest ET you can use with the above
> additional constraint.
>
> I notice that leftright confusability has gone out the window.
Not entirely (more comments to follow).
> But maybe that's ok, if we accept that this is not a notation for
> sightreading by performers.
I do want this to be a notation for sightreading. In order to get
all of the combinations required for the JI notation in single
symbols, you have to allow some lateral mirroring, but I don't think
it's going to occur often enough to cause a lot of trouble. There
will be a few instances that may be a bit tricky at first, but once
you learn what to watch out for (just as I learned to get the g's and
t's right in the word sagittal), it should be easy enough to deal
with.
I spent some time this past weekend figuring out how all of this was
going to translate into various ET's under 100, and every division I
tried could be notated without any lateral mirroring whatsoever.
(Even 58ET, which had given me problems before, now looks very good.)
> However, it is possible to improve the
> situation by making the leftright confusable pairs of symbols
either
> map to the same number of steps of 217ET or only differ by one
step,
> so a mistake will not be so disastrous. At the same time as we do
this
> we can reinstate your larger 11 and 13 commas, so you have both
sizes
> of these available. The 13 commas will have similar flags on left
and
> right, while the 11 commas will have dissimilar flags. It seems
better
> that the 11 commas should be confused with each other than the 13
> commas, since the 11 commas are closer together in size.
Good point! I'm glad to see that you have gotten involved in trying
to combine my two approaches. I also spent some time over the
weekend on this, and I notice that we did the 17 and 19 commas the
same way.
> To do this you make the following changes to your 217ETbased
scheme.
> 1. Swap the flags for the 17 comma and 19 comma (vL and vR)
In your subsequent message with the correction, you changed this back
to what I originally had. I chose that arrangement because I found
that the flags were more useful in combination that way (specifically
2176:2187). In my list of useful combinations I inadvertently
omitted my primary choice for 7deg217 as 2176:2187 plus 63:64, but
you figured it out (along with the alternate way to get 7 degrees).
> 2. Swap the flags for the 7 comma and 11assemisharp/5 comma. (xR
and sR)
This is the one (and only) thing that is different in our efforts at
achieving a combination of my two approaches and is something that I
hadn't considered. It has the effect of making both convex flags
larger than both straight flags. It also has other consequences,
which I will address below.
> 3. Make the xL flag the 11assemiflat/7 comma instead of the 17as
flat comma.
This restores a flag that was in my 17limit approach, and I also did
this in the combination of my two previous approaches.
> So we have [according to the second (corrected) posting]
> Steps Flags Commas
> 
> 1 vR 19comma 512:513
> 2 vL 17comma 2187:2176
> 3 vL+vR 17asflatcomma 4131:4096
> 4 sL 5comma 80:81
> 5 sR 7comma 63:64
> 6 xL or xR
> 7 vL+sR or xL+vR
> 8 vL+xR
> 9 sL+sR 13assemisharp comma 1024:1053
> 10 sL+xR 11assemisharp comma 32:33
> 11 xL+sR 11assemiflat comma 704:729
> 12 xL+xR 13assemiflat comma 26:27
> 21 apotome
I have given this quite a bit of study since yesterday, not only for
JI, but also to see how this translates to various ET notations, many
of which are affected by the exchange of the xRsR symbols. This
proposal changes things that I have become familiar with over the
past couple of months, but that is irrelevant inasmuch as our
objective is to make this notation as good as we possibly can. It is
best to do it right the first time so that we don't have to change it
later.
It didn't take me very long to reach a definite conclusion. I recall
that it was the issue of lateral confusibility that first led to the
adoption of a curved righthand flag for the 7comma alteration in
the 72ET notation. Before that all of the flags were straight.
Making the xRsR symbol exchange would once again give the 7comma
alteration a straight flag, which would negate the original reason
for the curved flag. The 72ET notation could still use curved right
flags, but they would no longer symbolize the 7comma alteration, but
the 54:55 alteration instead, which tends to obscure rather than
clarify the harmonic relationships. Also, since the JI notation
would use straight flags for both the 5 and 7comma alterations, then
lateral confusibility would make it more difficult to distinguish
between two of the most important prime factors, and we would be
giving this up without receiving anything of comparable benefit in
return.
Notating the 7comma with the xR (curved) flag, on the other hand,
makes a clear distinction between ratios of 5 and 7 in JI, 72ET, and
anywhere else that 80:81 and 63:64 are a different number of
degrees. It also minimizes the use of curved flags in the ET
notations, introducing them only as it is necessary or helpful: 1) to
avoid lateral confusibility (in 72ET); 2) to distinguish 32:33 from
1024:1053 (in 46 and 53ET, *without* lateral confusibility!); and 3)
to notate increments smaller than 80:81 (in 94ET). Lateral
confusibility enters the picture only when one goes above the 11
limit: In one instance one must learn to distinguish between
1024:1053 and 26:27 by observing which way the straight flag points
(leftward for the smaller ratio and rightward for the larger).
Another instance does not come up until the 19 limit, which involves
distinguishing the 17assharp flag from the 19 flag.
So I think we have enough reasons to stick with the convex curved
flag for the 7 comma. (I will also give one more reason below.)
By the way, something else I figured out over the weekend is how to
notate 13 through 20 degrees of 217 with single symbols, i.e., how to
subtract the 1 through 8degree symbols from the sagittal apotome
(/\). The symbol subtraction for notation of apotome complements
works like this:
For a symbol consisting of:
1) a left flag (or blank)
2) a single (or triple) stem, and
3) a right flag (or blank):
4) convert the single stem to a double (or triple to an X);
5) replace the left and right flags with their opposites according to
the following:
a) a straight flag is the opposite of a blank (and vice versa);
b) a convex flag is the opposite of a concave flag (and vice versa).
This produces a reasonable and orderly progression of symbols
(assuming that 63:64 is a curved convex flag; it does not work as
well with 63:64 as a straight flag) that is consistent with the
manner in which I previously employed the original sagittal symbols
for various ET's.
I will prepare a diagram illustrating the progression of symbols for
JI and for various ET's so we can see how all of this is going to
look.
Stay tuned!
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> Something we'll have to keep in mind is how much primal uniqueness
> should be traded off against human comprehension of the symbols. I
> think that the deciding factor should be in favor of the human, not
> the machine  software can be written to handle all sorts of
> complicated situations;
I agree. But it is also possible to disambiguate dual purpose flags by
say adding a blob to the end of the stroke for one use and not the
other.
> I had never given much thought to notating divisions above 100, but
I
> would like to see how well the JI notation will work with these.
> Which ones between 94 and 217 would you consider the most important
> to be covered by this notation (listed in order of importance)?
I don't know order of importance. 96, 105, 108, 111, 113, 121, 130,
144, 149, 152, 159, 166, 171, 183, 190, 198, 212.
> And if 217 seems suitable, then we should stick with
> it. (Over the weekend I happened to notice that it's 7 times 31 
> in effect a division built on meantone quartercommas!)
Yes. I noticed that too. But I'm not sure it matters much, since 31ET
will of course _not_ be notated the same as every seventh note of
217ET.
> I thought that, in the event somebody *absolutely must* have 23, one
> could allow a little bit of slack if the following were taken into
> account:
...
> and, in addition, 21, 25, and 27 are all
> consistent with that.
You're right. I'd even like to see if we can push it to 31limit,
consistent with 311ET, since we're so close, but this would use
additional flags (and/or additional schismas like 4095:4096 and
3519:3520) and not affect the existing 23limit, 217ET
correspondence.
You know I went thru the prime factorisation of all the
superparticulars in John Chalmer's list, and you've found the only two
that are useful for this purpose.
But I was wondering if we can somehow use 1539:1540 which says that
the 19flag is the difference between the 11/5 flag and the 7flag.
Probably not, since it involves a subtraction and this schisma is a
whole cent.
John's list only goes up to 23limit. I'd like to see a list of
all the 31limit superparticulars to be sure we're not missing
something. Gene? But then there's no guarantee that a useful schisma
like that will be superparticular.
> > the 5 and 7 commas must correspond to single flags.
>
> That's correct (referring to the constraint).
>
> > Neither of them
> > can map to a single flag in either 282ET or 311ET and so the
> mapping
> > of commas to arrows is just way too obscure.
>
> I don't know exactly what you mean by this. As single flags these
> would just have to indicate more degrees, and in order to fill in
the
> gaps I would have to come up with a 4th kind of flag and another
> comma to go with it (but to what purpose?). In any case, your
> conclusion stands:
What I meant was that you can actually cover all the values from 0
steps to the number of steps in the 13assemiflat comma in 311ET
using only 6 flags, but if you do that, the 5 and 7 commas cannot be
singleflag. As you say, we can always add more flags to fill in the
gaps.
> I spent some time this past weekend figuring out how all of this was
> going to translate into various ET's under 100, and every division I
> tried could be notated without any lateral mirroring whatsoever.
> (Even 58ET, which had given me problems before, now looks very
good.)
That's great.
> > 2. Swap the flags for the 7 comma and 11assemisharp/5 comma. (xR
> and sR)
>
> This is the one (and only) thing that is different in our efforts at
> achieving a combination of my two approaches and is something that I
> hadn't considered. It has the effect of making both convex flags
> larger than both straight flags.
Yes, this was the other thing that recommended it to me.
> It didn't take me very long to reach a definite conclusion. I
recall
> that it was the issue of lateral confusibility that first led to the
> adoption of a curved righthand flag for the 7comma alteration in
> the 72ET notation. Before that all of the flags were straight.
> Making the xRsR symbol exchange would once again give the 7comma
> alteration a straight flag, which would negate the original reason
> for the curved flag.
Yes. I was considering putting a blob on the end of the straight 7
flag, but no. I agree with you now. Keep the curved flag for the
7comma. It is most important to get the 11limit right. The rest is
just icing on the cake, and a little lateral confusability there can
be tolerated.
> By the way, something else I figured out over the weekend is how to
> notate 13 through 20 degrees of 217 with single symbols, i.e., how
to
> subtract the 1 through 8degree symbols from the sagittal apotome
> (/\). The symbol subtraction for notation of apotome complements
> works like this:
>
> For a symbol consisting of:
> 1) a left flag (or blank)
> 2) a single (or triple) stem, and
> 3) a right flag (or blank):
> 4) convert the single stem to a double (or triple to an X);
> 5) replace the left and right flags with their opposites according
to
> the following:
> a) a straight flag is the opposite of a blank (and vice versa);
> b) a convex flag is the opposite of a concave flag (and vice
versa).
You gotta admit this isn't exactly intuitive (particularly 5a). I'm
more interested in the singlestem saggitals used with the standard
sharpflat symbols, but it's nice that you can do that.
> I will prepare a diagram illustrating the progression of symbols for
> JI and for various ET's so we can see how all of this is going to
> look.
>
> Stay tuned!
Sure. This is fun.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> I don't know order of importance. 96, 105, 108, 111, 113, 121, 130,
> 144, 149, 152, 159, 166, 171, 183, 190, 198, 212.
This list gives me indigestionwhat happened to 99, 118 and 140, for
starters?
> John's list only goes up to 23limit. I'd like to see a list of
> all the 31limit superparticulars to be sure we're not missing
> something. Gene? But then there's no guarantee that a useful
schisma
> like that will be superparticular.
I don't have such a lit, but it would make more sense to look for
such up to a size limit, I think.
i In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > I thought that, in the event somebody *absolutely must* have 23,
one
> > could allow a little bit of slack if the following were taken
into
> > account:
> ...
> > and, in addition, 21, 25, and 27 are all
> > consistent with that.
>
> You're right. I'd even like to see if we can push it to 31limit,
> consistent with 311ET, since we're so close, but this would use
> additional flags (and/or additional schismas like 4095:4096 and
> 3519:3520) and not affect the existing 23limit, 217ET
> correspondence.
Considering that the semantics of the notation have already put us
past the 19 limit, that 217 is not 23limit consistent, and that 311
is such an excellent division, I'd say let's go for it!
> You know I went thru the prime factorisation of all the
> superparticulars in John Chalmer's list, and you've found the only
two
> that are useful for this purpose.
I guess I just have a knack for finding useful commas (even before I
start looking for them). Are you ready for the next one? It's a
honey: 20735:20736 (5*11*13*29:2^8*3^4, ~0.083 cents). And it turns
out that we don't need any new flags to get the 29 factor: its
defining interval is 256:261 (2^8:3^2*29, ~33.487 cents), and the
convex left flag that we already have (715:729) is ~33.571 cents.
> ... I was considering putting a blob on the end of the straight 7
> flag, but no. I agree with you now. Keep the curved flag for the
> 7comma. It is most important to get the 11limit right. The rest
is
> just icing on the cake, and a little lateral confusability there
can
> be tolerated.
As long as we are going for a higher prime limit that will almost
certainly require an additional kind of flag, perhaps that will
present an opportunity to deconfuse the situation a bit, but that
remains to be seen.
Here's something to keep in mind as we raise the prime limit. I am
sure that there are quite a few people who would think that making a
notation as versatile as this one promises to get is overkill. I
think that such a criticism is valid only if its complexity makes it
more difficult to do the simpler things. Let's try to keep it simple
for the ET's under 100 (as I believe we have been able to do so far),
keeping the advanced features in reserve for the powerJI composer
who wants a lot of prime numbers. If we build everything in from the
start and do it right, then there will be no need to revise it later
and upset a few people in the process.
> > By the way, something else I figured out over the weekend is how
to
> > notate 13 through 20 degrees of 217 with single symbols, i.e.,
how to
> > subtract the 1 through 8degree symbols from the sagittal apotome
> > (/\). The symbol subtraction for notation of apotome
complements
> > works like this:
> >
> > For a symbol consisting of:
> > 1) a left flag (or blank)
> > 2) a single (or triple) stem, and
> > 3) a right flag (or blank):
> > 4) convert the single stem to a double (or triple to an X);
> > 5) replace the left and right flags with their opposites
according to
> > the following:
> > a) a straight flag is the opposite of a blank (and vice versa);
> > b) a convex flag is the opposite of a concave flag (and vice
versa).
>
> You gotta admit this isn't exactly intuitive (particularly 5a). I'm
> more interested in the singlestem saggitals used with the standard
> sharpflat symbols, but it's nice that you can do that.
Believe it or not, the logic behind 5a) is pretty solid, while it is
5b) that is a bit contrived. The above is an expansion of what I
originally did for the 72ET notation before any curved flags were
introduced. Allow me to elaborate on this. Consider the following:
81:80 upward is a left flag: /
33:32 upward is both flags: /\
so 55:54 upward is 33:32 *less* a left flag: \
Since an apotome upward is two stems with both flags: /\
then an apotome *minus 81:80* is the apotome symbol *less a left
flag*: \
which illustrates how we arrive at a symbol for the apotome's
complement of 81/80 by changing / to \ according to 4) and 5a)
above.
Using curved flags in the 72ET native notation to alleviate lateral
confusibility complicates this a little when we wish to notate the
apotome's complement (4deg72) of 64/63 (2deg72), a single *convex
right* flag. I was doing it with two stems plus a *convex left*
flag, but the above rules dictate two stems with *straight left* and
*concave right* flags. As it turns out, the symbol having a single
stem with *concave left* and *straight right* flags is also 2deg72,
and its apotome complement is two stems plus a *convex left* flag
(4deg72), which gives me what I was using before for 4 degrees. So
with a little bit of creativity I can still get what I had (and
really want) in 72; the same thing can be done in 43ET. This is the
only bit of trickery that I have found any need for in divisions
below 100.
As you noted, it is nice that, given the way that we are developing
the symbols, this notation will allow the composer to make the
decision whether to use a singlesymbol approach or a singlesymbols
withsharpandflat approach. And the musical marketplace could
eventually make a final decision between the two. So while we can
continue to debate this point, we are under no pressure or obligation
to come to an agreement on it.
> > I will prepare a diagram illustrating the progression of symbols
for
> > JI and for various ET's so we can see how all of this is going to
> > look.
> >
> > Stay tuned!
>
> Sure. This is fun.
More fun (if more complicated) than I had ever expected!
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> i In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> Considering that the semantics of the notation have already put us
> past the 19 limit, that 217 is not 23limit consistent, and that 311
> is such an excellent division, I'd say let's go for it!
> I guess I just have a knack for finding useful commas (even before I
> start looking for them). Are you ready for the next one? It's a
> honey: 20735:20736 (5*11*13*29:2^8*3^4, ~0.083 cents). And it turns
> out that we don't need any new flags to get the 29 factor: its
> defining interval is 256:261 (2^8:3^2*29, ~33.487 cents), and the
> convex left flag that we already have (715:729) is ~33.571 cents.
256:261 is also the 29comma I had settled on.
That's awesome! But don't forget that the convex left flag also has
the meaning 45056:45927 (2^12*11:3^8*7) from its use in combination
with the convex right flag to give the large 11comma 704:729 (what
I've taken to abbreviating as the 11'comma. Fortunately this still
differs by less than 0.5 c (5103:5104) from the 29comma.
> As long as we are going for a higher prime limit that will almost
> certainly require an additional kind of flag, perhaps that will
> present an opportunity to deconfuse the situation a bit, but that
> remains to be seen.
>
> Here's something to keep in mind as we raise the prime limit. I am
> sure that there are quite a few people who would think that making a
> notation as versatile as this one promises to get is overkill.
I personally think primes beyond 11 are of very limited use musically,
but I know there are people who claim to have sucessfully used up to
31.
> I
> think that such a criticism is valid only if its complexity makes it
> more difficult to do the simpler things. Let's try to keep it
simple
> for the ET's under 100 (as I believe we have been able to do so
far),
> keeping the advanced features in reserve for the powerJI composer
> who wants a lot of prime numbers.
I totally agree.
I think we can take the 11limit (at least the semantics) as set in
stone now.
sL 80:81 21.51 c
sR 54:55 31.77 c
xL 45056:45927 33.15 c
xR 63:64 27.26 c
And the 13limit is set in stone in so far as it uses no new flags but
gives the existing ones additional meanings.
sL 65536:66339 21.08 c 6/311ET
sR 22113:22528 32.19 c 8/311ET
xL 715:729 33.57 c 9/311ET
xR 64:65 26.84 c 7/311ET
But I think we're still free to fiddle around with 17, 19, 23, 29, 31
with the proviso that we introduce no more than one new flag as we
introduce each new prime in order.
One thing that annoys me is that the 23comma that works so well re
nonewflags (16384:16767) is not necessarily the most useful one. I
prefer 729:736 since it spans the same number of fifths (6 instead of
+6) and is smaller by a pythagorean comma.
The 31comma I favour is 243:248 (3^5:2^3*31) 35.26 c.
We have a bunch of commas between 20 and 35 cents which can correspond
to a single flag. It really seems to me that the 17comma (8.73 c)
should be represented by something noticeably smaller. Using a concave
flag goes some of the way, but maybe not small enough. And certainly
the 19comma (3.38 c) should be represented by something fairly
insignificant in size, being 1/7th to 1/10th the size of the others
and 1/3rd the size of the 17comma.
What if we make the 19comma just a blob on the end of the shaft.
Neither right nor left but able to be combined with any flags. Then
maybe we can get from 19 to 31 with only the two concave flags.
What if we leave concaveleft as 2176:2187 (the 17comma) but make
concaveright 19683:19840, so that we have:
vL 2176:2187 8.73 c 3/311ET
vR 19683:19840 13.75 c 3/311ET
sL+vR 243:248 35.26 c 9/311ET
By the way, I think that two straight left flags, one above the other
on the same shaft, is the best thing for two 5commas. And do we
really need the 17'comma, 4096:4131 (14.73 c)?
The above doesn't give us all the steps of 311ET from 1 to 17, but I
don't think that matters. We don't need to actually be able to notate
311ET. The gaps are 2, 5 and 15 steps (and 12 if you don't accept my
suggestion for two 5commas).
> If we build everything in from the
> start and do it right, then there will be no need to revise it later
> and upset a few people in the process.
Indeed.
> Believe it or not, the logic behind 5a) is pretty solid, while it is
> 5b) that is a bit contrived.
...
> So
> with a little bit of creativity I can still get what I had (and
> really want) in 72; the same thing can be done in 43ET. This is
the
> only bit of trickery that I have found any need for in divisions
> below 100.
Yes. I follow that. Sounds ok.
> As you noted, it is nice that, given the way that we are developing
> the symbols, this notation will allow the composer to make the
> decision whether to use a singlesymbol approach or a
singlesymbols
> withsharpandflat approach. And the musical marketplace could
> eventually make a final decision between the two. So while we can
> continue to debate this point, we are under no pressure or
obligation
> to come to an agreement on it.
Agreed.
One of the possibilities of George's new sagittal notation is that one
can ignore the meaning of the individual flags and simply take it as a
set of symbols for 2^a*3^b*p commas, one per prime p from 5 to 31, to
be used in conjunction with # and b and to be simply placed sideby
side. As such it presently looks something like this (with the symbols
for the primes above 13 still being negotiated).
Joseph, I'm sorry to have to point out that these symbols bear a much
greater resemblance to the "European" symbols, than the Sims symbols.
I've also proposed singleASCIIcharacter substitutes for some of
them.
[If you're reading this on the yahoogroups website you will need to
choose Message Index, Expand Messages, to see the following symbols
rendered correctly.]
5comma 80;81
/
/ 
 / \


7comma 63;64
_
 \
 
 P L


11comma 32;33
/\
/  \
 ^ v


13comma 1024;1053
_
/ \
/  



17comma 2176;2187

_/



19comma 512;513
O

 * o


23'comma 16384;16767 (unfortunately not 729;736)
\
_/ \



29comma 256;261
_
/ 
 
 q d


31comma 243;248
/
/ \_



For the down versions of these, flip them vertically (don't rotate
them 180 degrees).
The smaller 23comma _can_ be rendered as, the unfortunately
complicated:
23comma 729;736
O
_/O



Notice that lateral confusability only occurs beyond the 23limit, and
this might be eliminated by adding blobs to the end of the curved
strokes of the 29 and 31 commas like this.
29comma 256;261
_
/ 
b 



31comma 243;248
/
/ \o



We also have optional symbols for larger 11, 13 and 17 commas.
11'comma 704;729
_ _
/  \
  
 ] [


13'comma 26;27
_
/ \
b  \



Only if you start combining multiple symbols into a single symbol, do
you begin to assume the vanishing of the following pair of
subhalfcent schismas: 4095:4096 (13limit), 3519:3520 (23limit
using large 23comma).
If you use the symbols for the large 11comma and large 13comma as
well as the small ones, you are also assuming the vanishing of the
subhalfcent schisma 5103:5104 (29limit).
This is all George Secor's work (apart from the liberties I've taken
with the post 13limit symbols), I'm just trying to explain one
delightful aspect of it.
 I wrote:
> We also have optional symbols for larger 11, 13 and 17 commas.
The astute reader will have noticed that I did not in fact give a
symbol for a larger 17comma (which would have been 4096:4131). That's
because it can't be done without either adding another type of
flag, disallowing the smaller 23comma, or assuming the vanishing of a
schisma of about 1.6 cents (1543503872:1544898987) which is too big
for a JI notation.
I don't really see a need for the larger 17 comma. It would merely
allow one to notate, for example, a 16:17 above C as a slightly raised
(pythagorean) Db instead of a slightly lowered (pythagorean) C#.
If we disallowed the smaller 23comma (729:736) the larger 17comma
could have a symbol like this. [Message Index, Expand messages]
17'comma 4096:4131

_/O



 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Joseph, I'm sorry to have to point out that these symbols bear a
much
> greater resemblance to the "European" symbols, than the Sims
symbols.
>
The one thing I've always found unjustifiable and now find
irredeemable about the Sims notation is the use of arrows with full
heads to indicate something smaller than the arrows with half heads. I
could almost make a version of this notation that is compatible with
the Sims notation, if it wasn't for the twelfthtone arrows.
Joseph, remind me what you don't like about slashes again, assuming
the up slash has a short vertical stroke thru the middle of it and the
down slash doesn't?
George, remind me why the 5flag is on the left and the 7flag on the
right, and why the 5flag is straight and the 7flag curved? Why
couldn't either of these properties be switched between 5 and 7?
Here's a more complete "singleASCIIcharacter substitutes" proposal.
Symbol
dn up Comma Abbrev. descr. of actual symbol

\ / 5comma 80;81 sL
L P 7comma 63;64 xR
v ^ 11comma 32;33 sL+sR
[ ] 11'comma 704;729 xL+xR
{ } 13comma 1024;1053 sL+xR
;  13'comma 26;27 xL+sR
j f 17comma 2176;2187 vL
* o 19comma 512;513 cO
w m 23comma 729;736 vL+cO+cI
W M 23'comma 16384;16767 vL+sR
q d 29comma 256;261 xL
y h 31comma 243;248 sL+vR
If it turns out we allow the large 17comma instead of the small
23comma then this could be:
J F 17'comma 4096;4131 vL+cI
The abbreviated descriptions above refer to singleshaft arrows with
various "flags" making up the arrow head:
sL straight left
sR straight right
xL convex left
xR convex right
vL concave left
vR concave right
cO small filled circle, outer (on tip of arrow)
cI small filled circle, inner (away from tip of arrow)
By the way, we can actually notate 311ET with combinations of these
flags, so that no note has more than one arrow next to it in addition
to a sharp or flat. Not that this is of any particular importance. The
values of the flags in steps of 311ET are:
sL 6
sR 8
xL 9
xR 7
vL 3
vR 3
cO 1
cI 1
On Mon, 01 Apr 2002 07:42:04 0000, "dkeenanuqnetau" <d.keenan@uq.net.au>
wrote:
>Here's a more complete "singleASCIIcharacter substitutes" proposal.
>
>Symbol
>dn up Comma Abbrev. descr. of actual symbol
>
>\ / 5comma 80;81 sL
>L P 7comma 63;64 xR
>v ^ 11comma 32;33 sL+sR
>[ ] 11'comma 704;729 xL+xR
>{ } 13comma 1024;1053 sL+xR
>;  13'comma 26;27 xL+sR
>j f 17comma 2176;2187 vL
>* o 19comma 512;513 cO
>w m 23comma 729;736 vL+cO+cI
>W M 23'comma 16384;16767 vL+sR
>q d 29comma 256;261 xL
>y h 31comma 243;248 sL+vR
On Windows systems, the  character appears as an unbroken vertical line.
It might make more sense to use : for the 13'comma up (Windows does have a
broken vertical line character, but it's not ASCII).
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Here's a more complete "singleASCIIcharacter substitutes" proposal.
>
> Symbol
> dn up Comma Abbrev. descr. of actual symbol
> 
> \ / 5comma 80;81 sL
> L P 7comma 63;64 xR
> v ^ 11comma 32;33 sL+sR
> [ ] 11'comma 704;729 xL+xR
> { } 13comma 1024;1053 sL+xR
> ;  13'comma 26;27 xL+sR
> j f 17comma 2176;2187 vL
> * o 19comma 512;513 cO
> w m 23comma 729;736 vL+cO+cI
> W M 23'comma 16384;16767 vL+sR
> q d 29comma 256;261 xL
> y h 31comma 243;248 sL+vR
I don't know if anyone cares about 12etcompatibility up to the
31limit, but in case they do here is what you get if you enforce compatibility with the "standard" h12 and h7 mappings:
2187/2048, 256/243, 81/80, 64/63, 729/704, 1053/1024, 4131/4096,
513/512, 16767/16384, 261/256, 67797/65536
 In tuningmath@y..., "genewardsmith" <genewardsmith@j...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Here's a more complete "singleASCIIcharacter substitutes"
proposal.
> >
> > Symbol
> > dn up Comma Abbrev. descr. of actual symbol
> > 
> > \ / 5comma 80;81 sL
> > L P 7comma 63;64 xR
> > v ^ 11comma 32;33 sL+sR
> > [ ] 11'comma 704;729 xL+xR
> > { } 13comma 1024;1053 sL+xR
> > ;  13'comma 26;27 xL+sR
> > j f 17comma 2176;2187 vL
> > * o 19comma 512;513 cO
> > w m 23comma 729;736 vL+cO+cI
> > W M 23'comma 16384;16767 vL+sR
> > q d 29comma 256;261 xL
> > y h 31comma 243;248 sL+vR
>
> I don't know if anyone cares about 12etcompatibility up to the
> 31limit, but in case they do here is what you get if you enforce
compatibility with the "standard" h12 and h7 mappings:
>
> 2187/2048, 256/243, 81/80, 64/63, 729/704, 1053/1024, 4131/4096,
> 513/512, 16767/16384, 261/256, 67797/65536
I'm not sure I know what you mean by "enforce compatibility with the
"standard" h12 and h7 mappings", because I can't see why you would get
67797/65536 as the 31comma when this contains 3^7 and 248/243 only
contains 3^5. This is the difference between a 1:31 from C being a Cb
or a B respectively (which are of course the same thing in 12tET).
I'm also unclear how this method chooses 729/704 over 8192/8019 for
the 11comma, and 16767/16384 over 736/729 for the 23comma. Since
both these choices involve 3^6 versus 3^6.
I'm guessing these have something to do with h7 or 7ET. Could you
please explain your method in more detail?
Perhaps instead of 7ET it would make more sense to use the C major
scale in 12ET.
It seems there is no argument over which commas to use for 3,5,7,19,29
 the same choices for these can be arrived at in any number of ways 
but 11,13,17,23,31 are not yet settled. Even if our final system has
symbols for more than one comma per prime, we will still want to
specify a "preferred" comma for each prime.
I believe we decided long ago that we were basing this notation on
pythagorean (i.e a chain of Just fifths) not 12ET, although we might
favour 12ET to the extent that all the odd numbers, up to an
oddlimit determined by our highest prime, should be expressible
without requiring enharmonics such as G# and Ab to be used
simultaneously, and more strictly, without requiring anyone to go
outside of a chain of fifths having 12 notes, e.g. Eb to G# or Ab to
C#. And further, not to use any commas (at least as "preferred"
commas) that fail to vanish in 12ET, such as 26:27, 16:17, 18:19 or
23:24. 32:33 is only considered to pass this test because it vanishes
in the case of 3:11 and 9:11 even though it doesn't vanish in the case
of 1:11.
This last requirement is more accurately expressed as requiring all
preferred commas to be smaller than half an apotome (i.e. smaller than
56.84 cents).
It might seem obvious that we should limit the 12 base notes to which
the commas must be applied, and hence the range of exponents of 3 in
these commas, to a range of either 6 to +5 fifths from the 1/1, or a
range of 5 to +6. However 25 is within our oddlimit and presumably
we want the "25comma" to simply be two 5commas, which means two
syntonic commas (6400:6561), which means 3^8, which means our range of
allowedexponentsof3 must be shifted at least as far as 3 to +8.
The only alternatives to this are either to go outside a 12 note chain
of fifths within our oddlimit, or to invoke a different comma for 25,
requiring its own symbol, such as the diaschisma (2025:2048) which has
3^4. A separate 25comma symbol seems like a bad idea to me, unless
it is obviously made up of two 5comma symbols, in which case it
should _be_ two 5commas.
Here's a table showing all the 2,3,prime commas (and 2,3,25 commas)
that could possibly be of any interest in this notational endeavour.
[On the website, use Message Index, Expand Messages to see the columns
formatted correctly.]
3 Note
exp Odd Comma Cents Odd Comma Cents

7 Cb 29 65536:63423 56.74 31 65536:67797 58.72
6 Gb 11 8192:8019 36.95 23 16384:16767 40.00
5 Db 17 4096:4131 14.73
4 Ab 13 1024:1053 48.35 25 2048:2025 19.55
3 Eb 19 512:513 3.38
2 Bb 7 64:63 27.26 29 256:261 33.49
1 F 11 32:33 53.27 21
0 C 1
1 G 3
2 D 9
3 A 27 13 27:26 65.34
4 E 5 81:80 21.51
5 B 31 243:248 35.26 15
6 F# 23 729:736 16.54 11 729:704 60.41
7 C# 17 2187:2176 8.73
8 G# 13 6561:6656 24.89 25 6561:6400 43.01
Within a range of 3 to +8 fifths we have the following commas for the
disputed primes, smaller than half an apotome.
11 32:33
13 6561:6656 (note: not 1024:1053)
17 2187:2176
23 729:736
31 243:248
I would be very sorry not to have 1024:1053 as the 13comma because:
(a) it has that neat subhalfcent schisma with the 5 and 7 commas
(4095:4096).
(b) 8:13 makes more sense as a neutral sixth than a superaugmented
fifth.
There is a completely unrelated reason why we should go to at least a
3exponent of 7, and that is so that we have 2187:2176 as the
preferred 17comma. We need a comma of about this size (8.7 cents) to
help us in notating the larger ETs, because it fills in a huge gap
between the 19comma of 3.4 cents and the 5comma of 21.5 cents. I
expect the other 17comma (4096:4131) at 14.7 cents would be nowhere
near as useful.
Within a range of 4 to +7 fifths we would have the same commas as 3
to +8 fifths, except for the 13comma, which would become the beloved
1024:1053. But rather than introduce the disachisma as a 25comma I'm
inclined to allow the 3exponents to range from 4 to +8 so that, from
C, an 8:11 is a variety of Ab (or A, using the nonpreferred 13comma)
and a 16:25 is a variety of G#. Too bad about the possibility of
simultaneous enharmonics. How do others feel about this?
 In tuningmath@y..., Herman Miller <hmiller@I...> wrote:
> On Mon, 01 Apr 2002 07:42:04 0000, "dkeenanuqnetau" <d.keenan@u...>
> wrote:
>
> >Here's a more complete "singleASCIIcharacter substitutes"
proposal.
> >
> >Symbol
> >dn up Comma Abbrev. descr. of actual symbol
> >
> >\ / 5comma 80;81 sL
> >L P 7comma 63;64 xR
> >v ^ 11comma 32;33 sL+sR
> >[ ] 11'comma 704;729 xL+xR
> >{ } 13comma 1024;1053 sL+xR
> >;  13'comma 26;27 xL+sR
> >j f 17comma 2176;2187 vL
> >* o 19comma 512;513 cO
> >w m 23comma 729;736 vL+cO+cI
> >W M 23'comma 16384;16767 vL+sR
> >q d 29comma 256;261 xL
> >y h 31comma 243;248 sL+vR
>
> On Windows systems, the  character appears as an unbroken vertical
line.
> It might make more sense to use : for the 13'comma up (Windows does
have a
> broken vertical line character, but it's not ASCII).
Thanks Herman,
I had forgotten that the vertical bar is in two pieces in some fonts
(where?). I proposed that pair of characters ;  for compatibility
with Scala's JI notation.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> But rather than introduce the disachisma as a 25comma I'm
> inclined to allow the 3exponents to range from 4 to +8 so that,
from
> C, an 8:11 is a variety of Ab (or A, using the nonpreferred
13comma)
> and a 16:25 is a variety of G#.
That should have been "... an 8:13 is a variety of Ab ...".
Dave wrote:
>v ^ 11comma 32;33 sL+sR
>[ ] 11'comma 704;729 xL+xR
Ai, I had recently removed v ^ from the JI
notation systems to make them exclusively
denote the diaschisma, and replaced them with
[ ] for the undecimal comma 33/32. They are also
used in the new E217 system which is probably
the highest ET system I'm going to support.
33/32 is smaller anyway so I feel it's more
logical to have the symbols exchanged.
Manuel
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > But rather than introduce the disachisma as a 25comma I'm
> > inclined to allow the 3exponents to range from 4 to +8 so that,
> from
> > C, an 8:11 is a variety of Ab (or A, using the nonpreferred
> 13comma)
> > and a 16:25 is a variety of G#.
>
> That should have been "... an 8:13 is a variety of Ab ...".
i'm much more comfortable with 8:13 as a variety of C:A. the dominant
13th chord in 12equal definitely gains a lot of tonalness because
13:11 and even 13:7 are represented by their best approximations in
12equal. and simpler chords in 12equal simply fail to evoke the 13
limit, under any circumstances.
 In tuningmath@y..., manuel.op.de.coul@e... wrote:
> Dave wrote:
> >v ^ 11comma 32;33 sL+sR
> >[ ] 11'comma 704;729 xL+xR
>
> Ai, I had recently removed v ^ from the JI
> notation systems to make them exclusively
> denote the diaschisma, and replaced them with
> [ ] for the undecimal comma 33/32.
I'm sorry. I wasn't aware of this. Maybe it's a good idea, but I would
have used u and n for the diaschisma since Rappoport's symbols are
convex arrowheads (with no shaft).
> They are also
> used in the new E217 system which is probably
> the highest ET system I'm going to support.
How many steps of E217 are they? I assume by "they" you mean [ ].
> 33/32 is smaller anyway so I feel it's more
> logical to have the symbols exchanged.
I don't follow this. v ^ seem like smaller symbols to me than [ ].
When considering the ASCII symbols alone, and other notation systems,
I wouldn't have a problem with using [ ] for 32:33 and v ^ for
704:729. It would be nice to have the preferred (smaller) 11 and 13
commas (dieses) be [] and {}. The trouble is that George's symbol for
32:33 is an arrow with straight headflags (an ordinary arrow) and the
symbol for 704:729 has convex headflags (like Rappoport's diaschisma
symbols with a vertical shaft added).
Anyway, things are still in flux, so lets not worry about it too much
just yet.
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > That should have been "... an 8:13 is a variety of Ab ...".
>
> i'm much more comfortable with 8:13 as a variety of C:A. the
dominant
> 13th chord in 12equal definitely gains a lot of tonalness because
> 13:11 and even 13:7 are represented by their best approximations in
> 12equal. and simpler chords in 12equal simply fail to evoke the
13
> limit, under any circumstances.
Paul,
Don't panic. I took that on board some time ago, thanks to you. The
complete corrected sentence reads.
"But rather than introduce the disachisma as a 25comma I'm
inclined to allow the 3exponents to range from 4 to +8 so
that, from C, an 8:13 is a variety of Ab (or A, using the
nonpreferred 13comma) and a 16:25 is a variety of G#."
Which is one reason we have both 26:27 ; , and 1024:1053 { } as
13commas. Are you saying we shouldn't have 1024:1053 at all? The
notation is pythagoreanbased not 12ET based and so to avoid multiple
accidentals one might wish to notate an 8:13 from A as A:F} rather
than A:F#;
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Which is one reason we have both 26:27 ; , and 1024:1053 { } as
> 13commas. Are you saying we shouldn't have 1024:1053 at all?
so the rule is that every comma and its 2187:2048 complement has a
unique symbol? if so, then the symbols should reflect that in a
natural way . . .
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Which is one reason we have both 26:27 ; , and 1024:1053 { } as
> > 13commas. Are you saying we shouldn't have 1024:1053 at all?
>
> so the rule is that every comma and its 2187:2048 complement has a
> unique symbol?
No, not every comma. So far George and I have only agreed on the
desirability of apotome complements of those commas which are close to
the halfapotome, say those between 1/3 and 2/3 apotome. So far that's
only the 11 and 13 commas (dieses).
> if so, then the symbols should reflect that in a
> natural way . . .
Read George Secor's two most recent posts to this forum (and mine with
the ASCII graphics of his symbols) and let us know whether you find
the system sufficiently natural.
The two 17commas that have been mentioned are pythagorean comma
complements, and of the two 23commas mentioned, one is a pythagorean
comma larger than the other; similarly the two 31commas mentioned ;
although I'm still waiting to hear from Gene (or anyone) why anyone
would want 65536:67797 (2^16:3^7*31) 59.7 c as a 31comma when we have
243:248 (3^5:2^3*31) 35.3 c.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
> >  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > Which is one reason we have both 26:27 ; , and 1024:1053 { }
as
> > > 13commas. Are you saying we shouldn't have 1024:1053 at all?
> >
> > so the rule is that every comma and its 2187:2048 complement has a
> > unique symbol?
>
> No, not every comma. So far George and I have only agreed on the
> desirability of apotome complements of those commas which are close
to
> the halfapotome, say those between 1/3 and 2/3 apotome. So far
that's
> only the 11 and 13 commas (dieses).
but these 'sizes' won't come out anything like that in many, if not
most, equal temperaments. right?
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > Considering that the semantics of the notation have already put
us
> > past the 19 limit, that 217 is not 23limit consistent, and that
311
> > is such an excellent division, I'd say let's go for it!
>
> > I guess I just have a knack for finding useful commas (even
before I
> > start looking for them). Are you ready for the next one? It's a
> > honey: 20735:20736 (5*11*13*29:2^8*3^4, ~0.083 cents). And it
turns
> > out that we don't need any new flags to get the 29 factor: its
> > defining interval is 256:261 (2^8:3^2*29, ~33.487 cents), and the
> > convex left flag that we already have (715:729) is ~33.571 cents.
>
> 256:261 is also the 29comma I had settled on.
>
> That's awesome! But don't forget that the convex left flag also has
> the meaning 45056:45927 (2^12*11:3^8*7) from its use in combination
> with the convex right flag to give the large 11comma 704:729 (what
> I've taken to abbreviating as the 11'comma. Fortunately this still
> differs by less than 0.5 c (5103:5104) from the 29comma.
With the 29 factor we have passed the point (in both 217 and 311)
where all of the ratios within the harmonic limit are a unique number
of degrees, so some bridging is inevitable. However, what's most
important is that the number of degrees for the 29 comma is
consistent with this flag in both 217 (6 deg) and 311 (9 deg).
> I personally think primes beyond 11 are of very limited use
musically,
> but I know there are people who claim to have sucessfully used up
to
> 31.
Perhaps you would change your mind if you tried a few isoharmonic
chords (i.e., chords having a common difference tone between
consecutive tones) containing higher primes, such as 16:19:22:25 or
17:21:25:29. They are consonant in the sense that, if you mistune
one of the tones in the chord, you will easily hear it go out of tune
due to the perception of difference tones beating against one
another. (The JI heptatonic scales that I refer to below were
constructed with this principle in mind.)
> I think we can take the 11limit (at least the semantics) as set in
> stone now.
>
> sL 80:81 21.51 c
> sR 54:55 31.77 c
> xL 45056:45927 33.15 c
> xR 63:64 27.26 c
>
> And the 13limit is set in stone in so far as it uses no new flags
but
> gives the existing ones additional meanings.
>
> sL 65536:66339 21.08 c 6/311ET
> sR 22113:22528 32.19 c 8/311ET
> xL 715:729 33.57 c 9/311ET
> xR 64:65 26.84 c 7/311ET
I prefer to consider the xL flag as 715:729 with the additional
meaning of 45056:45927. Not only is this a much simpler ratio, but
it also gives an exact 26:27 for the 13 diesis. This makes the
defining commas yield exact dieses for both 11 and 13 (rather than
two different 11 dieses). The 4095:4096 schisma then defines
alternate dieses for both 11 and 13, which better maintains the one
commaperprime (or in this case, oneflagperprime) concept.
I realize that 26:27 is more than half an apotome, but it is not that
much more, still falling within the neighborhood of half an apotome;
in any case, its symbol is a combination of two commaflags (rather
than a single flag), and we have already specified that it will never
be combined with a sharp or flat that alters in the opposite
direction.
Or am I just splitting hairs, inasmuch as conflating 4095:4096 would
allow us to look at it either way (or both ways at once)?
> But I think we're still free to fiddle around with 17, 19, 23, 29,
31
> with the proviso that we introduce no more than one new flag as we
> introduce each new prime in order.
Maybe. I have some comments about this below.
> One thing that annoys me is that the 23comma that works so well re
> nonewflags (16384:16767) is not necessarily the most useful one.
I
> prefer 729:736 since it spans the same number of fifths (6 instead
of
> +6) and is smaller by a pythagorean comma.
Upon reading this, I decided to go through some of my papers to find
all of the JI heptatonic scales in which I used 23, and I found that
I had three that (relative to C as 1/1) require an Fsharp and only
one that requires a Gflat, which does indeed make 729:736 more
useful. These are all *diatonic* usages, in which one has no liberty
to "misspell" a note by using its equivalent sharp or flat (as might
be allowable in a *chromatic* usage). But if there is even a single
case that uses the flat (which there is), then we need 16384:16767 as
well as 729:736.
Whatever the case, what is significant here is that neither comma
introduces a new flag unless we disallow 4096:4131 as a defining
comma for 17 (on the basis of one comma per prime) and replace the
definition of 3deg217 and/or 4deg311 with the 23comma, 729:736. We
then have the alternate 23 comma (16384:16767) as a freebie.
In the process of counting the number of sharpvs.flat occurrences
for 23, I also kept count for some other high oddnumber ratios and
came to some conclusions which I will give below.
> The 31comma I favour is 243:248 (3^5:2^3*31) 35.26 c.
I agree. I also found what should have been a useful schisma for
this (59024:59049, or 2^4*7*17*31:3^10, ~0.733 cents), but its usage
is inconsistent in both 217 and 311, since 243:248 is not the same
number of degrees as 238:243 (vL+xR) in either division. (So with 31
my luck with schismas has run out.)
However, with 31 we have reached the point where the defining comma
does not introduce an interval having a new number of degrees in
either 217 or 311; in other words, the 31 comma is totally
unnecessary for the notation. This should come as no surprise, since
with 31 we have passed the point where the ratios can be represented
uniquely (as both 29:30 and 30:31 are 15deg311). In fact, it
happened sooner than this, starting with 24:25 and 25:26 (which are
both 18deg311).
So, while we can consistently notate 31 in *either* 217 *or* 311, it
must be done with the realization that ratios of 31 are not unique
and that some intervals may occasionally bridge to other primes.
(This makes me begin to wonder why we would want to use 311 instead
of 217; more about this below.)
> We have a bunch of commas between 20 and 35 cents which can
correspond
> to a single flag. It really seems to me that the 17comma (8.73 c)
> should be represented by something noticeably smaller. Using a
concave
> flag goes some of the way, but maybe not small enough. And
certainly
> the 19comma (3.38 c) should be represented by something fairly
> insignificant in size, being 1/7th to 1/10th the size of the others
> and 1/3rd the size of the 17comma.
>
> What if we make the 19comma just a blob on the end of the shaft.
> Neither right nor left but able to be combined with any flags. Then
> maybe we can get from 19 to 31 with only the two concave flags.
I tried something of this sort for both the 17 and 19 commas prior to
presenting the 17 and 23limit approaches in message #3793, but I
didn't come up with any way of symbolizing them that was simple
enough to satisfy me, so I dropped it. Among the things I tried was
a small filledin triangle on the end of the shaft, but it just
didn't seem to stand out well enough to work. I also tried both open
and filledin triangles (which could also point in either direction)
on the other end of the shaft, but this required a longer shaft and
started to get a bit cumbersome (so that it might be unclear which
note of a chord is being altered if the symbol were too large
vertically). Perhaps you have a better idea?
> What if we leave concaveleft as 2176:2187 (the 17comma) but make
> concaveright 19683:19840, so that we have:
>
> vL 2176:2187 8.73 c 3/311ET
> vR 19683:19840 13.75 c 3/311ET
>
> sL+vR 243:248 35.26 c 9/311ET
That's interesting, but desiring a complete range of intervals in
311, I would prefer to see an interval of 2deg311, which in
combination with 2176:2187, would also supply one of 5deg311.
> By the way, I think that two straight left flags, one above the
other
> on the same shaft, is the best thing for two 5commas.
I was thinking about that also. The interval of 12deg311 that it
supplies is one that we need.
> And do we
> really need the 17'comma, 4096:4131 (14.73 c)?
I guess we could consider the 17asflat interval as approximated by
the sum of the 17 and 19 commas, as long as the number of system
degrees matches (as it does in both 217 and 311) and as long as we
could still notate that number of degrees as their combination. This
would also lead to the question, do we really need the 23comma
(729:736) of that same number of degrees?
> The above doesn't give us all the steps of 311ET from 1 to 17, but
I
> don't think that matters. We don't need to actually be able to
notate
> 311ET. The gaps are 2, 5 and 15 steps (and 12 if you don't accept
my
> suggestion for two 5commas).
I think that the gaps are unacceptable for a couple of reasons.
As I mentioned above, in the process of counting sharpvs.flat
occurrences for 23 in my JI heptatonic scales, I also counted the
number of sharpvs.flat occurrences for 17 (3#, 3b), 19 (4#, 5b), 25
(5#, 3b), and 29 (no#, 4b). I conclude that there has to be a
provision for spelling any interval in at least two different ways,
which is a compelling reason for providing a complete set of symbols
for whatever division we settle on for the JI notation.
In addition, being able to notate all of the degrees would ensure
that no matter how much modulation is done in JI, at least one would
never run out of symbols. If this were not done, then we would lose
one of the principal advantages of mapping the JI notation to a
specific division.
So I think that we are either going to have to fill in the gaps in
311 or go with 217.
To further complicate things in 311, I also noticed in my notes that,
while I favored using 19 as an Eflat (where 1/1 is C), there were
almost as many instances where a heptatonic scale called for a D
sharp (using the 19comma 19456:19683, 3^9:2^14*19, ~20.082 cents).
This usage turns out to be *inconsistent* in 311 (but consistent in
217), a problem that I didn't expect to find. While I compared the
inconsistency of a few ratios of 23 in 217 with driving a car
slightly onto the shoulder, the 311 problem (affecting all ratios of
19) is more like attempting to drive in the lesstraveled direction
on route 19 and finding yourself in the wrong lane moving against
traffic. So I am beginning to have serious doubts about going to 311.
Taking another look at 217, I see that it does uniquely represent all
of the 19limit consonances and, except for the slight inconsistency
previously noted with the 23 factor, is otherwise consistent through
the 31 limit. We have been able to define usable commas all the way
through the 29 limit (past the point where we even need them), and we
can even notate ratios of 31 consistently (if not uniquely) with
intervals of the appropriate numbers of degrees. And the next two
odd numbers are not primes, so we are actually getting a 35oddlimit
capability with 217.
So if 311 proves to be a bit unwieldly, at least I believe we have a
workable solution in 217. (By the way, I noticed that Manuel in
message #3930 indicated that 217ET is probably the highest system
that he is going to support.)
Something else in favor of 217: I noted earlier that it is 7 times
31. If you make instruments for 31ET (or train string players to
play 31ET), then your JI can be reckoned in alterations of +/ 1 to
3 increments of 217ET (i.e., multiples of ~5.5 cents), to a maximum
of 16.6 cents, which is not an unreasonable amount of intonation
adjustment for instruments of flexible pitch. This would be a decent
practical alternative to alterations reckoned relative to 12ET,
whether for (toocoarse) 72ET or Johnny Reinhard's (toosmall) one
cent increments. Plus you get 31ET in the bargain. And I should
mention that a series of 41 fifths in 217 (kept within an octave)
brings you only one degree away from your starting point, giving you
a very close approximation of 41ET.
So the question now becomes: Are we left with any good reason for
basing the JI notation on 311 instead of 217?
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Joseph, I'm sorry to have to point out that these symbols bear a
much
> > greater resemblance to the "European" symbols, than the Sims
symbols.
> >
>
> The one thing I've always found unjustifiable and now find
> irredeemable about the Sims notation is the use of arrows with full
> heads to indicate something smaller than the arrows with half
heads. I
> could almost make a version of this notation that is compatible
with
> the Sims notation, if it wasn't for the twelfthtone arrows.
>
> Joseph, remind me what you don't like about slashes again, assuming
> the up slash has a short vertical stroke thru the middle of it and
the
> down slash doesn't?
>
> George, remind me why the 5flag is on the left and the 7flag on
the
> right, and why the 5flag is straight and the 7flag curved? Why
> couldn't either of these properties be switched between 5 and 7?
One of the main features of the sagittal notation is the
incorporation of Bosanquet's slanted comma lines with the appropriate
slope indicating an up or down (Didymus) comma. While the lines by
themselves do not provide the easiest way to distinguish up from
down, when one of these is used in combination with at least one
vertical line (as it is in the sagittal notation), an arrow is
formed, which clearly points in the appropriate direction, thus
allowing us to be doubly sure of its meaning. This symbol is basic
to the notation, being used for this single purpose in the
overwhelming majority of ET's, as well as in the JI notation.
A straight flag is preferred over a curved 5flag for two reasons:
1) A curved line does not have a constant slope and therefore tends
to obscure the direction of pitch alteration that it is supposed to
be signifying.
2) Straight flags are simpler to make than curved flags, so they
should be used more frequently. This is consistent with my
guidelines (in message #3817) to keep things simple by:
<< [minimizing] the use of curved flags in the ET notations,
introducing them only as it is necessary or helpful: 1) to avoid
lateral confusibility (in 72ET); 2) to distinguish 32:33 from
1024:1053 (in 46 and 53ET, *without* lateral confusibility!); and 3)
to notate increments smaller than 80:81 (in 94ET). >>
Finally, the 5comma flag must be on the *left side* in order to have
the desired slope.
Note that the Sims notation uses these symbols (which are 1 degree in
the 72ET sagittal notation) for 2 degrees of 72. Ezra Sims' failure
to take into account any prior use of certain notational features
when he devised his 72tone notation is most unfortunate. However, I
feel no obligation to corrupt the sagittal notation in order to make
it Simscompatible, because I believe that he is the oddmanout in
this game. It would be far more reasonable to ask the users of the
Sims notation to change the symbols for 2deg72 by reversing them left
toright, which should cause them no disorientation, but I doubt that
anything like that is going to happen.
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >  In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
> > >  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > >
> > > > Which is one reason we have both 26:27 ; , and 1024:1053 { }
> as
> > > > 13commas. Are you saying we shouldn't have 1024:1053 at all?
> > >
> > > so the rule is that every comma and its 2187:2048 complement
has a
> > > unique symbol?
> >
> > No, not every comma. So far George and I have only agreed on the
> > desirability of apotome complements of those commas which are
close
> to
> > the halfapotome, say those between 1/3 and 2/3 apotome. So far
> that's
> > only the 11 and 13 commas (dieses).
>
> but these 'sizes' won't come out anything like that in many, if not
> most, equal temperaments. right?
Not really. When notating those strange and difficult ET's that I
must assume you have in mind (inasmuch as the twelve ET's under 100
that I have tried so far behave quite nicely, thank you*), you can
pick and choose the most appropriate symbols to use for the various
degrees of alteration. There are at present (in the 217ETbased
notation) 13 different symbols less than half an apotome in size from
which to select.
Only the two largest of these have dedicated apotomecomplement
symbols. These four symbols taken as a group are the four varieties
of "semisharps" or "semiflats" representing ratios of 11 and 13, and
their distinctive appearance would make it relatively easy to
translate them at sight from JI to ET notation  a primary reason
for providing the dedicated symbols.
By the way, the approximate 1/3 to 2/3 apotome range given above has
turned out to be narrower: more like 4/10 to 6/10 apotome.
George
*The twelve ET's are 22, 27, 34, 41, 43, 46, 50, 53, 58, 72, 94, and
96. I didn't bother to count the more trivial ones, such as 12, 17,
19, and 31.
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > No, not every comma. So far George and I have only agreed on the
> > desirability of apotome complements of those commas which are close
> to
> > the halfapotome, say those between 1/3 and 2/3 apotome. So far
> that's
> > only the 11 and 13 commas (dieses).
>
> but these 'sizes' won't come out anything like that in many, if not
> most, equal temperaments. right?
Right, but
(a) it's a JI notation too
(b) they will tend more and more to these sizes as we go to larger and
larger ETs
(c) they will tend to these sizes if averaged over a large number of
small ETs
 Dave Keenan
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > No, not every comma. So far George and I have only agreed on the
> > desirability of apotome complements of those commas which are close
> to
> > the halfapotome, say those between 1/3 and 2/3 apotome. So far
> that's
> > only the 11 and 13 commas (dieses).
>
> but these 'sizes' won't come out anything like that in many, if not
> most, equal temperaments. right?
Right, but
(a) it's a JI notation too
(b) they will tend more and more to these sizes as we go to larger and
larger ETs
(c) they will tend to these sizes if averaged over a large number of
small ETs
 Dave Keenan
 In tuningmath@y..., "annelizkeenan" <annelizkeenan@y...> wrote:
>
>  Dave Keenan
who's anneliz and does she like to annelize things as much as you
do? :)
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
>  In tuningmath@y..., "annelizkeenan" <annelizkeenan@y...> wrote:
>
> >
> >  Dave Keenan
>
> who's anneliz and does she like to annelize things as much as you
> do? :)
Tee hee.
She's Anne Elizabeth and she's my sister and no she certainly does
not analyze. If you saw her fingers flying on her violin fingerboard
during some of the Irish jigs she plays, you'd know that if she
thought for a millisecond about what she was actually doing, the whole
thing would just explode into a shower of wood, catgut and horsehair.
:)
InReplyTo: <a8fkev+5808@eGroups.com>
gdsecor wrote:
> Something else in favor of 217: I noted earlier that it is 7 times
> 31. If you make instruments for 31ET (or train string players to
> play 31ET), then your JI can be reckoned in alterations of +/ 1 to
> 3 increments of 217ET (i.e., multiples of ~5.5 cents), to a maximum
> of 16.6 cents, which is not an unreasonable amount of intonation
> adjustment for instruments of flexible pitch. This would be a decent
> practical alternative to alterations reckoned relative to 12ET,
> whether for (toocoarse) 72ET or Johnny Reinhard's (toosmall) one
> cent increments. Plus you get 31ET in the bargain. And I should
> mention that a series of 41 fifths in 217 (kept within an octave)
> brings you only one degree away from your starting point, giving you
> a very close approximation of 41ET.
31ET for the 31limit, great!
In fact, a temperament with a period of 1/31 octaves is already my top
inconsistent 21limit temperament:
1/3, 16.0 cent generator
basis:
(0.032258064516129031, 0.013300385023233607)
mapping by period and generator:
[(31, 0), (49, 0), (72, 0), (87, 0), (107, 0), (115, 1), (127, 1), (132,
1)]
mapping by steps:
[(62, 31), (98, 49), (144, 72), (174, 87), (214, 107), (229, 115), (253,
127), (263, 132)]
highest interval width: 1
complexity measure: 31 (62 for smallest MOS)
highest error: 0.009287 (11.145 cents)
Without a history of usage of such temperaments, it's really impossible to
say if that 11 cent error is good or bad. It's lower in absolute terms
than 12equal in the 5limit, which many put up with. But if the 21limit
is taken to mean the difference between 22:21 (80.5 cents) and 21:20
(84.5) cents is significant, 11 cents is huge. Of course, any
inconsistent temperament will remove such distinctions. Here are the
21limit equivalences for this particular temperament:
15:14 =~ 16:15
10:9 =~ 19:17 =~ 9:8
12:11 =~ 11:10
22:19 =~ 15:13
13:11 =~ 19:16
16:13 =~ 21:17
9:7 =~ 14:11
21:16 =~ 17:13
17:16 =~ 19:18
15:11 =~ 11:8 =~ 26:19
22:21 =~ 21:20
20:19 =~ 18:17
The online scripts and results won't handle the 31limit, but I can run it
locally (it's a trivial change). The temperament George describes is
roughly
>>> temper.Temperament(31, 217, temper.primes)
1/8, 5.3 cent generator
basis:
(0.032258064516129031, 0.0043924816210988982)
mapping by period and generator:
[(31, 0), (49, 1), (72, 0), (87, 0), (107, 2), (115, 2), (127, 2), (132,
2),
(140, 2), (151, 3), (154, 3)]
mapping by steps:
[(217, 31), (344, 49), (504, 72), (609, 87), (751, 107), (803, 115), (887,
127),
(922, 132), (982, 140), (1054, 151), (1075, 154)]
highest interval width: 6
complexity measure: 186 (217 for smallest MOS)
highest error: 0.002255 (2.706 cents)
217equal is only accurate to 3.2 cents, so there is some kind of
improvement here. The complexity may be too high for the tradeoff to be
worthwhile.
An interesting alternative is
>>> temper.Temperament(62,217,temper.primes)
4/9, 16.7 cent generator
basis:
(0.032258064516129031, 0.013944526171363311)
mapping by period and generator:
[(31, 0), (50, 2), (72, 0), (87, 0), (109, 4), (116, 3), (128, 3),
(133, 3)
, (142, 4), (151, 1), (154, 1)]
mapping by steps:
[(217, 62), (344, 98), (504, 144), (609, 174), (751, 214), (803, 229),
(887, 253
), (922, 263), (982, 280), (1054, 301), (1075, 307)]
highest interval width: 6
complexity measure: 186 (217 for smallest MOS)
highest error: 0.002263 (2.715 cents)
If you're prepared to forego ratios involving 27, the complexity is only
4*26, compared to 5*26 for the other version. One disadvantage is that
the 3:1 has a more complex approximation. But that may not be important,
as you can modulate by as many 31equal fifths as you like, and they're
good enough in another context. Again, there isn't enough (any!) history
of tempered 31limit music to pronounce on the importance of these
differences.
Graham
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> With the 29 factor we have passed the point (in both 217 and 311)
> where all of the ratios within the harmonic limit are a unique
number
> of degrees, so some bridging is inevitable. However, what's most
> important is that the number of degrees for the 29 comma is
> consistent with this flag in both 217 (6 deg) and 311 (9 deg).
I had mistakenly thought that 311ET was 31limitunique until Graham
corrected me. What's the smallest ET that has a 31limitunique mapping,
even if it's not consistent? Could it be 624ET. How about a 35limit
unique mapping? Never mind.
> I prefer to consider the xL flag as 715:729 with the additional
> meaning of 45056:45927.
...
> Or am I just splitting hairs, inasmuch as conflating 4095:4096 would
> allow us to look at it either way (or both ways at once)?
Yes. Splitting hairs. All I'm saying is, if someone only needs an 11limit
notation then they shouldn't have to know anything about xL also being
715:729 (13limit).
> As I mentioned above, in the process of counting sharpvs.flat
> occurrences for 23 in my JI heptatonic scales, I also counted the
> number of sharpvs.flat occurrences for 17 (3#, 3b), 19 (4#, 5b),
25
> (5#, 3b), and 29 (no#, 4b). I conclude that there has to be a
> provision for spelling any interval in at least two different ways,
> which is a compelling reason for providing a complete set of symbols
> for whatever division we settle on for the JI notation.
Oh dear, I think I've agreed with certain things you proposed which have
turned out to be the beginning of a slippery slope that I don't want to go
down. One was to accept certain schismas vanishing in order to minimise the
number of different flags making up the symbols. The other was to accept
the need for more than one comma for certain primes.
Now you say we need two commas for _every_ prime. Why stop there? Shouldn't
we provide for anyone who wants to notate the prime intervals from C as
varieties of any of the following:
5 E,Fb
7 A#,Bb
11 E#,F,F#,Gb
13 G#,Ab,A
17 C#,Db
19 D#,Eb
23 F#,Gb
29 A#,Bb,Cb
31 B,Cb,C
These all involve commas smaller than 3/5 apotome. There are _only_ 23 of
them. I am of course being facetious. I'm more inclined to go back to a
strict single comma per prime. In all the lower ETs where either an
11comma or a 13comma is a semisharp or semiflat, its apotome complement
is the same number of steps, so there isn't any need for both.
In rational tunings, I don't see why we have to cater for all the base
notes (powers of 3 to which the prime commas are applied) remaining within
a single diatonic scale. Actually it isn't even possible, unless you allow
commas much larger than 3/5 apotome. I see no problem with requiring a
chromatic scale for the base notes, or worse, a few enharmonics such as Ab
and G# together.
Remember I'm happy to use up to 3 symbols against a note (for rational
tunings and very large ETs) rather than have too many new symbols for
people to learn.
> In addition, being able to notate all of the degrees would ensure
> that no matter how much modulation is done in JI, at least one would
> never run out of symbols. If this were not done, then we would lose
> one of the principal advantages of mapping the JI notation to a
> specific division.
...
> So the question now becomes: Are we left with any good reason for
> basing the JI notation on 311 instead of 217?
From your point of view, I would say that you are better off with 217ET.
However I do not wish to base a JI (rational) notation on _any_ temperament
that has errors larger than 0.5 c. For me, 217ET and 311ET were merely a
way of looking for schismas that might be notationally usable (less than
0.5 c), and of checking that things were working sensibly, and it was nice
to actually be able to notate those ETs themselves. But I'm taking Johnny
Reinhard at his word when he says (or implies) that nothing less than
1200ET is good enough as an ETbased JI notation.
What is the first ET above 1200 that is 35limit consistent? 35limit unique?
But I am more interested in a 31limit (or 35limit) temperament (not equal
or linear or even planar, but maybe 6D or 7D or 8D) with as many
subhalfcent schismas as possible, vanishing. Hence the challenge which
noone's taken up yet, except in part.
I don't share your obsession with packing all the necessary information
into a single symbol. Some folks may well be willing to notate their JI
piece by mapping to the nearest degree of 217ET, and they may well be glad
of the possibility to do it with only one accidental per note, but that
won't be everyone's cup of tea. Given that it may involve errors of up to 3
cents, it must be a conscious decision, not something forced upon us by the
notation.
However, it seems that most would be willing to wear it if a 0.5 c error
were forced upon them by the notation. The only objections to this will be
"philosophical" ones, not ones that anyone can hear (except in something
like Lamonte Young's Dream House).
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
InReplyTo: <3.0.6.32.20020404214429.00b1bb90@uq.net.au>
David C Keenan wrote:
> What is the first ET above 1200 that is 35limit consistent? 35limit
> unique?
Above 1200? 1600. The only ones below that are 311 and 388. If I can
remember how I did uniqueness for the 11limit, I'll try a search on that
as well.
Anyway, the simplest linear temperament from consistent ETs is
236/699, 405.2 cent generator
basis:
(1.0, 0.3376257079318315)
mapping by period and generator:
[(1, 0), (8, 19), (23, 75), (18, 45), (46, 126), (1, 8), (29, 98),
(11, 20
), (42, 111), (16, 33), (61, 166)]
mapping by steps:
[(388, 311), (615, 493), (901, 722), (1089, 873), (1342, 1076), (1436,
1151), (1
586, 1271), (1648, 1321), (1755, 1407), (1885, 1511), (1922, 1541)]
highest interval width: 316
complexity measure: 316 (388 for smallest MOS)
highest error: 0.001078 (1.293 cents)
The next two are unique
41/497, 24.7 cent generator
basis:
(0.25, 0.020624732047905801)
mapping by period and generator:
[(4, 0), (7, 8), (13, 45), (1, 124), (6, 95), (20, 63), (18, 20), (8,
109),
(11, 86), (21, 19), (10, 119)]
mapping by steps:
[(1600, 388), (2536, 615), (3715, 901), (4492, 1089), (5535, 1342), (5921,
1436)
, (6540, 1586), (6797, 1648), (7238, 1755), (7773, 1885), (7927, 1922)]
highest interval width: 214
complexity measure: 856 (1212 for smallest MOS)
highest error: 0.000284 (0.341 cents)
unique
467/1911, 293.3 cent generator
basis:
(1.0, 0.24437509095621543)
mapping by period and generator:
[(1, 0), (27, 104), (38, 165), (49, 212), (56, 215), (45, 169), (79,
340),
(61, 267), (49, 182), (53, 197), (79, 303)]
mapping by steps:
[(1600, 311), (2536, 493), (3715, 722), (4492, 873), (5535, 1076), (5921,
1151),
(6540, 1271), (6797, 1321), (7238, 1407), (7773, 1511), (7927, 1541)]
highest interval width: 696
complexity measure: 696 (978 for smallest MOS)
highest error: 0.000298 (0.358 cents)
unique
which isn't enough to prove that 1600equal is unique.
Graham
I doublechecked the following and found it to be in error:
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>
> To further complicate things in 311, I also noticed in my notes
that,
> while I favored using 19 as an Eflat (where 1/1 is C), there were
> almost as many instances where a heptatonic scale called for a D
> sharp (using the 19comma 19456:19683, 3^9:2^14*19, ~20.082
cents).
> This usage turns out to be *inconsistent* in 311 (but consistent in
> 217), a problem that I didn't expect to find. While I compared the
> inconsistency of a few ratios of 23 in 217 with driving a car
> slightly onto the shoulder, the 311 problem (affecting all ratios
of
> 19) is more like attempting to drive in the lesstraveled direction
> on route 19 and finding yourself in the wrong lane moving against
> traffic. So I am beginning to have serious doubts about going to
311.
The 19comma 19456:19683 is 2^10*19:3^9 (~20.082 cents), and its
usage is consistent in *both* 217 and 311.
I'm sorry about the misinformation, but I'm very delighted that I was
wrong about this.
George
Dave wrote:
>I had mistakenly thought that 311ET was 31limitunique until Graham
>corrected me. What's the smallest ET that has a 31limitunique mapping,
>even if it's not consistent? Could it be 624ET.
Yes, 624 to 633tET are all 35limit unique.
1600tET is 37limit consistent and 55limit unique.
Manuel
 In tuningmath@y..., graham@m... wrote:
> InReplyTo: <a8fkev+5808@e...>
> gdsecor wrote:
>
> > Something else in favor of 217: I noted earlier that it is 7
times
> > 31. If you make instruments for 31ET (or train string players
to
> > play 31ET), then your JI can be reckoned in alterations of +/ 1
to
> > 3 increments of 217ET (i.e., multiples of ~5.5 cents), to a
maximum
> > of 16.6 cents, which is not an unreasonable amount of intonation
> > adjustment for instruments of flexible pitch. This would be a
decent
> > practical alternative to alterations reckoned relative to 12ET,
> > whether for (toocoarse) 72ET or Johnny Reinhard's (toosmall)
one
> > cent increments. Plus you get 31ET in the bargain. And I
should
> > mention that a series of 41 fifths in 217 (kept within an octave)
> > brings you only one degree away from your starting point, giving
you
> > a very close approximation of 41ET.
monzo was going to make midi mahler renditions in 217equal some time
ago, when we were discussing how 217 is the et that supports adaptive
5limit ji similar to vicentino's second tuning. 152equal does so as
well, but increases the shifts from about 1/4 comma to about 1/3
comma, which may actually be a serious disadvantage when you note
that the jnd can fall between these two values.
i don't think george has discussed adaptive ji, so i thought is was
worth jumping in with this.
Note: I posted this more than a couple of hours ago, but it still
hasn't shown up, so, after having made a correction, I am trying
again. Please disregard the duplicate, as I suspect that it will
eventually show up. gs
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > With the 29 factor we have passed the point (in both 217 and 311)
> > where all of the ratios within the harmonic limit are a unique
number
> > of degrees, so some bridging is inevitable. However, what's most
> > important is that the number of degrees for the 29 comma is
> > consistent with this flag in both 217 (6 deg) and 311 (9 deg).
>
> I had mistakenly thought that 311ET was 31limitunique until
Graham
> corrected me. ...
> > ... I conclude that there has to be a
> > provision for spelling any interval in at least two different
ways,
> > which is a compelling reason for providing a complete set of
symbols
> > for whatever division we settle on for the JI notation.
>
> Oh dear, I think I've agreed with certain things you proposed which
have
> turned out to be the beginning of a slippery slope that I don't
want to go
> down. One was to accept certain schismas vanishing in order to
minimise the
> number of different flags making up the symbols. The other was to
accept
> the need for more than one comma for certain primes.
>
> Now you say we need two commas for _every_ prime. Why stop there?
Shouldn't
> we provide for anyone who wants to notate the prime intervals from
C as
> varieties of any of the following:
> 5 E,Fb
> 7 A#,Bb
> 11 E#,F,F#,Gb
> 13 G#,Ab,A
> 17 C#,Db
> 19 D#,Eb
> 23 F#,Gb
> 29 A#,Bb,Cb
> 31 B,Cb,C
>
> These all involve commas smaller than 3/5 apotome. There are _only_
23 of
> them. I am of course being facetious. I'm more inclined to go back
to a
> strict single comma per prime. In all the lower ETs where either an
> 11comma or a 13comma is a semisharp or semiflat, its apotome
complement
> is the same number of steps, so there isn't any need for both.
That is correct if you are indicating "semisharp" and "semiflat" to
be exactly half an apotome in those ET's (or is this what you really
meant to say?). In 39, 46, and 53ET this is not the case, and you
either need to have both 11commas for the two different kinds of
semisharps/semiflats, or else have the 11 comma for one and the 13
comma for the other (although these would then be faux complements).
I think that I have probably misunderstood what you were trying to
say and that my going on about this is sort of pointless, considering
that we are getting the notation for the secondary 11 and 13 commas
as a freebie with the 4095:4096 schisma. I think that your point is
that each pair of 11 and 13 commas consists of apotomecomplements,
so that the larger of each could be notated as an apotome minus the
smaller; hence we really have only one comma for each prime in the
semantics of the notation; but for convenience in handling the
semisharps & semiflats we have symbols that make it appear as if
there were two.
Anyway, in pointing out the need for being able to respell ratios at
will, my primary objective was to indicate why we needed to notate
all of the degrees in the ET, not to open up a can of worms whereby
we need to have multiple commas per prime. One apiece should do.
> > ... So the question now becomes: Are we left with any good
reason for
> > basing the JI notation on 311 instead of 217?
>
> From your point of view, I would say that you are better off with
217ET.
This amounts, then, to a 19limitunique&consistent, polyphonic
readable sagittal notation with nonunique capability up to the 35
odd limit. That sounds like something that fulfills (and in some
ways exceeds) our original objective (as I understood it).
> However I do not wish to base a JI (rational) notation on _any_
temperament
> that has errors larger than 0.5 c. For me, 217ET and 311ET were
merely a
> way of looking for schismas that might be notationally usable (less
than
> 0.5 c), and of checking that things were working sensibly, and it
was nice
> to actually be able to notate those ETs themselves. But I'm taking
Johnny
> Reinhard at his word when he says (or implies) that nothing less
than
> 1200ET is good enough as an ETbased JI notation.
>
> What is the first ET above 1200 that is 35limit consistent? 35
limit unique?
Graham suggests 1600ET in his message #3947. It looks like a good
choice, inasmuch as:
1) It is 37limit consistent;
2) The largest error for any 37limit consonance is ~0.36441 cents
(for 19:25);
3) It conflates all three of my schismas: 4095:4096, 3519:3520, and
20735:20736 (but not the 31schisma that I tried, 59024:59049, which
was also unusable in 311);
4) And I strongly suspect that it will be found to be at least 37
limit unique, inasmuch as the largest superparticular ratios that are
not unique are 54:55 and 55:56 (both 42deg1600).
> But I am more interested in a 31limit (or 35limit) temperament
(not equal
> or linear or even planar, but maybe 6D or 7D or 8D) with as many
> subhalfcent schismas as possible, vanishing. Hence the challenge
which
> noone's taken up yet, except in part.
Well, good luck on that one!
> I don't share your obsession with packing all the necessary
information
> into a single symbol. Some folks may well be willing to notate
their JI
> piece by mapping to the nearest degree of 217ET, and they may well
be glad
> of the possibility to do it with only one accidental per note, but
that
> won't be everyone's cup of tea. Given that it may involve errors of
up to 3
> cents, it must be a conscious decision, not something forced upon
us by the
> notation.
You're talking about two separate issues here:
1) One altering symbol per note:
Since we seem to have concluded that the sagittal JI notation is
going with 217, we can now determine what each ET notation is going
to look like. This means that I can now go ahead and present my Sims
vs. sagittal comparison on the main tuning list without having to
issue a caveat that the 72ET sagittal notation might not be the
final version. In that comparison I will show an instance in which
double symbols (Sims or otherwise) could present some confusion. I
will also make the point that, once they are learned, single symbols
can be read more quickly than double ones (particularly in chords),
since there is less to read.
2) Mapping to 217ET:
There is a question that needs to be asked: are we notating JI or are
we notating 217ET? I understood that we were notating JI (mapped
onto 217 for convenience in understanding some of the size
relationships among the various ratios), which makes discussion about
3cent errors a bit irrelevant.
Now one may also want to make use of the 217 mapping as a convenience
in conceptualizing a way of arriving at the approximate pitches
represented by those ratios (which are in turn represented by symbols
that correlate with a 217 mapping), and without any finetuning (by
ear) you would be entitled to contemplate 3cent errors. (Come to
think of it, anyone coming within 3 cents  less than the Miracle
tuning minimax deviation  is doing pretty well by almost anybody's
standard.) But this is more of a matter of how the composer is going
to treat the notation:
1) Either sticking with a specific set of ratios (in which case the
ratios and symbols could, for reference, be listed in a table
alongside each other, with cents values, if that helps), in which
case the 217 mapping (and error thereof) would have little or no
relevance;
2) Or else freely employing whatever intervals are permitted by the
notation, with little regard to keeping track of ratios, in which
case it could very well turn into (at best) a 217ET performance.
Putting this another way: If I write a piece for 13limit JI using
just 12 tones per octave and map the tones (consistently) into 12,
specifying the ratios that I want for each position, would you be
entitled to claim that I would be getting errors close to 50 cents
for some of the tones if I used the 12ET notation?
I believe the problem is more of a matter of how the composer and
performer understand the notation than with the notation itself.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > In all the lower ETs where either an
> > 11comma or a 13comma is a semisharp or semiflat, its apotome
> complement
> > is the same number of steps, so there isn't any need for both.
>
> That is correct if you are indicating "semisharp" and "semiflat" to
> be exactly half an apotome in those ET's (or is this what you really
> meant to say?).
I was being sloppy. I didn't check the facts before I wrote. Sorry.
> In 39, 46, and 53ET this is not the case, and you
> either need to have both 11commas for the two different kinds of
> semisharps/semiflats, or else have the 11 comma for one and the 13
> comma for the other (although these would then be faux complements).
Thanks. I agree that 46 and 53 are the first where this is a problem.
39ET is 1,3,13inconsistent and one can choose a number of steps for
1024:1053 that is the apotome complement of 32:33, but this is not
possible for 46 and 53. So I guess you're assuming 26:27 as _the_ 13
comma. Or are you assumng 704:729 as _the_ 11 comma?
> I think that I have probably misunderstood what you were trying to
> say and that my going on about this is sort of pointless,
considering
> that we are getting the notation for the secondary 11 and 13 commas
> as a freebie with the 4095:4096 schisma. I think that your point is
> that each pair of 11 and 13 commas consists of apotomecomplements,
> so that the larger of each could be notated as an apotome minus the
> smaller; hence we really have only one comma for each prime in the
> semantics of the notation; but for convenience in handling the
> semisharps & semiflats we have symbols that make it appear as if
> there were two.
That wasn't my point. As I say, I was just careless. But it's a good
point. Thanks for making it.
> Anyway, in pointing out the need for being able to respell ratios at
> will, my primary objective was to indicate why we needed to notate
> all of the degrees in the ET, not to open up a can of worms whereby
> we need to have multiple commas per prime. One apiece should do.
Ok. Good. But hopefully we can have less than one apiece, with the
other subhalfcent schismas you found.
> > > ... So the question now becomes: Are we left with any good
> reason for
> > > basing the JI notation on 311 instead of 217?
> >
> > From your point of view, I would say that you are better off with
> 217ET.
>
> This amounts, then, to a 19limitunique&consistent, polyphonic
> readable sagittal notation with nonunique capability up to the 35
> odd limit. That sounds like something that fulfills (and in some
> ways exceeds) our original objective (as I understood it).
Sure. But I don't understand what 217ET or 311ET have to do with it.
217ET just happens to be the highest ET that you can notate with it.
The definitions of the symbols must be based on the commas, not the
degrees of 217ET.
What do you mean by "polyphonicreadable"? As opposed to what?
> > However I do not wish to base a JI (rational) notation on _any_
> temperament
> > that has errors larger than 0.5 c. For me, 217ET and 311ET were
> merely a
> > way of looking for schismas that might be notationally usable
(less
> than
> > 0.5 c), and of checking that things were working sensibly, and it
> was nice
> > to actually be able to notate those ETs themselves. But I'm taking
> Johnny
> > Reinhard at his word when he says (or implies) that nothing less
> than
> > 1200ET is good enough as an ETbased JI notation.
> >
> > What is the first ET above 1200 that is 35limit consistent? 35
> limit unique?
>
> Graham suggests 1600ET in his message #3947. It looks like a good
> choice, inasmuch as:
>
> 1) It is 37limit consistent;
>
> 2) The largest error for any 37limit consonance is ~0.36441 cents
> (for 19:25);
>
> 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and
> 20735:20736 (but not the 31schisma that I tried, 59024:59049, which
> was also unusable in 311);
>
> 4) And I strongly suspect that it will be found to be at least 37
> limit unique, inasmuch as the largest superparticular ratios that
are
> not unique are 54:55 and 55:56 (both 42deg1600).
>
> > But I am more interested in a 31limit (or 35limit) temperament
> (not equal
> > or linear or even planar, but maybe 6D or 7D or 8D) with as many
> > subhalfcent schismas as possible, vanishing. Hence the challenge
> which
> > noone's taken up yet, except in part.
>
> Well, good luck on that one!
You have in effect found one of these, with your 3 schismas. I'd just
like to be sure that there isn't a whole 'nother larger set of
31limit schismas that give an even better compression of the number
of flags, without exceeding the 0.5 cent.
But I guess I'll just assume 1600ET. In other words, any schisma that
vanishes in 1600ET is acceptable to be built into the notation, but
certainly not all the schismas that vanish in 217ET (or 311ET or
388ET).
So in that sense only, I could say that the notation is "based on"
1600ET, but there is certainly no desire to notate every degree of
1600ET using a single accidental per note, or even a single
accidental in addition to a sharp or flat. In fact there is no desire
to notate 1600ET at all, and it is fine that 217ET is the highest ET
that can be so notated.
> > I don't share your obsession with packing all the necessary
> information
> > into a single symbol. Some folks may well be willing to notate
> their JI
> > piece by mapping to the nearest degree of 217ET, and they may
well
> be glad
> > of the possibility to do it with only one accidental per note, but
> that
> > won't be everyone's cup of tea. Given that it may involve errors
of
> up to 3
> > cents, it must be a conscious decision, not something forced upon
> us by the
> > notation.
>
> You're talking about two separate issues here:
>
> 1) One altering symbol per note:
>
> Since we seem to have concluded that the sagittal JI notation is
> going with 217, we can now determine what each ET notation is going
> to look like. This means that I can now go ahead and present my
Sims
> vs. sagittal comparison on the main tuning list without having to
> issue a caveat that the 72ET sagittal notation might not be the
> final version.
Agreed. I'm pretty sure it's only the concave flags whose meaning
could still change.
> In that comparison I will show an instance in which
> double symbols (Sims or otherwise) could present some confusion. I
> will also make the point that, once they are learned, single symbols
> can be read more quickly than double ones (particularly in chords),
> since there is less to read.
>
> 2) Mapping to 217ET:
>
> There is a question that needs to be asked: are we notating JI or
are
> we notating 217ET? I understood that we were notating JI (mapped
> onto 217 for convenience in understanding some of the size
> relationships among the various ratios), which makes discussion
about
> 3cent errors a bit irrelevant.
OK. Good. So I wish you'd stop talking about it being "based on" or
"going with" 217ET, or any other ET with larger than 0.5 cent errors.
> Now one may also want to make use of the 217 mapping as a
convenience
> in conceptualizing a way of arriving at the approximate pitches
> represented by those ratios (which are in turn represented by
symbols
> that correlate with a 217 mapping), and without any finetuning (by
> ear) you would be entitled to contemplate 3cent errors. (Come to
> think of it, anyone coming within 3 cents  less than the Miracle
> tuning minimax deviation  is doing pretty well by almost anybody's
> standard.) But this is more of a matter of how the composer is
going
> to treat the notation:
>
> 1) Either sticking with a specific set of ratios (in which case the
> ratios and symbols could, for reference, be listed in a table
> alongside each other, with cents values, if that helps), in which
> case the 217 mapping (and error thereof) would have little or no
> relevance;
Yes. This is more fundamental to the notation.
> 2) Or else freely employing whatever intervals are permitted by the
> notation, with little regard to keeping track of ratios, in which
> case it could very well turn into (at best) a 217ET performance.
Yes, so 217ET is just one ET that could be used in this way. The
notation is not based on it. It just happens to be the highest one
that is fully notatable with single symbols.
> Putting this another way: If I write a piece for 13limit JI using
> just 12 tones per octave and map the tones (consistently) into 12,
> specifying the ratios that I want for each position, would you be
> entitled to claim that I would be getting errors close to 50 cents
> for some of the tones if I used the 12ET notation?
No.
> I believe the problem is more of a matter of how the composer and
> performer understand the notation than with the notation itself.
Yes. I'm glad we understand each other. Showing how the notation maps
to 217ET is no different from showing how it maps to 22ET. The
_definition_ of the symbols is in terms of the commas and schismas.
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and
> 20735:20736 (but not the 31schisma that I tried, 59024:59049, which
> was also unusable in 311);
59024:59049 (2^4*7*17*31:3^10) doesn't pass the Reinhard test anyway,
being 0.73 c, however it might tempt me if it could be combined with
other suitable schismas, as per my challenge.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > 3) It conflates all three of my schismas: 4095:4096, 3519:3520,
and
> > 20735:20736 (but not the 31schisma that I tried, 59024:59049,
which
> > was also unusable in 311);
>
> 59024:59049 (2^4*7*17*31:3^10) doesn't pass the Reinhard test
anyway,
> being 0.73 c, however it might tempt me if it could be combined with
> other suitable schismas, as per my challenge.
We can forget about that 31schisma. What's wrong with 253935:253952
(3^5*5*11*19 : 2^13*31) 0.12 cents. Consistent with 311ET 388ET
1600ET, but not 217ET.
31 comma = (11 comma  5 comma) + 19 comma
Since (11 comma  5 comma) is a single flag and 19 comma is a single
flag (or blob) then this 31 comma can be represented by a pair of
flags. The fact that it doesn't work in 217ET doesn't matter because
the notation is not "based on" 217ET and the 31 comma is not needed
in order to notate 217ET.
George,
Here's another pass at a full set of 31limit symbols, taken simply as one
symbol per prime from 5 to 31. Whadya think?
[If you're reading this on the yahoogroups website you will need to
choose Message Index, Expand Messages, to see the following symbols
rendered correctly.]
5comma 80:81
/
/ 
 \ /


7comma 63:64
_
 \
 
 L P


11comma 32:33
/\
/  \
 v ^


13comma 1024:1053
_
/ \
/  
 { } flags based on vanishing of schisma 4095:4096


17comma 2176:2187

_/
 j f


19comma 512:513
_
(_)

 o *


23comma 729:736

\_
 w m


29comma 256:261
_
/ 
 
 q d flag based on vanishing of schisma 20735:20736


31comma 243:248
_
(_)
 \
 y h flags based on vanishing of schisma 253935:253952


We also have optional symbols for larger 11, 13 and 23 commas.
11'comma 704:729
_ _
/  \
  
 [ ] flags based on vanishing of schisma 5103:5104


13'comma 26:27
_
/ \
  \
 ;  flags based on vanishing of schisma 20735:20736


23'comma 16384:16767
\
_/ \
 W M flags based on vanishing of schisma 3519:3520


 Dave Keenan
Brisbane, Australia
http://dkeenan.com
Or perhaps the 19 and 31 commas should be:
19comma 512:513
_
(_)




and 31comma 243:248
_
(_)\
 \



or 31comma 243:248
_
(_)
\
 \


The circle was always intended to be filled, and is now a kind of left
flag rather than central. This eliminates a lot of possible redundant
combinations, and the attendant lateral confusability, by making it
only combinable with right flags. It is also nice that the 17 and 19
flags look a little like the digits 7 and 9 respectively.
In case anyone cares; there's a 37schisma that vanishes in 1600ET (and
388ET) that lets us make a symbol for a 37comma, with no new flags.
The 37 comma is 999:1024 (3^3*37 : 2^10) 42.8 cents, so a 1:37 from C is a
lowered Eb.
The schisma is 570236193:570425344 (3^12*29*37 : 2^25*17) 0.57 cents, which
means the 37 arrowhead combines the 17flag and the 29flag. Only trouble
is, these are presently both left flags, concave and convex.
I believe we have a rule that says that at each prime limit, the symbols
should be as simple as possible and that no higher prime should be allowed
to "reach down" and cause us to change the way we do the flags for a lower
limit.
However, the only reason so far, to make the 17 flag a left flag is to make
a symbol for the large 23 comma by combining the 17 flag with the 115
flag. It isn't essential to have a symbol for the large 23 comma, so the 17
flag and (small) 23 flag (both concave) could swap sides. 8 steps of 217ET
could still be notated as either the 25 symbol (sL+sL) or 7 flag + 23 flag
(vL+xR).
1:37 from C is very close to halfway between D and Eb, a pythagorean limma
apart, so there is a good argument for needing the large 37 comma of
(36:37) 47.4 cents as well. There is a 1600ET schisma that gives us the
large 37 comma without new flags, provided we're willing to combine 3 of
them, 19 flag + 23 flag + 7 flag (cL+vL+xR), which actually do combine ok.
The schisma is 6992:6993 (2^4*19*23 : 3^3*7*37) 0.25 cents.
This is probably all pretty silly, catering for 37, and we should probably
just forget it and keep the large 23 comma symbol, but here's a pass at a
full set of 37limit symbols anyway.
[If you're reading this on the yahoogroups website you will need to
choose Reply or Message Index, Expand Messages, to see the following
symbols rendered correctly.]
5comma 80:81
/
/ 
 \ /


7comma 63:64
_
 \
 
 L P


11comma 32:33
/\
/  \
 v ^


13comma 1024:1053
_
/ \
/  
 { } flags based on vanishing of schisma 4095:4096


17comma 2176:2187

\_
 j f


19comma 512:513
_
(_)

 o *


23comma 729:736

_/
 w m


29comma 256:261
_
/ 
 
 q d flag based on vanishing of schisma 20735:20736


31comma 243:248
_
(_)\
 \
 y h flags based on vanishing of schisma 253935:253952


37comma 999:1024
_
/ 
 \_
 flags based on vanishing of schisma 570236193:570425344


We also have optional symbols for larger 11, 13 and 37 commas.
11'comma 704:729
_ _
/  \
  
 [ ] flags based on vanishing of schisma 5103:5104


13'comma 26:27
_
/ \
  \
 ;  flags based on vanishing of schisma 20735:20736


37'comma 36:37
_ _
(_) \
_/ 
 flags based on vanishing of schisma 6992:6993


If we really still wanted a symbol for the large 23 comma I guess we could
still combine the 17 and 115 flags like this:
23'comma 16384:16767

\_
 \ W M flags based on vanishing of schisma 3519:3520


 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > > > ... So the question now becomes: Are we left with any good
reason for
> > > > basing the JI notation on 311 instead of 217?
> > >
> > > From your point of view, I would say that you are better off
with 217ET.
> >
> > This amounts, then, to a 19limitunique&consistent, polyphonic
> > readable sagittal notation with nonunique capability up to the
35
> > odd limit. That sounds like something that fulfills (and in some
> > ways exceeds) our original objective (as I understood it).
>
> Sure. But I don't understand what 217ET or 311ET have to do with
it.
> 217ET just happens to be the highest ET that you can notate with
it.
> The definitions of the symbols must be based on the commas, not the
> degrees of 217ET.
>
> What do you mean by "polyphonicreadable"? As opposed to what?
As opposed to polyphonicconfusible or polyphonicdifficultorslow
toread. This was just my way of putting in another plea for single
symbol modifications to notes  my obsession, as you call it.
> > > However I do not wish to base a JI (rational) notation on
_any_ temperament
> > > that has errors larger than 0.5 c. For me, 217ET and 311ET
were merely a
> > > way of looking for schismas that might be notationally usable
(less than
> > > 0.5 c), and of checking that things were working sensibly, and
it was nice
> > > to actually be able to notate those ETs themselves. But I'm
taking Johnny
> > > Reinhard at his word when he says (or implies) that nothing
less than
> > > 1200ET is good enough as an ETbased JI notation.
So (as I see it) Johnny's obsession has become yours as well. As I
said before, I really don't think that an underlying ET needs to have
that much accuracy  it's going to take a great deal of skill and
concentration to hold a sustained pitch that steady on an instrument
of flexible pitch, and if it's of short duration, then it would be
pretty difficult to perceive an error of, say, 3 cents, except on
laboratory equipment (which I wouldn't expect anyone to bring to a
concert). This is why I feel that 217ET is adequate: it puts you
close enough for most purposes, and if that is not close enough
(meaning that can still hear that you're not close enough), then you
can make a superfine correction in intonation by ear. I should
emphasize that those intervals in which you are most likely to be
able to hear 2cent errors are the 5limit consonances, none of which
have an error greater than 1 cent in 217ET.
Anyway, I expect that we can allow for each other's obsessions and
can continue to work on this together to achieve both of our
objectives.
> > There is a question that needs to be asked: are we notating JI or
are
> > we notating 217ET? I understood that we were notating JI
(mapped
> > onto 217 for convenience in understanding some of the size
> > relationships among the various ratios), which makes discussion
about
> > 3cent errors a bit irrelevant.
>
> OK. Good. So I wish you'd stop talking about it being "based on" or
> "going with" 217ET, or any other ET with larger than 0.5 cent
errors.
How about a compromise in which we "go with" both 217 and 1600ET (37
limit), with a specific set of symbols for 217 and a superset for
1600? (This might also make it possible to notate 311ET using the
full set of symbols.) I am suggesting this in light of your
observation:
> Yes, so 217ET is just one ET that could be used in this way. The
> notation is not based on it. It just happens to be the highest one
> that is fully notatable with single symbols.
This is one point that has become all too apparent, as you have
proceeded (in your subsequent messages) to suggest changes in the
symbols that:
1) Go beyond the three types of flags (straight, convex, & concave)
that work so elegantly for 217 (remember that I said that something
that could be regarded as "overkill" was immune to criticism as long
as the additional complexity didn't make it more difficult to do the
simpler things; this introduces more complexity for 217ET);
2) Introduce new symbols that I have no idea how to incorporate into
a singlesymbol notation (this makes it difficult to do something
that I was previously able to do with 217ET); and
3) Employ semantics inconsistent with 217ET, as in the following:
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > > 3) It conflates all three of my schismas: 4095:4096, 3519:3520,
and
> > > 20735:20736 (but not the 31schisma that I tried, 59024:59049,
which
> > > was also unusable in 311);
> >
> > 59024:59049 (2^4*7*17*31:3^10) doesn't pass the Reinhard test
anyway,
> > being 0.73 c, however it might tempt me if it could be combined
with
> > other suitable schismas, as per my challenge.
>
> We can forget about that 31schisma. What's wrong with 253935:253952
> (3^5*5*11*19 : 2^13*31) 0.12 cents. Consistent with 311ET 388ET
> 1600ET, but not 217ET.
>
> 31 comma = (11 comma  5 comma) + 19 comma
>
> Since (11 comma  5 comma) is a single flag and 19 comma is a
single
> flag (or blob) then this 31 comma can be represented by a pair of
> flags. The fact that it doesn't work in 217ET doesn't matter
because
> the notation is not "based on" 217ET and the 31 comma is not
needed
> in order to notate 217ET.
I made it a point to think very carefully before replying to your
subsequent messages, because I know you spent a lot of time and
effort on the content and have come up with some very good things,
such as the 31schisma (above). During the two weeks or so that I
spent leading up to my 17limit (183tone) and 23limit (217tone)
approaches, I also spent a lot of time trying various things, and I
don't consider the time wasted that I spent on ideas that I
subsequently discarded. In the process of developing a notation such
as this, you want to try as many things as you can possibly think of,
because that best enables you to see why the method that is finally
chosen is the best one. I wanted to find a way to resolve this that
would satisfy both of our requirements.
Here is the compromise that I am proposing: Let's keep the 217ET
based symbols as they are, defining 2176:2187 as xL and 512:513 as
xR, with their combination allowed to represent either 4096:4131 or
729:736 as required (in 217ET or another ET, where consistent, but
incapable of being combined with anything else). Then, for the 1600
based notation, let's expand on that with a combination of the
following methods:
1) Allow two flags to appear on the same side, as was suggested for
6400:6561, the 25 comma. This would then allow us to use sR+vR (with
the concave flag at the top of an upwardpointing arrow) to notate
the 31comma 243:248, using the schisma 353935:253952. Also, the
alternate 37comma 999:1024 could be notated with xL+vL, using the
schisma 570236193:570425344. (We would have to experiment to see how
this would be done. With the convex flag at the end, the two would
form a sort of loop; or they might be made to interlock.)
2) Define one or more additional types of flags to notate new primes,
beginning with a new left one for the 23comma, 729:736. This would
then allow us to use newL+xR+vR to notate the 37comma 36:37, using
the schisma 6992:6993. (Thus, the symbols for the two 37commas both
contain a combination of a convex and concave flag on the same side,
which is most appropriate!)
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> This is probably all pretty silly, catering for 37, and we should
probably
> just forget it and keep the large 23 comma symbol, but here's a
pass at a
> full set of 37limit symbols anyway.
Silly or not, I think we should keep whatever capability we can, as
long as it is consistent. And I would prefer to keep *both* the
large 23 comma symbol and a full set of 37limit symbols, as with
this "compromise."
Overkill? Maybe, but it keeps the simpler things simple, while
serving those who want a lot of capability. And it does follow the
nomorethanonenewcommaper prime guideline throughout. Also,
there are some divisions between 100 and 217 that the 217notation
won't handle (such as 140), for which I would expect that the
extended set of symbols could be used.
So how does that grab you?
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Sure. But I don't understand what 217ET or 311ET have to do with
> it.
> > 217ET just happens to be the highest ET that you can notate with
> it.
> > The definitions of the symbols must be based on the commas, not
the
> > degrees of 217ET.
> >
> > What do you mean by "polyphonicreadable"? As opposed to what?
>
> As opposed to polyphonicconfusible or polyphonicdifficultorslow
> toread. This was just my way of putting in another plea for
single
> symbol modifications to notes  my obsession, as you call it.
And a very fine obsession it is. I do not want to deflect you from it in
the slightest. I would merely like it recognised that it is not the most
general use of the notation. The most general is: by using more than one
symbol at a time one can uniquely notate any rational pitch up to a 37
prime limit. So in this way of using it, it is not even "based on" 1600ET.
1600ET was merely used in determining the symbols for the prime commas,
after which the symbols are considered atomic.
> So (as I see it) Johnny's obsession has become yours as well.
Not personally, but I think it wise to recognise that his opinions are
widely respected in microtonal circles, and so if we hope for this notation
to achieve wide acceptance we might as well eliminate the possible
objection that it does not allow one to uniquely notate 19primelimit
rational pitches which may be _more_ than 3 cents apart.
e.g. 19/14 and 34/25 are the same in 217ET, but differ by 475:476 or 3.6
cents. Here's an extreme example, which I admit is unlikely to be
encountered in real life. The comma 29229255:29360128 (3^12*5*11 : 7*2^22
vanishes in 217ET but is 7.7 cents in rational tuning.
> As I
> said before, I really don't think that an underlying ET needs to
have
> that much accuracy  it's going to take a great deal of skill and
> concentration to hold a sustained pitch that steady on an instrument
> of flexible pitch, and if it's of short duration, then it would be
> pretty difficult to perceive an error of, say, 3 cents, except on
> laboratory equipment (which I wouldn't expect anyone to bring to a
> concert). This is why I feel that 217ET is adequate: it puts you
> close enough for most purposes, and if that is not close enough
> (meaning that can still hear that you're not close enough), then you
> can make a superfine correction in intonation by ear. I should
> emphasize that those intervals in which you are most likely to be
> able to hear 2cent errors are the 5limit consonances, none of
which
> have an error greater than 1 cent in 217ET.
Except for the last sentence, I have posted similar opinions to the tuning
list myself many times over the years. It's curious that I chose 2.8 cent
maximum error as my (fairly arbitrary) cutoff for what I consider a
"microtemperament", without ever considering it as a halfstep of 217ET.
For example, I consider 72ET to be a 7limit microtemperament, but not a
9limit or higher one. 217ET is therefore the smallest ET that is a
21limit microtemperament, and if that cutoff were bumped to 2.9 cents it
would be a 35limit microtemperament (max 37limit error is 4.6 cents).
311ET is a 45limit microtemperament and has no error greater than 1.9
cents in the 41limit. So 311ET is way more than we need from this point
of view, and 217ET is just right.
By the way, 1600ET gets us to the 45limit without exceeding 0.5 cents
error, but there is no way to get its 41 or 43 commas by combining existing
flag commas, not even 3 or more of them with multiple flags allowed per
side. Thank goodness! 37 is already more than we need.
> Anyway, I expect that we can allow for each other's obsessions and
> can continue to work on this together to achieve both of our
> objectives.
Absolutely. I am immensely enjoying working with you on this.
> > OK. Good. So I wish you'd stop talking about it being "based on"
or
> > "going with" 217ET, or any other ET with larger than 0.5 cent
> errors.
>
> How about a compromise in which we "go with" both 217 and 1600ET
(37
> limit), with a specific set of symbols for 217 and a superset for
> 1600? (This might also make it possible to notate 311ET using the
> full set of symbols.)
OK. Except I'd probably prefer to put it this way:
The notation is based on pythagorean AG,#,b, with the addition of a pair
of arrow symbols (up and down) for each prime number from 5 to 37. Each
pair of arrow symbols corresponds to a comma that is smaller than a
halfapotome (56.8 cents) and relates the prime number to a chain of
between 4 and 7 fifths, ignoring octaves. That's from Ab to C# relative to C.
This requires 10 new pairs of symbols, which might be hard to learn and
might result in some notes having a ridiculous number of accidentals before
them, except that the symbols are not atomic. They are themselves made up
of a vertical shaft with only 4 kinds of halfarrowhead or flag. Most of
these flags come in left and right varieties for a total of 7 kinds of flag
(ignoring up and down varieties).
These 7 flags correspond to the commas for the primes 5, 7, 11*, 17, 19,
23, 29. The symbols for the commas for 13, 31 and 37 and some optional
additional commas, are obtained by combining flags on the same shaft
according to an arithmetic which corresponds to simple addition of the
nearest 1/1600ths of an octave.
* The 11 comma is symbolised, not by a single flag but by a new flag
combined with the 5 flag, and so we refer to this new flag as the 115 flag.
Because we use 1600ET for this flag arithmetic, if we choose to combine
multiple symbols into a single symbol we can do so without introducing any
error greater than about half a cent.
The system is designed so that at each prime limit lower than 37, it is as
simple as possible. No higher prime has been allowed to complicate the
system for those who don't need it. Here are the numbers of different flags
that must be learnt at each prime limit
5 1
7 2
11 3
13 3
17 4
19 5
23 6
29 7
31 7
37 7
Although we've so far described this as a notation for purely rational
scales, it works beautifully for equal temperaments too. [explain how 
choose your fifth etc.]
In the case of equal temperaments we use only the symbols for the lowest
primes, or combinations thereof, that are necessary to notate each step. It
turns out that one doesn't need to go past 19limit to notate most ETs of
interest.
217ET is the largest ET that can be notated by this method, using only one
symbol per note (in addition to a possible sharp or flat symbol). 217ET
has no error greater than 2.9 cents in the 35limit, and so provided that
such errors are acceptable, we can use it to notate up to 35limit rational
scales using only one symbol per note.
So far we have assumed that the arrow symbols will be used in conjunction
with conventional sharp and flat symbols, but this is not necessary either.
The system includes additional arrow symbols, which use the same flags
(half arrowheads) but have multiple shafts to the arrow. These can cover
the range from a doubleflat to a doublesharp using single symbols.
> I am suggesting this in light of your
> observation:
>
> > Yes, so 217ET is just one ET that could be used in this way. The
> > notation is not based on it. It just happens to be the highest one
> > that is fully notatable with single symbols.
>
> This is one point that has become all too apparent, as you have
> proceeded (in your subsequent messages) to suggest changes in the
> symbols that:
>
> 1) Go beyond the three types of flags (straight, convex, & concave)
> that work so elegantly for 217 (remember that I said that something
> that could be regarded as "overkill" was immune to criticism as long
> as the additional complexity didn't make it more difficult to do the
> simpler things; this introduces more complexity for 217ET);
217ET only needs 19limit, correct? I don't understand why you consider
that changing the 19flag to something other than a concave flag is an
increase in complexity. The 5 limit uses only a straight left flag. We
didn't require that the 7 limit use the straight right flag but went to a
convex flag and didn't use the straight right until 11limit. This would be
similar; delaying the use of the other convex flag until 23 limit; and
could be justified on exactly the same grounds, namely eliminating lateral
confusability from the 19limit (and thereby greatly reducing it in 217ET).
> 2) Introduce new symbols that I have no idea how to incorporate into
> a singlesymbol notation (this makes it difficult to do something
> that I was previously able to do with 217ET); and
I think this is the big one, but I have a proposed solution. Later.
> 3) Employ semantics inconsistent with 217ET, ...
I don't see this as a problem because I don't think that anything employing
those semantics is required in order to notate 217ET
> I made it a point to think very carefully before replying to your
> subsequent messages, because I know you spent a lot of time and
> effort on the content and have come up with some very good things,
> such as the 31schisma (above). During the two weeks or so that I
> spent leading up to my 17limit (183tone) and 23limit (217tone)
> approaches, I also spent a lot of time trying various things, and I
> don't consider the time wasted that I spent on ideas that I
> subsequently discarded. In the process of developing a notation
such
> as this, you want to try as many things as you can possibly think
of,
> because that best enables you to see why the method that is finally
> chosen is the best one. I wanted to find a way to resolve this that
> would satisfy both of our requirements.
I totally agree.
> Here is the compromise that I am proposing: Let's keep the 217ET
> based symbols as they are, defining 2176:2187 as xL and 512:513 as
> xR, with their combination allowed to represent either 4096:4131 or
> 729:736 as required (in 217ET or another ET, where consistent, but
> incapable of being combined with anything else). Then, for the
1600
> based notation, let's expand on that with a combination of the
> following methods:
>
> 1) Allow two flags to appear on the same side, as was suggested for
> 6400:6561, the 25 comma. This would then allow us to use sR+vR
(with
> the concave flag at the top of an upwardpointing arrow) to notate
> the 31comma 243:248, using the schisma 353935:253952. Also, the
> alternate 37comma 999:1024 could be notated with xL+vL, using the
> schisma 570236193:570425344. (We would have to experiment to see
how
> this would be done. With the convex flag at the end, the two would
> form a sort of loop; or they might be made to interlock.)
I have no objection to using multiple flags on the same side, to notate
primes beyond 29. However I consider 999:1024 to be the standard 37 comma
because it is smaller than 36:37, also because it only requires 2
lowerprime flags instead of 3. Can you explain why you want 36:37 to be
the standard 37 comma?
> 2) Define one or more additional types of flags to notate new
primes,
> beginning with a new left one for the 23comma, 729:736.
Beginning and ending with a new 23flag. 7 flags is enough.
> This would
> then allow us to use newL+xR+vR to notate the 37comma 36:37, using
> the schisma 6992:6993. (Thus, the symbols for the two 37commas
both
> contain a combination of a convex and concave flag on the same side,
> which is most appropriate!)
Other combinations might have other kinds of appropriateness, such as one
containing the other flipped horizontally.
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > This is probably all pretty silly, catering for 37, and we should
> probably
> > just forget it and keep the large 23 comma symbol, but here's a
> pass at a
> > full set of 37limit symbols anyway.
>
> Silly or not, I think we should keep whatever capability we can, as
> long as it is consistent. And I would prefer to keep *both* the
> large 23 comma symbol and a full set of 37limit symbols, as with
> this "compromise."
OK. But I'd prefer a slightly different compromise where the 19 flag is the
one that is other than straight, convex or concave and gives the impression
of being smaller than any of them. So the following has the 19 and 23 flags
swapped relative to your suggestion.
17 vL
19 smallL
23 vR
23' vL + sR
31 smallL + sR
37 xL + vL (999:1024)
37' smallL + vR + xR (36:37)
Now to the problems that occur when you try to make this work for 217ET
with the full sagittal treatment, i.e. no # or b.
Here's what you wrote earlier about the notation of apotome complements:
>By the way, something else I figured out over the weekend is how to
>notate 13 through 20 degrees of 217 with single symbols, i.e., how to
>subtract the 1 through 8degree symbols from the sagittal apotome
>(/\). The symbol subtraction for notation of apotome complements
>works like this:
>
>For a symbol consisting of:
>1) a left flag (or blank)
>2) a single (or triple) stem, and
>3) a right flag (or blank):
>4) convert the single stem to a double (or triple to an X);
>5) replace the left and right flags with their opposites according to
>the following:
> a) a straight flag is the opposite of a blank (and vice versa);
> b) a convex flag is the opposite of a concave flag (and vice versa).
>
>This produces a reasonable and orderly progression of symbols
>(assuming that 63:64 is a curved convex flag; it does not work as
>well with 63:64 as a straight flag) that is consistent with the
>manner in which I previously employed the original sagittal symbols
>for various ET's.
The problem I have with this (even assuming _your_ suggested compromise) is
that, while the opposite of sL and sR must certainly be blanks if the
apotome is to be a doubleshafted sL+sR, the other opposites are entirely
arbitrary. What I dislike about the result of your choice is that, having
learnt that xL is larger than sL, I now find that when they have a double
shaft under them, the order of these two is reversed, while all the others
remain the same.
Why can't we simply give a fixed comma value to the second shaft (and so on
for subsequent shafts), so the ordering of flag combinations learnt for the
first halfapotome is simply repeated in the second halfapotome (and all
other halfapotomes). To do this, the second shaft need only be declared
equal in value to xL+xR.
Another advantage of this is that one does not need to use flags that
properly belong to higher limits in the second and subsequent halfapotomes
of lower limit rational notations, or of ET notations based on lower
limits. e.g. There will be no concave flags (or small flag) in 72ET. And
there will be no need for xL or vR in 217ET.
This also solves your problem number 2 above.
Objections?
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > How about a compromise in which we "go with" both 217 and 1600ET
> > (37limit), with a specific set of symbols for 217 and a superset
for
> > 1600? (This might also make it possible to notate 311ET using
the
> > full set of symbols.)
>
> OK. Except I'd probably prefer to put it this way:
>
> The notation is based on pythagorean AG,#,b, with the addition of
a pair
> of arrow symbols (up and down) for each prime number from 5 to 37.
Each
> pair of arrow symbols corresponds to a comma that is smaller than a
> halfapotome (56.8 cents) and relates the prime number to a chain of
> between 4 and 7 fifths, ignoring octaves. That's from Ab to C#
relative to C.
>
> This requires 10 new pairs of symbols, which might be hard to learn
and
> might result in some notes having a ridiculous number of
accidentals before
> them, except that the symbols are not atomic. They are themselves
made up
> of a vertical shaft with only 4 kinds of halfarrowhead or flag.
Most of
> these flags come in left and right varieties for a total of 7 kinds
of flag
> (ignoring up and down varieties).
>
> These 7 flags correspond to the commas for the primes 5, 7, 11*,
17, 19,
> 23, 29. The symbols for the commas for 13, 31 and 37 and some
optional
> additional commas, are obtained by combining flags on the same shaft
> according to an arithmetic which corresponds to simple addition of
the
> nearest 1/1600ths of an octave.
>
> * The 11 comma is symbolised, not by a single flag but by a new flag
> combined with the 5 flag, and so we refer to this new flag as the
115 flag.
>
> Because we use 1600ET for this flag arithmetic, if we choose to
combine
> multiple symbols into a single symbol we can do so without
introducing any
> error greater than about half a cent.
>
> The system is designed so that at each prime limit lower than 37,
it is as
> simple as possible. No higher prime has been allowed to complicate
the
> system for those who don't need it. Here are the numbers of
different flags
> that must be learnt at each prime limit
>
> 5 1
> 7 2
> 11 3
> 13 3
> 17 4
> 19 5
> 23 6
> 29 7
> 31 7
> 37 7
>
> Although we've so far described this as a notation for purely
rational
> scales, it works beautifully for equal temperaments too. [explain
how 
> choose your fifth etc.]
>
> In the case of equal temperaments we use only the symbols for the
lowest
> primes, or combinations thereof, that are necessary to notate each
step. It
> turns out that one doesn't need to go past 19limit to notate most
ETs of
> interest.
>
> 217ET is the largest ET that can be notated by this method, using
only one
> symbol per note (in addition to a possible sharp or flat symbol).
217ET
> has no error greater than 2.9 cents in the 35limit, and so
provided that
> such errors are acceptable, we can use it to notate up to 35limit
rational
> scales using only one symbol per note.
>
> So far we have assumed that the arrow symbols will be used in
conjunction
> with conventional sharp and flat symbols, but this is not necessary
either.
> The system includes additional arrow symbols, which use the same
flags
> (half arrowheads) but have multiple shafts to the arrow. These can
cover
> the range from a doubleflat to a doublesharp using single
symbols.
Okay, that sounds like a good description of what we are are very
close to achieving. I might prefer to call the 11comma a diesis
(although it is plain that you are using the term "comma" in a
broader sense here), which would further justify the introduction of
the 115 comma that is used in achieving it, just as the 13diesis is
also the (approximate) sum of two commas.
> > I am suggesting this in light of your
> > observation:
> >
> > > Yes, so 217ET is just one ET that could be used in this way.
The
> > > notation is not based on it. It just happens to be the highest
one
> > > that is fully notatable with single symbols.
> >
> > This is one point that has become all too apparent, as you have
> > proceeded (in your subsequent messages) to suggest changes in the
> > symbols that:
> >
> > 1) Go beyond the three types of flags (straight, convex, &
concave)
> > that work so elegantly for 217 (remember that I said that
something
> > that could be regarded as "overkill" was immune to criticism as
long
> > as the additional complexity didn't make it more difficult to do
the
> > simpler things; this introduces more complexity for 217ET);
>
> 217ET only needs 19limit, correct? I don't understand why you
consider
> that changing the 19flag to something other than a concave flag is
an
> increase in complexity. The 5 limit uses only a straight left flag.
We
> didn't require that the 7 limit use the straight right flag but
went to a
> convex flag and didn't use the straight right until 11limit. This
would be
> similar; delaying the use of the other convex flag until 23 limit;
and
> could be justified on exactly the same grounds, namely eliminating
lateral
> confusability from the 19limit (and thereby greatly reducing it in
217ET).
It was getting more complicated inasmuch as I was leading up to my
next point:
> > 2) Introduce new symbols that I have no idea how to incorporate
into
> > a singlesymbol notation (this makes it difficult to do something
> > that I was previously able to do with 217ET); and
>
> I think this is the big one, but I have a proposed solution. Later.
It doesn't work (see my reply below).
> > 3) Employ semantics inconsistent with 217ET, ...
>
> I don't see this as a problem because I don't think that anything
employing
> those semantics is required in order to notate 217ET
I had the impression that the 23flag used in combination with
something else defined another prime inconsistenly in 217, but that
one (for the 37comma 36:37) requires 3 flags, so it wouldn't be used
anyway.
> > I made it a point to think very carefully before replying to your
> > subsequent messages, because I know you spent a lot of time and
> > effort on the content and have come up with some very good
things,
> > such as the 31schisma (above). During the two weeks or so that
I
> > spent leading up to my 17limit (183tone) and 23limit (217
tone)
> > approaches, I also spent a lot of time trying various things, and
I
> > don't consider the time wasted that I spent on ideas that I
> > subsequently discarded. In the process of developing a notation
such
> > as this, you want to try as many things as you can possibly think
of,
> > because that best enables you to see why the method that is
finally
> > chosen is the best one. I wanted to find a way to resolve this
that
> > would satisfy both of our requirements.
>
> I totally agree.
>
> > Here is the compromise that I am proposing: Let's keep the 217
ET
> > based symbols as they are, defining 2176:2187 as xL and 512:513
as
> > xR, with their combination allowed to represent either 4096:4131
or
> > 729:736 as required (in 217ET or another ET, where consistent,
but
> > incapable of being combined with anything else).
In the preceding sentence it should be obvious to you that I meant to
say "defining 2176:2187 as *vL* and 512:513 as *vR*", but just so no
one else misunderstands, I am correcting this here.
> > Then, for the 1600
> > based notation, let's expand on that with a combination of the
> > following methods:
> >
> > 1) Allow two flags to appear on the same side, as was suggested
for
> > 6400:6561, the 25 comma. This would then allow us to use sR+vR
(with
> > the concave flag at the top of an upwardpointing arrow) to
notate
> > the 31comma 243:248, using the schisma 353935:253952. Also, the
> > alternate 37comma 999:1024 could be notated with xL+vL, using
the
> > schisma 570236193:570425344. (We would have to experiment to see
how
> > this would be done. With the convex flag at the end, the two
would
> > form a sort of loop; or they might be made to interlock.)
>
> I have no objection to using multiple flags on the same side, to
notate
> primes beyond 29. However I consider 999:1024 to be the standard 37
comma
> because it is smaller than 36:37, also because it only requires 2
> lowerprime flags instead of 3. Can you explain why you want 36:37
to be
> the standard 37 comma?
Using primes this high has more legitimacy, in my opinion, in otonal
chords than in utonal chords. If C is 1/1, then 37/32 would be D
(9/8) raised by 37:36. With 1024:999 the 37 factor is in the smaller
number of the ratio, which is not where I need it.
For a similar reason I regard 26:27 as the principal 13diesis.
Taking C as 1/1, to get 13/8 I want to lower A (27/16) by a semiflat
(26:27) instead of raising Aflat by a semisharp (1053:1024), even if
1053:1024 is the smaller diesis. But considering that 26:27 is more
than half an apotome (and that we are adequately representing both of
these in the notation anyway), I have no problem that you prefer to
state it the other way.
While we are on the subject of higher primes, I have one more
schisma, just for the record. This is one that you probably won't be
interested in, inasmuch as it is inconsistent in both 311 and 1600,
but consistent and therefore usable in 217. It is 6560:6561
(2^5*5*41:3^8, ~0.264 cents), the difference between 80:81 and 81:82,
the latter being the 41comma, which can be represented by the sL
flag. I don't think I ever found a use for any ratios of 37, but Erv
Wilson and I both found different practical applications for ratios
involving the 41st harmonic back in the 1970's, so I find it rather
nice to be able to notate this in 217.
> > 2) Define one or more additional types of flags to notate new
primes,
> > beginning with a new left one for the 23comma, 729:736.
>
> Beginning and ending with a new 23flag. 7 flags is enough.
Yes, in light of the additional schismas that you have found.
> > This would
> > then allow us to use newL+xR+vR to notate the 37comma 36:37,
using
> > the schisma 6992:6993. (Thus, the symbols for the two 37commas
both
> > contain a combination of a convex and concave flag on the same
side,
> > which is most appropriate!)
>
> Other combinations might have other kinds of appropriateness, such
as one
> containing the other flipped horizontally.
>
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > > This is probably all pretty silly, catering for 37, and we
should probably
> > > just forget it and keep the large 23 comma symbol, but here's a
pass at a
> > > full set of 37limit symbols anyway.
> >
> > Silly or not, I think we should keep whatever capability we can,
as
> > long as it is consistent. And I would prefer to keep *both* the
> > large 23 comma symbol and a full set of 37limit symbols, as with
> > this "compromise."
>
> OK. But I'd prefer a slightly different compromise where the 19
flag is the
> one that is other than straight, convex or concave and gives the
impression
> of being smaller than any of them. So the following has the 19 and
23 flags
> swapped relative to your suggestion.
>
> 17 vL
> 19 smallL
> 23 vR
> 23' vL + sR
> 31 smallL + sR
> 37 xL + vL (999:1024)
> 37' smallL + vR + xR (36:37)
Why are you requiring that the new type of flag (whether for 19 or
23) be smaller in size? I would have the new flag represent 23 on
the basis that it is a *higher prime* than 19. Then with 217ET
(which is unique only through 19 and completely consistent only
through 21) we need only the three types of flags that are used for
the 19limit notation, with a *newL* (differentlooking *left*) flag
for the 23 comma being foreign to all three: the 19limit, 217ET,
and the singlesymbol notation.
Otherwise, I would need to have a way to incorporate the new flag
into the singlesymbol notation, which will be discussed next.
> Now to the problems that occur when you try to make this work for
217ET
> with the full sagittal treatment, i.e. no # or b.
>
> Here's what you wrote earlier about the notation of apotome
complements:
>
> >By the way, something else I figured out over the weekend is how
to
> >notate 13 through 20 degrees of 217 with single symbols, i.e., how
to
> >subtract the 1 through 8degree symbols from the sagittal apotome
> >(/\). The symbol subtraction for notation of apotome
complements
> >works like this:
> >
> >For a symbol consisting of:
> >1) a left flag (or blank)
> >2) a single (or triple) stem, and
> >3) a right flag (or blank):
> >4) convert the single stem to a double (or triple to an X);
> >5) replace the left and right flags with their opposites according
to the following:
> > a) a straight flag is the opposite of a blank (and vice versa);
> > b) a convex flag is the opposite of a concave flag (and vice
versa).
> >
> >This produces a reasonable and orderly progression of symbols
> >(assuming that 63:64 is a curved convex flag; it does not work as
> >well with 63:64 as a straight flag) that is consistent with the
> >manner in which I previously employed the original sagittal
symbols
> >for various ET's.
>
> The problem I have with this (even assuming _your_ suggested
compromise) is
> that, while the opposite of sL and sR must certainly be blanks if
the
> apotome is to be a doubleshafted sL+sR, the other opposites are
entirely
> arbitrary. What I dislike about the result of your choice is that,
having
> learnt that xL is larger than sL, I now find that when they have a
double
> shaft under them, the order of these two is reversed, while all the
others
> remain the same.
The heart of the problem is that, in order to have a completely
consistent order of symbols, sL and xL should be swapped, so that
straight flags are *always* larger than curved flags. However, this
would make both the 5comma and 7comma flags convex, which re
introduces the problem of lateral confusibility, not only between
ratios of 5 and 7, but also for the two 11dieses, which I think is a
more serious issue. (In addition, a curved 5flag would not have a
constant slope, thereby obscuring the commaup meaning.)
Another inconsistency is that vLsR is a smaller interval than sR
(in effect making vL alter by 2 degrees when used with  ), but
this one is fortunately avoided in 217: vLsR does not have to be
used, inasmuch as it is the same number of degrees as sL. (And
vLxR can also be avoided, being almost the same size as xLvR.)
All of these problems are easily avoided in lesser divisions by a
judicious selection of symbols.
So I would consider this an example of a situation that is (to quote
a joke I once heard) "hopeless but not serious."
> Why can't we simply give a fixed comma value to the second shaft
(and so on
> for subsequent shafts), so the ordering of flag combinations learnt
for the
> first halfapotome is simply repeated in the second halfapotome
(and all
> other halfapotomes). To do this, the second shaft need only be
declared
> equal in value to xL+xR.
That's the way I did it way back (about 3 months ago) when life was
much simpler: I was using only straight flags and 72ET was the most
complicated system I had to deal with. The problem in doing that now
is that the ratios that we're trying to represent don't ascend in the
same order from a halfapotome (now what ratio is that anyway?) as
they do from a unison; instead they occur in reverse order from the
apotome downward. So scratch that idea.
> Another advantage of this is that one does not need to use flags
that
> properly belong to higher limits in the second and subsequent half
apotomes
> of lower limit rational notations, or of ET notations based on lower
> limits. e.g. There will be no concave flags (or small flag) in 72
ET. And
> there will be no need for xL or vR in 217ET.
I would want xL in 217 anyway, since it does handle ratios of 29.
After all, this is supposed to allow 35limit (nonunique) notation,
and it would be better not to have a new flag appearing out of the
blue, just for 29.
Now regarding 72ET, you will recall that I said this earlier:
<< Using curved flags in the 72ET native notation to alleviate
lateral confusibility complicates this a little when we wish to
notate the apotome's complement (4deg72) of 64/63 (2deg72), a single
*convex right* flag. I was doing it with two stems plus a *convex
left* flag, but the above rules dictate two stems with *straight
left* and *concave right* flags. As it turns out, the symbol having
a single stem with *concave left* and *straight right* flags is also
2deg72, and its apotome complement is two stems plus a *convex left*
flag (4deg72), which gives me what I was using before for 4 degrees.
So with a little bit of creativity I can still get what I had (and
really want) in 72; the same thing can be done in 43ET. This is the
only bit of trickery that I have found any need for in divisions
below 100. >>
By using a "faux complement," I can avoid using any concave flags for
both 72ET and 43ET. In fact, the only ET's under 100 that need
concave flags (that I have tried so far) are 50, 58, 94, and 96, and
none of the more important ones do.
I still need to prepare a diagram that illustrates the sequence of
symbols in various ET's, and I'd like to do a fulloctave diagram for
217 as well, just so we have a better idea of how everything comes
out.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> I still need to prepare a diagram that illustrates the sequence of
> symbols in various ET's, and I'd like to do a fulloctave diagram
for
> 217 as well, just so we have a better idea of how everything comes
> out.
>
> George
i think it would be cool if someone notated the adaptiveji version
of the chord progression
Cmajor > A minor > D minor > G major > C major
in 217equal. then we could all look at it and see if we have any
major problems with it.
Hi George,

The 19 flag

I don't require that the new type of flag be small irrespective of what it
is used for. I only want the flag for the 3.3 cent 19 comma to be smaller
than all the others, because it is less than half the size of any other
flag comma and less than 1/6th of the size of all but the 17 comma. If this
is allowed, then it follows that it must be a new kind of comma, not
convex, striaght or concave.
It seems, from an RT point of view, that the 19 comma flag could equally
well be a left flag or a right flag, I have no great attachment to either.
However in notating 217ET you need to use 19 flag + 17 flag to notate 3
steps and so the 19 flag would be best on the opposite side from the 17
flag. And if we want the large 23 comma not to have flags on the same side,
then the 17 comma must be on the opposite side from the 115 flag, which
means that the 17 flag must be a left flag and the 19 flag a right flag (I
mistakenly had 19 as a left flag in my previous message).

Priorities

It seems that there is a significant difference of priorities between an
approach
(a) that seeks to notate a particular large ET, which is
19oddlimitunique, using single symbols spanning from doubleflat to
doublesharp (or even just from flat to sharp), and use subsets of it to
notate lower ETs, and extend it to uniquely notate 19orhigherprimelimit
RTs (rational tunings),
and an approach
(b) that seeks to notate 19orhigherprimelimit RTs and use subsets of it
to notate low enough ETs, and extend it to allow those ETs to be notated
using single symbols spanning from doubleflat to doublesharp (or even
just from flat to sharp).
I believe I've understood your points, but I don't have any suggestions yet
that might satisfy us both, so I'm just going to put it in the too hard
basket for a while, or let it churn away in my subconscious.

The new flag type

In the meantime, let's try to agree on what the new type of flag should
look like, irrespective of what it is used for. I realise now that my
earlier suggestions of blobs or circles failed to take account of the need
to work with multiple shafts and X shafts. I believe the following proposal
does.
It resulted from asking myself the question "What could be more concave
than concave and yet still indicate a direction, and work with multiple
shafts?". Of course I also wanted it to look smaller (just in case it might
get used for the 19 comma :), but I figured that since straight looks
smaller than convex and concave looks smaller than straight, then "more
concave than concave" is bound to look smaller than concave.
I settled on a _rightangle_ flag. It indicates direction simply by being
close to one end of the shaft. Since none of our arrows have
"tailfeathers" there can be no confusion about which direction is meant,
and in any case I find that it invites the eye to complete a small 45
degree right triangle. But I don't want this triangle completed literally,
since it would then look too large, and would no longer be "more concave
than concave".
In addition to its angularity (not straight, not curved), its smallness is
part of what distinguishes it, at a glance, from a concave flag.
Here's my best attempt at showing, in ASCIIgraphics, all the possible
combinations for up arrows (with no more than one flag to a side). I
haven't bothered to show combinations which are merely left/right reversals
of those shown, and I've given no consideration to possible meanings of
flags or which combinations may be irrelevant.
_
/ 
 



/
/ 




_/




_




___
/  \
  



/\
/  \




_/\_



__




_
/ \
  \



/
/ \_



_
 \
_/ 



_
_/



_
_ \
 



_\
 \



___
/  
  
 
 
 
/
/ 
/ 
 
 

/
_/ 
 
 

__
 
 
 
_____
/   \
   
 
 
 
/ \
/ \
 
 
 

_/ \_
 
 
 
__
 
 
 
 
___
/  \
   \
 
 
 
/
/ 
/ \_
 
 
___
  \
_/  
 
 
 
_
/
_/ 
 
 
___
_  \
  
 
 
 
_\
 \
 \
 
 
___
/ 
 



/
/
/



/
_/



__



_____
/  \
  



/\
/\




_/\_



__




___
/ \
  \



/
/
/\_


___
 \
_/ 



_
/
_/


___
_ \
 



_\
\
\


___
/  
  
\ /
X
/ \
/
/ 
/\ /
X
/ \

/
_/ /
X
/ \

__
\ /
X
/ \
_____
/   \
   
\ /
X
/ \
/ \
/ \
\ /
X
/ \

_/ \_
\ /
X
/ \
__
 
\ /
X
/ \
____
/  \
   \
\ /
X
/ \
/
/ 
/\ /\_
X
/ \
___
  \
_/  
\ /
X
/ \
_
/
_/ /
X
/ \
___
_  \
  
\ /
X
/ \
_\
 \
\ /\
X
/ \
I think the fact that they can be made distinct using the extremely limited
resolution of the above ASCIIgraphics, bodes well for the real, high
resolution symbols.
Notice how a lot of problems are eliminated by bending the lines of the X
shafts so they become parallel near the head of the arrow.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > I still need to prepare a diagram that illustrates the sequence of
> > symbols in various ET's, and I'd like to do a fulloctave diagram
> for
> > 217 as well, just so we have a better idea of how everything comes
> > out.
> >
> > George
>
> i think it would be cool if someone notated the adaptiveji version
> of the chord progression
>
> Cmajor > A minor > D minor > G major > C major
>
> in 217equal. then we could all look at it and see if we have any
> major problems with it.
You mean like
C:E\:G
AJ:C*:EJ
Dj:Ff:Aj
Go:BL:Do
C:E\:G
Where L \ J j o * f stand for the following arrows (George and I
haven't agreed on all of these yet):


 L 5/217 oct down (7 comma down)
 
_/


 \ 4/217 oct down (5 comma down)
\ 
\


_  J 3/217 oct down (17 comma down + 19 comma down)
\_



_  j 2/217 oct down (17 comma down)
\



 o 1/217 oct down (19 comma down)
_

_

 * 1/217 oct up (19 comma up)



_/
 f 2/217 oct up (17 comma up)


 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>
[gs:]
> > > ... you have proceeded ... to suggest changes in the symbols
that: ...
> > > 2) Introduce new symbols that I have no idea how to incorporate
into
> > > a singlesymbol notation (this makes it difficult to do
something
> > > that I was previously able to do with 217ET); and
> >
[dk:]
> > I think this is the big one, but I have a proposed solution.
Later.
>
[gs:]
> It doesn't work (see my reply below).
I was too hasty in jumping to a conclusion about this. It does
work! (See below.)
[dk:]
> > Why can't we simply give a fixed comma value to the second shaft
(and so on
> > for subsequent shafts), so the ordering of flag combinations
learnt for the
> > first halfapotome is simply repeated in the second halfapotome
(and all
> > other halfapotomes). To do this, the second shaft need only be
declared
> > equal in value to xL+xR.
>
[gs:]
> That's the way I did it way back (about 3 months ago) when life was
> much simpler: I was using only straight flags and 72ET was the
most
> complicated system I had to deal with. The problem in doing that
now
> is that the ratios that we're trying to represent don't ascend in
the
> same order from a halfapotome (now what ratio is that anyway?) as
> they do from a unison; instead they occur in reverse order from the
> apotome downward. So scratch that idea.
I spent some time messing around with the symbols last night, trying
to see if I could find a way to make the apotome complements more
intuitive and to eliminate the inconsistency from the sequence, and,
lo and behold, I ended up with with exactly what you suggested, with
the ordering of flag combinations in the first halfapotome repeated
in the second halfapotome.
My profuse apologies  this works beautifully!!! When I read your
proposal yesterday, it sounded so wacky that I didn't even bother to
check it out.
I arrived at my somewhat arbitrary method for determining the apotome
complement symbols when I was still tentatively working them out in
183ET, and at that time the vL and vR symbols were both being used
for 17commas. The idea of making the sequence match in the two half
apotomes never occurred to me, because I hadn't settled on mapping
the symbols into any preferred division yet, and I (correctly)
assumed that any such sequence would be rather arbitrary.
As it turns out, the complements that you propose are, as a whole,
much more intuitive than what I had, with the flag arithmetic being
completely consistent for the symbols that we prefer for each degree
of 217 (which we will hopefully agree upon within the next few days
as the "standard 217ET set"). This standard set of 217ET symbols
could then be used for determining the notation for any sharp/flat
(or other) equivalents that may be required for any JI interval (for
which the composer should be strongly encouraged to indicate the
symbolratio association in the score).
There are rules for arriving at the new complement symbols (including
the nonstandard ones), and they are a bit more convoluted that what I
had. But inasmuch as the overall result is much better than what I
previously had, I think that this is something that we should adopt
without any further hesitation.
[dk:]
> > Another advantage of this is that one does not need to use flags
that
> > properly belong to higher limits in the second and subsequent
halfapotomes
> > of lower limit rational notations, or of ET notations based on
lower
> > limits. e.g. There will be no concave flags (or small flag) in 72
ET. And
> > there will be no need for xL or vR in 217ET.
>
[gs:]
> I would want xL in 217 anyway, since it does handle ratios of 29.
> After all, this is supposed to allow 35limit (nonunique) notation,
> and it would be better not to have a new flag appearing out of the
> blue, just for 29.
The xL flag is already used for both the 11 and 12 degree symbols, so
it is definitely not foreign to 217, but I think you mean that there
is no need to use the apotomecomplement symbol of one that has an xL
or vR flag. (These are among the "nonstandard" symbols whose
complements have flags that are inconsistent with their order.) I
wouldn't prohibit this altogether, should one want to indicate a
ratio of 29, for example  apotome minus xL equals xL is easy
enough to understand, but we will need to be careful to make others
aware that this is a "nonstandard" complement symbol that doesn't
occur in the expected order.
[gs:]
> Now regarding 72ET, you will recall that I said this earlier:
>
> << Using curved flags in the 72ET native notation to alleviate
> lateral confusibility complicates this a little when we wish to
> notate the apotome's complement (4deg72) of 64/63 (2deg72), a
single
> *convex right* flag. I was doing it with two stems plus a *convex
> left* flag, but the above rules dictate two stems with *straight
> left* and *concave right* flags. As it turns out, the symbol
having
> a single stem with *concave left* and *straight right* flags is
also
> 2deg72, and its apotome complement is two stems plus a *convex
left*
> flag (4deg72), which gives me what I was using before for 4
degrees.
> So with a little bit of creativity I can still get what I had (and
> really want) in 72; the same thing can be done in 43ET. This is
the
> only bit of trickery that I have found any need for in divisions
> below 100. >>
>
> By using a "faux complement," I can avoid using any concave flags
for
> both 72ET and 43ET. In fact, the only ET's under 100 that need
> concave flags (that I have tried so far) are 50, 58, 94, and 96,
and
> none of the more important ones do.
This faux complement (apotome minus xR equals xL ) could still be
used in 72ET (but not 43ET), but I now feel very strongly that I
would want to avoid this practice, inasmuch as the symbol for the
true complement (apotome minus xR equals xR ) is now highly
intuitive. Changing a convention for this one special case would be
counterproductive if the true complement were used everywhere else.
But in any case (as you note), there would be no concave or small
flag in 72ET.
I have tried the new complement symbols out with a few ET's (58, 94,
and 96), and I am delighted with the result, especially 94.
George
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > I still need to prepare a diagram that illustrates the sequence
of
> > symbols in various ET's, and I'd like to do a fulloctave diagram
> for
> > 217 as well, just so we have a better idea of how everything
comes
> > out.
> >
> > George
>
> i think it would be cool if someone notated the adaptiveji version
> of the chord progression
>
> Cmajor > A minor > D minor > G major > C major
>
> in 217equal. then we could all look at it and see if we have any
> major problems with it.
Once Dave and I agree on how to symbolize the 19comma (512:513, the
single degree of 217), then I would be happy to make a figure
illustrating this on a real staff.
George
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> Hi George,
>
> 
> The 19 flag
> 
> I don't require that the new type of flag be small irrespective of
what it
> is used for. I only want the flag for the 3.3 cent 19 comma to be
smaller
> than all the others, because it is less than half the size of any
other
> flag comma and less than 1/6th of the size of all but the 17 comma.
If this
> is allowed, then it follows that it must be a new kind of comma, not
> convex, striaght or concave.
Why don't you look at this way: The four straight and convex flags
are all greater than 20 cents. The 17 and 19 flags are both less
than 10 cents. The 23 flag, 729:736, is ~16.5 cents, which is closer
in size to the four larger flags than the two smaller ones. So, as I
see it, *both* the 17 and 19 flags should be smaller than the others,
which is why I have been making concave flags laterally narrower than
the others in my bitmap diagrams for the past several weeks. You
haven't seen them this way yet, but I think they look great. In
particular, the symbol with two concave flags is noticeably smaller
than ones with two straight/convex flags  more similar in size to
one with a single straight or convex flag.
So I don't think there is any real problem here. We could make the
new 23 flag intermediate in size between a (smaller) concave flag and
a (larger) straight flag. All we need to do is to decide on a shape.
> It seems, from an RT point of view, that the 19 comma flag could
equally
> well be a left flag or a right flag, I have no great attachment to
either.
> However in notating 217ET you need to use 19 flag + 17 flag to
notate 3
> steps and so the 19 flag would be best on the opposite side from
the 17
> flag. And if we want the large 23 comma not to have flags on the
same side,
> then the 17 comma must be on the opposite side from the 115 flag,
which
> means that the 17 flag must be a left flag and the 19 flag a right
flag (I
> mistakenly had 19 as a left flag in my previous message).
>
> 
> Priorities
> 
> It seems that there is a significant difference of priorities
between an
> approach
> (a) that seeks to notate a particular large ET, which is
> 19oddlimitunique, using single symbols spanning from doubleflat
to
> doublesharp (or even just from flat to sharp), and use subsets of
it to
> notate lower ETs, and extend it to uniquely notate 19orhigher
primelimit
> RTs (rational tunings),
> and an approach
> (b) that seeks to notate 19orhigherprimelimit RTs and use
subsets of it
> to notate low enough ETs, and extend it to allow those ETs to be
notated
> using single symbols spanning from doubleflat to doublesharp (or
even
> just from flat to sharp).
>
> I believe I've understood your points, but I don't have any
suggestions yet
> that might satisfy us both, so I'm just going to put it in the too
hard
> basket for a while, or let it churn away in my subconscious.
I apologize for having responded to your apotomecomplement symbol
proposal yesterday without having given it sufficient thought, but I
hope that today's messages clear things up a bit so we can continue
working on this without too much discouragement.
George
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
> >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > > I still need to prepare a diagram that illustrates the sequence
of
> > > symbols in various ET's, and I'd like to do a fulloctave
diagram
> > for
> > > 217 as well, just so we have a better idea of how everything
comes
> > > out.
> > >
> > > George
> >
> > i think it would be cool if someone notated the adaptiveji
version
> > of the chord progression
> >
> > Cmajor > A minor > D minor > G major > C major
> >
> > in 217equal. then we could all look at it and see if we have any
> > major problems with it.
>
> You mean like
>
> C:E\:G
> AJ:C*:EJ
> Dj:Ff:Aj
> Go:BL:Do
> C:E\:G
>
> Where L \ J j o * f stand for the following arrows (George and I
> haven't agreed on all of these yet):
>
> 
> 
>  L 5/217 oct down (7 comma down)
>  
> _/
>
> 
> 
>  \ 4/217 oct down (5 comma down)
> \ 
> \
>
> 
> 
> _  J 3/217 oct down (17 comma down + 19 comma down)
> \_
> 
>
> 
> 
> _  j 2/217 oct down (17 comma down)
> \
> 
>
> 
> 
>  o 1/217 oct down (19 comma down)
> _
> 
>
> _
> 
>  * 1/217 oct up (19 comma up)
> 
> 
>
> 
> _/
>  f 2/217 oct up (17 comma up)
> 
> 
right, but i'd like to see this actually notated, on a staff.
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > Hi George,
> >
> > 
> > The 19 flag
> > 
> > I don't require that the new type of flag be small irrespective of
> what it
> > is used for. I only want the flag for the 3.3 cent 19 comma to be
> smaller
> > than all the others, because it is less than half the size of any
> other
> > flag comma and less than 1/6th of the size of all but the 17
comma.
> If this
> > is allowed, then it follows that it must be a new kind of comma,
not
> > convex, striaght or concave.
>
> Why don't you look at this way: The four straight and convex flags
> are all greater than 20 cents. The 17 and 19 flags are both less
> than 10 cents. The 23 flag, 729:736, is ~16.5 cents, which is
closer
> in size to the four larger flags than the two smaller ones. So, as
I
> see it, *both* the 17 and 19 flags should be smaller than the
others,
Here are the relative sizes of the flag commas (shown using steps of 1600ET).
_19__
_____17_____
__________23__________
______________5______________
_________________7__________________
___________________115___________________
_____________________29______________________
You are right that the largest difference in the series occurs between 17
and 23. But the next largest difference occurs between 19 and 17, and
between 23 and 5. So it seems that following your argument to its
conclusion would require that the 19, 17 and 23 flags should be of _three_
different types. I think it will be too hard to find 3 types that look
smaller than convex and straight. We're having enough trouble agreeing on
two. Although the 19 to 17 and 23 to 5 differences are the same in steps of
1600ET, the 19 to 17 difference is slightly greater in cents.
But what if we look not at the _differences_ between sucessive comma sizes
but at the _ratios_ between them? This is equivalent to looking at the
differences of their logarithms. I don't think we can expect the apparent
flag size to correspond to comma size anyway, the largest being nearly 10
times the smallest, so why not consider making flag size roughly
proportional to the log of comma size.
Here are the logs shown graphically (natural log of cents, times 17).
________19___________
_________________17__________________
_______________________23_______________________
__________________________5_________________________
____________________________7___________________________
____________________________115___________________________
_____________________________29_____________________________
In this case it is clear that 19 to 17 is a more significant step than 23
to 5.
Maybe log is a bit too strong. What about square root, which could be
interpreted as making areas correspond?
Square root of cents, times 12.
_________19___________
________________17_________________
______________________23_________________________
__________________________5_____________________________
______________________________7________________________________
_______________________________115_________________________________
________________________________29___________________________________
19 to 17 is still more significant than 23 to 5, but about the same as 17
to 23.
> which is why I have been making concave flags laterally narrower
than
> the others in my bitmap diagrams for the past several weeks. You
> haven't seen them this way yet, but I think they look great. In
> particular, the symbol with two concave flags is noticeably smaller
> than ones with two straight/convex flags  more similar in size to
> one with a single straight or convex flag.
That's good. I look forward to seeing them.
> So I don't think there is any real problem here. We could make the
> new 23 flag intermediate in size between a (smaller) concave flag
and
> a (larger) straight flag. All we need to do is to decide on a
shape.
I am completely stumped as to what could possibly be intermediate between
smallconcave and straight and still be sufficiently distinct from both. I
don't think it is possible.
However this isn't really your requirement. It's too strict. All you really
need, to acheive this (which I'm not necessarily agreeing is a good idea),
is two sizes of flag smaller than straight and convex, where 17 and 19 are
of the smallest type and 23 alone is of the second smallest type.
I've already proposed a fourth type, with rightangle being smaller than
concave. So you could make both 17 and 19 right angle flags and 23 would be
the sole concave flag.
Note that rightangleness can happen in the other direction too. We
actually have 5 types of flag. From smallest to largest they are:
concave rightangle
concave quartercircle
straight
convex quartercircle
convex rightangle
If these are all imagined to fit in a square of side 2, then ignoring the
thickness ofthe lines, the area enclosed by each (with the shaft and an
invisible horizontal line as the other boundaries) is respectively 0, 1, 2,
3, 4 (taking pi to be 3).
Perhaps the best way to take line thickness into account is to try to
render them in an extremely small bitmap. If try to put the flags into a
2x3 bitmap, which could well be required of a 9 or 10 point version of the
symbol when displayed on a computer screen, we find that the two smallest
ones are not distinct as described above, which is why I said the concave
rightangle needed to be smaller, as well as being a right angle. We can do
these:
@@@@@@
@@ @@
@@ @@
@@
@@
@@
@@
@@@@
@@ @@
@@ @@
@@
@@
@@
@@
@@
@@@@
@@ @@
@@
@@
@@
@@
@@
@@
@@@@@@
@@
@@
@@
@@
@@
@@@@
@@
@@
@@
@@
@@
There are of course only 64 possible 2x3 bitmaps. I generated them all
below and then deleted those that were
(a) not connected,
(b) pointed in the wrong direction
(c) apeared as two flags overlaid. i.e. had two or more ends
I've left those whose direction is ambiguous because they can be made
unambiguous at higher resolution and their nearness to one end of the arrow
disambiguates them.
There are 29 left.
@@@@
@@
@@
@@
@@
@@
@@
@@@@@@
@@
@@
@@
@@
@@
@@
@@@@
@@ @@
@@
@@
@@
@@
@@
@@@@@@
@@ @@
@@
@@
@@
@@
@@
@@
@@@@
@@
@@
@@
@@
@@
@@@@
@@@@
@@
@@
@@
@@
@@
@@
@@@@@@
@@
@@
@@
@@
@@
@@@@
@@@@@@
@@
@@
@@
@@
@@
@@@@@@
@@@@@@
@@
@@
@@
@@
@@
@@@@
@@ @@
@@ @@
@@
@@
@@
@@
@@@@@@
@@ @@
@@ @@
@@
@@
@@
@@
@@
@@@@
@@ @@
@@
@@
@@
@@
@@@@
@@@@
@@ @@
@@
@@
@@
@@
@@
@@@@@@
@@ @@
@@
@@
@@
@@
@@@@
@@@@@@
@@ @@
@@
@@
@@
@@
@@@@@@
@@@@@@
@@ @@
@@
@@
@@
@@
@@
@@
@@@@
@@
@@
@@
@@
@@@@
@@ @@
@@@@
@@
@@
@@
@@
@@
@@@@
@@@@
@@
@@
@@
@@
@@@@
@@@@
@@@@
@@
@@
@@
@@
@@@@
@@@@@@
@@@@
@@
@@
@@
@@
@@
@@
@@@@@@
@@
@@
@@
@@
@@@@
@@ @@
@@@@@@
@@
@@
@@
@@
@@@@@@
@@ @@
@@@@@@
@@
@@
@@
@@
@@
@@@@
@@@@@@
@@
@@
@@
@@
@@@@
@@@@
@@@@@@
@@
@@
@@
@@
@@
@@@@@@
@@@@@@
@@
@@
@@
@@
@@@@
@@@@@@
@@@@@@
@@
@@
@@
@@
@@@@@@
@@@@@@
@@@@@@
@@
@@
@@
@@
> I apologize for having responded to your apotomecomplement symbol
> proposal yesterday without having given it sufficient thought, but I
> hope that today's messages clear things up a bit so we can continue
> working on this without too much discouragement.
Certainly. No apology required. I'm very glad that you found the same thing
but by completely different means.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> Okay, that sounds like a good description of what we are are very
> close to achieving. I might prefer to call the 11comma a diesis
> (although it is plain that you are using the term "comma" in a
> broader sense here), which would further justify the introduction of
> the 115 comma that is used in achieving it, just as the 13diesis
is
> also the (approximate) sum of two commas.
Yes. I like that idea.
> In the preceding sentence it should be obvious to you that I meant
to
> say "defining 2176:2187 as *vL* and 512:513 as *vR*", but just so no
> one else misunderstands, I am correcting this here.
To be honest I had just read them as vL and vR without noticing the typos.
That's a worry. :)
> > I have no objection to using multiple flags on the same side, to
> notate
> > primes beyond 29. However I consider 999:1024 to be the standard
37
> comma
> > because it is smaller than 36:37, also because it only requires 2
> > lowerprime flags instead of 3. Can you explain why you want 36:37
> to be
> > the standard 37 comma?
>
> Using primes this high has more legitimacy, in my opinion, in otonal
> chords than in utonal chords.
Certainly.
> If C is 1/1, then 37/32 would be D
> (9/8) raised by 37:36. With 1024:999 the 37 factor is in the
smaller
> number of the ratio, which is not where I need it.
But you would simply notate it as Eb lowered by 999:1024. The comma is in
the same direction as the flat. I don't see that this has anything to do
with otonal vs. utonal.
> For a similar reason I regard 26:27 as the principal 13diesis.
> Taking C as 1/1, to get 13/8 I want to lower A (27/16) by a semiflat
> (26:27) instead of raising Aflat by a semisharp (1053:1024), even
if
> 1053:1024 is the smaller diesis.
I find this more understandable since the comma is in the opposite
direction to the flat.
> But considering that 26:27 is more
> than half an apotome (and that we are adequately representing both
of
> these in the notation anyway), I have no problem that you prefer to
> state it the other way.
Right.
I see that the basic difference in our approaches is that you want the
simplest/smallest comma relative to a diatonic scale (Pythagorean7), which
you then subtract an apotome from if it is bigger than an apotome, while
I'm happy with the simplest/smallest comma relative to a chromatic scale
(Pythagorean12).
> While we are on the subject of higher primes, I have one more
> schisma, just for the record. This is one that you probably won't
be
> interested in, inasmuch as it is inconsistent in both 311 and 1600,
> but consistent and therefore usable in 217. It is 6560:6561
> (2^5*5*41:3^8, ~0.264 cents), the difference between 80:81 and
81:82,
> the latter being the 41comma, which can be represented by the sL
> flag. I don't think I ever found a use for any ratios of 37, but
Erv
> Wilson and I both found different practical applications for ratios
> involving the 41st harmonic back in the 1970's, so I find it rather
> nice to be able to notate this in 217.
You should definitely mention it wrt 217ET, but make it clear it is not
universal. Whatever application did you or Erv find for the 41st harmonic?
Sounds crazy to me.
I think I lost some schismas for alternate 17 and 19 commas you found. Can
you remind me of those?
> Why are you requiring that the new type of flag (whether for 19 or
> 23) be smaller in size? I would have the new flag represent 23 on
> the basis that it is a *higher prime* than 19.
When you added the (then) new type of flag, convex, to the existing
straight type, you didn't require that it represent 11 (or 115) on the
basis that it is a higher prime than 7. You used the new type for 7 so as
to eliminate lateral confusability from the 7limit and from 72ET.
I'm saying we should do the same thing with 19 and 23. i.e. use the new
type for 19 so as to eliminate (or at least greatly reduce) lateral
confusability from the 19limit and from 217ET.
The problem is of course the lateral confusability of the symbols for 1
step and 2 steps of 217ET. There isn't even a consistent rule about which
is bigger, left or right. We already have xL > xR and sL < sR.
It is possible to choose valid alternatives for notating 217ET so as to
completely eliminate lateral confusability, but only if 17 and 19 are
different types, which for convenience I'll call "v" and "n" respectively.
1 n
2 v
3 vn
4 s
5 sn (confusable alternative is x)
6 x (confusable alternative is s)
7 vx or xn
8 vs or ss
9 sx
10 ss
11 xx
12 n (confusable alternative is xs)
The worst thing about this is not using the 7comma x for 5 steps. The
schisma that says 7 comma = 5 comma + 19 comma, only works in 217ET, not
1600ET.
However, if we do this (changing only 5 and 6)
1 n
2 v
3 vn
4 s
5 x
6 s
7 vx or xn
8 vs or ss
9 sx
10 ss
11 xx
12 n
then at least we only have to deal with s and s. These will arguably be
the easiest to learn because they will occur most often, being of the
lowest prime limit.
If 17 and 19 commas are of the same type we have
1 v
2 v
3 vv
4 s
5 x
6 s
7 vx or xv
8 vs or ss
9 sx
10 ss
11 xx
12 v
where we have two confusable pairs, and we have v > v while s < s. Or
we could use
1 v
2 v
3 vv
4 s
5 x
6 x
7 vx or xv
8 vs or ss
9 sx
10 ss
11 xx
12 v
where at least we have v > v and x > x. They even differ by the same
amount.
But I still prefer zero or one confusable to 2 confusables.
On another matter: Can you tell me why the apotome symbol should not be
xx instead of ss?
> Then with 217ET
> (which is unique only through 19 and completely consistent only
> through 21) we need only the three types of flags that are used for
> the 19limit notation, with a *newL* (differentlooking *left*) flag
> for the 23 comma being foreign to all three: the 19limit, 217ET,
> and the singlesymbol notation.
Well yes, minimising the number of flagtypes is an advantage, but does it
sufficiently compensate us for the lateral confusability it allows? And if
so, why did we not consider it so in the 7limit when we introduced a new
convex type of flag rather than use the other straight flag?
> Otherwise, I would need to have a way to incorporate the new flag
> into the singlesymbol notation, which will be discussed next.
I understand you've solved that problem now and we can just take it that
the second half apotome will follow the same sequence of flags as the
first, no matter what those flags may be. Is that correct?
> I would want xL in 217 anyway, since it does handle ratios of 29.
> After all, this is supposed to allow 35limit (nonunique) notation,
> and it would be better not to have a new flag appearing out of the
> blue, just for 29.
When dealing with 217ET (or limits lower than 29) I think it's ok to
describe x as the 13'7 flag rather than the 29 flag. I expect you'd
prefer this.
> As it turns out, the complements that you propose are, as a whole,
> much more intuitive than what I had, with the flag arithmetic being
> completely consistent for the symbols that we prefer for each degree
> of 217 (which we will hopefully agree upon within the next few days
> as the "standard 217ET set"). This standard set of 217ET symbols
> could then be used for determining the notation for any sharp/flat
> (or other) equivalents that may be required for any JI interval (for
> which the composer should be strongly encouraged to indicate the
> symbolratio association in the score).
>
> There are rules for arriving at the new complement symbols
(including
> the nonstandard ones), and they are a bit more convoluted that what
I
> had. But inasmuch as the overall result is much better than what I
> previously had, I think that this is something that we should adopt
> without any further hesitation.
I'm sure glad you found them by another route. I think I proposed them
mainly out of ignorance!
> The xL flag is already used for both the 11 and 12 degree symbols,
so
> it is definitely not foreign to 217, but I think you mean that there
> is no need to use the apotomecomplement symbol of one that has an
xL
> or vR flag.
Yes. I was forgetting about the 11 step symbol xx. The 12 step one can be
replaced with a doubleshaft symbol if need be, but you're right, there's
no way to do 217ET without the 29 flag although we should probably call it
the 13'7 flag in this context, except when we describe the nonunique
35limit possibilities of 217ET.
> (These are among the "nonstandard" symbols whose
> complements have flags that are inconsistent with their order.) I
> wouldn't prohibit this altogether, should one want to indicate a
> ratio of 29, for example  apotome minus xL equals xL is easy
> enough to understand, but we will need to be careful to make others
> aware that this is a "nonstandard" complement symbol that doesn't
> occur in the expected order.
Would you please describe the symbols you think should be used for 1 thru
21 steps of 217ET? ss type notation will do fine.
> I have tried the new complement symbols out with a few ET's (58, 94,
> and 96), and I am delighted with the result, especially 94.
That's great news.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
Hi George,
Of all those 2x3 bitmaps I gave in the preceding post, the only one that
leaps out at me as being sufficiently arrowlike and sufficiently distinct
from the other 5, is
@@
@@@@@@
@@ @@
@@
@@
@@
@@
Which may be interpreted as concave followed by convex (concavoconvex). It
helps to stand back and squint :)
I find this to be between convex and straight in apparent size, but more
similar to straight. If one counts pixels (whether black or white), within
the 2x3 space of the flag, which are either black or are on the same
horizontal line as, and to the right of, a black pixel, then one gets the
following sizes.
@@@@@@
@@ @@ 6 (the maximum possible)
@@ @@
@@ convex rightangle
@@
@@
@@
@@@@
@@ @@ 5
@@ @@
@@ convex quadrant
@@
@@
@@
@@
@@@@@@ 4
@@ @@
@@ concavoconvex
@@
@@
@@
@@
@@@@ 3
@@ @@
@@ straight
@@
@@
@@
@@
@@ 2
@@@@@@
@@ concave quadrant
@@
@@
@@
@@
@@@@ 1 (the minimum possible)
@@
@@ small concave rightangle
@@
@@
@@
Notice that doing it the other way, convex followed by concave
(convexoconcave), doesn't work in 2x3 and has the same "size" as
concavoconvex.
@@@@
@@@@ 4
@@@@@@
@@ convexoconcave?
@@
@@
@@
Here are sharp # and flat for comparison
@@ @@
@@ @@
@@@@@@@@@@
@@ @@
@@@@@@@@@@
@@ @@
@@ @@
@@
@@
@@
@@@@
@@ @@
@@@@
@@
As far as I can tell there can be no others besides these 6.
Here they are in 3x4. If the previous was 10 point, this would be 12 point.
@@@@@@@@
@@ @@ 12
@@ @@
@@ @@ convex rightangle
@@
@@
@@
@@
@@
@@@@
@@ @@ 9
@@ @@
@@ @@ convex quadrant
@@
@@
@@
@@
@@
@@
@@@@@@ 8
@@ @@
@@ @@ concavoconvex
@@
@@
@@
@@
@@
@@
@@@@ 6
@@ @@
@@ @@ straight
@@
@@
@@
@@
@@
@@
@@ 5
@@@@@@
@@ @@ alternative concavoconvex
@@
@@
@@
@@
@@
@@
@@ 4
@@@@
@@@@ @@ concave quadrant
@@
@@
@@
@@
@@
@@
@@
@@ 2
@@@@@@
@@ small concave rightangle
@@
@@
@@
@@
@@
@@
@@ @@
@@ @@@@
@@@@@@@@
@@@@ @@@@
@@@@@@@@
@@@@ @@
@@ @@
@@
@@
@@
@@
@@
@@@@
@@ @@
@@ @@
@@@@
@@
I figure, by measuring some sharps and flats on scores, that at full size
the symbols will be 11 (computerscreen) pixels high, which would be 14
point the way I've been calling them. But maybe what I've called 10, 12 and
14 point, should be called 8, 10 and 12 point. Anyway, here they are:
@@@@@@@@@@
@@ @@ 20
@@ @@
@@ @@ convex rightangle
@@ @@
@@
@@
@@
@@
@@
@@
@@@@@@
@@ @@ 17
@@ @@
@@ @@ convex quadrant
@@ @@
@@
@@
@@
@@
@@
@@
@@@@
@@ @@ 14
@@ @@
@@ @@ alternative convex quadrant
@@ @@
@@
@@
@@
@@
@@
@@
@@
@@ 11
@@@@@@@@
@@ @@ concavoconvex
@@ @@
@@
@@
@@
@@
@@
@@
@@
@@@@ 10
@@ @@
@@ @@ straight
@@ @@
@@
@@
@@
@@
@@
@@
@@
@@ 7
@@@@
@@ @@ concave quadrant
@@@@ @@
@@
@@
@@
@@
@@
@@
@@
@@
@@ 5
@@
@@@@ alternative concave quadrant
@@@@@@ @@
@@
@@
@@
@@
@@
@@
@@
@@
@@ 3
@@
@@@@@@@@ small concave rightangle
@@
@@
@@
@@
@@
@@
@@
@@
@@ 2
@@@@@@
@@ alternative small concave rightangle
@@
@@
@@
@@
@@
@@
@@
@@
@@ @@
@@ @@@@
@@@@@@@@
@@@@ @@
@@ @@
@@ @@@@
@@@@@@@@
@@@@ @@
@@ @@
@@
@@
@@
@@
@@
@@
@@@@@@
@@ @@
@@ @@
@@ @@
@@@@
@@
It's tempting to eliminate the two extremes, the right angles, and use the
middle four, but that only gives us one type that is smaller than straight,
and we don't really want to use convexoconcave for the 5 comma just because
we want to have two types smaller than it.
I suppose if we tried real hard we could convince ourselves that
concavoconvex looks smaller than straight. Or we could deliberately make it
narrower, but only in the two larger point sizes I gave above, like:
@@
@@@@@@ 4
@@ @@
@@ small concavoconvex
@@
@@
@@
@@
@@
@@
@@ 8
@@@@@@
@@ @@ small concavoconvex
@@ @@
@@
@@
@@
@@
@@
@@
And make the concave quadrants smaller too as you suggest.
@@
@@ 3
@@@@
@@ @@ small concave quadrant
@@
@@
@@
@@
@@
@@
@@
@@ 4
@@
@@@@ small concave quadrant
@@@@ @@
@@
@@
@@
@@
@@
@@
@@
But I'd still want both 23 and 17 to be concavoconvex and 19 to be concave,
unless we made 23 concavoconvex, 17 concave quadrant and 19 small concave
rightangle.
Another suggestion: In the larger point sizes above, there are two subtly
different alternatives for some of the flags (and others are possible). Why
not make the left and right varieties of the same flagtype use these
subtly different alternatives (one for the right and the other for the
left) according to their relative sizes?
We're really getting into the fine details of font design here. What
software are you using to create the bitmap versions you've been giving?
How many pixels high are they?
Eventually we'd need to give a resolution independent description as in a
Truetype or Postscript font. Presumably you'd want to copy the style of
existing sharps and flats, such as making horizontal (or nearhorizontal)
strokes much thicker than vertical ones, (as if painted with a brush that
is about 3 times higher than it is wide) to avoid them getting lost against
the staff lines.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
Hi George (and anyone else who is still following this crazy thread),
I wrote that the 29 flag (convex left) should be referred to as the 13'7
flag in sub29limit situations. Of course it's actually the 11'7 flag.
There is no 13'7 flag.
Here are some possible accidentals as bitmaps at computerscreen
resolution, shown on the staff, both linecentred and spacecentred.
@@
@@@@@@@@@@@@@@@@@@@@
@@ @@@@
@@@@@@@@
@@@@ @@
@@@@@@@@@@@@@@@@@@@@ sharp
@@ @@@@
@@@@@@@@
@@@@ @@
@@@@@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@@@@@
@@
@@ @@
@@ @@@@
@@@@@@@@@@@@@@@@@@@@
@@@@ @@
@@ @@ sharp
@@ @@@@
@@@@@@@@@@@@@@@@@@@@
@@@@ @@
@@ @@
@@
@@@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@@@
@@ @@
@@ double sharp
@@ @@
@@@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@@@
@@ @@
@@ @@
@@@@@@@@@@@@@@@@@@@@ double sharp
@@ @@
@@ @@
@@@@@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@@@@@
@@ @@
@@@@@@@@
@@ @@
@@@@@@@@@@@@@@@@@@@@ natural
@@ @@
@@@@@@@@
@@ @@
@@@@@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@@@@@
@@
@@
@@ @@
@@@@@@@@@@@@@@@@@@@@
@@ @@
@@ @@ natural
@@ @@
@@@@@@@@@@@@@@@@@@@@
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@@@
@@
@@
@@@@@@@@@@@@@@@@
@@
@@@@@@
@@ @@
@@@@@@@@@@@@@@@@ flat
@@ @@
@@@@
@@
@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@
@@ @@
@@ @@ flat
@@ @@
@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@@@@@
@@ @@
@@ @@
@@@@@@@@@@@@@@@@@@ convex rightangle
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@@@@@
@@@@@@@@@@@@@@@@@@
@@ @@
@@ @@ convex rightangle
@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@
@@@@@@
@@ @@
@@ @@
@@@@@@@@@@@@@@@@@@ convex quadrant
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@
@@@@@@@@@@@@@@@@@@
@@ @@
@@ @@ convex quadrant
@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@
@@
@@@@
@@ @@
@@@@@@@@@@@@@@@@@@ straight
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@@@
@@ @@
@@ @@ straight
@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@
@@
@@
@@@@@@
@@@@@@@@@@@@@@@@@@ narrow concavoconvex
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@
@@ @@ narrow concavoconvex
@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@@ narrow concave quadrant
@@@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@@@ narrow concave quadrant
@@@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@@@ small concave rightangle
@@@@@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@
@@
@@ small concave rightangle
@@@@@@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@
One problem is that, at this resolution, a narrow concavoconvex flag (on a
single shaft) differs by only one pixel from a flat symbol when it is
righthand downpointing and spacecentred.
@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@
@@ @@
@@ @@ flat
@@ @@
@@@@@@@@@@@@@@@@
@@
@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@
@@ @@
@@ @@ narrow concavoconvex
@@@@@@
@@@@@@@@@@@@@@@@
@@
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
I wrote:
"I see that the basic difference in our approaches is that you want the
simplest/smallest comma relative to a diatonic scale (Pythagorean7), which
you then subtract an apotome from if it is bigger than an apotome, while
I'm happy with the simplest/smallest comma relative to a chromatic scale
(Pythagorean12)."
I should have said "... which you then subtract from an apotome if it is
bigger than a half apotome ..."
I have worked out what I think are the commas for both approaches. In the
chromatic approach I assume the 12 notes are from 4 to +7 fifths, i.e. Eb
to G# when G is 1/1. I assume the two approaches must agree on the commas
for 5 and 7, so the diatonic approach uses 7 notes from 2 to +4 fifths,
i.e. F to B when G is 1/1.
The two approaches agree on the commas for 5,7,11,13,29,41 and differ on
the commas for 17,19,23,31,37.
Here are the commas they agree on
5 80:81
7 63:64
11 32:33
13 1024:1053
29 256:261
41 81:82
and those that differ
chromatic diatonic
17 2176:2187 4096:4131
19 512:513 19456:19683
23 729:736 16384:16767
31 243:248 31:32
37 999:1024 36:37
Do you agree?
I'm hoping we can get all the diatonic commas from the chromatic ones using
schismas valid in 1600ET.
Notice that it is only the chromatic choices for 17 and 19 that enable us
to notate the higher ETs.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#3994]:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > > Hi George,
> > >
> > > 
> > > The 19 flag
> > > 
> > > I don't require that the new type of flag be small irrespective
of what it
> > > is used for. I only want the flag for the 3.3 cent 19 comma to
be smaller
> > > than all the others, because it is less than half the size of
any other
> > > flag comma and less than 1/6th of the size of all but the 17
comma. If this
> > > is allowed, then it follows that it must be a new kind of
comma, not
> > > convex, striaght or concave.
> >
> > Why don't you look at this way: The four straight and convex
flags
> > are all greater than 20 cents. The 17 and 19 flags are both less
> > than 10 cents. The 23 flag, 729:736, is ~16.5 cents, which is
closer
> > in size to the four larger flags than the two smaller ones. So,
as I
> > see it, *both* the 17 and 19 flags should be smaller than the
others,
>
> Here are the relative sizes of the flag commas (shown using steps
of 1600ET).
>
> _19__
> _____17_____
> __________23__________
> ______________5______________
> _________________7__________________
> ___________________115___________________
> _____________________29______________________
>
> You are right that the largest difference in the series occurs
between 17
> and 23. But the next largest difference occurs between 19 and 17,
and
> between 23 and 5. So it seems that following your argument to its
> conclusion would require that the 19, 17 and 23 flags should be of
_three_
> different types. I think it will be too hard to find 3 types that
look
> smaller than convex and straight. We're having enough trouble
agreeing on
> two. Although the 19 to 17 and 23 to 5 differences are the same in
steps of
> 1600ET, the 19 to 17 difference is slightly greater in cents.
We're dealing with two major issues here, distinct, yet related, each
of which requires a decision:
1) Whether to make the 19 or the 23comma the new flag; and
2) What will be the shape for the new flag.
How we resolve either one of these depends on how we resolve the
other one, so debating these one at a time is not really getting us
very far.
To further complicate this, we're each looking at this with a
different primary objective in mind:
a) Modifying12things/modulo1600; vs.
b) modifying7things/modulo217.
with the other being secondary. Both objectives are important, but
our priorities are different.
What I suggest that we do is to work on two different solutions
simultaneously:
A) Using the new flag for the 19comma; and
B) Using the new flag for the 23comma,
which will almost certainly require different solutions to issue 2).
In the process of evaluating possibilities for new flags, we can then
select our best choice(s) for both Plan A and Plan B.
After developing both plans so as to give the best possible outcome
for each (complete with actual bitmap examples of the symbols), we
can then discuss the advantages and disadvantages of each. That way,
we'll be evaluating the actual products as well as the concepts
governing them, instead of the concepts alone (as we have been doing
up to this point).
This is going to cost us more in time and effort (but fortunately not
money), but I think that it will be well worth the investment.
With the foregoing in mind, I will be deferring a reply to some of
the things you have brought up in your last three messages and will
address those things that need to be answered first.
> > which is why I have been making concave flags laterally narrower
> than
> > the others in my bitmap diagrams for the past several weeks. You
> > haven't seen them this way yet, but I think they look great. In
> > particular, the symbol with two concave flags is noticeably
smaller
> > than ones with two straight/convex flags  more similar in size
to
> > one with a single straight or convex flag.
>
> That's good. I look forward to seeing them.
We'll have to create directories in tuningmath/files for each of us
to put our examples for the other to retrieve and modify. Your ASCII
bitmaps are good up to a point, but they're no substitute for the
real thing.
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#3995]:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > Okay, that sounds like a good description of what we are are very
> > close to achieving. I might prefer to call the 11comma a diesis
> > (although it is plain that you are using the term "comma" in a
> > broader sense here), which would further justify the introduction
of
> > the 115 comma that is used in achieving it, just as the 13
diesis
> is
> > also the (approximate) sum of two commas.
>
> Yes. I like that idea.
There are also other things such as this that we should keep in mind
when it comes time to write an article formally introducing the
notation to the rest of the microtonal world. I presume that we
should coauthor this, inasmuch as you have gotten so heavily
involved in this project. (I have more details to discuss, and I
think we should continue any further discussion about this offlist.)
> > In the preceding sentence it should be obvious to you that I
meant to
> > say "defining 2176:2187 as *vL* and 512:513 as *vR*", but just so
no
> > one else misunderstands, I am correcting this here.
>
> To be honest I had just read them as vL and vR without noticing the
typos.
> That's a worry. :)
Not to worry.
> > > I have no objection to using multiple flags on the same side,
to notate
> > > primes beyond 29. However I consider 999:1024 to be the
standard 37 comma
> > > because it is smaller than 36:37, also because it only requires
2
> > > lowerprime flags instead of 3. Can you explain why you want
36:37 to be
> > > the standard 37 comma?
> >
> > Using primes this high has more legitimacy, in my opinion, in
otonal
> > chords than in utonal chords.
>
> Certainly.
>
> > If C is 1/1, then 37/32 would be D
> > (9/8) raised by 37:36. With 1024:999 the 37 factor is in the
smaller
> > number of the ratio, which is not where I need it.
>
> But you would simply notate it as Eb lowered by 999:1024. The comma
is in
> the same direction as the flat. I don't see that this has anything
to do
> with otonal vs. utonal.
Now that you mention it, it does have more to do with the sharpvs.
flat (or in this case naturalvs.flat) issue, in which case you can
use whatever else you think is appropriate to decide between the
two. So, for the reasons you gave, yes, you are entitled to have
999:1024 as the principal comma.
> ... I see that the basic difference in our approaches is that you
want the
> simplest/smallest comma relative to a diatonic scale (Pythagorean
7), which
> you then subtract an apotome from if it is bigger than an apotome,
while
> I'm happy with the simplest/smallest comma relative to a chromatic
scale
> (Pythagorean12).
This is related to your viewing the new symbols as modifying 12
pitches vs. mine as modifying 7. As long as the notation is
versatile enough to do what both of us require, then I think we'll be
okay.
> > While we are on the subject of higher primes, I have one more
> > schisma, just for the record. This is one that you probably
won't be
> > interested in, inasmuch as it is inconsistent in both 311 and
1600,
> > but consistent and therefore usable in 217. It is 6560:6561
> > (2^5*5*41:3^8, ~0.264 cents), the difference between 80:81 and
81:82,
> > the latter being the 41comma, which can be represented by the sL
> > flag. I don't think I ever found a use for any ratios of 37, but
Erv
> > Wilson and I both found different practical applications for
ratios
> > involving the 41st harmonic back in the 1970's, so I find it
rather
> > nice to be able to notate this in 217.
>
> You should definitely mention it wrt 217ET, but make it clear it
is not
> universal. Whatever application did you or Erv find for the 41st
harmonic?
> Sounds crazy to me.
Well, you have to be a little crazy to do microtonality in the first
place, and the more time you spend with it, the crazier you get.
Both Erv and I used 41 in different isoharmonic chords. I wrote some
notes further developing this last year, but I'll have to look for
them and get back to you about this.
> I think I lost some schismas for alternate 17 and 19 commas you
found. Can
> you remind me of those?
You didn't lose them; I never found any. I was wondering why you
didn't bring up the fact that (513/512) * (2187/2176) != (4131/4096)
when you proposed consolidating my 17limitin183 and 23limitin
217 approaches. Because of that, in notating a given ET I am
restricted to using only one (or only the other) of the symbols for a
17comma if the inequality doesn't vanish in that ET.
> On another matter: Can you tell me why the apotome symbol should
not be
> xx instead of ss?
The simplest ET notations (17, 22, 24, 31, 41, which require only 5
and 11comma symbols in their definition) use only straight flags, so
there is no point in confusing anyone with curved flags for the
apotome, which is twice ss in each of these (except 22, where ss
isn't used). Curved flags appear in the notation only when they are
necessary or helpful, etc., etc.
> When dealing with 217ET (or limits lower than 29) I think it's ok
to
> describe x as the 13'7 flag rather than the 29 flag. I expect
you'd
> prefer this.
Yes, inasmuch as it does produce an exact 26:27 diesis. This (taken
together with the low numbers in its ratio) is another reason why I
consider it the primary 13diesis, as opposed to 1024:1053, which is
only *approximated* with the 4095:4096 schisma.
> > As it turns out, the complements that you propose are, as a
whole,
> > much more intuitive than what I had, with the flag arithmetic
being
> > completely consistent for the symbols that we prefer for each
degree
> > of 217 (which we will hopefully agree upon within the next few
days
> > as the "standard 217ET set"). This standard set of 217ET
symbols
> > could then be used for determining the notation for any
sharp/flat
> > (or other) equivalents that may be required for any JI interval
(for
> > which the composer should be strongly encouraged to indicate the
> > symbolratio association in the score).
> >
> > There are rules for arriving at the new complement symbols
(including
> > the nonstandard ones), and they are a bit more convoluted that
what I
> > had. But inasmuch as the overall result is much better than what
I
> > previously had, I think that this is something that we should
adopt
> > without any further hesitation.
The fact that curved flags always convert to curved flags in arriving
at apotome complements (and never to straight ones) is a major reason
why I want a curved (concave) flag for the 19comma. Its apotome
complement symbol ( sx ) makes more sense that way. This is
something we can discuss further if we implement both plans A and B
above.
> I'm sure glad you found them by another route. I think I proposed
them
> mainly out of ignorance!
(Which is why you didn't take issue with my hasty  and equally
ignorant  rebuttal.) Well, in this case (your) ignorance is (my)
bliss. (And I'm glad your question prompted me to take another look
at what I had.)
> Would you please describe the symbols you think should be used for
1 thru
> 21 steps of 217ET? ss type notation will do fine.
Assuming Plan B for the moment, in which the 19comma is vR, the
standard symbol set is:
degs symbol ratio
  
1 v 512:513
2 v 2176:2187
3 vv 1114112:1121931
4 s 80:81
5 x 63:64
6 s 54:55
6' x 715:729, ~256:261 (not normally used separately)
7 vx 238:243
8 vs 4352:4455, ~16384:16767
9 sx 35:36, ~1024:1053
10 ss 32:33
11 xx 704:729, ~5005:5184
12 xs 26:27
The apotome complements:
13 v 2048:2187 less 4352:4455
14 vv 2048:2187 less 238:243
15 s 2048:2187 less 54:55
16 v 2048:2187 less 63:64
17 s 2048:2187 less 80:81
18 vx 2048:2187 less 1114112:1121931
19 vx 2048:2187 less 2176:2187
20 sx 2048:2187 less 512:513
21 ss 2048:2187 (the apotome)
The nonstandard combinations:
5' sv 40960:41553 (has v as complement)
6' x 715:729, ~256:261 (used separately as 29comma)
7' xv 366080:373977 (has vv as complement)
The only inconsistent apotome complement:
15' x 2048:2187 less 256:261 (used separately as /29comma)
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#3996]:
I have imported all of your symbols into MS Word (in courier font)
and will be looking at them in fullpage view, which reduces them
considerably.
> We're really getting into the fine details of font design here. What
> software are you using to create the bitmap versions you've been
giving?
It's strictly lowbudget, but it does the job: the "Paint" program
that is found among the "Accessories" that are shipped as a part of
the Windows operating system. If you use this, I would caution you
to save your work under a different name the first time you try
saving in monochrome format (which results in a considerably smaller
file size). In one version of Windows (2000) the block copy doesn't
work with a monochrome file, so you have to do all the work in full
color format, then save the final version as monochrome with a
different name.
> How many pixels high are they?
I set them up for use with singlepixel staff lines 8 pixels apart,
and each sagittal symbol is 10 pixels high. The conventional sharp
symbol that I made for comparative purposes is 14 pixels high, the
conventional flat 11 pixels, and the natural symbol 13 pixels. I
also did the Sims symbols  the square root sign and its inverse are
both 16 pixels high.
I have also made a bolder set of sagittal symbols that are wider (but
no higher), so that they may be easily read in poor light at music
stand distance; the widest of these, xx, is 16 pixels. In the non
bold set this symbol is 13 pixels wide.
> Eventually we'd need to give a resolution independent description
as in a
> Truetype or Postscript font. Presumably you'd want to copy the
style of
> existing sharps and flats, such as making horizontal (or near
horizontal)
> strokes much thicker than vertical ones, (as if painted with a
brush that
> is about 3 times higher than it is wide) to avoid them getting lost
against
> the staff lines.
Yes, but in my bold sagittal symbols I made the vertical lines
thicker as well, in response to Ted Mook's complaint about the
difficulty of distinguishing the Tartini fractional sharps from one
another.
Till next time, keep in tune!
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> To further complicate this, we're each looking at this with a
> different primary objective in mind:
>
> a) Modifying12things/modulo1600; vs.
>
> b) modifying7things/modulo217.
>
> with the other being secondary. Both objectives are important, but
> our priorities are different.
I think you'd better decide which one is more important! I'm also not clear why you need to introduce modulo anything. Why introduce approximations which may not be appropriate for the particular system you end up notating?
Here's a summary of what we've found re commas and schismas. Here are what
I understand as all the prime commas we'd like to have symbols for.
5 80:81
7 63:64
11 32:33 11' 704:729
13 1024:1053 13' 26:27
17 2176:2187 17' 4096:4131
19 512:513 19' 19456:19683
23 729:736 23' 16384:16767
29 256:261
31 243:248 31' 31:32
37 999:1024 37' 36:37
41 81:82
Taking the primes (and primed primes) as representing their commas, we have
a flag for each of the following commas.
5
7
(115)
17
19
23
29
Now I list the effect of all the notationallyuseful 1600ET schismas we've
found. In other words, how to make a symbol for each comma, using the flags
for the above commas.
And by the way, many thanks to Graham Breed for telling me that 1600ET was
what I was looking for. We can forget my 31limit challenge now.
symb lftflgs rtflgs

5 = 5
7 = 7
11 = 5 + (115)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (115)
17 = 17
17' = [none except 19 + 19 + 19 + 19]
19 = 19
19' = 23 + 19
23 = 23 [also 17 + 19 + 19]
23' = 17 + (115)
29 = 29
31 = 19 + (115)
31' = 5 + 23 + 23 [also 29 + 41]
37 = 29 + 17 [also 5 + 41]
37' = 5 + 17 + 23 [also 5 + 5 + 19, also 19 + 7 + 23]
41 = [none]
So if we want 17' or 41 in the RT (rational tuning) symbols, we'd need to
add new flags for them, which I think would be a bad idea.
Those schismas above, for primes 29 and below, which are not in square
brackets, are also valid in 217ET. Also the one for 31'.
The following are valid in 217ET, but not 1600ET. Note that 217ET only
gives unique mappings up to the 19oddlimit. (Are we sure it isn't only 17?)
17' = 23 = 17 + 19
41 = 5 = 19' = 19 + 23
Have I missed anything?
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "genewardsmith" <genewardsmith@j...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > To further complicate this, we're each looking at this with a
> > different primary objective in mind:
> >
> > a) Modifying12things/modulo1600; vs.
> >
> > b) modifying7things/modulo217.
> >
> > with the other being secondary. Both objectives are important,
but
> > our priorities are different.
>
> I think you'd better decide which one is more important!
We may need to do so eventually, but so far we are doing astoundingly
well at satsifying both of these. We are at present in complete
agreement up to the 17 limit. We really only need to agree to the
19limit since beyond that is not required for 217ET or any other ETs
that we plan to notate.
In fact I'm comitted to first ensuring that the 19limit and 217ET
notation is as good as it can be, ignoring any higherlimits or higher
ETs.
Our current disagreement re the 19 comma symbol is not a chromatic vs.
diatonic thing or a 217ET vs. 1600ET thing or a 19limit vs. higher
limits thing. It is a question of whether it is better (at the
19limit and in 217ET) to
(a) have only 3 styles of flag making up the symbols
and have apotome complement rules that are more intuitive (according
to George). (I haven't looked at this yet)
and have more lateral confusability
and have the 17 and 19 commas represented by samestyle flags even
though one is 2.6 times the size of the other, or
(b) have 4 styles of flag making up the symbols
and have apotome complement rules which are less intuitive
and have little or no lateral confusability
and have 17 and 19 commas represented by two different styles of flag
which give some indication of their relative sizes.
> I'm also
not clear why you need to introduce modulo anything. Why introduce
approximations which may not be appropriate for the particular system
you end up notating?
>
This is a very valid point. You have raised it before and I thought I
had answered it. But I'll take this opportunity to explain it in more
detail why the problem doesn't exist.
There is no modulo anything forced upon you by the notation. We have
always kept open the option of a strictly rational
onesymbolforonecommaperprime use of the symbols. In this way of
using it, you get a singleshaft arrow symbol for each prime comma (to
be used with a chromatic scale, not diatonic). Some have single flags
(half arrowheads) and some have double flags (full arrowheads). The
doubleflag symbols are generally larger than the single flag.
1600ET has merely been used as a logical way of constructing the
multiflag symbols. It wouldn't matter if no one ever knew this. You're
just given a bunch of symbols, one per prime. Provided you treat these
symbols as atomic there is no approximation whatsoever. It is only if
you start combining multiple symbols into a single symbol that you
start introducing approximations.
For example if a note is flattened by both a 5comma and a 7comma
then provided you have two separate arrows for the 5 and the 7 there
is no approximation, but if you choose to combine the 5 flag and 7
flag on a single shaft, then you find that you have made the symbol
for the 13comma and you have introduced an approximation.
In rational tunings this approximation is kept to inaudible levels by
basing it on 1600ET. In some ETs this approximation will of course be
quite large and quite audible, e.g. a whole step of the ET. But you
are still not forced to make the approximation at all.
When we look at the ETs where 5comma + 7comma =/= 13comma (among
those we intend to notate) we find in most cases that we only need to
use two of the 3 commas in notating the ET. e.g. In 27ET the 7comma
vanishes and we use the 5 and 13 comma symbols. In 50ET the 5comma
vanishes and we use the 7 and 13 comma symbols.
37ET is a case I'm not too sure about. Here we have the 5comma being
2 steps, the 13 comma being 3 steps and the 7comma vanishing. There
is no prime comma within the 41limit that is consistently equal to 1
step (11comma is 2 steps, same as 5comma). We could notate 1 step as
13comma up and 5comma down, but if we insist on single symbols, is
it ok to use the 7comma symbol to mean 13comma  5 comma? Or should
we use the 19comma symbol for one step, even though it's
1,3,pinconsistent?
Anyway, I hope you understand now that it is quite possible to use the
symbols without any approximations at all (except those native to the
scale being notated).
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> Here's a summary of what we've found re commas and schismas. Here are what
> I understand as all the prime commas we'd like to have symbols for.
>
> 5 80:81
> 7 63:64
> 11 32:33 11' 704:729
> 13 1024:1053 13' 26:27
> 17 2176:2187 17' 4096:4131
> 19 512:513 19' 19456:19683
> 23 729:736 23' 16384:16767
> 29 256:261
> 31 243:248 31' 31:32
> 37 999:1024 37' 36:37
> 41 81:82
Here are the corresponding 5 and 7 et mappings of the nonprime collection of commas:
[5, 8, 12, 14, 17, 18, 21, 21, 23, 24, 25, 26, 27]
[7, 11, 16, 20, 24, 26, 28, 30, 31, 34, 34, 37, 37]
These can be modifed to give the 41limit interval the notation, considered as JI notation, would be notating.
Here are the "standard" mappings, by way of comparison:
[5, 8, 12, 14, 17, 19, 20, 21, 23, 24, 25, 26, 27]
[7, 11, 16, 20, 24, 26, 29, 30, 32, 34, 35, 36, 38]
 In tuningmath@y..., "genewardsmith" <genewardsmith@j...> wrote:
These commas don't seem to be h217uniquewas there some 217 mapping which you were looking at, or do I have the wrong set, or the wrong idea?
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>We're dealing with two major issues here, distinct, yet related, each
>of which requires a decision:
>
>1) Whether to make the 19 or the 23comma the new flag; and
There is (at least) another possibility. We could make both 17 and 23 be
the new style of flag.
>2) What will be the shape for the new flag.
>
>How we resolve either one of these depends on how we resolve the
>other one, so debating these one at a time is not really getting us
>very far.
Agreed. So I ask you to consider a system where 19 is concave right (as you
desire), 17 is narrow concavoconvex left and 23 is (not so narrow)
concavoconvex right.
I believe this acheives all our objectives except the one that says you'd
prefer to have only 3 styles of flag in the 19 limit rather than 4. It has
flags that indicate relative size. It reduces lateral confusability. It has
simple and visually intuitive rules for apotome complements. I believe it
is more intuitive even than your current favourite.
>To further complicate this, we're each looking at this with a
>different primary objective in mind:
>
>a) Modifying12things/modulo1600; vs.
>
>b) modifying7things/modulo217.
>
>with the other being secondary. Both objectives are important, but
>our priorities are different.
I'm happy to try to ensure b) is as good as possible before returning to a).
>What I suggest that we do is to work on two different solutions
>simultaneously:
>
>A) Using the new flag for the 19comma; and
>
>B) Using the new flag for the 23comma,
>
>which will almost certainly require different solutions to issue 2).
>In the process of evaluating possibilities for new flags, we can then
>select our best choice(s) for both Plan A and Plan B.
>
>After developing both plans so as to give the best possible outcome
>for each (complete with actual bitmap examples of the symbols), we
>can then discuss the advantages and disadvantages of each. That way,
>we'll be evaluating the actual products as well as the concepts
>governing them, instead of the concepts alone (as we have been doing
>up to this point).
>
>This is going to cost us more in time and effort (but fortunately not
>money), but I think that it will be well worth the investment.
I'm hoping, with this message, to convince you that developing both A) and
B) is unnecessary, because you will prefer C) when you see how well it works.
C) Using new concavoconvex flags for both 17 and 23 commas.
>We'll have to create directories in tuningmath/files for each of us
>to put our examples for the other to retrieve and modify. Your ASCII
>bitmaps are good up to a point, but they're no substitute for the
>real thing.
Sure.
>There are also other things such as this that we should keep in mind
>when it comes time to write an article formally introducing the
>notation to the rest of the microtonal world. I presume that we
>should coauthor this, inasmuch as you have gotten so heavily
>involved in this project. (I have more details to discuss, and I
>think we should continue any further discussion about this offlist.)
Thanks. Yes I'd be delighted to coauthor it with you. Do you mean that we
should take all further discussion of the notation offlist, or just
discussions re the article?
>Now that you mention it, it does have more to do with the sharpvs.
>flat (or in this case naturalvs.flat) issue, in which case you can
>use whatever else you think is appropriate to decide between the
>two. So, for the reasons you gave, yes, you are entitled to have
>999:1024 as the principal comma.
OK.
>This is related to your viewing the new symbols as modifying 12
>pitches vs. mine as modifying 7. As long as the notation is
>versatile enough to do what both of us require, then I think we'll be
>okay.
It's certainly shaping up well, to do just that.
>You didn't lose them; I never found any. I was wondering why you
>didn't bring up the fact that (513/512) * (2187/2176) != (4131/4096)
That's 19 + 17 =/= 17'
>when you proposed consolidating my 17limitin183 and 23limitin
>217 approaches. Because of that, in notating a given ET I am
>restricted to using only one (or only the other) of the symbols for a
>17comma if the inequality doesn't vanish in that ET.
Can you live with that? Is it a significant problem? Can you see any easy
way around it? As it stands, you won't have a 17' comma symbol for rational
tunings, but you may have various symbols that correspond to it in various
ETs. It may be that wherever it is required in an ET, it always happens to
be the same as the 23 comma (as it is in 217ET).
There is a similar problem with the 19' comma, except it _can_ be notated
with two flags on the same side, namely 19 + 23.
>> On another matter: Can you tell me why the apotome symbol should
>not be
>> xx instead of ss?
>
>The simplest ET notations (17, 22, 24, 31, 41, which require only 5
>and 11comma symbols in their definition) use only straight flags, so
>there is no point in confusing anyone with curved flags for the
>apotome, which is twice ss in each of these (except 22, where ss
>isn't used). Curved flags appear in the notation only when they are
>necessary or helpful, etc., etc.
I'm convinced.
>> When dealing with 217ET (or limits lower than 29) I think it's ok
>to
>> describe x as the 13'7 flag rather than the 29 flag. I expect
>you'd
>> prefer this.
>
>Yes, inasmuch as it does produce an exact 26:27 diesis. This (taken
>together with the low numbers in its ratio) is another reason why I
>consider it the primary 13diesis, as opposed to 1024:1053, which is
>only *approximated* with the 4095:4096 schisma.
As I later corrected, it is of course _not_ a (13'7) flag. It is either an
11'7 flag or (what's relevant here) a (13'(115)) flag. Yes 26:27 is
certainly the primary 13diesis from a diatonic point of view.
>The fact that curved flags always convert to curved flags in arriving
>at apotome complements (and never to straight ones) is a major reason
>why I want a curved (concave) flag for the 19comma. Its apotome
>complement symbol ( sx ) makes more sense that way. This is
>something we can discuss further if we implement both plans A and B
>above.
With plan C) you can keep this property, which I agree is desirable, and
you will not need any new complement rules when the 23 flag is introduced.
You can keep the 19comma as a concave flag.
>Assuming Plan B for the moment, in which the 19comma is vR, the
>standard symbol set is:
>
>degs symbol ratio
>  
> 1 v 512:513
> 2 v 2176:2187
> 3 vv 1114112:1121931
> 4 s 80:81
> 5 x 63:64
> 6 s 54:55
> 6' x 715:729, ~256:261 (not normally used separately)
> 7 vx 238:243
> 8 vs 4352:4455, ~16384:16767
> 9 sx 35:36, ~1024:1053
> 10 ss 32:33
> 11 xx 704:729, ~5005:5184
> 12 xs 26:27
>
>The apotome complements:
> 13 v 2048:2187 less 4352:4455
> 14 vv 2048:2187 less 238:243
> 15 s 2048:2187 less 54:55
> 16 v 2048:2187 less 63:64
> 17 s 2048:2187 less 80:81
> 18 vx 2048:2187 less 1114112:1121931
> 19 vx 2048:2187 less 2176:2187
> 20 sx 2048:2187 less 512:513
> 21 ss 2048:2187 (the apotome)
>
>The nonstandard combinations:
> 5' sv 40960:41553 (has v as complement)
> 6' x 715:729, ~256:261 (used separately as 29comma)
> 7' xv 366080:373977 (has vv as complement)
>
>The only inconsistent apotome complement:
> 15' x 2048:2187 less 256:261 (used separately as /29comma)
I believe there are a number of typos in the above: The first 6' line
should be deleted. The 16 line should have x, not v. The 19 line should
have vs, not vx. 5' should say " has x as complement".
So we have plan B):
1 v +19
2 v 17+
3 vv 17+19
4 s 5+
5 x +7
6 s +(115)
7 vx 17+7
8 vs 17+(115)
9 sx 5+7
10 ss 5+(115)
11 xx (11'7)+7
12 xs (11'7)+5 ~= (13'(115))+(115)
13 v
14 vv
15 s
16 x
17 s
18 vx
19 vs
20 sx
21 ss
So the complementation rules are:
s <> (irrespective of whether it is left or right)
vL <> vL
vR <> xR
xL has no complement, or at least its complement is a vL pointing in the
opposite direction (up/down) so it is well that it can be avoided.
Note the inconsistency where a left v is its own complement while a right v
goes to an x.
Note that, if we include the 23 comma (3 steps of 217ET) as a new type of
right flag nR, we'll need an additional complementation rule.
nR <> nR
I assume you've noticed that a left flag and its complement must add up to
sL (in this case 4 steps), and a right flag and its complement must add up
to sR (in this case 6 steps).
It should also be noted that, for 5 steps and its complement 16, these
rules give valid alternatives, but they don't give _the_ answer that makes
the second halfapotome the same as the first. I'm guessing we need a
different kind of rule to deal with any degree that falls between sL and sR
in number of steps. This remains the same for Plan C). In many lower ETs
there will be no such degree.
There are 4 pairs of lateral confusables in the above.
Now I'll show plan C) (actually 2 options). I will use "c" to stand for the
new concavoconvex flag type. I would like the lefthand concavoconvex flag
(the 17 flag) to be narrower than the righthand one (the 23 flag), but
will use "c" for both. Here's option C1)
1 v +19
2 c 17+
3 cv 17+19
4 s 5+
5 x +7
6 s +(115)
7 cx 17+7
8 cs 17+(115)
9 sx 5+7
10 ss 5+(115)
11 xx (11'7)+7
12 xs (11'7)+5 ~= (13'(115))+(115)
13 c
14 cv
15 s
16 x
17 s
18 cx
19 cs
20 sx
21 ss
So the complementation rules are:
s <>
c <> c
vR <> xR
Notice that we've eliminated that inconsistency where a left v was its own
complement while a right v went to an x. And we only have 3 sets of lateral
confusables.
Here's option C2). The only change is to use the 23flag for 3 steps, so
that the number of flags is monotonic with degree size.
1 v +19
2 c 17+
3 c 23+
4 s 5+
5 x +7
6 s +(115)
7 cx 17+7
8 cs 17+(115)
9 sx 5+7
10 ss 5+(115)
11 xx (11'7)+7
12 xs (11'7)+5 ~= (13'(115))+(115)
13 c
14 c
15 s
16 x
17 s
18 cx
19 cs
20 sx
21 ss
And the complementation rules are:
s <>
c <> c
vR <> xR
No new rule is required for the 23 flag!
We appear to have 5 pairs of lateral confusables in this case, but remember
that I want the 23flag wider than the 17flag, although they are the same
shape (concavoconvex). This brings us back to 3 pairs. The straight and
convex flags could be given that treatment too, i.e. make the
largerincents of each pair, slightly wider than the other. However, when
they are combined on the same stem they should be the same size so such
symbols are symmetrical.
In those cases where there is lateral confusability between single s's and
c's, there is at least a consistent rule that left flags are smaller than
right flags.

I can state the algorithm used for the two options above. It should result
in a good set of symbols for any tractable ET.
Calculate the number of steps for each of the 7 flag commas.
Calculate the number of steps for all 12 combinations of a left and a right
flag.
Sort these 19 symbols according to number of steps.
Eliminate any symbol containing xL if it has fewer steps than the 5+(115)
symbol (ss) and there are other options for that number of steps.
Option 1
For each number of steps, choose the symbol that has the lowest prime
limit. If there is more than one with the lowest prime limit, then consider
their second highest primes, etc. For this purpose the 29 flag should be
considered to be the (11'7) flag.
Option 2
For each number of steps, choose the symbol that has the fewest flags. If
there is more than one with the fewest flags, then take the one with the
lowest prime limit etc.

I currently favour option 2. At least for 217ET I like the fact that it
doesn't give a doubleflag symbol for something as small as 3 steps.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "genewardsmith" <genewardsmith@j...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > Here's a summary of what we've found re commas and schismas. Here
are what
> > I understand as all the prime commas we'd like to have symbols
for.
> >
> > 5 80:81
> > 7 63:64
> > 11 32:33 11' 704:729
> > 13 1024:1053 13' 26:27
> > 17 2176:2187 17' 4096:4131
> > 19 512:513 19' 19456:19683
> > 23 729:736 23' 16384:16767
> > 29 256:261
> > 31 243:248 31' 31:32
> > 37 999:1024 37' 36:37
> > 41 81:82
>
> Here are the corresponding 5 and 7 et mappings of the nonprime
collection of commas:
>
> [5, 8, 12, 14, 17, 18, 21, 21, 23, 24, 25, 26, 27]
> [7, 11, 16, 20, 24, 26, 28, 30, 31, 34, 34, 37, 37]
Thanks for looking at this, but I don't understand. Why should we care
about their 5 and 7ET mappings, and what do the numbers above mean? I
don't understand why any of them are greater than 1.
> These can be modifed to give the 41limit interval the notation,
considered as JI notation, would be notating.
>
> Here are the "standard" mappings, by way of comparison:
>
> [5, 8, 12, 14, 17, 19, 20, 21, 23, 24, 25, 26, 27]
> [7, 11, 16, 20, 24, 26, 29, 30, 32, 34, 35, 36, 38]
Again, I don't understand what this means, but I'd like to, including
why the standard is a standard. All I can figure out is that the 5ET
one disagrees on 13 and 17, and the 7ET one disagrees on
17,23,31,37,41, about something.
> These commas don't seem to be h217uniquewas there some 217
> mapping which you were looking at, or do I have the wrong set, or >
the wrong idea?
No. I understand that 217ET is only unique up to the 19limit. But I
understand they are all unique in 1600ET. I'd be pleased if you have
an easy way to check these.
Also, I wonder if there are any other ETs between say 1000 and 1600ET
that are 41limit unique, or 37 limit unique, or even 31limit unique.
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote
[#3994]:
>
> > I think I lost some schismas for alternate 17 and 19 commas you
> found. Can
> > you remind me of those?
>
> You didn't lose them; I never found any. I was wondering why you
> didn't bring up the fact that (513/512) * (2187/2176) !=
(4131/4096)
> when you proposed consolidating my 17limitin183 and 23limitin
> 217 approaches. Because of that, in notating a given ET I am
> restricted to using only one (or only the other) of the symbols for
a
> 17comma if the inequality doesn't vanish in that ET.
But I thought about it this afternoon and found one for the 19as
sharp comma, 19456:19683. The schisma is 531392:531441
(2^6*19^2*23:3^12, ~0.160 cents). This occurs when the 19assharp
comma is approximated by adding the main 19comma (512:513) to the 23
comma (729:736). It vanishes in both 217 and 1600ET, but not in 311
ET.
George
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Here are the corresponding 5 and 7 et mappings of the nonprime
> collection of commas:
> >
> > [5, 8, 12, 14, 17, 18, 21, 21, 23, 24, 25, 26, 27]
> > [7, 11, 16, 20, 24, 26, 28, 30, 31, 34, 34, 37, 37]
>
> Thanks for looking at this, but I don't understand. Why should we care
> about their 5 and 7ET mappings, and what do the numbers above mean? I
> don't understand why any of them are greater than 1.
They are mappings to primes, from 2 to 41. The use of them in part is that they tell you how to go about transforming your notation to and from 41limit JI.
> > These can be modifed to give the 41limit interval the notation,
> considered as JI notation, would be notating.
> >
> > Here are the "standard" mappings, by way of comparison:
> >
> > [5, 8, 12, 14, 17, 19, 20, 21, 23, 24, 25, 26, 27]
> > [7, 11, 16, 20, 24, 26, 29, 30, 32, 34, 35, 36, 38]
>
> Again, I don't understand what this means, but I'd like to, including
> why the standard is a standard.
It's not actually an accepted standard, certainly not by Paul, but it seems to be the default meaning lurking in some discussionthat the mapping to primes is simply gotten by rounding off to the nearest integer.
> Also, I wonder if there are any other ETs between say 1000 and 1600ET
> that are 41limit unique, or 37 limit unique, or even 31limit unique.
I'll run a search for "standard" ones, at any rate, and report the results.
Hi George, please see:
/tuningmath/files/Dave/SagittalSingle217
C2DK.bmp
/tuningmath/files/Dave/SagittalMulti217C
2DK.bmp
They show my implementation of sagittal notation plan C2.
They show respectively singlesymbol and multisymbol (i.e. with
conventional sharps and flats) notation of all the steps of 217ET
from a doubleflat to a doublesharp.
Here's the association of symbols with prime commas again. I'm using
"w" for "wavy" now instead of "c" for "concavoconvex" (which was too
much of a mouthful).
Legend:
x convex
s straight
w wavy
v concave
1 v +19
2 w 17+
3 w 23+
4 s 5+
5 x +7
6 s +(115)
7 wx 17+7
8 ws 17+(115)
9 sx 5+7
10 ss 5+(115)
11 xx (11'7)+7
12 v [single] xs [multi] (11'7)+5 ~= (13'(115))+(115)
13 w
14 w
15 s
16 x
17 s
18 wx
19 ws
20 sx
21 ss
The flag apotomecomplementation rules are:
s <>
w <> w
vR <> xR
except that x <> x
All criticism and suggestions gratefully received. Feel free to edit
these bitmaps for your own purposes.
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote
[#3994]:
>
> > I think I lost some schismas for alternate 17 and 19 commas you
> found. Can
> > you remind me of those?
>
> You didn't lose them; I never found any. I was wondering why you
> didn't bring up the fact that (513/512) * (2187/2176) !=
(4131/4096)
> when you proposed consolidating my 17limitin183 and 23limitin
> 217 approaches. Because of that, in notating a given ET I am
> restricted to using only one (or only the other) of the symbols for
a
> 17comma if the inequality doesn't vanish in that ET.
>
> But I thought about it this afternoon and found one for the 19as
> sharp comma, 19456:19683. The schisma is 531392:531441
> (2^6*19^2*23:3^12, ~0.160 cents). This occurs when the 19assharp
> comma is approximated by adding the main 19comma (512:513) to the
23
> comma (729:736). It vanishes in both 217 and 1600ET, but not in
311
> ET.
Judging from your message #4008, you found this one also:
symb lftflgs rtflgs

19' = 23 + 19
but I don't think you mentioned it specifically.
The one thing that still bothers me is that there are two useful 17
commas, 2176:2187 and 4096:4131, and neither one is arrived at by a
schisma. Would you consider adding a flag for 4096:4131? If the
flags for the two 17commas are used together, we would then have a
simple way to notate tones modified by a Pythagorean comma, which
might be more useful to theorists than composers, but useful
nonetheless.
George
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Hi George, please see:
>
> /tuning
math/files/Dave/SagittalSingle217
> C2DK.bmp
> /tuning
math/files/Dave/SagittalMulti217C
> 2DK.bmp
>
> They show my implementation of sagittal notation plan C2. ...
>
> All criticism and suggestions gratefully received. Feel free to
edit
> these bitmaps for your own purposes.
You have really been busy over the weekend! I am going to have to
give all of this a bit of study before replying, so please have
patience.
However, in reply to what you have done with the symbols, I have
posted a file here:
/tuning
math/files/secor/notation/symbols1.bmp
(The URL broke into two lines, so you'll have to rejoin it before
using it to access the file. Also, only you have rights to write to
a file in a folder that you created, so you'll have to write to your
own files, while I write to mine.)
I noticed that what I was previously using for conventional symbols
was a bit different from what is commonly used, so I also made an
attempt over the weekend to improve on that. I prefer what you have
for the conventional sharp and flat symbols, but I suggest a wider
natural symbol (as shown in the 4th chord, first staff). You have
two different double sharp symbols (for the 6th chord); I also came
up with the same as your second one, which looks good on both a line
and a space (7th chord).
Per Ted Mook's criticism (found in your own message #24012 of May 30,
2001 on the main tuning list) about reading new symbols in poor
lighting at music stand distance, I made the symbols bolder in both
dimensions, and you will find (for the straight flag symbols only)
yours (on the second staff) compared with my latest version (on the
third staff). I saw no need to make the vertical strokes as long as
yours, which enables two new symbols altering notes a fifth apart to
be placed one above the other (first chord on the third staff). I
also put the symbols above the staves, making it easier to isolate
them for study.
Notice that I tried adding some nubs to the right flags to alleviate
the lateral confusibility problem. This could also be done with the
other flags for the larger alteration in each pair.
Another altitude consideration is at the upper right: uparrows used
with flats use up a lot of vertical space when the new symbols have
long vertical lines.
Let's see what agreement we can come to about how the straightflag
symbols should look before doing any more with the rest of them.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> Judging from your message #4008, you found this one also:
>
> symb lftflgs rtflgs
> 
> 19' = 23 + 19
>
> but I don't think you mentioned it specifically.
Only recently. I also pointed out that the 19' symbol therefore consists of
two flags on the same side.
> The one thing that still bothers me is that there are two useful 17
> commas, 2176:2187 and 4096:4131, and neither one is arrived at by a
> schisma. Would you consider adding a flag for 4096:4131?
You may have missed one of my messages. I actually asked how much that
bothered you.
> If the
> flags for the two 17commas are used together, we would then have a
> simple way to notate tones modified by a Pythagorean comma, which
> might be more useful to theorists than composers, but useful
> nonetheless.
I figure if we're gonna add yet another flag (is this getting ridiculous?
:) then it had better give us something else besides just 17'.
There are 3 possible values for an extra flag that would give us the 17'
comma (4096:4131). It could be 17' directly, or 17'17, or 17'19.
I had hoped one of these might give us 23 (and all the others that depend
on 23), but none of them do (in 1600ET).
They might give us a 41comma symbol.
They might fall near the middle of a big gap in flagcomma sizes.
17' doesn't fall in any big gaps (it's very near 23). 17'19 falls near the
middle between 17 and 23. 17'17 falls near the middle between 19 and 17.
Only one of the three gives us a 41comma symbol, albeit with 3 flags on
the same side! That's 17'17, which is 288:289, 6.0008 cents.
The (17'17) comma is the same size as the 19 comma (1 step) in 217ET,
253ET and 311ET. So it doesn't help us notate any higher ET than 217. But
who cares.
Here's how we name the commas:
Name Comma Name Comma

5 80:81
7 63:64
11 32:33 11' 704:729
(115) 54:55
13 1024:1053 13' 26:27
17 2176:2187 17' 4096:4131
(17'17) 288:289
19 512:513 19' 19456:19683
23 729:736 23' 16384:16767
29 256:261
31 243:248 31' 31:32
37 999:1024 37' 36:37
41 81:82
43 128:129
47 47:48
Assume we have a flag for each of the following 8 commas.
5
7
(115)
17
(17'17)
19
23
29
Now here's how to make a symbol for each comma, using the flags for the
above commas, and schismas valid in 1600ET, with a possible allocation to
left and right flags.
Symbol Left Right
for flags flags

5 = 5
7 = 7
11 = 5 + (115)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (115)
17 = 17
17' = 17 + (17'17) [2 left flags]
19 = 19
19' = 23 + 19 [2 right flags)
23 = 23 [also 17 + 19 + 19]
23' = 17 + (115)
29 = 29
31 = 19 + (115) [2 right flags]
31' = 5 + 23 + 23 [also 5 + 7 + (17'17)]
37 = 29 + 17
37' = 5 + 5 + 19 [also 5 + 17 + 23, also 19 + 7 + 23]
41 = 17 + (17'17) + (17'17) [3 left flags]
43 =(17'17) + 19 + 19
47 = 19 + 23 + 23 [also 5 + 17 + (17'17)]
In plan C, as symbolised in my recent bitmaps, there's one flag that hasn't
been used. That's a concave left flag (vL). I propose we use that for
(17'17).
So my proposal for the flags is
 Left Right
+
Convex  29 7
Straight  5 (115)
Wavy  17 23
Concave  (17'17) 19
Want to propose an alternative?
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> I noticed that what I was previously using for conventional symbols
> was a bit different from what is commonly used, so I also made an
> attempt over the weekend to improve on that. I prefer what you have
> for the conventional sharp and flat symbols, but I suggest a wider
> natural symbol (as shown in the 4th chord, first staff).
You're absolutely right. I'll use your natural.
> You have
> two different double sharp symbols (for the 6th chord); I also came
> up with the same as your second one, which looks good on both a line
> and a space (7th chord).
Ditto.
> Per Ted Mook's criticism (found in your own message #24012 of May
30,
> 2001 on the main tuning list) about reading new symbols in poor
> lighting at music stand distance, I made the symbols bolder in both
> dimensions, and you will find (for the straight flag symbols only)
> yours (on the second staff) compared with my latest version (on the
> third staff).
I still believe this is a problem, but I haven't yet found an acceptable
solution. I believe the problem is greatest between the 2, 3 (and in my
version, the X shaft symbols).
I think there should be a strong family resemblance between the standard
symbols and our new ones, or they will not be found unacceptable on visual
aesthetic grounds.
This is one reason why I find the bold vertical strokes an unacceptable
solution to 23 confusability.
Frankly, I think the best solution is to use two symbols side by side
instead of the 3 and X shaft symbols, the one nearest the notehead being a
whole sharp or flat (either sagittal or standard). I think we're packing so
much information into these accidentals that we can't afford to try to also
pack in the number of apotomes. I suggest we provide single symbols from
flat to sharp and stop there. At least you have provided the doubleshaft
symbols so you never have to have the two accidentals pointing in opposite
directions.
The fact that in all the history of musical notation, a single symbol for
doubleflat was never standardised, tells me that it isn't very important,
and we could easily get by without a single symbol for doublesharp too.
With your bold vertical strokes you're taking up so much width anyway, why
not just use two symbols? This would require far less interpretation.
Also, the X tail suggests to me that one should start with a double sharp
and add or subtract whatever is represented by the flags, which is of
course not the intended meaning at all.
> I saw no need to make the vertical strokes as long as
> yours, which enables two new symbols altering notes a fifth apart to
> be placed one above the other (first chord on the third staff).
You're right. I used the standard flat symbol as my model. You do see flats
on scores directly above one another, a fifth apart. They just let the
stems overlap. But I agree ours should be shorter.
To do it while maintaining the family resemblance, I've now taken the
standard natural symbol as my model, and shaved two pixels off the tails of
all my symbols. But this still leaves them 2 pixels further from the centre
than yours.
I also think our flags (except possibly the smallest ones), when
spacecentered, should overlap the line above and below by one pixel, just
as the body of the flat symbol does. i.e. The flags should in general be 11
pixels high. Note that the body of the standard natural and sharp symbols
overlap the lines by 2 pixels. I think some overlap is important so as not
to lose the detail of the symbol where the staff lines pass thru it.
> I
> also put the symbols above the staves, making it easier to isolate
> them for study.
Good idea.
> Notice that I tried adding some nubs to the right flags to alleviate
> the lateral confusibility problem. This could also be done with the
> other flags for the larger alteration in each pair.
Yes. That's definitely worth a try. But I'd like to see it done for all
four flag types, to ensure that it doesn't interfere with making the types
distinct from each other. e.g. the straights you've shown with nubs tend to
look a little bit concave.
> Another altitude consideration is at the upper right: uparrows used
> with flats use up a lot of vertical space when the new symbols have
> long vertical lines.
Yes. Good point. I'll meet you halfway on that one. My symbols are now 17
pixels high (while yours are 13), which means they overlap each other by
one pixel when they are a fifth apart. I'm sure if you look hard enough
you'll find some scores that overlap flats like this. They won't overlap
naturals or sharps a fifth apart, because that would form a phantom
accidental in between, but there's no danger of anything like that with our
flags.
> Let's see what agreement we can come to about how the straightflag
> symbols should look before doing any more with the rest of them.
Well I don't think it is possible to consider the straight flags in
isolation from all the others. The full set has to be seen together to be
sure they are sufficiently distinct from one another. You are throwing away
a lot of horizontal resolution when you make those vertical strokes wider.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > Judging from your message #4008, you found this one also:
> >
> > symb lftflgs rtflgs
> > 
> > 19' = 23 + 19
> >
> > but I don't think you mentioned it specifically.
>
> Only recently. I also pointed out that the 19' symbol therefore
consists of
> two flags on the same side.
>
> > The one thing that still bothers me is that there are two useful
17
> > commas, 2176:2187 and 4096:4131, and neither one is arrived at by
a
> > schisma. Would you consider adding a flag for 4096:4131?
>
> You may have missed one of my messages. I actually asked how much
that
> bothered you.
You were concerned about keeping one comma per prime, and I didn't
want to pursue the issue any further at the time, because I wan't
sure how many more of these would be needed. As it turns out, 17
looks like the only prime with this situation, and I personally feel
that the 17asflat scale function (hence 4096:4131) is musically
more useful (Margo Schulter uses the 14:17:21 triad) than the 17as
sharp function (2176:2187). However, we have seen that 2176:2187 is
more useful in combination with other flags, while a use for
4096:4131 in combination has yet to be found. (I'm not trying to
make a case concerning which of the two is more important, but rather
to make the point that each one is as important as the other.)
Since we have 4 flag types and 7 ratios for flags, then using the
remaining flag for 4096:4131 (either directly, or using an
alternative such as those you gave in the rest of your message) would
make the 8th one, giving us four pairs of flags.
So far this is in agreement with what you proposed in the rest of
your message. I haven't had time to work through all the details of
that yet (as well as some of the previous things that you sent
recently about this), but I was expecting to see the 23 flag on the
left. So I need to take my time and study this carefully before
replying.
> So my proposal for the flags is
>
>  Left Right
> +
> Convex  29 7
> Straight  5 (115)
> Wavy  17 23
> Concave  (17'17) 19
>
> Want to propose an alternative?
Maybe  and maybe not. We shall see.
George
I'll assume for the moment that you accept the addition of a (17'17) flag
as the best way of giving us 17', because it's the only such choice that
also gives us 41 (assuming we can only use 1600ET schismas).
Now if we ignore for the moment the relative sizes of the commas, and
therefore ignore which pairs we might want to have as left and right
varieties of the same type, I get the following leftright assignment of
flags as being the one that minimises flagsonthesameside for as far
down the list of prime commas as possible.
By the way, if this list has only one comma for a given prime, the reason
is that the same comma is optimal for both diatonicbased (F to B relative
to G) and chromaticbased (Eb to G# relative to G) notations.
Symbol Left Right
for flags flags

5 = 5
7 = 7
11 = 5 + (115)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (115)
17 = 17
17' = 17 + (17'17)
19 = 19
19' = 19 + 23
23 = 23
23' = 17 + (115)
29 = 29
31 = 19 + (115)
31' = 5 + (17'17) + 7
or 5 + 23 + 23
37 = 29 + 17
37' = 19 + 23 + 7
or 5 + 17 + 23
or 5+5+19
41 = 17 + (17'17) + (17'17)
43 = 19 + 19 + (17'17)
47 = 19 + 23 + 23
or 5 + 17 + (17'17)
The story goes like this. The 5 comma must be a left flag so that it works
like the Bosanquet comma slash. Given 5 as a left flag, the 13 symbol then
says the 7 comma must be a right flag and the 11 symbol says (115) must
also be a right flag. Given 7 and (115) right, both 11' and 13' then say
29 must be left. We accepted this much long ago. Now for the rest.
Given (115) right, 23' says 17 must be left and 31 says 19 must be left.
Given 17 left, 17' then says (17'17) must be right. Given 19 left, 19'
then says 23 must be right.
That's all 8 flags assigned, and it gets us to 31 limit with minimal
sameside flags. There is no other assignment of flags to left and right,
that will do that. Notice that the 37 comma is the only one forced to be
nonminimal by this assignment. It's very convenient that it gives use 4 on
each side.
So my new proposal for the flags is
 Left Right
+
Convex  29 7
Straight  5 (115)
Wavy  17 23
Concave  19 (17'17)
Which just swaps the two concave flags from what I had before.
The only possible alternatives involve choosing which of 17 and 19 is wavy
and which concave, and likewise for 23 and (17'17), unless you use other
types. The above proposal makes the two larger ones wavy and the two
smaller ones concave.
Note that the above samesideminimisation process ignores any flag
combinations not listed above, that might be wanted for 217ET (and other
ETs). This seems right to me, since I take the prime commas as fundamental,
rather than 217ET. But rest assured that it works out just fine for 217ET.
Now why did I have 19 as a right flag before?
That was necessary when we wanted to have the 17 and 19 commas being of the
same type. It was also necessary when we were planning to notate 3 steps of
217ET as 17+19. But now I'm proposing we notate 3 steps as 23 since that
avoids a double flag for such a small increment, and lets us have
numberofflags increasing monotonically with numberofsteps. i.e. we only
jump from one flag to two at one point in the sequence.
And why did you have 23 as a left flag?
Presumably only because you had 19 as a right flag for the above reasons,
and in that case 19' says 23 should be a left flag. Now that we see that 19
can and should be a left flag, we can see that 23 should be right.
Another consideration for determining which type of flag to use for each
flagcomma is the intuitiveness of the apotomecomplement rules in 217ET,
when the second halfapotome is made to follow the first.
Lets look at the size of all the flags in steps of 217ET, assuming the
optimum leftright asignment given above, but ignoring my proposed
wavyconcave assignment.
Complementary
Flag Size Size Flag
comma in steps of comma
name 217ET name

Left

29 6 2 none available with same side and direction
5 4 0 blank
17 2 2 17
19 1 3 none available with same side and direction
Right

7 5 1 (17'17)
(115) 6 0 blank
23 3 3 23
(17'17) 1 5 7
Assigning 17 and 23 to wavy and 19 and (17'17) to concave mean that wavy
is always its own complement and, on the right at least, concave and convex
are complements.
On the left, just as we must avoid using the 29flag below the
halfapotome, we should also avoid using the 19flag (since it doesn't have
a direct flagcomplement in 217ET). This is easy since we can use the
(17'17) flag for 1 step. The rule about using the lowest possible prime
would tell us to do this anyway.
What this means is that although I've switched the meaning of concave right
from 19 to (17'17), my "plan C2" 217ET notation proposal doesn't change.
Of course you may want to forget about 41 and go back and look at what
happens with those other two choices for a new flag to give 17', i.e.
(17'19) or 17' itself.
Here's the optimum leftright assignment using a (17'19) flag.
Symbol Left Right
for flags flags

5 = 5
7 = 7
11 = 5 + (115)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (115)
17 = 17
17' = 19 + (17'19)
19 = 19
19' = 19 + 23
or 17 + (17'19)
23 = 23
23' = 17 + (115)
29 = 29
31 = 19 + (115)
31' = 5 + 5 + (17'19)
or 5 + 23 + 23
or (17'19) + 7 + 23
37 = 29 + 17
or (17'19) + (115)
37' = 19 + 23 + 7
or 5 + 17 + 23
or 17 + 7 + (17'19)
or 5+5+19
Now lets look at the flag complements for this.
Complementary
Flag Size Size Flag
comma in steps of comma
name 217ET name

Left

29 6 2 none available with same side and direction
5 4 0 blank
17 2 2 17
19 1 3 none available with same side and direction
Right

7 5 1 none available with same side and direction
(115) 6 0 blank
23 3 3 23
(17'19) 2 4 none available with same side and direction
Looks bad.
Here's the optimum leftright assignment using a 17' flag.
Symbol Left Right
for flags flags

5 = 5
7 = 7
11 = 5 + (115)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (115)
17 = 17
17' = 17'
19 = 19
19' = 19 + 23
23 = 23
23' = 17 + (115)
29 = 29
31 = 19 + (115)
31' = 5 + 23 + 23
37 = 29 + 17
37' = 19 + 23 + 7
or 5 + 17 + 23
or 5+5+19
The only reason for putting 17' on the right here is to give 3 left and 3
right flags, since 17' doesn't actually combine with anything else.
Now lets look at the flag complements for this.
Complementary
Flag Size Size Flag
comma in steps of comma
name 217ET name

Left

29 6 2 none available with same side and direction
5 4 0 blank
17 2 2 17
19 1 3 none available with same side and direction
Right

7 5 1 none available with same side and direction
(115) 6 0 blank
23 3 3 17' or 23
17' 3 3 17' or 23
That also looks bad.
It looks to me that including a flag to give us 17' has narrowed our
choices considerably. This is not a bad thing, given that the one choice it
leaves us, works so well (at least in 217ET).
I think we should describe it as basically 29limit, and just list the
morethanoneflagperside symbols for 31, 37, 41, 43, 47 (and possibly
higher, I haven't checked) just once, near the end, as a curiosity.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > Per Ted Mook's criticism (found in your own message #24012 of May
> 30,
> > 2001 on the main tuning list) about reading new symbols in poor
> > lighting at music stand distance, I made the symbols bolder in
both
> > dimensions, and you will find (for the straight flag symbols
only)
> > yours (on the second staff) compared with my latest version (on
the
> > third staff).
>
> I still believe this is a problem, but I haven't yet found an
acceptable
> solution. I believe the problem is greatest between the 2, 3 (and
in my
> version, the X shaft symbols).
>
> I think there should be a strong family resemblance between the
standard
> symbols and our new ones, or they will not be found unacceptable on
visual
> aesthetic grounds.
Yes, that is a very valid point.
> This is one reason why I find the bold vertical strokes an
unacceptable
> solution to 23 confusability.
I added some more to this file:
/tuning
math/files/secor/notation/symbols1.bmp
I think the problem with reading them under poor conditions is a
combination of factors  vertical lines that are rather thin *and*
vertical lines that are too close together, with the second factor
being more of a problem than the first. I redid the straightflag
symbols on the fourth staff using singlepixel vertical lines with
enough space between them to make them legible at a distance. I also
put a couple of them in combination with conventional sharps and
flats at the upper right, for aesthetic evaluation.
Just above the "conventional accidentals" staff I also added my
conventional sharp to the left of yours. Mine (more than yours)
looks more like what I found in printed music, and I suspect that
Tartini fractional sharps constructed (or written) with toonarrow
spacing between the vertical lines (such as we have here) are what
led to Ted Mook's observation. (And I do prefer your version.)
> Frankly, I think the best solution is to use two symbols side by
side
> instead of the 3 and X shaft symbols, the one nearest the notehead
being a
> whole sharp or flat (either sagittal or standard). I think we're
packing so
> much information into these accidentals that we can't afford to try
to also
> pack in the number of apotomes. I suggest we provide single symbols
from
> flat to sharp and stop there. At least you have provided the double
shaft
> symbols so you never have to have the two accidentals pointing in
opposite
> directions.
What? Did I understand this correctly? Are you considering using
the doubleshaft symbols??? (Or are you suggesting that I should do
this and forget about the  and X symbols?)
> The fact that in all the history of musical notation, a single
symbol for
> doubleflat was never standardised, tells me that it isn't very
important,
> and we could easily get by without a single symbol for doublesharp
too.
I think that it's because two sharps placed together looks a little
weird, but not two flats. If I remember correctly, I think that
doubleflats are placed in contact with one another, as I put them
above the staff (but I will need to check on this.)
> With your bold vertical strokes you're taking up so much width
anyway, why
> not just use two symbols? This would require far less
interpretation.
I put in a couple of singlesymbol equivalents at the upper right,
just to show how little space they occupy in comparison to two
symbols, even if you put them right up against one another. (And
even with the bolder vertical strokes, the single symbols took up
less space than the double symbols without any bold vertical strokes.)
> Also, the X tail suggests to me that one should start with a double
sharp
> and add or subtract whatever is represented by the flags, which is
of
> course not the intended meaning at all.
This would represent something within 4/10 of an apotome to a double
sharp, but something like this would rarely be used.
> > I saw no need to make the vertical strokes as long as
> > yours, which enables two new symbols altering notes a fifth apart
to
> > be placed one above the other (first chord on the third staff).
>
> You're right. I used the standard flat symbol as my model. You do
see flats
> on scores directly above one another, a fifth apart. They just let
the
> stems overlap. But I agree ours should be shorter.
>
> To do it while maintaining the family resemblance, I've now taken
the
> standard natural symbol as my model, and shaved two pixels off the
tails of
> all my symbols. But this still leaves them 2 pixels further from
the centre
> than yours.
>
> I also think our flags (except possibly the smallest ones), when
> spacecentered, should overlap the line above and below by one
pixel, just
> as the body of the flat symbol does. i.e. The flags should in
general be 11
> pixels high. Note that the body of the standard natural and sharp
symbols
> overlap the lines by 2 pixels. I think some overlap is important so
as not
> to lose the detail of the symbol where the staff lines pass thru it.
I did make my new symbols (in the fourth staff) one pixel longer (at
the tip of the arrow), which can be seen only when the note is on a
line. I didn't think that it was wise to overlap the flag in the
other direction, because this would make the nubs at the ends of the
flags less visible if a staff line were to pass through them. Where
I now have them, the nubs (actually 3x3 pixel squares) are in both
cases immediately adjacent to staff lines.
> > Let's see what agreement we can come to about how the straight
flag
> > symbols should look before doing any more with the rest of them.
>
> Well I don't think it is possible to consider the straight flags in
> isolation from all the others. The full set has to be seen together
to be
> sure they are sufficiently distinct from one another.
Of course. However, at this point we're still establishing general
dimensions, etc., and it is easier to change a few symbols than many
of them.
And you have just stated the reason why I don't want to commit to
flag shapes for the higher primes  I need to see what the final
product looks like, which is why I want to do it more than one way.
> You are throwing away
> a lot of horizontal resolution when you make those vertical strokes
wider.
Now that I have made them narrower, we have one more pixel of
resolution.
By the way, if you look at the concave flags in my earlier figures,
you will see that one part of the curving flag has postive and
another part negative slope (and I am still making them this way); a
nub on the end of this sort of flag can be seen very easily.
George
It turns out that I'm replying to my own message, inasmuch as it took
over 16 hours to appear on the list after I posted it.
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > I think there should be a strong family resemblance between the
standard
> > symbols and our new ones, or they will not be found unacceptable
on visual
> > aesthetic grounds.
>
> Yes, that is a very valid point.
I did some more work on the symbols in a second file:
> /tuning
math/files/secor/notation/symbols2.bmp
which I will put out there, once Yahoo gets over its cranky spell and
lets me upload it.
The more I look at your symbols, the more I like their style, so
(assuming that the file is out there) please follow along with me.
The fifth staff is a synthesis of features from both of our efforts
above that. I made the sesequisharp () and doublesharp (X) group
of symbols intermediate in width between what each of us had, while
the semisharp () and sharp groups () are either the same as or
very close to your symbols. The biggest problem I had was with the
nubs (which I made rather large and ugly) still tending to get lost
in the staff lines. I tried one symbol (in the middle of the staff)
with a triangular nub, which looks a little neater, I think.
To the right of that I copied three of your symbols so I can comment
on them. In all three of them the concave or wavy flag is
significantly lower than the line or space for its note. I propose
using instead the concave style of flag that I described before, for
which I prepared a set of symbols on the 7th staff. (Note that the
nubs don't get lost, even though they are quite small.)
I like your wavy flag, but I would propose waving it a little higher,
as I did in the symbols just to the right of yours (back on the 5th
staff); I seemed to be getting a better result with a thinner flag,
which would also serve to avoid confusion of the wavy with the convex
flag. Perhaps the wavy flag would now be most appropriate for the
smallest intervals.
I put a set of convex flag symbols on the 6th staff, which (like the
straightflag symbols) combines features from both of our previous
efforts).
At the top right (under altitude considerations) I put my latest
version of the symbols in combination with conventional sharps and
flats, with singlesymbol equivalents included (above the staff).
Let me know what you think.
I will be going through your latest reply relating commas to flags,
now that I have a much better idea what these symbols are going to
look like.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> I did some more work on the symbols in a second file:
>
> > /tuning
> math/files/secor/notation/symbols2.bmp
>
> which I will put out there, once Yahoo gets over its cranky spell
and
> lets me upload it.
I'm still waiting to see that, but in the meantime I've put up my
latest versions with changes based on several of your suggestions, and
one innovation.
/tuningmath/files/Dave/SagittalSingle217
C2DK.bmp
/tuningmath/files/Dave/SagittalMulti217C
2DK.bmp
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> I added some more to this file:
>
> /tuning
> math/files/secor/notation/symbols1.bmp
>
> I think the problem with reading them under poor conditions is a
> combination of factors  vertical lines that are rather thin *and*
> vertical lines that are too close together, with the second factor
> being more of a problem than the first. I redid the straightflag
> symbols on the fourth staff using singlepixel vertical lines with
> enough space between them to make them legible at a distance. I
also
> put a couple of them in combination with conventional sharps and
> flats at the upper right, for aesthetic evaluation.
I like this better, but part of the family resemblance of the existing
symbols is that they are all ectomorphs, except for the rarely seen
doublesharp being a mesomorph. They are not endomorphs like your  and X
symbols. Even some of my  symbols are pushing it.
> Just above the "conventional accidentals" staff I also added my
> conventional sharp to the left of yours. Mine (more than yours)
> looks more like what I found in printed music, and I suspect that
> Tartini fractional sharps constructed (or written) with toonarrow
> spacing between the vertical lines (such as we have here) are what
> led to Ted Mook's observation. (And I do prefer your version.)
Maybe, but part of the problem is just that it is hard to tell 2 identical
sidebyside things from 3 identical sidebyside things with the same
spacing. I've made my  and X's wider now, but not as wide as yours, and
shortened the middle tail of the 3 by 3 pixels relative to the others, so
they are not 3 identical things any more.
> > Frankly, I think the best solution is to use two symbols side by
> side
> > instead of the 3 and X shaft symbols, the one nearest the notehead
> being a
> > whole sharp or flat (either sagittal or standard). I think we're
> packing so
> > much information into these accidentals that we can't afford to
try
> to also
> > pack in the number of apotomes. I suggest we provide single
symbols
> from
> > flat to sharp and stop there. At least you have provided the
double
> shaft
> > symbols so you never have to have the two accidentals pointing in
> opposite
> > directions.
>
> What? Did I understand this correctly? Are you considering using
> the doubleshaft symbols???
No.
> (Or are you suggesting that I should do
> this and forget about the  and X symbols?)
Yes.
> > The fact that in all the history of musical notation, a single
> symbol for
> > doubleflat was never standardised, tells me that it isn't very
> important,
> > and we could easily get by without a single symbol for
doublesharp
> too.
>
> I think that it's because two sharps placed together looks a little
> weird, but not two flats.
You're just strengthening my case. I don't find that two of our symbols
side by side have the same problem as the conventional sharp symbol (making
a third phantom symbol in between). I find them to be more like the
conventional flat symbol, for which no singlesymbol double has ever been
seen as necessary.
> If I remember correctly, I think that
> doubleflats are placed in contact with one another, as I put them
> above the staff (but I will need to check on this.)
Those I have found have not been touching. I do not propose to have ours
touching either.
> > With your bold vertical strokes you're taking up so much width
> anyway, why
> > not just use two symbols? This would require far less
> interpretation.
>
> I put in a couple of singlesymbol equivalents at the upper right,
> just to show how little space they occupy in comparison to two
> symbols, even if you put them right up against one another. (And
> even with the bolder vertical strokes, the single symbols took up
> less space than the double symbols without any bold vertical
strokes.)
OK. This wasn't a valid point against the singlesymbol sesquis and
doubles. But other points still stand.
> > Also, the X tail suggests to me that one should start with a
double
> sharp
> > and add or subtract whatever is represented by the flags, which is
> of
> > course not the intended meaning at all.
>
> This would represent something within 4/10 of an apotome to a double
> sharp, but something like this would rarely be used.
I may not have explained my point well enough. I mean that the musician
trained to see an X as a double sharp will tend to see your upward pointing
X symbols as a doublesharp _plus_ something, and your downward pointing X
symbols as a double sharp minus something. It will be tough for them to
unlearn the previous meaning of X. In your notation, the X effectively
means _either_ a sesqui sharp or a sesqui flat.
I just thing this is too confusing, and the  and X symbols are not
required anyway.
> I did make my new symbols (in the fourth staff) one pixel longer (at
> the tip of the arrow), which can be seen only when the note is on a
> line. I didn't think that it was wise to overlap the flag in the
> other direction, because this would make the nubs at the ends of the
> flags less visible if a staff line were to pass through them. Where
> I now have them, the nubs (actually 3x3 pixel squares) are in both
> cases immediately adjacent to staff lines.
I don't see what's wrong with the nubs straddling the line in one case, and
being in free space in the other case. i.e. put them, not one, but 2 pixels
further out than you have them now.
> Of course. However, at this point we're still establishing general
> dimensions, etc., and it is easier to change a few symbols than many
> of them.
Easier, yes. But inconclusive.
> And you have just stated the reason why I don't want to commit to
> flag shapes for the higher primes  I need to see what the final
> product looks like, which is why I want to do it more than one way.
Fine.
> By the way, if you look at the concave flags in my earlier figures,
> you will see that one part of the curving flag has postive and
> another part negative slope (and I am still making them this way); a
> nub on the end of this sort of flag can be seen very easily.
True.
> The more I look at your symbols, the more I like their style, so
> (assuming that the file is out there) please follow along with me.
>
> The fifth staff is a synthesis of features from both of our efforts
> above that. I made the sesequisharp () and doublesharp (X)
group
> of symbols intermediate in width between what each of us had,
Agreed.
> while
> the semisharp () and sharp groups () are either the same as or
> very close to your symbols. The biggest problem I had was with the
> nubs (which I made rather large and ugly) still tending to get lost
> in the staff lines. I tried one symbol (in the middle of the staff)
> with a triangular nub, which looks a little neater, I think.
I think the nubs will be fine if you make the flags the same height as the
body of the standard flat symbol. The line will then pass thru the middle
of the nub.
> To the right of that I copied three of your symbols so I can comment
> on them. In all three of them the concave or wavy flag is
> significantly lower than the line or space for its note.
You mean the upward pointing ones? Concave I can understand, but wavy? The
horizontally inflected part of the wavy flag is always exactly centred
relative to the center of the notehead.
> I propose
> using instead the concave style of flag that I described before, for
> which I prepared a set of symbols on the 7th staff. (Note that the
> nubs don't get lost, even though they are quite small.)
>
> I like your wavy flag, but I would propose waving it a little
higher,
> as I did in the symbols just to the right of yours (back on the 5th
> staff); I seemed to be getting a better result with a thinner flag,
> which would also serve to avoid confusion of the wavy with the
convex
> flag. Perhaps the wavy flag would now be most appropriate for the
> smallest intervals.
I look forward to seing it.
> I put a set of convex flag symbols on the 6th staff, which (like the
> straightflag symbols) combines features from both of our previous
> efforts).
>
> At the top right (under altitude considerations) I put my latest
> version of the symbols in combination with conventional sharps and
> flats, with singlesymbol equivalents included (above the staff).
I hope you show some down pointing ones next to a flat, because I think
they look strange if their tails are too much shorter than the flat's tail.
In fact I'd be in favour of making the tails of downpointing arrows longer
than uppointing ones.
> Let me know what you think.
Will do.
> I will be going through your latest reply relating commas to flags,
> now that I have a much better idea what these symbols are going to
> look like.
Great!
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > > I think there should be a strong family resemblance between the
> standard
> > > symbols and our new ones, or they will not be found unacceptable
> on visual
> > > aesthetic grounds.
> >
> > Yes, that is a very valid point.
Of course you knew I meant to write "will be found unacceptable" or "will
not be found acceptable".
> I did some more work on the symbols in a second file:
>
> > /tuning
> math/files/secor/notation/symbols2.bmp
>
> which I will put out there, once Yahoo gets over its cranky spell
and
> lets me upload it.
Got it at last. Thanks.
> The more I look at your symbols, the more I like their style,
I've just uploaded a MsWord document containing drawings that show how I
conceive of these flags in a resolutionindependent manner, so as to
produce that style. You will see how the style is designed to be compatible
with the conventional symbols, in particular the conventional sharp and
flat symbols which the sagittals will most often have to appear next to.
/tuningmath/files/Dave/Flags.doc
I suggest you print it and take a hiliter pen and colour in the parts that
actually make up the flags and stems. I couldn't figure any easy way to do
that in Word. I'm sure you'll figure out what needs colouring. Then turn
the second page upside down and hold each flag in turn, beside the standard
flat and then the standard sharp.
Notice that the prototype convex and concave flags are exact 180 degree
rotations of each other, and wavy is an exact 180 degree rotation of
itself. This was partly intended to help with flag complementation in 217ET.
> so
> (assuming that the file is out there) please follow along with me.
>
> The fifth staff is a synthesis of features from both of our efforts
> above that. I made the sesequisharp () and doublesharp (X)
group
> of symbols intermediate in width between what each of us had,
Ok. We agree on the linethickness and overall width of all the tails now,
5 pixels for , 7 pixels for both  and X. I hope you like the idea of
shortening the middle stem of the  by 3 pixels so it's more like '. I
think that having that ^ shape in the tail tends to put them
psychologically in the same apotome as the X tails.
We also agree on how far the tail projects away from the centreline of the
corresponding notehead. That's 11 pixels not including the pixel that's
_on_ the centreline. That's the same as a sharp or natural, but two pixels
shorter than a flat. These agreements are good.
But we still don't agree on the height of the X's. Your X's are not
constant. They vary according to what flags they have on them, and are
often not laterally symmetrical. My X's are all the same height as they are
wide (7 pixels) and are laterally symmetrical. They just meet the concave
flags, but for other flag types they are extended by two parallel lines at
the same spacing as the outer two of the '. If nothing else, it certainly
simplifies symbol construction, not to have to design a new X tail for
every possible combination of flags. And if we get into using more than one
flag on the same side (e.g. for 25) with these X tails, I figure we're
gonna need those parallel sides.
> while
> the semisharp () and sharp groups () are either the same as or
> very close to your symbols. The biggest problem I had was with the
> nubs (which I made rather large and ugly) still tending to get lost
> in the staff lines. I tried one symbol (in the middle of the staff)
> with a triangular nub, which looks a little neater, I think.
Yes it looks neater, but I fear it is out of character with the standard
accidentals. I even think that maybe _any_ nubs are outofcharacter. Of
course we have the precedent of the doublesharp symbol, but I tend to
think of _it_ as being outofcharacter with the other 3 standard symbols.
I suspect it is more often seen as the unpitched notehead than as an
accidental.
By the way, I'm finding Elton John and Bernie Taupin's 'Goodbye Yellow
Brick Road' songbook to contain examples of just about everything with
regard to accidentals.
My recommendation is to make the nubs 4 x 4 with the corner pixels knocked
out. More round, less square.
@@@@
@@@@@@@@
@@@@@@@@
@@@@
> To the right of that I copied three of your symbols so I can comment
> on them. In all three of them the concave or wavy flag is
> significantly lower than the line or space for its note.
Again, I don't understand this statement in regard to the wavy flags. But I
do notice you're missing one pixel from your copy of one of my wavys, which
makes it look a little bit lower. I hope the drawings in the flags.doc file
will help you understand where I'm coming from on the wavy and concave
flags. Unfortunately, in this conception, the concaves do not lend
themselves to the addition of nubs, because they are already quite thick on
the ends.
> I propose
> using instead the concave style of flag that I described before, for
> which I prepared a set of symbols on the 7th staff. (Note that the
> nubs don't get lost, even though they are quite small.)
I'm not averse to a slight recurve on the concaves, but I'm afraid I find
some of those in symbols2.bmp, so extreme in this regard, that they are
quite ambiguous in their direction. With a mental switch akin to the Necker
cube illusion, I can see them as either a recurved concave pointing upwards
or a kind of wavy pointing down. Apart from any nub, I don't think that
they should go more than one pixel back in the "wrong" direction. Those at
the extreme lower left of the page look ok.
I'm guessing that you need the huge recurve to convince yourself that
concave can represent larger commas than wavy?
I would have agreed that, if you want the set of 3 flag types that are
maximally distinct from one another, (to be used for the lowest primes) it
is probably {concave, straight, convex}. However in typing those curly
brackets above, I had the thought: Isn't it interesting that our character
set includes brackets that correspond to some of our flags (in like pairs
turned sideways). It has
( convex,
< straight, and
{ wavy,
but _not_ concave.
And of course we also have
[ convex right angle, but my feeling is that it would be hard to make
those fit the style of the standard accidentals.
> I like your wavy flag, but I would propose waving it a little
higher,
> as I did in the symbols just to the right of yours (back on the 5th
> staff); I seemed to be getting a better result with a thinner flag,
> which would also serve to avoid confusion of the wavy with the
convex
> flag.
It seems to me that you have increased the possibility of confusion of wavy
with convex, by waving it higher.
> Perhaps the wavy flag would now be most appropriate for the
> smallest intervals.
I'd need to know what they all mean re commas or see a complete set, in
order, for the first 12 degrees of 217ET. I thought it made sense for
apotomecomplements, that wavy should be its own complement, and convex and
concave should be complements, when they _have_ complements (which is only
on the right).
> I put a set of convex flag symbols on the 6th staff, which (like the
> straightflag symbols) combines features from both of our previous
> efforts).
These all look fine to me, except I'd leave off the nubs for those with two
flags of the same type.
I don't have a clear preference yet for nubs versus changeofwidth, for
indicating relative size while reducing lateral confusability. Maybe we can
perfect both, then present them and ask folks to vote on them.
On a single shaft I made sL 5 pixels (including the shaft) and sR 7 pixels.
But when I combine the two I make them both 6 pixels wide for a total
symbol width of 11. I did the same thing with xL 7 pixels (not shown) and
xR 5 pixels. I made wL 4 pixels and wR 5 pixels. wR represents a smaller
comma than sL, so it couldn't be more than 5. A 3 pixel wavy wouldn't work,
but when they are both on the same shaft I would make them both 4 pixels.
Both vL and vR would be 4 pixels because they represent smaller commas than
the wavys, and I figure you just can't go narrower than 4 pixels. I want to
make all the flags as narrow as is reasonable so that the double flag
symbols are not getting too wide and outofcharacter with the standard
symbols.
> At the top right (under altitude considerations) I put my latest
> version of the symbols in combination with conventional sharps and
> flats, with singlesymbol equivalents included (above the staff).
As I said before, it would be good to show some downpointing ones with
conventional flats.
> Let me know what you think.
Done.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
See my latest attempt at the variablewidth nubfree style. I've
included a symbol for every one of our prime commas up to 47, and
shown the purely sagittal doublesymbol notation between sharp and
doublesharp.
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> I'll assume for the moment that you accept the addition of a (17'
17) flag
> as the best way of giving us 17', because it's the only such choice
that
> also gives us 41 (assuming we can only use 1600ET schismas).
Some other things I like about it is that:
1) It fills in a size gap between the 19 and 17 commas (e.g, giving
2deg494);
2) It gives 2deg311 (added to the 19comma), should someone want to
notate that division.
> Now if we ignore for the moment the relative sizes of the commas,
and
> therefore ignore which pairs we might want to have as left and right
> varieties of the same type, I get the following leftright
assignment of
> flags as being the one that minimises flagsonthesameside for as
far
> down the list of prime commas as possible.
>
> By the way, if this list has only one comma for a given prime, the
reason
> is that the same comma is optimal for both diatonicbased (F to B
relative
> to G) and chromaticbased (Eb to G# relative to G) notations.
>
> Symbol Left Right
> for flags flags
> 
> 5 = 5
> 7 = 7
> 11 = 5 + (115)
> 11' = 29 + 7
> 13 = 5 + 7
> 13' = 29 + (115)
> 17 = 17
> 17' = 17 + (17'17)
> 19 = 19
> 19' = 19 + 23
> 23 = 23
> 23' = 17 + (115)
> 29 = 29
> 31 = 19 + (115)
> 31' = 5 + (17'17) + 7
> or 5 + 23 + 23
> 37 = 29 + 17
> 37' = 19 + 23 + 7
> or 5 + 17 + 23
> or 5+5+19
> 41 = 17 + (17'17) + (17'17)
> 43 = 19 + 19 + (17'17)
> 47 = 19 + 23 + 23
> or 5 + 17 + (17'17)
> ... So my new proposal for the flags is
>
>  Left Right
> +
> Convex  29 7
> Straight  5 (115)
> Wavy  17 23
> Concave  19 (17'17)
This looks very workable, and I am about 99 percent sold on it.
(Just give me some more time.)
> Complementary
> Flag Size Size Flag
> comma in steps of comma
> name 217ET name
> 
> Left
> 
> 29 6 2 none available with same side and direction
> 5 4 0 blank
> 17 2 2 17
> 19 1 3 none available with same side and direction
>
> Right
> 
> 7 5 1 (17'17)
> (115) 6 0 blank
> 23 3 3 23
> (17'17) 1 5 7
Likewise.
In your table of symbols:
Symbol Left Right
for flags flags

23' = 17 + (115)
31' = 5 + (17'17) + 7
or 5 + 23 + 23
37 = 29 + 17
options can be added for the following:
23' = 17 + (115)
or 29 + (17'17)
31' = 5 + (17'17) + 7
or 5 + 23 + 23
or 7 + 7
37 = 29 + 17
or 5 + 5
These 5+5 option for the 37comma uses a much smaller schisma
(6553600:6554439, ~0.222 cents) than what you have. But the problem
with these three options that I have given is that none of the
schismas vanish in 1600ET.
Should we rethink the question of whether it is really necessary for
these schismas to vanish in 1600ET, because I don't see any good
reason. While it is nice to have everything come out exact using
1600 as a frame of reference, do you think anyone is actually going
to be able to use it in a performance to produce pitches? The
increments are much smaller than 1 cent, and the pitches can't be
related easily to 12ET, as Johnny Reinhard is doing. (i.e., not a
subdivision, as is 1200ET), ) So if we're trying to accommodate him
with this notation, all that's really necessary is to keep the
schismas small and provide the number of cents somewhere on the
score, at least in a table with the symbols.
George
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > I added some more to this file:
> >
> > /tuning
> > math/files/secor/notation/symbols1.bmp
> >
> > I think the problem with reading them under poor conditions is a
> > combination of factors  vertical lines that are rather thin
*and*
> > vertical lines that are too close together, with the second
factor
> > being more of a problem than the first. I redid the straight
flag
> > symbols on the fourth staff using singlepixel vertical lines
with
> > enough space between them to make them legible at a distance. I
also
> > put a couple of them in combination with conventional sharps and
> > flats at the upper right, for aesthetic evaluation.
>
> I like this better, but part of the family resemblance of the
existing
> symbols is that they are all ectomorphs, except for the rarely seen
> doublesharp being a mesomorph. They are not endomorphs like your
 and X
> symbols. Even some of my  symbols are pushing it.
The symbols get fatter as the alterations become larger, which is
only logical. And I even put the fattest ones on a diet, and now
none of them is wider than its height. So what is the problem?
> > Just above the "conventional accidentals" staff I also added my
> > conventional sharp to the left of yours. Mine (more than yours)
> > looks more like what I found in printed music, and I suspect that
> > Tartini fractional sharps constructed (or written) with too
narrow
> > spacing between the vertical lines (such as we have here) are
what
> > led to Ted Mook's observation. (And I do prefer your version.)
>
> Maybe, but part of the problem is just that it is hard to tell 2
identical
> sidebyside things from 3 identical sidebyside things with the
same
> spacing. I've made my  and X's wider now, but not as wide as
yours, and
> shortened the middle tail of the 3 by 3 pixels relative to the
others, so
> they are not 3 identical things any more.
I believe that shortening the middle line makes it more difficult to
see it, thereby making it *more* difficult to distinguish three from
two. This is particularly true when the symbol modifies a note on a
line and the middle line terminates at a staff line (so you see only
two lines sticking out). In fact, after looking at this again, I
think I would be in favor of shorting all of the symbols from 17 to
16 pixels so that no vertical line would terminate at a staff line.
(This would also keep symbols modifying notes a fifth apart from
colliding. But you made a comment below regarding how the length of
a new symbol looks when placed beside a conventional flat, so I need
to evaluate this further.)
> > > Frankly, I think the best solution is to use two symbols side
by side
> > > instead of the 3 and X shaft symbols, the one nearest the
notehead being a
> > > whole sharp or flat (either sagittal or standard). I think
we're packing so
> > > much information into these accidentals that we can't afford to
try to also
> > > pack in the number of apotomes. I suggest we provide single
symbols from
> > > flat to sharp and stop there. At least you have provided the
doubleshaft
> > > symbols so you never have to have the two accidentals pointing
in opposite
> > > directions.
> >
> > What? Did I understand this correctly? Are you considering
using
> > the doubleshaft symbols???
>
> No.
>
> > (Or are you suggesting that I should do
> > this and forget about the  and X symbols?)
>
> Yes.
>
> > > The fact that in all the history of musical notation, a single
symbol for
> > > doubleflat was never standardised, tells me that it isn't very
important,
> > > and we could easily get by without a single symbol for double
sharp too.
> >
> > I think that it's because two sharps placed together looks a
little
> > weird, but not two flats.
>
> You're just strengthening my case. I don't find that two of our
symbols
> side by side have the same problem as the conventional sharp symbol
(making
> a third phantom symbol in between). I find them to be more like the
> conventional flat symbol, for which no singlesymbol double has
ever been
> seen as necessary.
>
> > If I remember correctly, I think that
> > doubleflats are placed in contact with one another, as I put
them
> > above the staff (but I will need to check on this.)
>
> Those I have found have not been touching. I do not propose to have
ours
> touching either.
Yes, I saw that, once I found an example.
> ... I just thing this is too confusing, and the  and X symbols
are not
> required anyway.
>
> > I did make my new symbols (in the fourth staff) one pixel longer
(at
> > the tip of the arrow), which can be seen only when the note is on
a
> > line. I didn't think that it was wise to overlap the flag in the
> > other direction, because this would make the nubs at the ends of
the
> > flags less visible if a staff line were to pass through them.
Where
> > I now have them, the nubs (actually 3x3 pixel squares) are in
both
> > cases immediately adjacent to staff lines.
>
> I don't see what's wrong with the nubs straddling the line in one
case, and
> being in free space in the other case. i.e. put them, not one, but
2 pixels
> further out than you have them now.
They bigger they get, the uglier they look. I eventually realized
that the reason why they had to be so big for the straight flags is
that the straight lines are of constant thickness, whereas the others
get thinner at the ends, making the nubs easier to see.
In your latest figures I notice that you are making a noticeable
difference in width between the left and right flags, which is very
effective with the straight flags. Perhaps this will be the best way
to distinguish left from right. A very small nub could still be used
at the end of the larger of each pair of curved flags as a stylistic
embellishment.
> > By the way, if you look at the concave flags in my earlier
figures,
> > you will see that one part of the curving flag has postive and
> > another part negative slope (and I am still making them this
way); a
> > nub on the end of this sort of flag can be seen very easily.
>
> True.
With your concave flags, half of the length of the curve is
coincident with the vertical arrow shaft, which makes it difficult to
tell that this was intended to be a concave curve. The portion of
the curve with least slope is much thicker, and taken together with
the overall lateral narrowness of the flag, it comes out looking more
like a blob than a curved line.
>
> > ... I copied three of your symbols so I can comment
> > on them. In all three of them the concave or wavy flag is
> > significantly lower than the line or space for its note.
>
> You mean the upward pointing ones? Concave I can understand, but
wavy? The
> horizontally inflected part of the wavy flag is always exactly
centred
> relative to the center of the notehead.
As with the concave flag, the top part of the curve is coincident
with the arrow shaft, so it (i.e., the version on which I was
commenting) tends to look like a smaller and lower convex flag that
is modifying a note one staff position lower. Your latest version
(19 April) of the wavy flag is identical to what I now have, except
that I have made the (vertical) extremity of the flag one pixel
shorter. Why shorter? I think that the concave and wavy flags
should be smaller than the convex and straight flags  both in
length and thickness.
> > I propose
> > using instead the concave style of flag that I described before,
for
> > which I prepared a set of symbols on the 7th staff. (Note that
the
> > nubs don't get lost, even though they are quite small.)
I would further like to modify what I have for these by using
different lateral widths (left vs. right), so I still have some work
to do on the symbols before putting a new file out there.
> > At the top right (under altitude considerations) I put my latest
> > version of the symbols in combination with conventional sharps
and
> > flats, with singlesymbol equivalents included (above the staff).
>
> I hope you show some down pointing ones next to a flat, because I
think
> they look strange if their tails are too much shorter than the
flat's tail.
> In fact I'd be in favour of making the tails of downpointing
arrows longer
> than uppointing ones.
Okay, I'll try this and let you know what I think. (But I always
thought that the tails of conventional flats were too long anyway.)
Slowly, but surely, we are making progress.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  Left Right
> > +
> > Convex  29 7
> > Straight  5 (115)
> > Wavy  17 23
> > Concave  19 (17'17)
>
> This looks very workable, and I am about 99 percent sold on it.
> (Just give me some more time.)
Sure. We want to be sure we've explored every option thoroughly.
> In your table of symbols:
>
> Symbol Left Right
> for flags flags
> 
>
> 23' = 17 + (115)
>
> 31' = 5 + (17'17) + 7
> or 5 + 23 + 23
>
> 37 = 29 + 17
>
> options can be added for the following:
>
> 23' = 17 + (115)
> or 29 + (17'17)
>
> 31' = 5 + (17'17) + 7
> or 5 + 23 + 23
> or 7 + 7
>
> 37 = 29 + 17
> or 5 + 5
>
> These 5+5 option for the 37comma uses a much smaller schisma
> (6553600:6554439, ~0.222 cents) than what you have. But the problem
> with these three options that I have given is that none of the
> schismas vanish in 1600ET.
>
> Should we rethink the question of whether it is really necessary for
> these schismas to vanish in 1600ET, because I don't see any good
> reason.
It doesn't have to be 1600ET. It doesn't even have to be an ET. It
might be a linear or planar or whatever temperament.
But I feel it is highly desirable to know that the schismas we are
using do not somewhere add up to something considerably more than 0.5
cents. i.e. I want to know what maximum error (over all the intervals
in our highest odd limit) is implied by our choice of notational
schismas. If we don't know what temperament it is based on, we may
happen to have two near 0.5 cent schismas that "pull in opposite
directions".
> While it is nice to have everything come out exact using
> 1600 as a frame of reference, do you think anyone is actually going
> to be able to use it in a performance to produce pitches?
Not at all. But it is significant that (in the simplest example) our
single symbols for 13 and 35 are identical. We are presenting the
composer with a choice. Either use a pair of separate symbols for 35,
or accept that the performer will read it as a 13 diesis (or the
corresponding number of cents) and introduce a certain error. We're
trying to keep that error below 0.5 cents, although I think we've
already got one of 0.6 c.
> The
> increments are much smaller than 1 cent, and the pitches can't be
> related easily to 12ET, as Johnny Reinhard is doing. (i.e., not a
> subdivision, as is 1200ET), )
Although it may be convenient that its divisions are exactly 3/4 of a
cent.
> So if we're trying to accommodate
him
> with this notation, all that's really necessary is to keep the
> schismas small and provide the number of cents somewhere on the
> score, at least in a table with the symbols.
That's right. But "keep the schismas small" means also the
effective _combination_ schismas for all the intervals between pairs
of odd numbers, not just the odd numbers themselves. Actually, I'm
not sure if I know what I'm talking about here. At least with 1600ET
we knew where we stood. The thing may be to find another ET above
1000 or some other system that accomodates all the schismas we want.
Gene was looking at these for us and he found 1600 was the best 31
limit unique ET of all those less than or equal to it in size, but
hasn't gone to higher primes yet. He may have lost interest.
There are 11 odd primes up to 37, if we have 11 independent schismas,
and express them as primeexponent vectors and take the determinant of
the resulting square matrix, I understand we'll get the cardinality of
the corresponding ET. However I'm pretty sure we don't have 11
independent schismas and I get a little hazy here about how to find
generator mappings. I'm hoping Graham Breed or Gene Smith can help us
here. I think we just need to give the vector for every schisma we're
interested in. Then ask them to tell us the maximum error implied by
various sets of these.
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> The symbols get fatter as the alterations become larger, which is
> only logical.
Sure.
> And I even put the fattest ones on a diet, and now
> none of them is wider than its height. So what is the problem?
I'm ignoring the tails. With the standard symbols the _body_ of the symbol
is never wider than it is high. But hey, I can live with it.
> I believe that shortening the middle line makes it more difficult to
> see it, thereby making it *more* difficult to distinguish three from
> two. This is particularly true when the symbol modifies a note on a
> line and the middle line terminates at a staff line (so you see only
> two lines sticking out).
Good point. How about making the middle one only 2 pixels shorter than the
outer ones. That will solve the latter problem.
> In fact, after looking at this again, I
> think I would be in favor of shorting all of the symbols from 17 to
> 16 pixels so that no vertical line would terminate at a staff line.
Then I think the sagittals will look odd with sharps too, not just flats.
And it will worsen the aspectratio problem. I believe flats have such long
tails, precisely to give them a similar aspect ratio to sharps and naturals.
> (This would also keep symbols modifying notes a fifth apart from
> colliding. But you made a comment below regarding how the length of
> a new symbol looks when placed beside a conventional flat, so I need
> to evaluate this further.)
I don't see a problem with them colliding. Have you found examples of flats
doing that yet? I have.
> In your latest figures I notice that you are making a noticeable
> difference in width between the left and right flags, which is very
> effective with the straight flags.
They were like that from the start. For straight and concave I have 5
pixels wide versus 7 and for wavy I have 4 versus 5, but concave are both 4
pixels.
> Perhaps this will be the best
way
> to distinguish left from right. A very small nub could still be
used
> at the end of the larger of each pair of curved flags as a stylistic
> embellishment.
I agree it would help with the lateral confusability. But from a purely
aesthetic point of view, I think I'd prefer not.
> With your concave flags, half of the length of the curve is
> coincident with the vertical arrow shaft, which makes it difficult
to
> tell that this was intended to be a concave curve. The portion of
> the curve with least slope is much thicker, and taken together with
> the overall lateral narrowness of the flag, it comes out looking
more
> like a blob than a curved line.
You're absolutely right. The concaves just don't work at only 4 pixels
wide. It's interesting how the knowledge of what it's supposed to look like
can blind one to alternative interpretations. That's why it's so good to
cooperate the way we are.
Trouble is, I just can't accept a 19 comma flag that's wider than 4 pixels
(including shaft) since it represents a barely perceptiple comma of about 3
cents. I'd really prefer to make it only 3 pixels, but that seems too low res.
How about we forget abour concave and _make_ it a (circular or semicircular
or triangular) blob? And move it up the shaft as you suggest, to center the
blob on the notehead. Too bad about the convex/concave complementarity.
> As with the concave flag, the top part of the curve is coincident
> with the arrow shaft, so it (i.e., the version on which I was
> commenting) tends to look like a smaller and lower convex flag that
> is modifying a note one staff position lower. Your latest version
> (19 April) of the wavy flag is identical to what I now have, except
> that I have made the (vertical) extremity of the flag one pixel
> shorter. Why shorter? I think that the concave and wavy flags
> should be smaller than the convex and straight flags  both in
> length and thickness.
Yes. The wavy doesn't work at 4 pixels wide, and apparently you find it
only barely works at 5 pixels. I like your idea of making both concave and
wavy vertically shorter than the others too. And I agree that the vertical
position should be a sort of compromise between centering the flag
_including_ the part coincident with the shaft, and centering it
_excluding_ the part coincident with the shaft.
> I would further like to modify what I have for these by using
> different lateral widths (left vs. right), so I still have some work
> to do on the symbols before putting a new file out there.
I look forward to it.
> Okay, I'll try this and let you know what I think. (But I always
> thought that the tails of conventional flats were too long anyway.)
I believe flats have such long tails, precisely to give them a similar
aspect ratio to sharps and naturals.
> Slowly, but surely, we are making progress.
Yes indeed. :)
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
At 01:26 22/04/02 0000, George Secor wrote:
>In your table of symbols:
>
>Symbol Left Right
>for flags flags
>
>
>23' = 17 + (115)
>
>31' = 5 + (17'17) + 7
> or 5 + 23 + 23
>
>37 = 29 + 17
>
>options can be added for the following:
>
>23' = 17 + (115)
> or 29 + (17'17)
>
>31' = 5 + (17'17) + 7
> or 5 + 23 + 23
> or 7 + 7
>
>37 = 29 + 17
> or 5 + 5
>
>These 5+5 option for the 37comma uses a much smaller schisma
>(6553600:6554439, ~0.222 cents) than what you have. But the problem
>with these three options that I have given is that none of the
>schismas vanish in 1600ET.
>
>Should we rethink the question of whether it is really necessary for
>these schismas to vanish in 1600ET ... ?
I believe we can forget 1600ET. It was just a handy place to look for
suitably small notational schismas. I've been foolishly failing to check
the size in cents of some of the more recent 1600ET schismas.
As you point out,
c37 = c29 + c17 involves a largish schisma of 0.57 cents, but one
alternative I gave,
c37' = c5 + c5 + c19, is completely unconscionable at 1.04 cents.
Also c43 = c19 + c19 + c(17'17), 0.72 cents, which I don't consider
usable either.
I've now exhaustively searched all combinations of up to 3 of our flags.
Here's what I end up with.
Symbol Left Right Schisma
for flags flags (cents)

5 = 5 0
7 = 7 0
11 = 5 + (115) 0
11' = 29 + 7 0.34
13 = 5 + 7 0.42
13' = 29 + (115) 0.08
17 = 17 0
17' = 17 + (17'17) 0
19 = 19 0
19' = 19 + 23 0.16
23 = 23 0
23' = 17 + (115) 0.49
or 29 + (17'17) 0.52 *
29 = 29 0
31 = 19 + (115) 0.12
31' = 29 + 5 0.03 *
or 7 + 7 0.44 *
or 5 + (17'17) + 7 0.19
or 5 + 23 + 23 0.37
37 = 5 + 5 0.22 *
or 29 + 17 0.57
37' = 19 + 23 + 7 0.25
or 5 + 17 + 23 0.65
41 = 5 0.26 *
or 17 + (17'17) + (17'17) 0.51
43 = 19 + 19 + (17'17) 0.72 [schisma too big]
47 = 17 + 7 0.45
or 19 + 29 0.42
or 19 + 23 + 23 0.02
or 5 + 17 + (17'17) 0.21
pythagorean
comma = 17 + 17 + (17'17) 0
diaschisma = 19 + 23 0.37 [same symbol as 19']
diesis = 17 + (115) 0.56 [same symbol as 23']
* doesn't vanish in 1600ET.
So, in addition to c37 = c5 + c5, there are some other schismas available
to us, that don't vanish in 1600ET and are smaller than those that do.
Namely:
31' = 29 + 5 0.03 cents
41 = 5 0.26 cents
We should definitely stop at prime 41, since there is no way to get 43 with
sufficient accuracy using our 8 existing flags. We're under the half cent
otherwise.
In the application you (or Erv) found for 41, would a 0.26 cent error in
the 41 have rendered it useless? Why not simply reuse the 5 comma as the 41
comma?
If we do that we eliminate one major reason for choosing (17'17) as our
final comma (over 17'19 or simply 17'). No other comma symbols depend on
it. But it is the only one that has good complementation rules in 217ET.
Actually, it might be better to stop at 31, since symbols with more than 2
flags (e.g. 37') are getting too difficult, for my liking.
I've uploaded a new version of
/tuningmath/files/Dave/SymbolsBySize.bmp
based on the first option for each symbol, up to the prime 41, in the table
above.
I realised recently that some of those alternate commas (the primed ones
that are intended for a diatonicbased notation) should not really be
defined as they currently are, but as their apotome complements, because
that's how they will be used. They are 17', 19', 23' and 25. Let's call the
apotome complements of these 17", 19", 23" and 25". For diatonicbased
purposes, these should be defined as 17:18, 18:19, 23:24 and 24:25
respectively, and should be assigned appropriate doubleshaft symbols.
The question is, can their symbols be sensibly based on the complementation
rules which we derived in the context of 217ET?
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > The symbols get fatter as the alterations become larger, which is
> > only logical.
>
> Sure.
>
> > And I even put the fattest ones on a diet, and now
> > none of them is wider than its height. So what is the problem?
>
> I'm ignoring the tails. With the standard symbols the _body_ of the
symbol
> is never wider than it is high. But hey, I can live with it.
Okay.
> > I believe that shortening the middle line makes it more difficult
to
> > see it, thereby making it *more* difficult to distinguish three
from
> > two. This is particularly true when the symbol modifies a note
on a
> > line and the middle line terminates at a staff line (so you see
only
> > two lines sticking out).
>
> Good point. How about making the middle one only 2 pixels shorter
than the
> outer ones. That will solve the latter problem.
The problem is that that middle line needs to be noticed as much as
the other two, so that we can see that there are three of them, and
making it shorter tends to deemphasize it.
> > In fact, after looking at this again, I
> > think I would be in favor of shorting all of the symbols from 17
to
> > 16 pixels so that no vertical line would terminate at a staff
line.
>
> Then I think the sagittals will look odd with sharps too, not just
flats.
> And it will worsen the aspectratio problem. I believe flats have
such long
> tails, precisely to give them a similar aspect ratio to sharps and
naturals.
>
> > (This would also keep symbols modifying notes a fifth apart from
> > colliding. But you made a comment below regarding how the length
of
> > a new symbol looks when placed beside a conventional flat, so I
need
> > to evaluate this further.)
>
> I don't see a problem with them colliding. Have you found examples
of flats
> doing that yet? I have.
Why don't we just make all three of the arrow shafts the same length,
and I'll forget about making the symbols shorter than 17 pixels.
> > In your latest figures I notice that you are making a noticeable
> > difference in width between the left and right flags, which is
very
> > effective with the straight flags.
>
> They were like that from the start. For straight and concave I have
5
> pixels wide versus 7 and for wavy I have 4 versus 5, but concave
are both 4
> pixels.
>
> > Perhaps this will be the best
> way
> > to distinguish left from right. A very small nub could still be
> used
> > at the end of the larger of each pair of curved flags as a
stylistic
> > embellishment.
>
> I agree it would help with the lateral confusability. But from a
purely
> aesthetic point of view, I think I'd prefer not.
So would you then be satisfied with a difference in width alone to
aid in making the lateral distinction?
> > With your concave flags, half of the length of the curve is
> > coincident with the vertical arrow shaft, which makes it
difficult
> to
> > tell that this was intended to be a concave curve. The portion
of
> > the curve with least slope is much thicker, and taken together
with
> > the overall lateral narrowness of the flag, it comes out looking
> more
> > like a blob than a curved line.
>
> You're absolutely right. The concaves just don't work at only 4
pixels
> wide. It's interesting how the knowledge of what it's supposed to
look like
> can blind one to alternative interpretations. That's why it's so
good to
> cooperate the way we are.
>
> Trouble is, I just can't accept a 19 comma flag that's wider than 4
pixels
> (including shaft) since it represents a barely perceptiple comma of
about 3
> cents. I'd really prefer to make it only 3 pixels, but that seems
too low res.
>
> How about we forget abour concave and _make_ it a (circular or
semicircular
> or triangular) blob? And move it up the shaft as you suggest, to
center the
> blob on the notehead. Too bad about the convex/concave
complementarity.
Why not just go with my version of the concave symbols:
/tuning
math/files/secor/notation/symbols2.bmp
(see upper right, top staff)? The left flag is 3 pixels wide, and
the right flag is 4 pixels wide, yet they are clearly identifiable.
(I also threw in a complement symbol.)
> > As with the concave flag, the top part of the curve is coincident
> > with the arrow shaft, so it (i.e., the version on which I was
> > commenting) tends to look like a smaller and lower convex flag
that
> > is modifying a note one staff position lower. Your latest
version
> > (19 April) of the wavy flag is identical to what I now have,
except
> > that I have made the (vertical) extremity of the flag one pixel
> > shorter.
This observation applied to your (wavy) flag for the 23 comma, not
the one for the 17comma.
> > Why shorter? I think that the concave and wavy flags
> > should be smaller than the convex and straight flags  both in
> > length and thickness.
>
> Yes. The wavy doesn't work at 4 pixels wide, and apparently you
find it
> only barely works at 5 pixels. I like your idea of making both
concave and
> wavy vertically shorter than the others too. And I agree that the
vertical
> position should be a sort of compromise between centering the flag
> _including_ the part coincident with the shaft, and centering it
> _excluding_ the part coincident with the shaft.
I wouldn't make the vertical arrow shaft shorter, though.
To the right of our convex symbols are our latest versions of the
wavy flags for comparison. I made the left wavy flag 4 pixels wide,
like the concave right flag, and the right wavy flag is 5 pixels
wide. Both of our wL+wR symbols have flags 4 pixels wide on each
side. As with the concave symbols, I also threw in a complement
symbol.
I also experimented with taking the curves out of the wavy symbols,
making them rightangle symbols, which I put at the far right. (The
left vs. right line lengths are different in both the horizontal and
vertical directions to aid in telling them apart.) We already have
two kinds of curvedline symbols, and substituting these for the wavy
symbols would give us two kinds of straightline symbols as well.
It's not that I don't like the wavy symbols (I do like them), but I
thought that this would make it easier  both to remember and to
distinguish them. (This one's your call.)
> > I would further like to modify what I have for these by using
> > different lateral widths (left vs. right), so I still have some
work
> > to do on the symbols before putting a new file out there.
>
> I look forward to it.
I copied your symbols (unaltered) into the second staff. Below that
I put my versions of the symbols for comparison.
I found that when I draw *convex* flags freehand that I tend to
curve the end of the flag inward slightly to make sure that it isn't
mistaken for a straight flag, and I have been doing something on this
order for some time with my bitmap symbols as well. I have modified
these also to reflect this, and you can let me know what you think.
(I notice that the right flag of your 47.4cent symbol has this sort
of feature  was that a mistake?) Or possibly only the left convex
flag could be given this feature to further distinguish it from the
convex right flag.
Also, observe my 43cent and 55cent symbols  the ones with two
flags on the same side.
> > Okay, I'll try this and let you know what I think. (But I always
> > thought that the tails of conventional flats were too long
anyway.)
>
> I believe flats have such long tails, precisely to give them a
similar
> aspect ratio to sharps and naturals.
Yes, good point, and one reason why I'm not reluctant to discard the
idea of making the symbols any shorter than 17 pixels. When I put a
5commadown symbol next to a flat the new symbol has a shorter stem
than the flat. I don't think that this is inappropriate, inasmuch as:
1) the two symbols are in about the same proportion lengthtowidth;
and
2) the difference in height is the same as that in the two vertical
lines of a conventional sharp symbol.
> > Slowly, but surely, we are making progress.
>
> Yes indeed. :)
And I can't imagine that anyone else has ever worked out a notation
in this much detail.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>
> Why not just go with my version of the concave symbols:
>
> /tuning
> math/files/secor/notation/symbols2.bmp
Sorry  wrong file!!! This is the new one:
/tuning
math/files/secor/notation/SymAllSz.bmp
George
In light of your recent difficulties with recognizable concave
symbols (and possibly the wavy ones), I'll do my best to respond to
some of the things in this message (#4117):
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> [dk:]
> I've just uploaded a MsWord document containing drawings that show
how I
> conceive of these flags in a resolutionindependent manner, so as to
> produce that style. You will see how the style is designed to be
compatible
> with the conventional symbols, in particular the conventional sharp
and
> flat symbols which the sagittals will most often have to appear
next to.
>
> /tuningmath/files/Dave/Flags.doc
>
> I suggest you print it and take a hiliter pen and colour in the
parts that
> actually make up the flags and stems. I couldn't figure any easy
way to do
> that in Word. I'm sure you'll figure out what needs colouring. Then
turn
> the second page upside down and hold each flag in turn, beside the
standard
> flat and then the standard sharp.
>
> Notice that the prototype convex and concave flags are exact 180
degree
> rotations of each other, and wavy is an exact 180 degree rotation of
> itself. This was partly intended to help with flag complementation
in 217ET.
This is somewhat different than what I had envisioned. In your plan
each curve changes direction by 90 degrees, whereas, in drawing the
symbols freehand, I found that it was most natural to exceed 90
degrees for the convex symbols and even to exceed 180 degrees for the
concave symbols. I found that, in order for the narrower concave
symbols to be recognizable, I had to curve the line more sharply the
closer I approached the end of the flag.
Regarding my comments yesterday about wavy vs. rightangle (i.e.,
straightenedwavy) symbols:
<< I also experimented with taking the curves out of the wavy
symbols, making them rightangle symbols, which I put at the far
right. (The left vs. right line lengths are different in both the
horizontal and vertical directions to aid in telling them apart.) We
already have two kinds of curvedline symbols, and substituting these
for the wavy symbols would give us two kinds of straightline symbols
as well. It's not that I don't like the wavy symbols (I do like
them), but I thought that this would make it easier  both to
remember and to distinguish them. (This one's your call.) >>
After trying out a few things with rightangle symbols, I would have
to say that I'm in favor of the wavy flags.
Also, regarding this comment:
<< (I notice that the right flag of your 47.4cent symbol has this
sort of feature  was that a mistake?) >>
I didn't notice until later that it is actually a 3flag symbol with
a combination of convex and wavy flags on the right side, which would
indicate that the combination as you have it isn't recognizable.
> Ok. We agree on the linethickness and overall width of all the
tails now,
> 5 pixels for , 7 pixels for both  and X.
Yes.
> We also agree on how far the tail projects away from the centreline
of the
> corresponding notehead. That's 11 pixels not including the pixel
that's
> _on_ the centreline. That's the same as a sharp or natural, but two
pixels
> shorter than a flat. These agreements are good.
Yes.
> But we still don't agree on the height of the X's. Your X's are not
> constant. They vary according to what flags they have on them, and
are
> often not laterally symmetrical. My X's are all the same height as
they are
> wide (7 pixels) and are laterally symmetrical. They just meet the
concave
> flags, but for other flag types they are extended by two parallel
lines at
> the same spacing as the outer two of the '. If nothing else, it
certainly
> simplifies symbol construction, not to have to design a new X tail
for
> every possible combination of flags. And if we get into using more
than one
> flag on the same side (e.g. for 25) with these X tails, I figure
we're
> gonna need those parallel sides.
I tried this out myself and ended up with exactly what you have for
these. I don't like the way the x's look with the straight and
convex flags for a couple of reasons:
1) The x appears too remote or detached from the flag(s), and
2) The x would seem to be indicating an alteration to a note on a
line or space two steps away from the note actually being altered,
which would tend to be confusing. (The Sims square root symbol also
has this problem.)
When I draw the Xsymbols freehand, I imagine that I am constructing
the diagonals of a trapezoid having 3 right angles. The size and
shape of the symbol is the same as the corresponding one with 3 arrow
shafts, and the four corners of the trapezoid are determined by the
two ends of the outside shafts and the points of intersection of
those shafts with a flag. This is the way I would construct the X's
for a scalable font.
> > while
> > the semisharp () and sharp groups () are either the same as or
> > very close to your symbols. The biggest problem I had was with
the
> > nubs (which I made rather large and ugly) still tending to get
lost
> > in the staff lines. I tried one symbol (in the middle of the
staff)
> > with a triangular nub, which looks a little neater, I think.
>
> Yes it looks neater, but I fear it is out of character with the
standard
> accidentals. I even think that maybe _any_ nubs are outof
character. Of
> course we have the precedent of the doublesharp symbol, but I tend
to
> think of _it_ as being outofcharacter with the other 3 standard
symbols.
> I suspect it is more often seen as the unpitched notehead than as an
> accidental.
I hope, then, that by agreeing on appropriate distinctions in size
between left and right flags we can eliminate them entirely.
> > I propose
> > using instead the concave style of flag that I described before,
for
> > which I prepared a set of symbols on the 7th staff. (Note that
the
> > nubs don't get lost, even though they are quite small.)
>
> I'm not averse to a slight recurve on the concaves, but I'm afraid
I find
> some of those in symbols2.bmp, so extreme in this regard, that they
are
> quite ambiguous in their direction. With a mental switch akin to
the Necker
> cube illusion, I can see them as either a recurved concave pointing
upwards
> or a kind of wavy pointing down. Apart from any nub, I don't think
that
> they should go more than one pixel back in the "wrong" direction.
Those at
> the extreme lower left of the page look ok.
You would have to think of the end of the flag as pointing and not
the curve of the flag. Putting a nub on the end might even help in
that regard. But I can't see how I can get away from using a rise of
several pixels to indicate the line or space of the note being
altered. The part of the curve coincident with the arrow shaft just
isn't going to accomplish that.
> I'm guessing that you need the huge recurve to convince yourself
that
> concave can represent larger commas than wavy?
No, forget about that. I want to keep the concave flags for the two
smallest alterations. The convextoconcave conversion in the
complementary symbols is a concept that I wouldn't want to discard.
> > I like your wavy flag, but I would propose waving it a little
> higher,
> > as I did in the symbols just to the right of yours (back on the
5th
> > staff); I seemed to be getting a better result with a thinner
flag,
> > which would also serve to avoid confusion of the wavy with the
> convex
> > flag.
>
> It seems to me that you have increased the possibility of confusion
of wavy
> with convex, by waving it higher.
We'll just have to evaluate our latest efforts to see if that's a
problem, but I don't think it is.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>
> When I draw the Xsymbols freehand, I imagine that I am
constructing
> the diagonals of a trapezoid having 3 right angles.
Oops! Make that "the diagonals of a trapezoid having three of its
sides at right angles (i.e., having two right angles) with the
vertical sides parallel." (Come to think of it, don't all trapezoids
have two right angles?)
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > When I draw the Xsymbols freehand, I imagine that I am
> constructing
> > the diagonals of a trapezoid having 3 right angles.
>
> Oops! Make that "the diagonals of a trapezoid having three of its
> sides at right angles (i.e., having two right angles) with the
> vertical sides parallel." (Come to think of it, don't all
trapezoids
> have two right angles?)
>
> George
.....
..../.....\
.../.......\
../.........\
./...........\
/.............\
?
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > When I draw the Xsymbols freehand, I imagine that I am
> constructing
> > the diagonals of a trapezoid having 3 right angles.
>
> Oops! Make that "the diagonals of a trapezoid having three of its
> sides at right angles (i.e., having two right angles) with the
> vertical sides parallel." (Come to think of it, don't all
trapezoids
> have two right angles?)
No, they don't have to have any right angles, but they always have
two parallel sides. (It only took 3 messages to get it right; I must
be having a bad trapezoid day!)
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > >
> > > When I draw the Xsymbols freehand, I imagine that I am
> > constructing
> > > the diagonals of a trapezoid having 3 right angles.
> >
> > Oops! Make that "the diagonals of a trapezoid having three of
its
> > sides at right angles (i.e., having two right angles) with the
> > vertical sides parallel." (Come to think of it, don't all
> trapezoids
> > have two right angles?)
>
> No, they don't have to have any right angles, but they always have
> two parallel sides. (It only took 3 messages to get it right; I
must
> be having a bad trapezoid day!)
>
> George
Or you could say I fell into that trapezoid!
gs
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
> .....
> ..../...
> .../...,'
> ../..,'
> ./.,'
> /,'
> ?
never mind  this is not a trapezoid.
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>The problem is that that middle line needs to be noticed as much as
>the other two, so that we can see that there are three of them, and
>making it shorter tends to deemphasize it.
I think we need a third (and fourth and fifth ...) opinion on this one.
From a performer who sight reads.
>Why don't we just make all three of the arrow shafts the same length,
>and I'll forget about making the symbols shorter than 17 pixels.
I have found our cooperation on this notation to be remarkable egoless,
with both of us concerned only with what will be best for the enduser, and
not concerned with "getting our own way". But we've always given reasons
for rejecting the other's proposal, so as to avoid any hurt feelings. I
feel that any compromises we have made so far, e.g in lengths, widths,
thicknesses or curvatures, have been made because we believe the best
option most likely lies in between our two extremes.
Now I may be reading it wrong, but the above seems to be suggesting a
tradeoff of two completely unrelated things, purely on the basis, "you let
me have my way on this and I'll let you have your way on that". If we can't
agree on something, I'd prefer to seek other opinions, rather than engage
in such a tradeoff.
>So would you then be satisfied with a difference in width alone to
>aid in making the lateral distinction?
Yes.
>Why not just go with my version of the concave symbols:
>
>/tuning
>math/files/secor/notation/SymAllSz.bmp
As I wrote in
/tuningmath/message/4117
I'm not averse to a slight recurve on the concaves, but I'm afraid I find
your current proposals so extreme in this regard, that they are quite
ambiguous in their direction. With a mental switch akin to the Necker cube
illusion, I can see them as either a recurved concave pointing upwards or a
kind of wavy pointing down. Apart from any nub, I don't think that they
should go more than one pixel back in the "wrong" direction. Those at the
extreme lower left of Symbols2.bmp look ok.
>(see upper right, top staff)? The left flag is 3 pixels wide, and
>the right flag is 4 pixels wide, yet they are clearly identifiable.
>(I also threw in a complement symbol.)
These are 4 and 5 pixels wide by my reckoning (including the part
coincident with the shaft). One must define a flag as including a part
coincident with the shaft so one knows what it will look like when it is
sharing a  or X shaft with another flag. But I did overstate the case
when I said that 4 pixels wide doesn't work. We now have agreement that the
concaves on a single shaft should both be 4 pixels wide.
Maybe I went too far in reducing the height of the wavy and concave to 7
pixels (including shortening the shaft at the pointy end). I see that your
concaves are 9 pixels high, and your wavys are 10 pixels high, only one
pixel shorter than the straight and concaves. In fact when it isn't used
with the left wavy, your right wavy is the full 11 pixels in height and 6
pixels wide. I think these lead to too many symbols whose apparent visual
size is too far out of keeping with their size in cents.
I find that:
9.4 14.7 20.1 all look bigger than 21.5
27.5 and 30.6 look bigger than 31.8
I am proposing something between yours and mine. See
/tuningmath/files/Dave/SymbolsBySize.bmp
I don't think we actually need any lateral distinction between the two
concaves because in rational tunings the (17'17) flag will never occur on
its own, and I don't think any ETs of interest below 217ET will need to
use both 19 and (17'17). What do you think?
But it wouldn't hurt if they were distinct. The biggest problem (for me) is
trying to make the 19 comma (left concave) look as small as it really is
without it disappearing. If its width was in proportion to the width of the
5 comma flag, you wouldn't see it for the shaft! If we look at areas and
ignore the part coincident with the shaft, the 5 comma flag is 4 pixels by
11 pixels. The 19 comma flag would have to fit in a rectangle 7 pixels in
area. In my 22Apr proposal I've allowed those 7 pixels to blow out to 12,
3 wide by 4 high (excluding shaft). Yours is 18 pixels, 3 wide by 6 high.
If we simply count black pixels (excluding shaft) we find that the 5 comma
flag has 15, which means the 19 comma flag should only have 2.4, which we
might generously round up to 3 black pixels. Mine has 7, yours has 9.
So my (17'17) (right concave) flag is about the right size, but my 19 flag
is about double the size it should be. I can live with double, but I'n not
sure I can handle triple.
Now you probably think I'm being too literal with this representative size
stuff, but the problems occur when you have the 19 flag combined with
another flag and the result looks much bigger than some single flag that it
should be much smaller than. In particular
9.4 looking bigger than 21.5,
20.1 looking bigger than 27.3.
I suppose we can have a 19 comma flag that is lrger when used alone than
when combined with others, but I'd prefer not.
>I wouldn't make the vertical arrow shaft shorter, though.
OK.
>To the right of our convex symbols are our latest versions of the
>wavy flags for comparison. I made the left wavy flag 4 pixels wide,
>like the concave right flag, and the right wavy flag is 5 pixels
>wide. Both of our wL+wR symbols have flags 4 pixels wide on each
>side.
>As with the concave symbols, I also threw in a complement
>symbol.
Ah, but what exactly are they complements _of_?
I assume it was an oversight that left the wavy side of the 36.0 symbol
unmodified.
>I also experimented with taking the curves out of the wavy symbols,
>making them rightangle symbols, which I put at the far right. (The
>left vs. right line lengths are different in both the horizontal and
>vertical directions to aid in telling them apart.) We already have
>two kinds of curvedline symbols, and substituting these for the wavy
>symbols would give us two kinds of straightline symbols as well.
>It's not that I don't like the wavy symbols (I do like them), but I
>thought that this would make it easier  both to remember and to
>distinguish them. (This one's your call.)
You're right about them being more distinct, but the aesthetics are the
killer. Given more resolution, I'd go for something in between the existing
wavys and these rightangle ones, but not these totally sharp corners.
>I copied your symbols (unaltered) into the second staff. Below that
>I put my versions of the symbols for comparison.
>
>I found that when I draw *convex* flags freehand that I tend to
>curve the end of the flag inward slightly to make sure that it isn't
>mistaken for a straight flag, and I have been doing something on this
>order for some time with my bitmap symbols as well. I have modified
>these also to reflect this, and you can let me know what you think.
I think they look good, aesthetically speaking. The trouble is it makes the
down versions look too much like flats and backward flats. Also, you
decreased the size difference between the 7 flag and the 29 flag by adding
curvature on the outside of the 7 flag and the inside of the 29 flag.
I find the fact that the convex flags start off at rightangles to the
shaft and end parallel to the shaft, sufficient to make them distinct from
straight flags, without tending towards flats.
>(I notice that the right flag of your 47.4cent symbol has this sort
>of feature  was that a mistake?)
That's 37' = 19 + 23 + 7 = vL + wR + xR, so what you saw resulted from
mindlessly overlaying wR and xR. Being 37', my heart wasn't in it. I've had
a better go at it now, based on what you did for 25 and 31'.
>Or possibly only the left convex
>flag could be given this feature to further distinguish it from the
>convex right flag.
That would at least retain the full 2 pixel difference in width between XL
and xR, but still has the problem of looking too much like a backwards flat.
There is a way to make the convex more distinct from straight without
taking them closer to flats. We make them closer to being rightangles,
i.e. reduce the radius of the corner. I've shown comparisons with straight
flags and flats at top right of my latest bitmap.
>Also, observe my 43cent and 55cent symbols  the ones with two
>flags on the same side.
Yes. I wasn't very happy with mine. I like yours better, but I've modified
them very slightly. Tell me what you think.
Notice that it's OK for 31' down to look like a backwards flat, because it
_is_ a halfflat.
>Yes, good point, and one reason why I'm not reluctant to discard the
>idea of making the symbols any shorter than 17 pixels. When I put a
>5commadown symbol next to a flat the new symbol has a shorter stem
>than the flat. I don't think that this is inappropriate, inasmuch as:
>
>1) the two symbols are in about the same proportion lengthtowidth;
>and
>
>2) the difference in height is the same as that in the two vertical
>lines of a conventional sharp symbol.
Good points. OK. I'll forget the idea of giving down arrows longer shafts
than up arrows. Are we agreed then that all sagittals should be 17 pixels
high?
>And I can't imagine that anyone else has ever worked out a notation
>in this much detail.
Me neither.
It will of course be rejected out of hand by others, for reasons we haven't
even considered. :)
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
Ho George,
Sorry about my previous message in this thread. It should have been posted
16 hours ago, but I managed to email it to myself (replying to my own
forward of your message from the Yahoo website) and only discovered my
oversight the next morning (Australian time). So I sent it as soon as I
discovered it.
I see now, that you had already addressed some of the points in it.
First a correction. I wrote:
"We now have agreement that the concaves on a single shaft should both be 4
pixels wide."
That was wrong. At that stage we only had agreement that vL should be 4
pixels wide. But since then I've also agreed with you that vR can be 5
pixels wide, as shown in the latest SymbolsBySize.bmp (which incidentally
was sitting there for those 16 hours but you had no way of knowing it).
At 00:21 24/04/02 0000, you wrote:
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>In light of your recent difficulties with recognizable concave
>symbols (and possibly the wavy ones), I'll do my best to respond to
>some of the things in this message (#4117):
>
> In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>> [dk:]
>> I've just uploaded a MsWord document containing drawings that show
>how I
>> conceive of these flags in a resolutionindependent manner, so as to
>> produce that style. You will see how the style is designed to be
>compatible
>> with the conventional symbols, in particular the conventional sharp
>and
>> flat symbols which the sagittals will most often have to appear
>next to.
>>
>> /tuningmath/files/Dave/Flags.doc
>>
>> I suggest you print it and take a hiliter pen and colour in the
>parts that
>> actually make up the flags and stems. I couldn't figure any easy
>way to do
>> that in Word. I'm sure you'll figure out what needs colouring. Then
>turn
>> the second page upside down and hold each flag in turn, beside the
>standard
>> flat and then the standard sharp.
>>
>> Notice that the prototype convex and concave flags are exact 180
>degree
>> rotations of each other, and wavy is an exact 180 degree rotation of
>> itself. This was partly intended to help with flag complementation
>in 217ET.
>
>This is somewhat different than what I had envisioned. In your plan
>each curve changes direction by 90 degrees, whereas, in drawing the
>symbols freehand, I found that it was most natural to exceed 90
>degrees for the convex symbols
I have of course addressed the problems I see with that, in my previous
message and shown it in SymbolsBySize.bmp. I expect the
confusionwithflats problem didn't occur to you because you have, in the
past, only been interested in a totally sagittal notation, and had been
drawing them significantly smaller than flats.
>and even to exceed 180 degrees for the
>concave symbols. I found that, in order for the narrower concave
>symbols to be recognizable, I had to curve the line more sharply the
>closer I approached the end of the flag.
That may be OK for the resindependent description, but unfortunately it's
impossible to indicate it at the resolution we're using, while still
keeping the concaves as small as they need to be, or sufficiently distinct
from wavy's pointing the other way.
And of course it spoils the visual complementarity of convex and concave,
although the fact that the concaves need to be so much smaller than the
convex already does that to some degree. I guess their visual
complementarity could still be seen as a flip about the diagonal, rather
than a rotation of 180 degrees, however that doesn't work for the wavy. A
diagonal flip of wavy isn't the same kind of wavy, but something new.
Anyway, visual complementarity cues are nowhere as important as
distinctness from other symbols, and not just other accidental symbols.
There's also the quaverrest symbol to consider. I've just added
comparisons for that, to
/tuningmath/files/Dave/SymbolsBySize.bmp
>After trying out a few things with rightangle symbols, I would have
>to say that I'm in favor of the wavy flags.
Good.
>I didn't notice until later that it is actually a 3flag symbol with
>a combination of convex and wavy flags on the right side, which would
>indicate that the combination as you have it isn't recognizable.
No. It was garbage. If you haven't already seen my new c37' symbol, it
would be interesting if you would have a go at one yourself and then see
how similar they might be.
>> But we still don't agree on the height of the X's. Your X's are not
>> constant. They vary according to what flags they have on them, and
>are
>> often not laterally symmetrical. My X's are all the same height as
>they are
>> wide (7 pixels) and are laterally symmetrical. They just meet the
>concave
>> flags, but for other flag types they are extended by two parallel
>lines at
>> the same spacing as the outer two of the '. If nothing else, it
>certainly
>> simplifies symbol construction, not to have to design a new X tail
>for
>> every possible combination of flags. And if we get into using more
>than one
>> flag on the same side (e.g. for 25) with these X tails, I figure
>we're
>> gonna need those parallel sides.
>
>I tried this out myself and ended up with exactly what you have for
>these. I don't like the way the x's look with the straight and
>convex flags for a couple of reasons:
>
>1) The x appears too remote or detached from the flag(s), and
>
>2) The x would seem to be indicating an alteration to a note on a
>line or space two steps away from the note actually being altered,
>which would tend to be confusing. (The Sims square root symbol also
>has this problem.)
All good points. I really don't like the 's or X's anyway, for reasons
I've given before, and that I don't feel you've addressed. I'm just sort of
going along for the ride on those, assuming you're going to have them
anyway, and trying to make the best of it.
>When I draw the Xsymbols freehand, I imagine that I am constructing
>the diagonals of a trapezoid having 3 right angles. The size and
>shape of the symbol is the same as the corresponding one with 3 arrow
>shafts, and the four corners of the trapezoid are determined by the
>two ends of the outside shafts and the points of intersection of
>those shafts with a flag. This is the way I would construct the X's
>for a scalable font.
I understood what you meant, despite the "3" right angles. I was more
disappointed to find that the following 3 messages from you and 2 from
Paul, in this thread, were only trapezoid trivia. :)
Here's some more. The U.S. definitions of trapezoid and trapezium are
exactly swapped relative to the British/Australian definitions.
In the OED and Macquarie dictionaries, a trapezium has only one pair of
sides parallel, while a trapezoid has none. Websters has it the other way
'round. There's no requirement for any rightangles anywhere.
At least it's good to know Paul's reading the thread. I've been wondering
whether noone else was contributing because
(a) they think we're doing such a wonderful job without them, or
(b) they have no interest whatsoever in the topic, and think we're a couple
of looneys?
Hey, I've become so obsessed about this notation that I was lying in bed
this morning thinking how my various sleeping postures could be read as
various sagittal symbols. I was imagining children being taught them
kinaesthetically. Sagittal aerobic workout videos by Jane Fonda! :)
>I hope, then, that by agreeing on appropriate distinctions in size
>between left and right flags we can eliminate them entirely.
Yes.
>> I'm not averse to a slight recurve on the concaves, but I'm afraid
>I find
>> some of those in symbols2.bmp, so extreme in this regard, that they
>are
>> quite ambiguous in their direction. With a mental switch akin to
>the Necker
>> cube illusion, I can see them as either a recurved concave pointing
>upwards
>> or a kind of wavy pointing down. Apart from any nub, I don't think
>that
>> they should go more than one pixel back in the "wrong" direction.
>Those at
>> the extreme lower left of the page look ok.
>
>You would have to think of the end of the flag as pointing and not
>the curve of the flag.
Well then why shouldn't I think of the end of the wavy flag as pointing,
and we're back where we started. Same problem. Of course we're talking "at
a distance in poor light" here.
>Putting a nub on the end might even help in
>that regard.
A nub on which one. The wavy or the concave? I just don't think there is
room in a concave symbol, (even at _twice_ the size it should be relative
to other symbols), for nubs or curlicues.
> But I can't see how I can get away from using a rise of
>several pixels to indicate the line or space of the note being
>altered. The part of the curve coincident with the arrow shaft just
>isn't going to accomplish that.
So how about my latest attempt? Unfortunately it probably looks the most
like a quaverrest of any of them.
>I want to keep the concave flags for the two
>smallest alterations. The convextoconcave conversion in the
>complementary symbols is a concept that I wouldn't want to discard.
OK. I like that too. But don't forget that it only works on the right hand
side.
Those complementation rules that work for 217ET, do not work for rational
tunings. I've been investigating it in some depth. It seems it is not
possible to assign consistent values to the various flags when sitting atop
a double shaft, so as to get apotome complements of all of 17, 17', 19,
19', 23, 23', while maintaining the same ordering of flagcombinations in
the second half apotome as in the first.
I'm at a loss what to do about this, except for one offthewall
suggestion, which is to run the order of flagcombinations in the _reverse_
direction in the second half apotome. This of course means that /\ would
be the 11' comma symbol and we'd have to find a new symbol for sharp. The
apotome complement of the natural would of course be a natural with two
tails (and no antenna), a rhombus on stilts. And of course the same thing
flipped vertically for a flat.
The advantage of this would be that a symbol and its complement would
always have exactly the same flags. The disadvantage is that, having worked
so hard to get the single shaft symbols to be sizerepresentative, we'd
have to try to make them work in exactly the _opposite_ way when used with
two shafts. A twoshaft single concave would have to look the biggest and a
twoshaft doublestraight /\, the smallest.
I hope to post more about the possible compromises with the original /\ =
# scheme, in future.
>> It seems to me that you have increased the possibility of confusion
>of wavy
>> with convex, by waving it higher.
>
>We'll just have to evaluate our latest efforts to see if that's a
>problem, but I don't think it is.
No. It isn't a problem now.
Actually I don't think it's a problem if a reader interprets the concave
symbols as short straight flags that start part way down the shaft, and
similarly if the wavys look like short convex flags that start part way
down the shaft. Their shortness and their partwaydowntheshaftness are
quite sufficient to distinguish them. What does it matter if they don't
actually look like concave and wavy. The important thing is that they are
distinct from the others and each other. We could even call them
shortstraight and shortconvex (or just short curved).
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
Regarding the problem of apotome complement symbols for rational
tunings, please see
/tuningmath/files/Dave/Complements.bmp
It should be selfexplanatory.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Regarding the problem of apotome complement symbols for rational
> tunings, please see
> /tuningmath/files/Dave/Complements.bmp
> It should be selfexplanatory.
I just uploaded a new version of that file, which now contains not
only a statement of the problem, but a solution, which turns out to be
related to 453ET. It requires the addition of one new symbol as the
complement of the 25comma symbol. The new symbol is a combination of
left wavy and left concave flags, and at around 12.1 cents, it goes in
the middle of the largest remaining gap.
So now everything that needs a complement has one. There are no simple
complementation rules beyond 13 limit, but I can live with that. Those
symbols that don't have complements should be avoided.
There are some alternatives for complements in some cases. The 17
comma symbol (wavy left) has several other 3flag options for its
complement besides the 37' comma symbol (vL+xR+wR). These are
xL+vR+vR, wL+sR+vR, sL+wL+wR, which don't yet exist.
I haven't checked whether this system of complements lets us give each
flag a constant value when it occurs on a doubleshaft. An examination
of this might cause one to choose some different alternative to those
I have chosen.
I've also uploaded a new version of
/tuningmath/files/Dave/SymbolsBySize.bmp
showing the new symbol, with all the others, on the staff.
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> At 01:26 22/04/02 0000, George Secor wrote:
[DK, msg #4148:]
> I've now exhaustively searched all combinations of up to 3 of our
flags.
>
> Here's what I end up with.
>
> Symbol Left Right Schisma
> for flags flags (cents)
> 
> 5 = 5 0
> 7 = 7 0
> 11 = 5 + (115) 0
> 11' = 29 + 7 0.34
> 13 = 5 + 7 0.42
> 13' = 29 + (115) 0.08
> 17 = 17 0
> 17' = 17 + (17'17) 0
> 19 = 19 0
> 19' = 19 + 23 0.16
> 23 = 23 0
> 23' = 17 + (115) 0.49
> or 29 + (17'17) 0.52 *
> 29 = 29 0
> 31 = 19 + (115) 0.12
> 31' = 29 + 5 0.03 *
> or 7 + 7 0.44 *
> or 5 + (17'17) + 7 0.19
> or 5 + 23 + 23 0.37
> 37 = 5 + 5 0.22 *
> or 29 + 17 0.57
> 37' = 19 + 23 + 7 0.25
> or 5 + 17 + 23 0.65
> 41 = 5 0.26 *
> or 17 + (17'17) + (17'17) 0.51
> 43 = 19 + 19 + (17'17) 0.72 [schisma too big]
> 47 = 17 + 7 0.45
> or 19 + 29 0.42
> or 19 + 23 + 23 0.02
> or 5 + 17 + (17'17) 0.21
Okay, I'm with you 100 percent on this now. (I haven't checked all
of these schismas, but trust that you have been thorough with this.)
Something that I especially like is that everything through the 29
limit works without requiring two flags on the same side.
> pythagorean
> comma = 17 + 17 + (17'17) 0
> diaschisma = 19 + 23 0.37 [same symbol as 19']
> diesis = 17 + (115) 0.56 [same symbol as 23']
>
> * doesn't vanish in 1600ET.
Very nice!
> So, in addition to c37 = c5 + c5, there are some other schismas
available
> to us, that don't vanish in 1600ET and are smaller than those that
do.
> Namely:
>
> 31' = 29 + 5 0.03 cents
> 41 = 5 0.26 cents
>
> We should definitely stop at prime 41, since there is no way to get
43 with
> sufficient accuracy using our 8 existing flags. We're under the
half cent
> otherwise.
I see no point in going to 43, so I agree.
> In the application you (or Erv) found for 41, would a 0.26 cent
error in
> the 41 have rendered it useless? Why not simply reuse the 5 comma
as the 41
> comma?
My original regarding this was in message #3985:
<< While we are on the subject of higher primes, I have one more
schisma, just for the record. This is one that you probably won't be
interested in, inasmuch as it is inconsistent in both 311 and 1600,
but consistent and therefore usable in 217. It is 6560:6561
(2^5*5*41:3^8, ~0.264 cents), the difference between 80:81 and 81:82,
the latter being the 41comma, which can be represented by the sL
flag. I don't think I ever found a use for any ratios of 37, but Erv
Wilson and I both found different practical applications for ratios
involving the 41st harmonic back in the 1970's, so I find it rather
nice to be able to notate this in 217. >>
Inasmuch as the sL flag *is* the 5comma, what you now suggest is
exactly what I originally proposed to do for 217ET. So, yes, we are
in agreement on this. (And I don't see how anybody could have a
problem with an error of only 0.26 cents.)
> If we do that we eliminate one major reason for choosing (17'17)
as our
> final comma (over 17'19 or simply 17'). No other comma symbols
depend on
> it. But it is the only one that has good complementation rules in
217ET.
In addition to this, I would argue in favor of the 17'17 comma in
that it nicely fills the size gap between the 19 and 17 commas.
(Although the 17'19 comma does fill the size gap between the 17 and
17' commas, the combination of 17+19 can also do this.) Who knows
what interval someone might want in the future (e.g., to notate
2deg224 as vR or 2deg311 as vL+vR), and having the 17'7 comma just
might make their day.
> Actually, it might be better to stop at 31, since symbols with more
than 2
> flags (e.g. 37') are getting too difficult, for my liking.
At least we could list these as possiblilities for applications in
which precise higherprime ratios are desired (e.g., for computer
music in which ASCII versions of the commasymbols might be used as
input to achieve the appropriate frequencies)  just to say we've
covered as many of the bases as possible.
> I've uploaded a new version of
> /tuning
math/files/Dave/SymbolsBySize.bmp
> based on the first option for each symbol, up to the prime 41, in
the table
> above.
>
> I realised recently that some of those alternate commas (the primed
ones
> that are intended for a diatonicbased notation) should not really
be
> defined as they currently are, but as their apotome complements,
because
> that's how they will be used. They are 17', 19', 23' and 25. Let's
call the
> apotome complements of these 17", 19", 23" and 25". For diatonic
based
> purposes, these should be defined as 17:18, 18:19, 23:24 and 24:25
> respectively, and should be assigned appropriate doubleshaft
symbols.
>
> The question is, can their symbols be sensibly based on the
complementation
> rules which we derived in the context of 217ET?
Before answering this question, let me present a rationale for
selection of a standard set of 217ET symbols.
In the standard (or preferred) set of symbols for 217ET, we will
want to follow the complementation rules strictly. We will also want
to use the same sequence of flags in the second halfapotome as
occurs in the corresponding (i.e., 2to10degree) portion of the
first halfapotome. There are two ways in which this can be
accomplished (with the differences indicated by asterisks next to the
degree number in the first column):
deg Plan A Plan B

1 v v
2 w w
3* w wv
4 s s
5 x x (or sv)
6 s s
7* sw wx
8 ws ws
9 sx sx
10 ss ss
11 xx xx
12 xs xs
13 w w
14* w wv
15 s s
16 x x (or sv)
17 s s
18* sw wx
19 ws ws
20 sx sx
21 ss ss
Note: The symbols x and sv, which convert to complements of sv
and x, respectively, are virtual equivalents of one another,
differing by the schisma 163840:163863, ~0.243 cents. This enables
x to be used (in either plan) as both the 217ET and the JI
complement of x.
Plan A is essentially different from plan B *only* in the symbol
chosen for 3deg: w vs. wv. The other differences are derived from
from this as follows:
1) The aptotome complements (or 20deg) for 3deg in plan A and plan B
are sw and wx, respectively.
2) Keeping a uniform flag sequence between the halfapotomes, the
flags for 14deg must match those for 3deg, i.e., w and wv,
respectively.
3) Keeping a uniform flag sequence between the halfapotomes, the
flags for 7deg must match those for 18deg, i.e., sw and wx,
respectively.
Plan A has four more pairs of laterally confusible symbols than does
plan B: between 2 and 3deg, 7 and 8deg, 13 and 14deg, and between 18
and 19deg. This would make plan A less desirable than plan B.
Although it might be considered more desirable to use a singleflag
rather than a doubleflag symbol for 3deg, the combination (as the
sum of the 1deg and 2deg symbols) is easier to remember.
The sequence of symbols in plan B beginning with 5deg and continuing
through 12deg (and likewise for 16 through 21deg) is rather simple to
memorize, since the right flags alternate between convex and
straight, while the left flags change every second degree. The
sequence in plan A appears more random.
It is also interesting to note that plan B uses the lowest possible
prime symbols, avoiding altogether those that define the 19 and 23
commas.
For this reason, I would consider plan B as the standard set of 217
ET symbols.
Of course, the 23comma (wR) flag would still follow the
complementation rules that you gave earlier (in msg. #4071), with the
flags being:
 Left Right
+
Convex  29 7
Straight  5 (115)
Wavy  17 23
Concave  19 (17'17)
and the complementation rules being:
Complementary
Flag Size Size Flag
comma in steps of comma
name 217ET name

Left

29 6 2 none available with same side and direction
5 4 0 blank
17 2 2 17
19 1 3 none available with same side and direction
Right

7 5 1 (17'17)
(115) 6 0 blank
23 3 3 23
(17'17) 1 5 7
By modifying the complementation rules slightly, the following
additional pairs of JI and auxiliary 217ET complements may be
defined having the vL and xL flags:
apotome  v = xw
apotome  vw = x
apotome  x = vw
apotome  xw = v
Note that the right wavy (23comma) flag involved here is not used in
the standard set of symbols in plan B, so it would be a simple matter
to remember that any complements involving this flag are not among
the standard 217ET set.
Now, to repeat your question:
<< I realised recently that some of those alternate commas (the
primed ones that are intended for a diatonicbased notation) should
not really be defined as they currently are, but as their apotome
complements, because that's how they will be used. They are 17', 19',
23' and 25. Let's call the apotome complements of these 17", 19", 23"
and 25". For diatonicbased purposes, these should be defined as
17:18, 18:19, 23:24 and 24:25 respectively, and should be assigned
appropriate doubleshaft symbols.
The question is, can their symbols be sensibly based on the
complementation rules which we derived in the context of 217ET? >>
Yes, three of the four will convert consistently, as follows:
apotome  17' = 17:18, by converting wv to wx
apotome  19' = 18:19, by converting vw to x
apotome Â– 23' = 23:24, by converting ws to w
And the fourth one, which is not a new prime, can still be
represented as:
apotome  25 = 24:25, by w (nonunique, but consistent)
which should be okay, since 217ET is unique only through the 19
limit anyway.
George
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4161]:
> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >The problem is that that middle line needs to be noticed as much
as
> >the other two, so that we can see that there are three of them,
and
> >making it shorter tends to deemphasize it.
>
> I think we need a third (and fourth and fifth ...) opinion on this
one.
> From a performer who sight reads.
>
> >Why don't we just make all three of the arrow shafts the same
length,
> >and I'll forget about making the symbols shorter than 17 pixels.
>
> I have found our cooperation on this notation to be remarkable ego
less,
> with both of us concerned only with what will be best for the end
user, and
> not concerned with "getting our own way". But we've always given
reasons
> for rejecting the other's proposal, so as to avoid any hurt
feelings. I
> feel that any compromises we have made so far, e.g in lengths,
widths,
> thicknesses or curvatures, have been made because we believe the
best
> option most likely lies in between our two extremes.
>
> Now I may be reading it wrong, but the above seems to be suggesting
a
> tradeoff of two completely unrelated things, purely on the
basis, "you let
> me have my way on this and I'll let you have your way on that". If
we can't
> agree on something, I'd prefer to seek other opinions, rather than
engage
> in such a tradeoff.
If you look at my reason for suggesting that the symbols be shortened
to less than 17 pixels, you will then see that the two things are
closely related (from my message #4133):
<< I believe that shortening the middle line makes it more difficult
to see it, thereby making it *more* difficult to distinguish three
from two. This is particularly true when the symbol modifies a note
on a line and the middle line terminates at a staff line (so you see
only two lines sticking out). In fact, after looking at this again,
I think I would be in favor of shorting all of the symbols from 17 to
16 pixels so that no vertical line would terminate at a staff line.
(This would also keep symbols modifying notes a fifth apart from
colliding. But you made a comment below regarding how the length of
a new symbol looks when placed beside a conventional flat, so I need
to evaluate this further.) >>
In other words, if vertical lines terminate at a staff line, they
might not appear to "stick out" as much as they would if they fell
one pixel short of the staff line. However, while a (shorter) middle
line *would not* terminate at a staff line in instances where the
outer lines *do*, it *would* in instances where the outer ones
*don't*, making it doubly obscure by the point of its termination
*and* by its shorter length. What I was advocating in the above
paragraph was both *shorter* and *equallength* lines, and if I had
to concede one of the two, then it would be the shortness, but not
the equal length.
For the life of me, I just can't understand how you are so insistent
that something can be made more noticeable by making it *smaller* or
*shorter*, especially when you *don't even want* symbols with triple
shafts or X's. Would Ted Mook have been able to read a Tartini
sesquisharp more easily by making its center vertical line shorter?
I would think that the change would make it more confusible with a
conventional sharp. I have done quite a bit of sightreading in my
time, both on keyboard and wind instruments, and I think that I'm
arguing in the best interest of the enduser.
Quote for the day: "Be reasonable  do it my way."
Anyway, if we can't agree on this, and if you think I haven't given
good enough reasons, then we should get some opinions from a few
other people.
> >So would you then be satisfied with a difference in width alone to
> >aid in making the lateral distinction?
>
> Yes.
Okay, that's one more thing on which we can agree!
> >Why not just go with my version of the concave symbols:
> >
> >/tuning
> >math/files/secor/notation/SymAllSz.bmp
>
> As I wrote in
> /tuningmath/message/4117
>
> I'm not averse to a slight recurve on the concaves, but I'm afraid
I find
> your current proposals so extreme in this regard, that they are
quite
> ambiguous in their direction. With a mental switch akin to the
Necker cube
> illusion, I can see them as either a recurved concave pointing
upwards or a
> kind of wavy pointing down. Apart from any nub, I don't think that
they
> should go more than one pixel back in the "wrong" direction. Those
at the
> extreme lower left of Symbols2.bmp look ok.
I don't understand this  the symbols that you seem to be referring
to each have the curve going upward 6 pixels from its lowest point,
yet you think that they are okay? Or perhaps you are referring to
the "wrong" direction laterally?
In my subsequent file SymAllSz I made the left flag symbol one pixel
narrower and the nub on the right flag symbol smaller (which I think
we would both consider an improvement, even if that has nothing to do
with the "wrong" direction). My next comment refers to this file:
> >(see upper right, top staff)? The left flag is 3 pixels wide, and
> >the right flag is 4 pixels wide, yet they are clearly
identifiable.
> >(I also threw in a complement symbol.)
>
> These are 4 and 5 pixels wide by my reckoning (including the part
> coincident with the shaft). One must define a flag as including a
part
> coincident with the shaft so one knows what it will look like when
it is
> sharing a  or X shaft with another flag. But I did overstate the
case
> when I said that 4 pixels wide doesn't work. We now have agreement
that the
> concaves on a single shaft should both be 4 pixels wide.
I don't think we do. Referring again to SymAllSz, at the right end
of the bottom staff my vL flag is 4 pixels and my vR flag is 5, and
the combination of 4+5 is a total of 8 (the center pixel being
occupied by both). If you reduce the total width by 1 pixel by
putting a mirror image of the left pixel on the right side, you will
find that the result looks like a Ushaped curve rather than two
separate curves, so I would not recommend this. (However, see what I
have to say below.)
> Maybe I went too far in reducing the height of the wavy and concave
to 7
> pixels (including shortening the shaft at the pointy end). I see
that your
> concaves are 9 pixels high, and your wavys are 10 pixels high, only
one
> pixel shorter than the straight and concaves. In fact when it isn't
used
> with the left wavy, your right wavy is the full 11 pixels in height
and 6
> pixels wide. I think these lead to too many symbols whose apparent
visual
> size is too far out of keeping with their size in cents.
>
> I find that:
> 9.4 14.7 20.1 all look bigger than 21.5
> 27.5 and 30.6 look bigger than 31.8
>
> I am proposing something between yours and mine. See
>
> /tuning
math/files/Dave/SymbolsBySize.bmp
Okay, those look good, including the vL+vR symbol. Let's go with
them!
> I don't think we actually need any lateral distinction between the
two
> concaves because in rational tunings the (17'17) flag will never
occur on
> its own, and I don't think any ETs of interest below 217ET will
need to
> use both 19 and (17'17). What do you think?
I'd keep them distinct  you'll never know what somebody is going to
want to use this notation for (assuming that anyone is going to use
it at all).
> But it wouldn't hurt if they were distinct. The biggest problem
(for me) is
> trying to make the 19 comma (left concave) look as small as it
really is
> without it disappearing. If its width was in proportion to the
width of the
> 5 comma flag, you wouldn't see it for the shaft! If we look at
areas and
> ignore the part coincident with the shaft, the 5 comma flag is 4
pixels by
> 11 pixels. The 19 comma flag would have to fit in a rectangle 7
pixels in
> area. In my 22Apr proposal I've allowed those 7 pixels to blow out
to 12,
> 3 wide by 4 high (excluding shaft). Yours is 18 pixels, 3 wide by 6
high.
>
> If we simply count black pixels (excluding shaft) we find that the
5 comma
> flag has 15, which means the 19 comma flag should only have 2.4,
which we
> might generously round up to 3 black pixels. Mine has 7, yours has
9.
>
> So my (17'17) (right concave) flag is about the right size, but my
19 flag
> is about double the size it should be. I can live with double, but
I'n not
> sure I can handle triple.
>
> Now you probably think I'm being too literal with this
representative size
> stuff, but the problems occur when you have the 19 flag combined
with
> another flag and the result looks much bigger than some single flag
that it
> should be much smaller than. In particular
> 9.4 looking bigger than 21.5,
> 20.1 looking bigger than 27.3.
>
> I suppose we can have a 19 comma flag that is lrger when used alone
than
> when combined with others, but I'd prefer not.
I think that the way you have it is fine  just so it's big enough
to see. After all, it's not going to be confused with anything else.
> >I wouldn't make the vertical arrow shaft shorter, though.
>
> OK.
>
> >To the right of our convex symbols are our latest versions of the
> >wavy flags for comparison. I made the left wavy flag 4 pixels
wide,
> >like the concave right flag, and the right wavy flag is 5 pixels
> >wide. Both of our wL+wR symbols have flags 4 pixels wide on each
> >side.
>
> >As with the concave symbols, I also threw in a complement
> >symbol.
>
> Ah, but what exactly are they complements _of_?
I didn't have anything in mind at the time  I just wanted to
illustrate the curve with 2 shafts. (However, my last posting *does*
include this one as the complement of xw.)
> I assume it was an oversight that left the wavy side of the 36.0
symbol
> unmodified.
Yes. I was a bit hasty.
> >I also experimented with taking the curves out of the wavy
symbols,
> >making them rightangle symbols, which I put at the far right.
(The
> >left vs. right line lengths are different in both the horizontal
and
> >vertical directions to aid in telling them apart.) We already
have
> >two kinds of curvedline symbols, and substituting these for the
wavy
> >symbols would give us two kinds of straightline symbols as well.
> >It's not that I don't like the wavy symbols (I do like them), but
I
> >thought that this would make it easier  both to remember and to
> >distinguish them. (This one's your call.)
>
> You're right about them being more distinct, but the aesthetics are
the
> killer. Given more resolution, I'd go for something in between the
existing
> wavys and these rightangle ones, but not these totally sharp
corners.
You're right about the aesthetics  that's the reason why I also
prefer wavy to rightangle symbols. If you're happy with what you
have now (they look like mine from the staff above), then we can go
with them.
> >I copied your symbols (unaltered) into the second staff. Below
that
> >I put my versions of the symbols for comparison.
> >
> >I found that when I draw *convex* flags freehand that I tend to
> >curve the end of the flag inward slightly to make sure that it
isn't
> >mistaken for a straight flag, and I have been doing something on
this
> >order for some time with my bitmap symbols as well. I have
modified
> >these also to reflect this, and you can let me know what you
think.
>
> I think they look good, aesthetically speaking. The trouble is it
makes the
> down versions look too much like flats and backward flats. Also, you
> decreased the size difference between the 7 flag and the 29 flag by
adding
> curvature on the outside of the 7 flag and the inside of the 29
flag.
>
> I find the fact that the convex flags start off at rightangles to
the
> shaft and end parallel to the shaft, sufficient to make them
distinct from
> straight flags, without tending towards flats.
Okay, we can leave them as they were. It was just a suggestion.
> >(I notice that the right flag of your 47.4cent symbol has this
sort
> >of feature  was that a mistake?)
>
> That's 37' = 19 + 23 + 7 = vL + wR + xR, so what you saw resulted
from
> mindlessly overlaying wR and xR. Being 37', my heart wasn't in it.
I've had
> a better go at it now, based on what you did for 25 and 31'.
Yes, now I can tell what it's supposed to be.
> >Or possibly only the left convex
> >flag could be given this feature to further distinguish it from
the
> >convex right flag.
>
> That would at least retain the full 2 pixel difference in width
between XL
> and xR, but still has the problem of looking too much like a
backwards flat.
>
> There is a way to make the convex more distinct from straight
without
> taking them closer to flats. We make them closer to being right
angles,
> i.e. reduce the radius of the corner. I've shown comparisons with
straight
> flags and flats at top right of my latest bitmap.
I like the original version (DK22) better. I wasn't having any
problem with the bitmap distinction  it was just when I was drawing
them freehand that sometimes they didn't look as different from
straight flags as I would have liked them to. But that's no reason
to change the bitmap version; what you originally had looks better,
so leave well enough alone!
> >Also, observe my 43cent and 55cent symbols  the ones with two
> >flags on the same side.
>
> Yes. I wasn't very happy with mine. I like yours better, but I've
modified
> them very slightly. Tell me what you think.
The 43cent one is good (it looks like the one I did). For the 55
cent symbol, why don't you try removing the top straight flag from
the 43cent symbol and adding a reversed 27.3cent symbol to it, so
that the two flags cross. (Also try the same thing with one of my
27.3cent symbols reversed and see if you that the effect is even
better, since the two tend to cross more at right angles.)
> Notice that it's OK for 31' down to look like a backwards flat,
because it
> _is_ a halfflat.
Sure!
> >Yes, good point, and one reason why I'm not reluctant to discard
the
> >idea of making the symbols any shorter than 17 pixels. When I put
a
> >5commadown symbol next to a flat the new symbol has a shorter
stem
> >than the flat. I don't think that this is inappropriate, inasmuch
as:
> >
> >1) the two symbols are in about the same proportion lengthto
width;
> >and
> >
> >2) the difference in height is the same as that in the two
vertical
> >lines of a conventional sharp symbol.
>
> Good points. OK. I'll forget the idea of giving down arrows longer
shafts
> than up arrows. Are we agreed then that all sagittals should be 17
pixels
> high?
Okay, then, even if we don't agree about the equal vs. unequal 3
vertical lines.
> >And I can't imagine that anyone else has ever worked out a
notation
> >in this much detail.
>
> Me neither.
>
> It will of course be rejected out of hand by others, for reasons we
haven't
> even considered. :)
But of course. We can't think of everything or please everybody, can
we? We just do the best we can.
George
At 22:15 24/04/02 0000, George Secor wrote:
>Okay, I'm with you 100 percent on this now. (I haven't checked all
>of these schismas, but trust that you have been thorough with this.)
>Something that I especially like is that everything through the 29
>limit works without requiring two flags on the same side.
Yes. That is more than I would have expected if you'd asked me at the
start. It's certainly a nice vindication of your sagittal idea.
>> pythagorean
>> comma = 17 + 17 + (17'17) 0
>> diaschisma = 19 + 23 0.37 [same symbol as 19']
>> diesis = 17 + (115) 0.56 [same symbol as 23']
>>
>> * doesn't vanish in 1600ET.
>
>Very nice!
I gave up on trying to make an actual symbol for the pythagorean comma
based on the above identity. Maybe you want to have a go.
>Inasmuch as the sL flag *is* the 5comma, what you now suggest is
>exactly what I originally proposed to do for 217ET. So, yes, we are
>in agreement on this.
Great! Sorry I forgot your original proposal re 41.
> (And I don't see how anybody could have a
>problem with an error of only 0.26 cents.)
Just don't say that too loudly around here.
>> If we do that we eliminate one major reason for choosing (17'17)
>as our
>> final comma (over 17'19 or simply 17'). No other comma symbols
>depend on
>> it. But it is the only one that has good complementation rules in
>217ET.
>
>In addition to this, I would argue in favor of the 17'17 comma in
>that it nicely fills the size gap between the 19 and 17 commas.
Yes.
>(Although the 17'19 comma does fill the size gap between the 17 and
>17' commas, the combination of 17+19 can also do this.)
Yes. As you have probably read by now, I am proposing precisely that; a
17+19 symbol (2 left flags) to serve as the rational complement of 25.
>Who knows
>what interval someone might want in the future (e.g., to notate
>2deg224 as vR or 2deg311 as vL+vR), and having the 17'7 comma just
>might make their day.
Yes. Or 2deg453 as vR and 4deg453 as vL+wL.
There is no doubt in my mind now, that (17'17) is the best choice for the
last flag. We're in complete agreement now on what the 8 flags mean (when
on a single shaft).
>> Actually, it might be better to stop at 31, since symbols with more
>than 2
>> flags (e.g. 37') are getting too difficult, for my liking.
>
>At least we could list these as possiblilities for applications in
>which precise higherprime ratios are desired (e.g., for computer
>music in which ASCII versions of the commasymbols might be used as
>input to achieve the appropriate frequencies)  just to say we've
>covered as many of the bases as possible.
Yes we should list them, but beyond 31 we do not have unique symbols. 35 is
also 13, 37 is also 25, 41 is also 5, so the above application wouldn't work.
One minor point to note in connection with relegating primes above 31 to
secondclass citizen status is that the 37' symbol _is_ unique, and I'm
currently using it as the rational complement of the 17 symbol. But there's
no need to call it 37' in that context, and anyway a different 3flag
symbol may turn out to be better as the rational complement of the 17 symbol.
>In the standard (or preferred) set of symbols for 217ET, we will
>want to follow the complementation rules strictly. We will also want
>to use the same sequence of flags in the second halfapotome as
>occurs in the corresponding (i.e., 2to10degree) portion of the
>first halfapotome.
Agreed.
>There are two ways in which this can be
>accomplished (with the differences indicated by asterisks next to the
>degree number in the first column):
>
>deg Plan A Plan B
>
> 1 v v
> 2 w w
> 3* w wv
> 4 s s
> 5 x x (or sv)
> 6 s s
> 7* sw wx
> 8 ws ws
> 9 sx sx
>10 ss ss
>11 xx xx
>12 xs xs
>13 w w
>14* w wv
>15 s s
>16 x x (or sv)
>17 s s
>18* sw wx
>19 ws ws
>20 sx sx
>21 ss ss
>
>Note: The symbols x and sv, which convert to complements of sv
>and x, respectively, are virtual equivalents of one another,
>differing by the schisma 163840:163863, ~0.243 cents. This enables
>x to be used (in either plan) as both the 217ET and the JI
>complement of x.
Agreed.
>Plan A is essentially different from plan B *only* in the symbol
>chosen for 3deg: w vs. wv. The other differences are derived from
>from this as follows:
>
>1) The aptotome complements (or 20deg) for 3deg in plan A and plan B
>are sw and wx, respectively.
sw works as a rational complement to w, but wx doesn't work as
rational complement to wv. Instead I propose ws (or possibly xv) as
rational complement of wv.
However this is fairly irrelevant since neither plan A nor plan B can agree
with the rational complement rules, since rational complementation must
deny that wavy left is its own flagcomplement.
>2) Keeping a uniform flag sequence between the halfapotomes, the
>flags for 14deg must match those for 3deg, i.e., w and wv,
>respectively.
>
>3) Keeping a uniform flag sequence between the halfapotomes, the
>flags for 7deg must match those for 18deg, i.e., sw and wx,
>respectively.
>
>Plan A has four more pairs of laterally confusible symbols than does
>plan B: between 2 and 3deg, 7 and 8deg, 13 and 14deg, and between 18
>and 19deg. This would make plan A less desirable than plan B.
>
>Although it might be considered more desirable to use a singleflag
>rather than a doubleflag symbol for 3deg, the combination (as the
>sum of the 1deg and 2deg symbols) is easier to remember.
>
>The sequence of symbols in plan B beginning with 5deg and continuing
>through 12deg (and likewise for 16 through 21deg) is rather simple to
>memorize, since the right flags alternate between convex and
>straight, while the left flags change every second degree. The
>sequence in plan A appears more random.
>
>It is also interesting to note that plan B uses the lowest possible
>prime symbols, avoiding altogether those that define the 19 and 23
>commas.
>
>For this reason, I would consider plan B as the standard set of 217
>ET symbols.
I'm convinced. Plan B it is.
>Of course, the 23comma (wR) flag would still follow the
>complementation rules that you gave earlier (in msg. #4071), with the
>flags being:
>
>  Left Right
>+
>Convex  29 7
>Straight  5 (115)
>Wavy  17 23
>Concave  19 (17'17)
>
>and the complementation rules being:
>
> Complementary
>Flag Size Size Flag
>comma in steps of comma
>name 217ET name
>
>Left
>
>29 6 2 none available with same side and direction
>5 4 0 blank
>17 2 2 17
>19 1 3 none available with same side and direction
>
>Right
>
>7 5 1 (17'17)
>(115) 6 0 blank
>23 3 3 23
>(17'17) 1 5 7
>
>By modifying the complementation rules slightly, the following
>additional pairs of JI and auxiliary 217ET complements may be
>defined having the vL and xL flags:
>
>apotome  v = xw
>apotome  vw = x
>apotome  x = vw
>apotome  xw = v
These all agree with my proposed rational complements.
>Note that the right wavy (23comma) flag involved here is not used in
>the standard set of symbols in plan B, so it would be a simple matter
>to remember that any complements involving this flag are not among
>the standard 217ET set.
Right.
>Now, to repeat your question:
>
><< I realised recently that some of those alternate commas (the
>primed ones that are intended for a diatonicbased notation) should
>not really be defined as they currently are, but as their apotome
>complements, because that's how they will be used. They are 17', 19',
>23' and 25. Let's call the apotome complements of these 17", 19", 23"
>and 25". For diatonicbased purposes, these should be defined as
>17:18, 18:19, 23:24 and 24:25 respectively, and should be assigned
>appropriate doubleshaft symbols.
>
>The question is, can their symbols be sensibly based on the
>complementation rules which we derived in the context of 217ET? >>
>
>Yes, three of the four will convert consistently, as follows:
>
>apotome  17' = 17:18, by converting wv to wx
I propose instead that 17:18 should be ws.
>apotome  19' = 18:19, by converting vw to x
Agreed.
>apotome Â– 23' = 23:24, by converting ws to w
I propose instead that 23:24 should be wv. This is the inverse of the 17'
complement.
>And the fourth one, which is not a new prime, can still be
>represented as:
>
>apotome  25 = 24:25, by w (nonunique, but consistent)
>
>which should be okay, since 217ET is unique only through the 19
>limit anyway.
By now I guess you've read my rational complement proposal based on 453ET.
I'd prefer 24:25 to have a unique symbol and have proposed a new symbol
wv for this.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
At 22:17 24/04/02 0000, you wrote:
>If you look at my reason for suggesting that the symbols be shortened
>to less than 17 pixels, you will then see that the two things are
>closely related (from my message #4133):
My apologies.
>For the life of me, I just can't understand how you are so insistent
>that something can be made more noticeable by making it *smaller* or
>*shorter*, especially when you *don't even want* symbols with triple
>shafts or X's. Would Ted Mook have been able to read a Tartini
>sesquisharp more easily by making its center vertical line shorter?
>I would think that the change would make it more confusible with a
>conventional sharp. I have done quite a bit of sightreading in my
>time, both on keyboard and wind instruments, and I think that I'm
>arguing in the best interest of the enduser.
You're certainly more qualified than me in that regard. It was the way the
middle stroke is always shortened in an uppercase E that got me thinking.
Also, I find that something with an apparent V notched out of its tail is
somewhat distinct from something with a square tail, no matter the number
of shafts.
>Quote for the day: "Be reasonable  do it my way."
Good one. ;)
>Anyway, if we can't agree on this, and if you think I haven't given
>good enough reasons, then we should get some opinions from a few
>other people.
I've emailed Ted Mook for his opinion. You should have received a copy.
>I don't understand this  the symbols that you seem to be referring
>to each have the curve going upward 6 pixels from its lowest point,
>yet you think that they are okay? Or perhaps you are referring to
>the "wrong" direction laterally?
Sorry. I must have screwed up here. I was probably looking at a version of
Symbols2.bmp I had already edited myself and forgotten. None of the
concaves in it look ok to me. The ""wrong" direction" was referring to
vertical direction only.
>In my subsequent file SymAllSz I made the left flag symbol one pixel
>narrower and the nub on the right flag symbol smaller (which I think
>we would both consider an improvement, even if that has nothing to do
>with the "wrong" direction).
Yes I found it to be an improvement.
>> I am proposing something between yours and mine. See
>>
>> /tuning
>math/files/Dave/SymbolsBySize.bmp
>
>Okay, those look good, including the vL+vR symbol. Let's go with
>them!
Yikes! OK.
>> I don't think we actually need any lateral distinction between the
>two
>> concaves because in rational tunings the (17'17) flag will never
>occur on
>> its own, and I don't think any ETs of interest below 217ET will
>need to
>> use both 19 and (17'17). What do you think?
>
>I'd keep them distinct  you'll never know what somebody is going to
>want to use this notation for (assuming that anyone is going to use
>it at all).
OK. Yeah.
>> I suppose we can have a 19 comma flag that is lrger when used alone
>than
>> when combined with others, but I'd prefer not.
>
>I think that the way you have it is fine  just so it's big enough
>to see. After all, it's not going to be confused with anything else.
OK. Great.
>You're right about the aesthetics  that's the reason why I also
>prefer wavy to rightangle symbols. If you're happy with what you
>have now (they look like mine from the staff above), then we can go
>with them.
Yes. They are yours. Except I think I took one pixel off the end of the
right hand ones so they are the same height as the left ones.
>> That's 37' = 19 + 23 + 7 = vL + wR + xR, so what you saw resulted
>from
>> mindlessly overlaying wR and xR. Being 37', my heart wasn't in it.
>I've had
>> a better go at it now, based on what you did for 25 and 31'.
>
>Yes, now I can tell what it's supposed to be.
Good.
>> >Or possibly only the left convex
>> >flag could be given this feature to further distinguish it from
>the
>> >convex right flag.
>>
>> That would at least retain the full 2 pixel difference in width
>between XL
>> and xR, but still has the problem of looking too much like a
>backwards flat.
>>
>> There is a way to make the convex more distinct from straight
>without
>> taking them closer to flats. We make them closer to being right
>angles,
>> i.e. reduce the radius of the corner. I've shown comparisons with
>straight
>> flags and flats at top right of my latest bitmap.
>
>I like the original version (DK22) better. I wasn't having any
>problem with the bitmap distinction  it was just when I was drawing
>them freehand that sometimes they didn't look as different from
>straight flags as I would have liked them to. But that's no reason
>to change the bitmap version; what you originally had looks better,
>so leave well enough alone!
I was getting to like the squarer DK23 ones, as more distinct from flats,
but OK. Does this mean we now have a full set of singleshaft symbols that
we both find acceptable? I think maybe we're still tinkering with some of
the twoflagsonthesameside ones. What do you think of the new vw
symbol, the complement to the ss symbol?
>The 43cent one is good (it looks like the one I did).
Yes I only moved a few pixels so it looks smoother in all alignments.
>For the 55
>cent symbol, why don't you try removing the top straight flag from
>the 43cent symbol and adding a reversed 27.3cent symbol to it, so
>that the two flags cross. (Also try the same thing with one of my
>27.3cent symbols reversed and see if you that the effect is even
>better, since the two tend to cross more at right angles.)
I prefer them meeting, rather than crossing. Why do you like the crossing?
What I imagine happening when two flags are combined on one side, is that
the two flags are scaled down to about two thirds of their height including
scaling the vertical line thickness. And they are scaled up (out) in the
horizontal direction slightly to compensate for the loss of area due to the
vertical scaledown. Then one of them is moved to the top of the available
space and the other to the bottom, and overlaid. Then a little bit of
license is used to make it look like something sensible and be sufficiently
distinct from everything else.
I thought I extracted that from what you were doing.
>But of course. We can't think of everything or please everybody, can
>we? We just do the best we can.
Indeed.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>
> ... I was more
> disappointed to find that the following 3 messages from you and 2
from
> Paul, in this thread, were only trapezoid trivia. :)
Well, sorry to disappoint you again today. I'm having a time trying
to keep up with answering all of this (besides being a week behind in
reading the digests from 3 different tuning lists). I need a day
off, so I'm sending no big/serious message(s) today.
> Here's some more. The U.S. definitions of trapezoid and trapezium
are
> exactly swapped relative to the British/Australian definitions.
>
> In the OED and Macquarie dictionaries, a trapezium has only one
pair of
> sides parallel, while a trapezoid has none. Websters has it the
other way
> 'round. There's no requirement for any rightangles anywhere.
Interesting. Well, at least we understand what we're both talking
about, even if it doesn't come out quite right sometimes. (By the
way, do you know any more trapezoid jokes, or do you think I should
just leave that subject and shut my trapezoid?)
> At least it's good to know Paul's reading the thread. I've been
wondering
> whether noone else was contributing because
> (a) they think we're doing such a wonderful job without them, or
> (b) they have no interest whatsoever in the topic, and think we're
a couple
> of looneys?
Speaking of Paul (at least I didn't put all seriousness aside and
say "speaking off looneys", which would have been most unkind!), now
that we have made such wonderful (and unexpected) progress agreeing
on singleshaft standard 217ET symbols, I realized that Paul's
request to see an adaptiveJI progression (message #3950) can be
filled. Would you care to do the honors, or shall I?
> Hey, I've become so obsessed about this notation that I was lying
in bed
> this morning thinking how my various sleeping postures could be
read as
> various sagittal symbols. I was imagining children being taught them
> kinaesthetically. Sagittal aerobic workout videos by Jane Fonda! :)
Now that's scary. It sounds like you need a day off, too.
> >> I'm not averse to a slight recurve on the concaves, but I'm
afraid I find
> >> some of those in symbols2.bmp, so extreme in this regard, that
they are
> >> quite ambiguous in their direction. With a mental switch akin to
the Necker
> >> cube illusion, I can see them as either a recurved concave
pointing upwards
> >> or a kind of wavy pointing down. ...
Whoa! This sort of thing is too convoluted for me today. Hopefully,
I'll be back tomorrow for more  more serious, that is.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> Interesting. Well, at least we understand what we're both talking
> about, even if it doesn't come out quite right sometimes. (By the
> way, do you know any more trapezoid jokes, or do you think I should
> just leave that subject and shut my trapezoid?)
Son, I think you oughta make like one a' them trapezoid monks I heered
about.
> > At least it's good to know Paul's reading the thread. I've been
> wondering
> > whether noone else was contributing because
> > (a) they think we're doing such a wonderful job without them, or
> > (b) they have no interest whatsoever in the topic, and think we're
> a couple
> > of looneys?
>
> Speaking of Paul (at least I didn't put all seriousness aside and
> say "speaking off looneys", which would have been most unkind!), now
> that we have made such wonderful (and unexpected) progress agreeing
> on singleshaft standard 217ET symbols, I realized that Paul's
> request to see an adaptiveJI progression (message #3950) can be
> filled. Would you care to do the honors, or shall I?
Since you're taking the day off. I guess I'd better do it.
> > Hey, I've become so obsessed about this notation that I was lying
> in bed
> > this morning thinking how my various sleeping postures could be
> read as
> > various sagittal symbols. I was imagining children being taught
them
> > kinaesthetically. Sagittal aerobic workout videos by Jane Fonda!
:)
>
> Now that's scary. It sounds like you need a day off, too.
Yeah.
> > >> I'm not averse to a slight recurve on the concaves, but I'm
> afraid I find
> > >> some of those in symbols2.bmp, so extreme in this regard, that
> they are
> > >> quite ambiguous in their direction. With a mental switch akin
to
> the Necker
> > >> cube illusion, I can see them as either a recurved concave
> pointing upwards
> > >> or a kind of wavy pointing down. ...
>
> Whoa! This sort of thing is too convoluted for me today. Hopefully,
> I'll be back tomorrow for more  more serious, that is.
We've already dealt with that one. So you can relax.

The continuing search for the ideal rational complement symbols

Hi George,
I was wrong about being able to notate 453ET. It would need the addition
of a 22 step symbol sx, which would be like the c31' symbol flipped
horizontally, and would look way too much like a conventional flat.
Fortunately notating 453ET wasn't the point.
The point is that whatever rational complements we decide on, should also
be the true complements in some ET (I think). It doesn't matter whether we
have a symbol for every degree of that ET, in fact it's probably better if
we don't. I think the higher that ET is, the better, except that if we go
too high we find that too many symbols don't _have_ a complement.
We know 217ET doesn't work for rational complements because it is only
19limit unique and so doesn't provide enough unique complements.
I proposed 453ET and found I needed an additional symbol vw to get a
complement for c25. But 453ET isn't that great. It would be nice to use an
ET that was 31limit unique, like our symbols.
Thanks to Gene Smith's search for good 31limit unique ETs, I tried 653ET
and found that it works! It needed only the same additional symbol vw,
but this time it is the complement of c23', not c25.
You can see the 653ET complements in the latest version of
/tuningmath/files/Dave/Complements.bmp
4095:4096 doesn't actually vanish in 653ET. It seems to be the only one of
our subsymbol schismas that doesn't. So the complementarity of sx (c13)
and v, (v is c(17'17)), is based on sx being 653ET's best 13comma,
not on it being the sum of the 5 and 7 commas.
One thing to note is that in 653ET x and x are not complements.
All these rational complement schemes seem very unsatisfactory to me. And
the thing is, I don't think anyone will use them. Not even the relatively
simple ones in 217ET. Just like I don't think anyone will use the  and
X shaft symbols. Expecting people to learn the singleshaft symbols is more
than enough. A sharp with a down symbol next to it is going to be way more
easily parsed than some double shaft symbol with a combination of flags
that they are used to associating with some other prime when on a single
shaft (or more likely have never seen before), and if taken as simply a
number of cents, must be added onto, not an actual half apotome, but an
11'diesis.
So basically, I've now got what I wanted from this, and what I think the
microtonal world might want, but if you are still determined to persue
multishaft symbols I'm still willing to comment on your proposals.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
> right, but i'd like to see this actually notated, on a staff.
Here it is.
/tuningmath/files/Dave/AdaptiveJI.bmp
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
> > right, but i'd like to see this actually notated, on a staff.
>
> Here it is.
> /tuningmath/files/Dave/AdaptiveJI.bmp
pretty wild! you should post this over to the tuning list and see
what others think, like bob wendell for instance.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
> > right, but i'd like to see this actually notated, on a staff.
>
> Here it is.
> /tuningmath/files/Dave/AdaptiveJI.bmp
***What on earth is going on here?
Could we move some of this over to the "main" list for our
appreciation??
jp
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> At 22:15 24/04/02 0000, George Secor wrote:
This is just a quick comment on the singleshaft symbols.
I was just noticing how large a few of the symbols are in comparison
to the conventional sharp and flat symbols. I suggest making the
convex left flag one pixel narrower for the 33.5, 39.5, 50.0, and
65.3cent symbols. (I tried it by replacing the left halves with the
left half of the 60.4cent symbol. The 55.0 and 60.4cent symbols
can remain the way they are.) I think that this alleviates the
conventionalsaggital size disparity somewhat, in addition to making
a better size progression, but is not enough of a change to cause
lateral confusibility.
And while I am on the subject of finetuning symbols:
> I gave up on trying to make an actual symbol for the pythagorean
comma
> based on the above identity. Maybe you want to have a go.
Try this: Make a copy of the 17' symbol (wL+vR). Move all of the
pixels in the 4 leftmost columns up 1 position (thus raising the wavy
flag by 1). Then copy these and paste them so that you have a second
wavy flag 4 pixels lower than the top one. The two wavy flags
together are considerably smaller than a single 29 flag (even with my
proposed reduction in size for the latter), and together they clearly
indicate the staff position of the note being altered.
> >> Actually, it might be better to stop at 31, since symbols with
more
> >than 2
> >> flags (e.g. 37') are getting too difficult, for my liking.
> >
> >At least we could list these as possiblilities for applications in
> >which precise higherprime ratios are desired (e.g., for computer
> >music in which ASCII versions of the commasymbols might be used
as
> >input to achieve the appropriate frequencies)  just to say we've
> >covered as many of the bases as possible.
>
> Yes we should list them, but beyond 31 we do not have unique
symbols. 35 is
> also 13, 37 is also 25, 41 is also 5, so the above application
wouldn't work.
I don't see this as a problem  the whole point of identifying the
schismas is to minimize the number of symbols required for the
notation, which by its very nature reduces and eventually eliminates
uniqueness once the harmonic limit reaches a certain size. With the
schismas as small as they are, we are entitled to assert that, for
all practical purposes, it is impossible to tell these intervals
apart, whereby they no longer have separate and distinct identities
(i.e., bridging at the point of inaudibility). It is entirely
appropriate for the notation to reflect this reality, and there
should be no need to apologize for it.
> One minor point to note in connection with relegating primes above
31 to
> secondclass citizen status is that the 37' symbol _is_ unique, and
I'm
> currently using it as the rational complement of the 17 symbol. But
there's
> no need to call it 37' in that context, and anyway a different 3
flag
> symbol may turn out to be better as the rational complement of the
17 symbol.
I'll need to look at this further.
> > ... For this reason, I would consider plan B as the standard set
of 217
> >ET symbols.
>
> I'm convinced. Plan B it is.
It's my turn to say yikes! OK.
And with that, I hope that Paul likes the adaptive JI figure that you
made. (I notice that the 17comma wavy symbols possess the same
slope directionality as the 5comma symbols, which makes an effective
emphasis of the direction of pitch alteration, particularly in the D
minor triad. (This same directionality cue also works with the wavy
flags in the 3degree symbols; and there would be little problem with
misinterpreting the direction from the slope of the singledegree
concave symbols, since a concave flag slopes both ways.) These new
symbols may take a little bit of time to get used to, but everything
fits together so logically that it should be a relatively easy matter
to learn the 12 symbols  e.g., relative to learning the alphabet.
(Your reply continued with a discussion of rational complements, but
I should leave off here until I have studied the rest of your
messages.)
George
 In tuningmath@y..., "jpehrson2" <jpehrson@r...> wrote:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> /tuningmath/message/4174
>
>
> >  In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
> > > right, but i'd like to see this actually notated, on a staff.
> >
> > Here it is.
> > /tuning
math/files/Dave/AdaptiveJI.bmp
>
> ***What on earth is going on here?
Something really amazing  In the course of our notational dicussion
& debate, Dave Keenan and I have finally agreed over the past couple
of days on quite a few things, and we now have something to show for
it.
> Could we move some of this over to the "main" list for our
> appreciation??
>
> jp
If there's still any file space. It's your move, Dave!
George
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>> At 22:15 24/04/02 0000, George Secor wrote:
>
>This is just a quick comment on the singleshaft symbols.
>
>I was just noticing how large a few of the symbols are in comparison
>to the conventional sharp and flat symbols. I suggest making the
>convex left flag one pixel narrower for the 33.5, 39.5, 50.0, and
>65.3cent symbols. (I tried it by replacing the left halves with the
>left half of the 60.4cent symbol. The 55.0 and 60.4cent symbols
>can remain the way they are.) I think that this alleviates the
>conventionalsaggital size disparity somewhat, in addition to making
>a better size progression, but is not enough of a change to cause
>lateral confusibility.
I've done it. See
/tuningmath/files/Dave/SymbolsBySize3.bmp
It _does_ increase lateral confusability somewhat.
Which is one thing that causes me to repropose the convex with the
slightly squarer corners. Your only comment about them has been to "leave
well enough alone". Can you give a more detailed reason for rejecting them?
>And while I am on the subject of finetuning symbols:
>
>> I gave up on trying to make an actual symbol for the pythagorean
>comma
>> based on the above identity. Maybe you want to have a go.
>
>Try this: Make a copy of the 17' symbol (wL+vR). Move all of the
>pixels in the 4 leftmost columns up 1 position (thus raising the wavy
>flag by 1). Then copy these and paste them so that you have a second
>wavy flag 4 pixels lower than the top one. The two wavy flags
>together are considerably smaller than a single 29 flag (even with my
>proposed reduction in size for the latter), and together they clearly
>indicate the staff position of the note being altered.
See the above file for my best attempt. Not precisely what you suggested,
but close.
I don't want to jump the gun and go to the main list just yet, and when I
do, I'll want a staff showing the odd harmonics of G up to 41, including
all optional spellings (using single shaft symbols with conventional sharps
and flats), as well as the 217ET notation and a couple of other ETs.
You might want to check out
http://shareware.search.com/search?cat=247&tag=ex.sa.sr.srch.sa_all&q=truety
pe+font+editor
and
to get the free download which is fully functional except for save.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4179]:
> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> >> At 22:15 24/04/02 0000, George Secor wrote:
> >
> >I was just noticing how large a few of the symbols are in
comparison
> >to the conventional sharp and flat symbols. I suggest making the
> >convex left flag one pixel narrower for the 33.5, 39.5, 50.0, and
> >65.3cent symbols. (I tried it by replacing the left halves with
the
> >left half of the 60.4cent symbol. The 55.0 and 60.4cent symbols
> >can remain the way they are.) I think that this alleviates the
> >conventionalsaggital size disparity somewhat, in addition to
making
> >a better size progression, but is not enough of a change to cause
> >lateral confusibility.
>
> I've done it. See
> /tuning
math/files/Dave/SymbolsBySize3.bmp
>
> It _does_ increase lateral confusability somewhat.
But there's stil quite a difference, so I don't think that it would
be a problem. Note that the change does make the vertical shaft look
more centered in that large 65.3cent symbol.
By the way, it's been bugging me that we've yet to agree on the
spelling of confusable vs. confusible. I finally looked up the able
vs. ible rules. There were two that applied (source: _The Grammar
Bible_, Strumpf & Douglas, Knowledgeopolis, 1999):
Rule 2: If the base itself is a complete English word, use the
suffix able. Examples: changeable, flyable
Which would result in "confuseable". However, see
Rule 4: If you can add the suffix ion to the base to make a
legitimate English word, then you should use the suffix ible.
Examples: corruptible (corruption), perfectible (perfection)
Which results in "confusible".
I hope that rule 4 will end the confusion, even if it doesn't
eliminate all of the confusibility.
> Which is one thing that causes me to repropose the convex with the
> slightly squarer corners. Your only comment about them has been
to "leave
> well enough alone". Can you give a more detailed reason for
rejecting them?
Just an aesthetic consideration: the flags with the squarer corners
tend to look like rightangle flags with rounded corners, as opposed
to flags that are curved along their entire length. (I notice that
you did make a difference for this in the 55cent symbol also, which
is good; I later realized that I was mistaken in suggesting that it
didn't need to be changed to conform to the others.)
> >And while I am on the subject of finetuning symbols:
> >
> >> I gave up on trying to make an actual symbol for the pythagorean
> >comma
> >> based on the above identity. Maybe you want to have a go.
> >
> >Try this: Make a copy of the 17' symbol (wL+vR). Move all of the
> >pixels in the 4 leftmost columns up 1 position (thus raising the
wavy
> >flag by 1). Then copy these and paste them so that you have a
second
> >wavy flag 4 pixels lower than the top one. The two wavy flags
> >together are considerably smaller than a single 29 flag (even with
my
> >proposed reduction in size for the latter), and together they
clearly
> >indicate the staff position of the note being altered.
>
> See the above file for my best attempt. Not precisely what you
suggested,
> but close.
It is appropriate that you limited the downward travel of the lower
flag on the left side to conform to the rest of the symbols. I was a
little hesitant to put the top wavy flag any higher, lest it be
confused with a convex flag, but I now realize that the two are so
different in size that this wouldn't be a problem. What you have
looks good.
> I don't want to jump the gun and go to the main list just yet, and
when I
> do, I'll want a staff showing the odd harmonics of G up to 41,
including
> all optional spellings (using single shaft symbols with
conventional sharps
> and flats), as well as the 217ET notation and a couple of other
ETs.
Then I think that we should decide on standard (or preferred) sets of
symbols for as many ET's as we can before doing this. I would also
like to get the rest of the single symbols taken care of, too. (The
question about the length of the middle shaft of the sesquisymbols
shouldn't hold us back from designing the flags. There also remains
the question about the design of the Xsymbols  I don't recall that
you replied to my diagonalsofatrapezoid answer; we were both
getting a little punchy from overwork, and this is something that we
need to get back to.)
> You might want to check out
> http://shareware.search.com/search?
cat=247&tag=ex.sa.sr.srch.sa_all&q=truety
> pe+font+editor
It looks like there are a few packages that could be used. Do you
have any suggestions or preferences?
> and
>
> http://www.sibelius.com
>
> to get the free download which is fully functional except for save.
>  Dave Keenan
This, I presume, would give us a chance to see how a new font would
work with their product.
> Brisbane, Australia
> http://dkeenan.com
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4179]:
>But there's stil quite a difference, so I don't think that it would
>be a problem. Note that the change does make the vertical shaft look
>more centered in that large 65.3cent symbol.
OK. I accept the smaller width for the left convex flag.
>By the way, it's been bugging me that we've yet to agree on the
>spelling of confusable vs. confusible. I finally looked up the able
>vs. ible rules. There were two that applied (source: _The Grammar
>Bible_, Strumpf & Douglas, Knowledgeopolis, 1999):
>
>Rule 2: If the base itself is a complete English word, use the
>suffix able. Examples: changeable, flyable
>
>Which would result in "confuseable". However, see
>
>Rule 4: If you can add the suffix ion to the base to make a
>legitimate English word, then you should use the suffix ible.
>Examples: corruptible (corruption), perfectible (perfection)
>
>Which results in "confusible".
>
>I hope that rule 4 will end the confusion, even if it doesn't
>eliminate all of the confusibility.
I've got bad news for you. ;)
It would seem to me that one should only apply such rules when the word
itself cannot be found in any dictionary, or when dictionaries disagree,
and as such, rule 4 is a good one, since it predicts the dictionary
spelling for most such words.
Unfortunately I find "confusability" and not "confusibility" in my Shorter
Oxford. And the Australian English dictionary that comes with Microsoft
Word accepts confusable and confusability, but not confusible or
confusibility. Of course it's possible that a US dictionary may have
"ible". Let me know if you find one. I couldn't easily figure out how to
switch my Microsoft Word to use a US English dictionary.
I have no objection if you wish to continue spelling it "ible".
>> Which is one thing that causes me to repropose the convex with the
>> slightly squarer corners. Your only comment about them has been
>to "leave
>> well enough alone". Can you give a more detailed reason for
>rejecting them?
>
>Just an aesthetic consideration:
Thought it might be.
>the flags with the squarer corners
>tend to look like rightangle flags with rounded corners, as opposed
>to flags that are curved along their entire length.
Would that be such a bad thing, if it makes them more distinct from other
types?
I'll go with your preference for these bitmaps, because I agree they are
more pleasing to the eye, but when it comes to designing an outline font
I'd be tempted to do something in between the two.
>> I don't want to jump the gun and go to the main list just yet, and
>when I
>> do, I'll want a staff showing the odd harmonics of G up to 41,
>including
>> all optional spellings (using single shaft symbols with
>conventional sharps
>> and flats), as well as the 217ET notation and a couple of other
>ETs.
>
>Then I think that we should decide on standard (or preferred) sets of
>symbols for as many ET's as we can before doing this.
What would be even better is, after doing a few very different ones the
hard way, and therefore thinking about what the issues are, if we could
simply give an algorithm for choosing the notation for any ET. I gave two,
earlier. They are both undoubtedly too simple. The difference between them
was exactly the difference between plan A and Plan B for 217ET, i.e.
whether to favour fewer flags or lower primes. It seems we've decided in
favour of lower primes so far. Lets see how that pans out for a few other ETs.
One thing we need to decide is how we are going to decide when NET's best
fifth isn't good enough and instead notate it as every nth step of n*NET.
I propose that we not accept any notational fifth for which either the
apotome or the pythagorean limma vanishes, or is a negative number of steps.
This excludes from using their "native" fifth, only the ETs 2 thru 11, 13
thru 16, 18,20,21,23,25,28,30 and 35.
I also feel that, if we are using a comma for a prime greater than 9 to
notate an ET, then the user would be justified in assuming that the best
4:9s in the tuning are notated as ..., Bb:C, F:G, C:D, G:A, D:E, A:B, E:F#,
... etc.
If this is not the case then I suggest that the ET's native fifth is not
acceptable for notation, since we have no "9comma" symbol. This further
excludes
32,33,37,40,42,44,45,47,49,52,54,57,59,61,62,64,66,69,71,73,74,76,78,81,83,8
5,86,88,90,93,95,97,98,100,102,103,105,107,110,112,114,115,117,119,122,1124,
126,127,129,131,134,136,138,139,141,143,146,148 and about half of all ETs
from then on.
Perhaps it simpler if I just list the ones that we _do_ need to try to
notate, and beside them list the others that they give us as subsets.
12 (6 4 3 2)
17
19
22 (11)
24 (8)
26 (13)
27 (9)
29
31
34
36 (18)
38
39
41
43
46 (23)
48 (16)
50 (5 10 25)
51
53
55
56 (7 14 28)
58
60 (15 20 30)
63 (21)
65
67
68
70 (35)
72
80 (40)
84 (42)
94 (47)
96 (32)
99 (33)
104 (52)
108 (54)
111 (37)
118 (59)
128 (64)
132 (44 66)
135 (45)
142 (71)
147 (49)
152 (76)
171 (57)
183 (61)
186 (62)
207 (69)
217
This includes every ET up to 72.
Here are the first few, showing the accidentals I think are required in
addition to # and b in the twosymbol approach.
ET 12 17 19 22 24 26 27 29 31 34 36 38 39
steps symbols
1 ss x s ss x s s x s x ss s
2 sx ss w ss
ET 41 43 46 48 50 51 53 55 56 58 60 63 65
steps symbols
1 s x s s sx x s x x s s x s
2 ss ss ss ss x vx ss x s s x s x
3 sx ss ss ss ss
ET 67 68 70 72 80 84 94 96 99 104 108 111
steps symbols
1 x x s s x s w s w v s w
2 x s s x s x s x s x ss s
3 ss ss ss ss x sx x x x x x s
4 sx ss ss ss sx wx vs ws
5 ss sx ss
I've mostly used a "lowest prime" algorithm but there are definitely
problems with this in 38, 50, 51, 55, 56, 67, 68, 99, 108, where either a
two flag symbol is fewer steps than one of its component single flags, or
symbols represent sizes that are out of order relative to their JI sizes,
or a comma has been used that are not 1,3,pconsistent (99ET), or other
problems.
>I would also
>like to get the rest of the single symbols taken care of, too. (The
>question about the length of the middle shaft of the sesquisymbols
>shouldn't hold us back from designing the flags. There also remains
>the question about the design of the Xsymbols  I don't recall that
>you replied to my diagonalsofatrapezoid answer; we were both
>getting a little punchy from overwork, and this is something that we
>need to get back to.)
Diagonals of trapezoid is as good as anything, but the main problem with
the X's is what you have pointed out yourself; that the crossing of the X
seems to be referring to a different note. I also agree that the essential
problem of a triple tail is not addressed by making the middle shaft
shorter. I suggest you consider other possible tails that have no such
extra distracting intersection point and no triples. There are V tails and
wavy tails (singly or in pairs with parallel waves or counter waves), and
other kinds of curved tails both single and double.
>> You might want to check out
>> http://shareware.search.com/search?
>cat=247&tag=ex.sa.sr.srch.sa_all&q=truety
>> pe+font+editor
>
>It looks like there are a few packages that could be used. Do you
>have any suggestions or preferences?
Font Lab is the easiest to use and has the most features, but I can't
justify paying for it. The demo is limited to saving 20 glyphs and when you
actually generate font files, half of those glyphs will be modified to
include an "FL" logo.
Font Creator (fcreap) is bare bones, but the price is right.
>> http://www.sibelius.com
>>
>> to get the free download which is fully functional except for save.
>>  Dave Keenan
>
>This, I presume, would give us a chance to see how a new font would
>work with their product.
Yes. You got it right with the 8 pixels between staff lines. The outline
fonts are designed for 128 units between staff lines, so it is 32 font
units per bitmap pixel.
I'm sorry but I'm going to have to take an extended holiday (a week or two)
from this stuff until I catch up on a lot of other work I'm supposed to
have done. I look forward to great progress when I return.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> Yes. You got it right with the 8 pixels between staff lines. The
outline
> fonts are designed for 128 units between staff lines, so it is 32
font
> units per bitmap pixel.
There's a bit of an arithmetic problem there. I should have written "256
units between staff lines", e.g. from 128 to +128. 32 font units per
bitmap pixel is correct.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > Yes. You got it right with the 8 pixels between staff lines. The
> outline
> > fonts are designed for 128 units between staff lines, so it is 32
> font
> > units per bitmap pixel.
>
> There's a bit of an arithmetic problem there. I should have
written "256
> units between staff lines", e.g. from 128 to +128. 32 font units
per
> bitmap pixel is correct.
Yes, that's more reasonable. A scalable font should work very nicely
with that amount of resolution.
> I'm sorry but I'm going to have to take an extended holiday (a week
or two)
> from this stuff until I catch up on a lot of other work I'm
supposed to
> have done. I look forward to great progress when I return.
We've been working on this pretty intensively, and it will do you
good to get away from it for a little while. In the meantime I have
plenty to keep me busy, including catching up on a backlog of tuning
list digests.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > > Yes. You got it right with the 8 pixels between staff lines.
The
> > outline
> > > fonts are designed for 128 units between staff lines, so it is
32
> > font
> > > units per bitmap pixel.
> >
> > There's a bit of an arithmetic problem there. I should have
> written "256
> > units between staff lines", e.g. from 128 to +128. 32 font units
> per
> > bitmap pixel is correct.
>
> Yes, that's more reasonable. A scalable font should work very
nicely
> with that amount of resolution.
>
> > I'm sorry but I'm going to have to take an extended holiday (a
week
> or two)
> > from this stuff until I catch up on a lot of other work I'm
> supposed to
> > have done. I look forward to great progress when I return.
>
> We've been working on this pretty intensively, and it will do you
> good to get away from it for a little while. In the meantime I
have
> plenty to keep me busy, including catching up on a backlog of
tuning
> list digests.
>
> George
***It might be fun to post a "summary" on what has been going on here
on the "Main" list... Looks like a lot of interesting stuff has been
going on.
J. Pehrson
 In tuningmath@y..., "jpehrson2" <jpehrson@r...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>
> /tuningmath/message/4193
>
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > > I'm sorry but I'm going to have to take an extended holiday (a
week or two)
> > > from this stuff until I catch up on a lot of other work I'm
supposed to
> > > have done. I look forward to great progress when I return.
> >
> > We've been working on this pretty intensively, and it will do you
> > good to get away from it for a little while. In the meantime I
have
> > plenty to keep me busy, including catching up on a backlog of
tuning
> > list digests.
> >
> > George
>
> ***It might be fun to post a "summary" on what has been going on
here
> on the "Main" list... Looks like a lot of interesting stuff has
been
> going on.
>
> J. Pehrson
Dave & I need to get a few more things straightened out before we do
that. It's been a long and complicated task to agree on as many
things as we already have, but it is still premature to put this out
on the main tuning list. There are still things that we haven't
discussed in sufficient detail, and bringing others into the fray at
this point would be counterproductive.
Remember, patience comes to those who wait for it.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > > We've been working on this pretty intensively, and it will do
you
> > > good to get away from it for a little while. In the meantime I
> have
> > > plenty to keep me busy, including catching up on a backlog of
> tuning
> > > list digests.
And responding to some of my posts in this thread regarding
rational apotome complements.
> Remember, patience comes to those who wait for it.
Tee hee.
I never did say how much I enjoyed your Justin Tenacious story. I
didn't guess what the Leprechauns' solution would be, and of course
it's debatable. But fun.
On the subject of fiction: I was thinking that mathematics is so
unfashionable as a justification for anything musical that we should
invent some mythology to introduce our notation. I think the notation
should somehow be given by the gods rather than designed (or found
mathematically), and of course there's some truth in that if you take
numbers (at least the rationals) to be godgiven, or as an
atheistmystic like me might prefer, built into the fabric of the
universe.
I have images of Olympian gods throwing arrows at us in a dream. Or
we've discovered a lost parchment from Atlantis or something. Any
ideas?
Back to mathematics:
My obsession is to have no notational schisma greater than 0.5 c and
yours is to be able to notate practically everything with a single
accidental, and we're doing fairly well at acheiving both. However
there seems to be a big hole in the latter when it comes to the
relatively common rational subdiminished fifth or augmented fourth
5:7. How will you notate a 5:7 up from C? And its inversion 7:10?
I imagine you might want to be able to notate the entire 11limit
diamond (and Partch's extensions of it) (rationally, not in 72ET or
miracle temperament) with single accidentals, so 7:11 and 11:14 might
be a problem too.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
I wrote:
"I imagine you might want to be able to notate the entire 11limit
diamond (and Partch's extensions of it) (rationally, not in 72ET or
miracle temperament) with single accidentals, so 7:11 and 11:14 might
be a problem too."
Actually, it looks like 11/5 might be more of a problem than either 11/7 or
7/5.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4204]:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > > > We've been working on this pretty intensively, and it will do
you
> > > > good to get away from it for a little while. In the meantime
I have
> > > > plenty to keep me busy, including catching up on a backlog of
tuning
> > > > list digests.
>
> And responding to some of my posts in this thread regarding
> rational apotome complements.
It looks as if this will occur at least partially in this message.
> I never did say how much I enjoyed your Justin Tenacious story. I
> didn't guess what the Leprechauns' solution would be, and of course
> it's debatable. But fun.
It's been weeks since I put that out there, and you're the first one
to say anything about it. I was wondering whether anyone had read it.
There are lots of solutions:
13:15:17:20:23 or 12:14:16:18:21 or 26:30:34:39:45 can simulate 5ET
17:19:21:23:25:28:31 or 18:20:22:24:27:30:33 can simulate 7ET
And here's an application for the 41st harmonic:
22:24:26:28:30:32:35:38:41 or 24:26:28:30:33:36:39:42:45 can
simulate 9ET
The lower the ET, the lower the numbers in the scale ratio. As I
recall, when I tried these on the Scalatron, I thought that the most
successful ones were the 5 and 7ET ones with the higher primes. As
for how well it works? About as well as you could expect,
considering that this came from a tricky leprechaun. But I found
that I liked the way these sound better than the ET's.
> On the subject of fiction: I was thinking that mathematics is so
> unfashionable as a justification for anything musical that we
should
> invent some mythology to introduce our notation. I think the
notation
> should somehow be given by the gods rather than designed (or found
> mathematically), and of course there's some truth in that if you
take
> numbers (at least the rationals) to be godgiven, or as an
> atheistmystic like me might prefer, built into the fabric of the
> universe.
>
> I have images of Olympian gods throwing arrows at us in a dream. Or
> we've discovered a lost parchment from Atlantis or something. Any
> ideas?
Yes  considerably more than you bargained for, and I didn't even
have to invent any ideas. From my perspective the pursuit of truth
is better than fiction, but even more unfashionable than
mathematics. I grew up as an atheist/skeptic and in my late teens
came to faith in a personal God by wrestling with the problem of
accounting for intricate design in the universe. (I'll try to keep
this on topic.) I could not accept the notion that the great variety
and complexity of lifeforms in existence are nothing more than the
product of random processes, i.e., purely by chance. For me this
opened up the possibility of belief in a CreatorGod that might have
revealed himself to us in the course of history. After careful
consideration of the writings of the Hebrew prophets and Christian
apostles, I concluded that there were more problems in rejecting
their testimony than in accepting it, and I became a Christian.
And a year or so later I became a microtonalist.
Last August I read an article about persons who sustained injury to
the part of the brain that processes music, so that even the simplest
tunes were now incomprehensible. By this I realized that music is
more than just our ability to hear sounds  it is something that we
were designed to be able to enjoy  nothing less than a gift of God.
When Margo Schulter first wrote me last September about my 17tone
welltemperament (only shortly after I had read that article), she
included the following words of appreciation:
> It has often been written that music is a gift both priceless and
> divine, and your welltemperament is to me a musical offering most
> precious, a gift which I hope to honor through use in much joyous
> musicmaking.
And my reply (10 Sep 2001) included the following:
<< Yes, music is a gift, both priceless and divine, and I have never
been more keenly aware of that than now. In writing my book, I have
found that some of my best ideas just seem to appear suddenly at my
mental doorstep, and I sometimes wonder whether I really thought of
those things myself or whether God gave me some sort of special I
loveyou note to help me along and to remind me that, when I agreed
to trust him with everything in my life, that he would guide my steps
and direct my path. But then I realize that everything good in this
world is ultimately his doing. Whatever gifts we may possess,
whether musical or otherwise, as we use them for even the best
reasons and highest motivation, it is always good to be reminded that
we should not forget to honor the Giver of those gifts. (If this
sounds anything like a lecture, then believe me Â– I needed to hear it
as much as anyone.) >>
Anyway, back to musical notation.
> Back to mathematics:
>
> My obsession is to have no notational schisma greater than 0.5 c
and
> yours is to be able to notate practically everything with a single
> accidental, and we're doing fairly well at acheiving both. However
> there seems to be a big hole in the latter when it comes to the
> relatively common rational subdiminished fifth or augmented fourth
> 5:7. How will you notate a 5:7 up from C? And its inversion 7:10?
>
> I imagine you might want to be able to notate the entire 11limit
> diamond (and Partch's extensions of it) (rationally, not in 72ET
or
> miracle temperament) with single accidentals, so 7:11 and 11:14
might
> be a problem too.
For 7:5 above C it is fortunate that the (17'17) comma (~6.001
cents) is very close to the difference between the 7 and 5 commas,
5103:5120 (3^6*7:2^10*5, ~5.758 cents). So, according to the
complementation rules, 7/5 of C is G lowered by sx.
Likewise, for 11:7 above C the 29 comma (~33.487 cents) is very close
to the difference between a Pythagorean Gsharp and 11/7, so that
11/7 of C is G raised by vw, according to the complementation rules.
I don't think that either of these complements is at issue in your
concern about rational complementation.
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4205]:
>
> Actually, it looks like 11/5 might be more of a problem than either
11/7 or 7/5.
I will see about 11/10 when I get a chance.
I've been spending a lot of time lately on the rational
complementation problem and have made considerable progress. I still
need to reread some of your recent messages so that I can get some
additional perspective on my latest work before I post it. (All of
this attention to detail is going to have to pay off in the long run.)
George
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>It's been weeks since I put that out there, and you're the first one
>to say anything about it. I was wondering whether anyone had read it.
When something is perfect, what can anyone say? :)
>There are lots of solutions:
>
> 13:15:17:20:23 or 12:14:16:18:21 or 26:30:34:39:45 can simulate 5ET
> 17:19:21:23:25:28:31 or 18:20:22:24:27:30:33 can simulate 7ET
>
>And here's an application for the 41st harmonic:
>
> 22:24:26:28:30:32:35:38:41 or 24:26:28:30:33:36:39:42:45 can
>simulate 9ET
I get the picture. As you said in the story, it's a novel idea to look for
harmonic series pieces that approximate ETs instaed of the other way
'round. When I said it's debatable whether the request was satisfied: It's
unclear whether Justin wanted all the intervals (dyads) to be justly
intoned (I don't find 11:12 to be justly intoned) and if he did, whether he
will be satisfied with only 5 notes and that 8:9 and 6:7 are in the same
interval class. But I'm splitting hairs.
>The lower the ET, the lower the numbers in the scale ratio. As I
>recall, when I tried these on the Scalatron, I thought that the most
>successful ones were the 5 and 7ET ones with the higher primes. As
>for how well it works? About as well as you could expect,
>considering that this came from a tricky leprechaun. But I found
>that I liked the way these sound better than the ET's.
OK.
>Yes  considerably more than you bargained for, and I didn't even
>have to invent any ideas. From my perspective the pursuit of truth
>is better than fiction, but even more unfashionable than
>mathematics. I grew up as an atheist/skeptic and in my late teens
>came to faith in a personal God by wrestling with the problem of
>accounting for intricate design in the universe. (I'll try to keep
>this on topic.) I could not accept the notion that the great variety
>and complexity of lifeforms in existence are nothing more than the
>product of random processes, i.e., purely by chance. For me this
>opened up the possibility of belief in a CreatorGod that might have
>revealed himself to us in the course of history. After careful
>consideration of the writings of the Hebrew prophets and Christian
>apostles, I concluded that there were more problems in rejecting
>their testimony than in accepting it, and I became a Christian.
Yikes! That is seriously offtopic. And random processes vs. creator god =
personal god, seriously fails to exhaust the possibilities. I'll limit my
reply to suggesting two brilliant books. "Darwin's Dangerous Idea" by
Daniel Dennett and "A Brief History of Everything" by Ken Wilber.
>And a year or so later I became a microtonalist.
>
>Last August I read an article about persons who sustained injury to
>the part of the brain that processes music, so that even the simplest
>tunes were now incomprehensible. By this I realized that music is
>more than just our ability to hear sounds  it is something that we
>were designed to be able to enjoy  nothing less than a gift of God.
We were designed to enjoy it, yes. But to assume that _must_ imply the
latter, is either poetic (which is fine), or a serious failure of the
imagination.
[More seriously offtopic stuff deleted]
If you promise to lay off the monotheistic dualism (look where _that's_ got
the planet at the moment), I promise to lay off the atheistic mysticism
(not that I ever layed any on).
What I had in mind for the introduction of the notation was some light
hearted fiction that was obviously fiction, preferably not involving any
real extant religion.
>For 7:5 above C it is fortunate that the (17'17) comma (~6.001
>cents) is very close to the difference between the 7 and 5 commas,
>5103:5120 (3^6*7:2^10*5, ~5.758 cents). So, according to the
>complementation rules, 7/5 of C is G lowered by sx.
>
>Likewise, for 11:7 above C the 29 comma (~33.487 cents) is very close
>to the difference between a Pythagorean Gsharp and 11/7, so that
>11/7 of C is G raised by vw, according to the complementation rules.
>
>I don't think that either of these complements is at issue in your
>concern about rational complementation.
That's correct. These complements are uncontroversial.
>
> In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4205]:
>>
>> Actually, it looks like 11/5 might be more of a problem than either
>11/7 or 7/5.
>
>I will see about 11/10 when I get a chance.
>
>I've been spending a lot of time lately on the rational
>complementation problem and have made considerable progress. I still
>need to reread some of your recent messages so that I can get some
>additional perspective on my latest work before I post it. (All of
>this attention to detail is going to have to pay off in the long run.)
Thanks.
My current thinking is that the rational complements should be based on
665ET, an ET with an extremely good 1:3 so there is no danger of any size
crossovers with any pairs of symbols.
We only need to introduce a vv symbol (instead of my earlier proposed vw)
as the complement to ss, the 25 comma symbol.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
George and Dave,
(see, I read this stuff! I'm just not wellversed enough to contribute...)
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> What I had in mind for the introduction of the notation was some
> lighthearted fiction that was obviously fiction, preferably not
> involving any real extant religion.
Dave, I remember you doing something like this once before  was it in connection with the Excel/3D tuning object project? In any event, George, Dave had a great mockmyth behind the stuff (whatever theory/piece/project it was) that was very tongueincheek. Take him up on the offer!
Cheers,
Jon
I wrote:
"My current thinking is that the rational complements should be based on
665ET, an ET with an extremely good 1:3 so there is no danger of any size
crossovers with any pairs of symbols, existing or future.
We only need to introduce a vv symbol (instead of my earlier proposed vw)
as the complement to ss, the 25 comma symbol."
The last paragraph was wrong. It seems that at least one other 3 flagger
must be introduced as the complement of the 17 flag, and possibly some
others, as shown below.
Here's my latest proposal for rational apotome complements.
Symbol Comp Comma name Comments

v xw 19
v sx (17'17)
w wwx 17 (or vws, less preferred)
wv ws 17'
w sw 23
vw x 19'
s s 5
wwv vx pythag comma (comp probably not required)
x x 7
s s (115)
x vw 29 or (11'7)
vs vwv 31 (comp probably not required)
sw w (prob not required, 5 comma + 23 comma)
vwx none 11/5 (hope comp is not required)
ws wv 23'
ss vv 25
vwx vv 37' (comp probably not required)
sx v 13
ss xx 11
sx sx 31'
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
[Regarding my fictional story about Justin Tenacious]
> >It's been weeks since I put that out there, and you're the first
one
> >to say anything about it. I was wondering whether anyone had read
it.
>
> When something is perfect, what can anyone say? :)
Well, you've made my day! In a conversation, if nobody says anything
in response to something that was said, one can usually assume that:
1) They didn't hear and/or understand it; or
2) It was of no interest; or
3) It was either so incredibly ignorant, stupid, or trivial that it
would be best to say nothing.
So I was a little puzzled during those weeks of silence.
> > ... From my perspective the pursuit of truth
> > is better than fiction, but even more unfashionable than
> > mathematics.
>
> Yikes! That is seriously offtopic. And random processes vs.
creator god =
> personal god, seriously fails to exhaust the possibilities. I'll
limit my
> reply to suggesting two brilliant books. "Darwin's Dangerous Idea"
by
> Daniel Dennett and "A Brief History of Everything" by Ken Wilber.
And in return, may I suggest _Darwin on Trial_ by Phillip E. Johnson
and _Intelligent Design: The Bridge Between Science & Theology_ by
William Dembski.
> [More seriously offtopic stuff deleted]
>
> If you promise to lay off the monotheistic dualism (look where
that's_ got
> the planet at the moment), I promise to lay off the atheistic
mysticism
> (not that I ever layed any on).
I was just indulging in a little introspection illustrating how one's
philosophical outlook might be related to the creative process in
music (or in this case, music theory).
I also submit that it is not religious belief that causes turmoil and
misery so much as the inability of those with strongly held beliefs
(religious, political, or otherwise controversial) to respect the
rights of those who differ, particularly when they are so insecure in
their beliefs that they respond irrationally when challenged. Even
microtonality has its fanatics, and whatever our passion, we just
have to learn to get along.
But that's enough of that.
 In tuningmath@y..., "jonszanto" <jonszanto@y...> wrote:
> George and Dave,
>
> (see, I read this stuff! I'm just not wellversed enough to
contribute...)
>
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > What I had in mind for the introduction of the notation was some
> > lighthearted fiction that was obviously fiction, preferably not
> > involving any real extant religion.
>
> Dave, I remember you doing something like this once before  was it
in connection with the Excel/3D tuning object project? In any event,
George, Dave had a great mockmyth behind the stuff (whatever
theory/piece/project it was) that was very tongueincheek. Take him
up on the offer!
>
> Cheers,
> Jon
Yes, it's a wonderful idea. Even before reading these latest
responses, I showed Dave's suggestion last night to my daughter (who
is very knowledgeable about classical mythology), and she offered up
the following bit of information: One source of our inspiration
could be Apollo, the Greek god of music. Apollo happens to have a
certain distinction in classical mythology: most of the Greek gods
had a Roman counterpart, but not Apollo  the Romans had to adopt
him from the Greeks.
I'll have to ask her if there were any mathematical gods in the Greek
pantheon. After all, just as we have been collaborating on this
notation, I don't think it is approrpriate that Apollo would be
working on this one all by himself.
My daughter also loves to write (she has some RPG fan fiction out on
the web) and would probably like to get involved in this also.
So, Dave, where and when do we get started? (Offlist, I would say.
Tuningmath just doesn't seem to be the right place.)
George
Dave,
I've put out a file containing my latest proposal for symbols for
alterations above the halfapotome.
/tuning
math/files/secor/notation/Symbols3.bmp
I've paid particular attention to scaling the width of the 2 & 3
shaft and X symbols. For these I didn't think it was appropriate to
make the concave flags as small as you did in your examples
(comparing the size of the symbols at the left extreme with those at
the right extreme of the line above), and I find that these and the
wavy flags are quite readable this way. I realize that a few of
these symbols won't be used the way we presently have things figured
out, but I did all of these just to get a sense of continuity in the
progression of size moving vertically.
I also tried my hand at ss, ss, and ssX symbols at the far
right, just to see how those might look. (I hope that the meanings
of x and X don't get too confusing.)
George
There were mistakes in my latest proposal for rational apotome complements.
The 17' and 23' comma complements were wrong. I'll give the whole thing
again with corrections and additions.
Symbol Comp Comma name Comments

v xw 19
v sx (17'17)
w wwx 17
vv vws
wv xv 17'
w sw 23
vw x 19'
s s 5
wwv vx pythag comma (comp probably not required)
x x 7
vx wwv (probably not required)
s s (115)
x vw 29 or (11'7)
vs vwv 31 (comp probably not required)
wx ww (probably not required)
sw w (prob not required, 5 comma + 23 comma)
vwx none 11/5 (hope comp is not required)
xv wv alt 23' (comp is good reason to make this standard 23')
ws vw 23'
ss vv 25
vwx vv 37' (comp probably not required)
sx v 13
ss xx 11
sx sx 31'
The above complements correspond to flags being the following numbers of
steps of 665ET.
v 2 3 v
w 5 9 w
s 12 15 x
x 19 18 s
By the way, 217ET isn't the largest ET we can notate using symbols having
no more than one flag per side. We can do 306ET as follows. I think it is
the largest.
1 v
2 v
3 vv
4 w or wv
5 s
6 ww
7 x
8 vx
9 s
10 vs
11 xv
12 sx
13 xw
14 ss
15 xx
16 v
17 v
18 vv
19 w or wv
20 s
21 ww
22 x
23 vx
24 s
25 vs
26 xv
27 sx
28 xw
29 ss
I haven't shown _all_ the alternatives above. Flag values are
v 1 2 v
w 2 4 w
s 5 7 x
x 9 9 s
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
I've added one more rational complement, for vww, which may be of use as
an alternate 7/5 comma.
Symbol Comp Comma name Comments

v xw 19
v sx (17'17)
w wwx 17
vv vws
wv xv 17'
w sw 23
vw x 19'
s s 5
wwv vx pythag comma (comp probably not required)
x x 7
vww ww (7/5)'
vx wwv (probably not required)
s s (115)
x vw 29 or (11'7)
vs vwv 31 (comp probably not required)
wx ww (probably not required)
sw w (prob not required, 5 comma + 23 comma)
vwx none 11/5 (hope comp is not required)
xv wv alt 23' (comp is good reason to make this standard 23')
ws vw 23'
ss vv 25
vwx vv 37' (comp probably not required)
sx v 13
ss xx 11
sx sx 31'
The above complements correspond to flags being the following numbers of
steps of 665ET.
v 2 3 v
w 5 9 w
s 12 15 x
x 19 18 s
I note that, apart from a few exceptions below the resolution of 665ET, we
have the following complementary pairs of flags on the same side.
Left side
v ww
w vw
s (blank)
x (none)
Right side
v x
w w
s (blank)
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> Dave,
>
> I've put out a file containing my latest proposal for symbols for
> alterations above the halfapotome.
>
> /tuning
> math/files/secor/notation/Symbols3.bmp
The  and  shaft symbols look great, but I'm afraid the whole
concept of  and X shaft symbols will have to be a minority report.
I'd rather just stack a ss beside the  and  symbols.
What did you think of my suggestion to use V tails or single and
double wavy tails?
> I've paid particular attention to scaling the width of the 2 & 3
> shaft and X symbols. For these I didn't think it was appropriate to
> make the concave flags as small as you did in your examples
> (comparing the size of the symbols at the left extreme with those at
> the right extreme of the line above), and I find that these and the
> wavy flags are quite readable this way.
Agreed. I may want to fiddle with a pixel here and there if I get
time. But otherwise I think they are great.
> I realize that a few of
> these symbols won't be used the way we presently have things figured
> out, but I did all of these just to get a sense of continuity in the
> progression of size moving vertically.
>
> I also tried my hand at ss, ss, and ssX symbols at the far
> right, just to see how those might look.
ss looks OK, but maybe you should try omitting the part of one shaft
that appears between the two flags.
Wanna try some of the other twoflagsononeside symbols I've
proposed as rational complements? e.g. wwx (17), vws (other
possibility for 17), vw (23'), vv (25).
> (I hope that the meanings
> of x and X don't get too confusing.)
I read them just fine.
I'm keen to finalise the rational complement relationships and the
single 11/5 comma symbol if any.
Should we look at possible single symbols for 13/5, 13/7, 13/11 commas too,
or is this getting too silly?
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> So, Dave, where and when do we get started? (Offlist, I would say.
> Tuningmath just doesn't seem to be the right place.)
What about spiritualtuning?
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > Dave,
> >
> > I've put out a file containing my latest proposal for symbols for
> > alterations above the halfapotome.
> >
> > /tuning
math/files/secor/notation/Symbols3.bmp
>
> The  and  shaft symbols look great, but I'm afraid the whole
> concept of  and X shaft symbols will have to be a minority
report.
> I'd rather just stack a ss beside the  and  symbols.
I think that looks almost as bad as putting a couple of conventional
sharp symbols next to each other  there's a reason why the
conventional doublesharp symbol was invented. In fact, what you're
suggesting is worse than two conventional sharps next to each other:
you've got one symbol with various flags and the other with two
straight flags. In the latter symbol the flags are irrelevant  the
only information that the player really needs is given by the two
shafts, so the straight flags are just extraneous information which
can only generate annoyance and confusion.
On the singleshaft symbols concave and wavy flags are rather tiny,
whereas on a threeshaft symbol they are larger and, therefore,
easier to read. Double symbols create a lot of clutter, particularly
in keyboard music, where the process of reading the notation must be
made as efficient as possible.
Have I given enough reasons for single symbols?
> What did you think of my suggestion to use V tails or single and
> double wavy tails?
I don't see any advantage in the V tails. And I see a real problem
with wavy tails  a performer would already be required to identify
three different types of curved flags (in addition to straight
flags), and wavy tails would probably generate confusion by hindering
that process. I think that the straight shafts (and X) are
sufficient to communicate what they are called upon to do without
drawing undue attention to themselves.
> > I've paid particular attention to scaling the width of the 2 & 3
> > shaft and X symbols. For these I didn't think it was appropriate
to
> > make the concave flags as small as you did in your examples
> > (comparing the size of the symbols at the left extreme with those
at
> > the right extreme of the line above), and I find that these and
the
> > wavy flags are quite readable this way.
>
> Agreed. I may want to fiddle with a pixel here and there if I get
> time. But otherwise I think they are great.
Wonderful!
> > I realize that a few of
> > these symbols won't be used the way we presently have things
figured
> > out, but I did all of these just to get a sense of continuity in
the
> > progression of size moving vertically.
> >
> > I also tried my hand at ss, ss, and ssX symbols at the far
> > right, just to see how those might look.
>
> ss looks OK, but maybe you should try omitting the part of one
shaft
> that appears between the two flags.
My rationale for leaving it has to do with the ssX symbol: I tried
making that one with the top of the X occurring at the lower flag.
This left the two flags disjointed, so I connected them with parallel
vertical lines. It didn't look good, and since I didn't think there
was any problem in reading the symbol with the X terminating at the
top flag, that was what I used. Extending all of the lines in the
other symbols up to the top flag makes them consistent with the X
symbol.
> Wanna try some of the other twoflagsononeside symbols I've
> proposed as rational complements? e.g. wwx (17), vws (other
> possibility for 17), vw (23'), vv (25).
First I have to see what all of these are about, which will involve
spending more time with the rational complements.
> > (I hope that the meanings
> > of x and X don't get too confusing.)
>
> I read them just fine.
>
> I'm keen to finalise the rational complement relationships and the
> single 11/5 comma symbol if any.
I'll start to answer some of this in my next posting.
> Should we look at possible single symbols for 13/5, 13/7, 13/11
commas too,
> or is this getting too silly?
I'm seriously wondering whether anyone is going to use any of those 2
flags/side symbols. Our whole reason for notating 217 was to get 19
limitunique capability (which would handle these eventualities), but
now that we have it we're still off on a quest into the great beyond 
 311, 1600, 453, 653, 665 (with which I find a serious problem that
I will address in my next posting)  what next?
Stay tuned!
George
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4213]:
> There were mistakes in my latest proposal for rational apotome
complements.
> The 17' and 23' comma complements were wrong. I'll give the whole
thing
> again with corrections and additions.
(You updated this in your next posting, which I will respond to
below.)
> The above complements correspond to flags being the following
numbers of
> steps of 665ET.
>
> v 2 3 v
> w 5 9 w
> s 12 15 x
> x 19 18 s
I have a serious problem with using 665ET as a basis for anything.
It is only 9limit consistent  the 11 factor falls almost midway
between degrees. Among other things, this causes xL+sR to be 37
degrees, whereas it should be 36.
Below I will also give another problem with using 665 as a basis for
rational complementation.
> By the way, 217ET isn't the largest ET we can notate using symbols
having
> no more than one flag per side. We can do 306ET as follows. I
think it is
> the largest.
Even if it isn't a very enticing one, with the 5 factor coming almost
midway between degrees and the 11 factor almost as bad.
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4214]:
> I've added one more rational complement, for vww, which may be of
use as
> an alternate 7/5 comma.
>
> Symbol Comp Comma name Comments
> 
> v xw 19
> v sx (17'17)
> w wwx 17
> vv vws
> wv xv 17'
> w sw 23
> vw x 19'
> s s 5
> wwv vx pythag comma (comp probably not required)
> x x 7
> vww ww (7/5)'
> vx wwv (probably not required)
> s s (115)
> x vw 29 or (11'7)
> vs vwv 31 (comp probably not required)
> wx ww (probably not required)
> sw w (prob not required, 5 comma + 23 comma)
> vwx none 11/5 (hope comp is not required)
> xv wv alt 23' (comp is good reason to make this standard
23')
> ws vw 23'
> ss vv 25
> vwx vv 37' (comp probably not required)
> sx v 13
> ss xx 11
> sx sx 31'
Working only with symbols that have no more than one flag per side, I
came up with the following:
symbol complement comma offset comments

v xw 19 0.14 cents
v sx (17'17) 1.50 653inconsistent
w ws 17 4.05 653 & 665inconsis.
vv none
wv sw 17' 0.49 665inconsistent
w wx 23 0.73 665inconsistent
vw x 19' 0.14
s s 5 0.00
ww sv 0.49 665inconsistent
x x 7 1.26 653inconsistent
sv ww 0.49 665inconsistent
vx none
s s (115) 0.00
x vw 29 0.14
vs none 31
wx w 0.73 665inconsistent
sw wv 0.49 665inconsistent
xv none
ws w 23' 4.05 653 & 665inconsis.
sx v 13 1.50 653inconsistent
xw v 0.14
These were the possibilities that I didn't use:
symbol complement comma offset comments

vv ws 3.40 653 & 665inconsistent
wv xv 17' 0.95
w sw 23 1.32 653inconsistent
ww vx 2.64 217, 653, 665inconsis.
vx ww 2.64 217, 653, 665inconsis.
sw w 1.32 653inconsistent
xv wv 0.95
ws vv 3.40 653 & 665inconsistent
The term "offset" requires some explanation.
I did a spreadsheet evaluating these complements based on your
Complements.bmp illustration:
/tuning
math/files/secor/notation/GSCompls.xls
in which I bolded my selections.
I made a copy of it:
/tuning
math/files/secor/notation/DKCompls.xls
in which I bolded your selections (not including any with 2
flags/side, which you may add if you wish).
The column labeled "udcompl cents" is the number of cents in the
unidecimal diesis minus the singleshaft complementary symbol, as in
your bitmap figure. The offset is the difference in cents between
the original symbol and the udminuscomplement.
You can change the number in cell A8 to any ET to view the
complementation inconsistencies of that ET in column I.
You passed up some nice smalloffset complements that are 665
inconsistent. The only ones that I was forced to pass up in 217ET
have an offset of over 2.6 cents. And you can see that 653ET also
has a number of inconsistent complements. This is due in large part
to the fact that 653 and 665 are much finer divisions, so this is not
surprising.
However, this is a good reason not to base rational complementation
on a particular division of the octave, but rather on the basis of a
small offset.
George
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> There were mistakes in my latest proposal for rational apotome
complements.
> ... I'll give the whole thing
> again with corrections and additions.
>
> Symbol Comp Comma name Comments
> 
> ...
> x vw 29 or (11'7)
> ...
>
> The above complements correspond to flags being the following
numbers of
> steps of 665ET.
>
> v 2 3 v
> w 5 9 w
> s 12 15 x
> x 19 18 s
We need to define the xL flag strictly as 11'7 (715:729), or we are
going to run into problems with ET's in which the number of degrees
is different than with the 29 (256:261) definition, which is quite a
few of them (e.g., 27, 46, 53, 99, 140, and 152ET, which all require
the xL flag in their notation).
In 665ET xL should be 18 degrees. This is the real reason why xL+sR
comes out as 37 degrees in 665 (whereas it should be 36), which is
different from the one I gave in posting #4220, in which I
atttributed the problem to the poor representation of the 11 factor.
However, I still think using 665ET as a basis for rational
complementation is inadvisable because of its inconsistency. (This
just makes one less reason.)
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote [#4220]:
> We need to define the xL flag strictly as 11'7 (715:729), or we
are
> going to run into problems with ET's in which the number of degrees
> is different than with the 29 (256:261) definition ...,
>
> In 665ET xL should be 18 degrees. ...
In addition, this affects the complementation table that I gave in
message #4220, which results in additional inconsistencies both for
653 and 665. The table now should read:
symbol complement comma offset comments

v xw 19 0.14 cents 653 & 665inconsis.
v sx (17'17) 1.50 653inconsistent
w ws 17 4.05 653 & 665inconsis.
vv none
wv sw 17' 0.49 665inconsistent
w wx 23 0.73 665inconsistent
vw x 19' 0.14 653 & 665inconsis.
s s 5 0.00
ww sv 0.49 665inconsistent
x x 7 1.26 653inconsistent
sv ww 0.49 665inconsistent
vx none
s s (115) 0.00
x vw 29 0.14 653 & 665inconsis.
vs none 31
wx w 0.73 665inconsistent
sw wv 0.49 665inconsistent
xv none
ws w 23' 4.05 653 & 665inconsis.
sx v 13 1.50 653inconsistent
xw v 0.14 653 & 665inconsis.
These were the possibilities that I didn't use:
symbol complement comma offset comments

vv ws 3.40 653 & 665inconsistent
wv xv 17' 0.95 653 & 665inconsistent
w sw 23 1.32 653inconsistent
ww vx 2.64 217, 653, 665inconsis.
vx ww 2.64 217, 653, 665inconsis.
sw w 1.32 653inconsistent
xv wv 0.95 653 & 665inconsistent
ws vv 3.40 653 & 665inconsistent
I have also modified the files:
/tuning
math/files/secor/notation/GSCompls.xls
and
/tuning
math/files/secor/notation/DKCompls.xls
to reflect this change.
George
At 22:17 9/05/02 0000, you wrote:
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>I have a serious problem with using 665ET as a basis for anything.
>It is only 9limit consistent  the 11 factor falls almost midway
>between degrees. Among other things, this causes xL+sR to be 37
>degrees, whereas it should be 36.
True, but an error of a whole step of 665ET is still 33% smaller than an
error of a half step of 217ET. Consistency relates to accuracy relative to
step size, but surely absolute accuracy is more relevant here? I wasn't
intending to notate 665ET, it was just a way of ensuring that the flags
(and the second shaft) could consistently be assigned fixed values
(different kind of consistency) while minimising offsets.
But you'll be pleased to know that I've abandoned 665ET (and all ETs) as a
basis for rational complements.
I agree 306ET is not very enticing.
Thanks for those spreadsheets. I like the idea of ignoring ETs and just
trying to minimise the offsets.
We can take a set of symbol complements and treat them as a system of
linear equations which can then be solved to obtain values in cents for the
individual flags. It is possible to make a set that has no solution. This
is a different (and more serious) kind of inconsistency than the kind we
talk about when we say a certain ET is nlimit inconsistent. I think it is
important that they be consistent in this sense.
Of course the glaring problem with your recent proposal is the 4 cent
offset for w <> ws. All the others are less than half that. I'd be
much happier if we could keep the max offset to 1.5 cents or less. But I
also agree that the use of symbols with twoflagsaside should be a last
resort.
The 25 comma symbol really does need a complement, e.g. C:G#\\. I don't
think there's any problem with _its_ complement having two flags a side, in
fact I think it would be expected. My favourite complement for
ss is vv. This works in your system as well as mine.
For our system of linear equations, we can write this as
ss + vv = 113.685 cents
If we insist on consistency (as in having a solution for the flag sizes)
then the above implies
sv + sv = 113.685
It seems very desirable to have
x + x = 113.685
We agree on that.
Taken with the above, that implies
sv = x
and means that sv is not available as a complement for anything else. I
assume we want our rational complements to be uniqu. I would simply outlaw
sv and sv. You have sv as the complement of ww. I don't think we
actually need a complement for ww. Do we?
The above implies that
sv + x = 113.685
which implies
v + sx = 113.685
another complement that we agree on.
Of course there's no question that
s + s = 113.685
Another equivalent pair that we agree on is
v + xw = 113.685
vw + x = 113.685
with its 0.14 cent offset.
I set up a speadsheet that allows one to enter these equations and then
solves them for the size of each flag in cents. I've made the value of the
second shaft (the difference between ss and ss) a free variable too,
which is the equivalent of allowing the top and bottom parts of my
complement.bmp diagrams to slide against each other to minimise offsets.
Based on the solution for all the flags, I calculate the errors in all the
commas (including some alternate symbols for 23', 31 and 37') and find the
maximum error over all of them.
I find that with all the complement equations I want, I still have two
degrees of freedom left over. I use these to specify the values in cents of
the 5 and 7 comm flags. I adjust these to minimise the maximumabsolute
error. I found a set of complement relationships, which is a mixture of
yours and my earlier ones, that lets me get the maximum error in any comma
(i.e. sum of flag values minus comma value) down to 1.12 cents. This
includes every comma up to 41 and the ones for 11/7 and 7/5.
The best I can do with your complements is a max error of 2.0 cents. It
makes sense that the max error would be half the max offset.
Unfortunately mine requires that the complement of w has 3 flags, xvv.
This is a consequence of the complement of wv being xv. I'm hoping you
can find a set of complements that either have a lower max error, or a
similar max error without needing a 3 flag symbol as the complement of a
oneflag symbol, but it doesn't look too hopeful.
Here's the system I'm talking about. I've put an asterisk against those
that differ from yours.
symbol complement comma offset

v xw 19 0.14 cents
v sx (17'17) 1.50
* w xvv 17
vv none
* wv xv 17'
w wx 23 0.73
vw x 19' 0.14
s s 5 0.00
* ww none
x x 7 1.26
* sv not used equiv to x and so not used
vx none
s s (115) 0.00
x vw 29 0.14
vs none 31
wx w alt 31 0.73
* sw none
* xv wv 23' was alt 23'
* ws none was 23'
*ss vv 25
sx v 13 1.50 alt 37' (replacing 3flag symbol)
xw v 0.14
>You passed up some nice smalloffset complements that are 665
>inconsistent. The only ones that I was forced to pass up in 217ET
>have an offset of over 2.6 cents. And you can see that 653ET also
>has a number of inconsistent complements. This is due in large part
>to the fact that 653 and 665 are much finer divisions, so this is not
>surprising.
Yes. Consistency is irrelevant here. It's the offsets (or errors) in cents
that matter.
>However, this is a good reason not to base rational complementation
>on a particular division of the octave, but rather on the basis of a
>small offset.
Agreed.
The spreadsheets I used for solving the two sets of equations are at
/tuningmath/files/Dave/DKCompSolve.xls
/tuningmath/files/Dave/GSCompSolve.xls
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
I wrote:
"Consistency is irrelevant here. It's the offsets (or errors) in cents that
matter."
Of course I meant nlimit consistency of ETs is irrelevant here. I think
consistency of the system of linear equations representing the complement
rules is very important.
Another problem I have with your proposal is a crossover of symbol sizes
between single and double shafts when one includes the obvious complement
for the 25 comma symbol. It is related to the 4 cent offset. Assuming we
take the complement of ss to be vv, and you have w complement is ws,
then in order of increasing size (of commas represented, not of solutions
to equations) we have
w vv ... ws ss
but in order of increasing size, the complements go
vv w ... ss ws
Maybe what this is trying to tell us is that we should consider making ss
the complement of w (17 comma), and w the complement of ss (25 comma).
When I substitute that for the w rule in your system I can get the max
error down to 1.21 cents, provided I use wx for 31 and sx for 37'.
I've put up the spreadsheet as
/tuningmath/files/Dave/GS2CompSolve.xls
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
I realised that the system of my previous message is no good because it
didn't give a complement for the 23' comma with either the standard or
alternate symbol. But now I think I've cracked it.
I've found a system where every comma that needs a complement has one, and
no new symbols are required, and the maximum error is 1.23 cents according
to my spreadsheet. The maximum offset is 1.53 cents according to your
spreadsheet. The system happens to be consistent with 494ET. You can see
it in
/tuningmath/files/Dave/DK2Compls.xls
and
/tuningmath/files/Dave/DK2CompSolve.xls
The 23' comma symbol is now xv, not ws, because ws has no
oneflagperside complement in this system. This involves a 0.52 cent
schisma. It also frees ws to be used as purely a 125diesis symbol if we
want (0.56 cent schisma).
The 31 comma symbol is now wx, not vs, and the 37' symbol is sx, not
vwx. These involve schismas of 0.73 cents and 0.88 cents respectively, but
I consider these a price worth paying for the reduced number of symbols and
the complete rational complements without any 3flag symbols.
As a bonus, all the symbols that do not have complements are not needed at
all. We could take ww and sv as complementary but I don't think we need
either of them. I believe we only need 20 singleshaft symbols and 16
doubleshaft symbols. None of these symbols have more than 2 flags and only
4 have two flags on the same side (the 25 and 31' symbols ss and sx and
the complements of the 17 and 31' symbols ss and xs).
Here it is. The differences from your most recent proposal are shown with
asterisks.
symbol complement comma offset (cents)

natural ss apotome 0.00
v xw 19 0.14
v sx (17'17) 1.50
* w ss 17 1.53
* wv xv 17' 1.03
w wx 23 0.73
vw x 19' 0.14
s s 5 0.00
x x 7 1.26
s s (115) 0.00
x vw 29 0.14
wx w 31 0.73
* xv wv 23' 1.03
*ss w 25 1.53
sx v 13 1.50
sx xs 13 1.50 alternative singleshaft complement
xw v 0.14
ss xx 11 0.00
sx sx 31' 0.00
In 494ET the flags correspond to the following numbers of steps.
v 1 2 v
w 4 7 w
s 9 11 x
x 14 13 s
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> I realised that the system of my previous message is no good
because it
> didn't give a complement for the 23' comma with either the standard
or
> alternate symbol. But now I think I've cracked it.
I just got this one when I was about to post my latest, so you can
look at my proposal and I'll look at yours, and we'll see where that
gets us.
I would like to see the complementation used in 217ET (and available
for use in other ET's) compatible with the rational complementation
scheme, i.e., if at all possible, all of the rational complements
would be valid in 217ET. It bothers me that (the way things are at
present) wL doesn't have a decent 217ET complement  as you noted,
the 4cent offset of wL+sR is excessive, and I want to do better than
that. But (as I believe you also indicated) I would prefer not to
have a 3flag complement in the 217ET (or any other ET) notation.
I wondered whether redefining one of the flags would help us to
accomplish this.
The following is a complementation scheme I determined by redefining
the wR flag as 2319:
symbol complement comma offset comments

v xvw 19 0.22 cents Complement requires 3 flags
v sx (17'17) 1.50
w xw 17 2.19 Better than before!
vv ss 0.88
w ws 2319 0.39 The new wR flag
wv xv 17' 1.03
vw wx 0.73
vvw x 19' 0.14 19' requires 3 flags
s s 5 0.00
ww vx 0.73
x x 7 1.26
sv sv 1.74 We probably won't need this
vx ww 0.73
s s (115) 0.00
x vvw ~29 0.22 Complement requires 3 flags
sw vw 0.57 Not usable in 217
vs vwv 31 0.02 Not usable in 217
wx vw 0.73
xv wv 1.03
ws w 0.39
ss vv 0.88
xw w 2.19
sx v 13 1.50
xvw v 0.22
I also did another speadsheet for this:
/tuning
math/files/secor/notation/GSComp2.xls
With this change, the 23 comma (the former wR flag) is available as
vw, and previous 2flag combinations using the wR flag could still
be achievable using a 3flag symbol. We would have to see whether
either of the two 3flag wR combinations that you presently have are
achievable by other means.
I also updated my file:
/tuning
math/files/secor/notation/Symbols3.bmp
in which I added some 3flag symbols at the far right (and also fixed
a few mistakes). I came to the conclusion that 4flag symbols (2
flags per side) would probably not be a good idea (too difficult to
read), but 3 flags are okay (provided, of course, that not all 3
flags are on the same side).
Two complements that formerly could be achieved with 2flag symbols
now require 3flag symbols, but these were not (and still would not
be) standard 217ET symbols. In fact, 217ET could still be done
with the standard symbols that we previously agreed on (since none of
them contained a wR flag), although two pairs of these would be _faux
complements_ that are consistent in 217ET (but are not rational
complements). Your previous complementation rules could then be
called "217ET faux complementation rules" for these standard
symbols, since we would probably desire to keep things as simple as
possible in 217.
At present both concave flags in 217 are one degree, and with the wR
flag as (2319), both wavy flags in 217 would be two degrees, which
would further simplify things in 217.
Do you see any problems with this proposal?
George
At 22:41 13/05/02 0000, you wrote:
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>I would like to see the complementation used in 217ET (and available
>for use in other ET's) compatible with the rational complementation
>scheme, i.e., if at all possible, all of the rational complements
>would be valid in 217ET.
You'll be pleased to know that my latest proposal has the above property.
However, it is not consistent with the plan B notation for 217ET that we
agreed on earlier. Nor is it consistent with plan A. In particular, my
proposal has no rational complement for ws (ws is no longer the 23'
symbol, xv is).
8 steps of 217ET would need to be notated as ss the 25 comma. I don't
have a problem with that since it involves a lower prime and still has only
2 flags (it's just that they unfortunately have to be on the same side
because they are the same flag).
Also, the 217ET 7 step symbol would need to become xv to agree with the
rational complement of wv. Alternatively the 3 step symbol could be
changed to w and the 7 step symbol could remain as wx. But the latter
pair represent higher primes and introduce one more lateral confusable. But
at least it doesn't introduce 2 more like the old plan A, and I still like
the idea of not having a doubleflag for 3 steps, when 4, 5 and 6 are
single flags. What do you think?
>It bothers me that (the way things are at
>present) wL doesn't have a decent 217ET complement  as you noted,
>the 4cent offset of wL+sR is excessive, and I want to do better than
>that.
I believe the answer is to use ss as the complement of w in 217ET (and
rationally).
>But (as I believe you also indicated) I would prefer not to
>have a 3flag complement in the 217ET (or any other ET) notation.
I totally agree about no 3flaggers in any ETs. What's more I don't want
any 3flaggers in the rational notation (including complements)!, now that
I know it is possible to do so with only a tiny increase in the 23'
schisma, and larger but still modest increases in the 31 and 37' schismas.
Having the Reinhard property hold up to the 29 limit is good enough for me.
>I wondered whether redefining one of the flags would help us to
>accomplish this.
Well it helped, but not enough. And I don't think it is necessary.
...
>Do you see any problems with this proposal?
Only that it needs symbols with 3 flags. I hope I have shown that this is
not necessary. But you should go over my proposal with a finetoothed comb.
I've managed to fool myself into believing that various schemes would work
so many times only to discover later that they wouldn't, that I no longer
trust my own checking.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
I've updated
/tuningmath/files/Dave/Complements.bmp
to show my latest proposal.
Assuming you find it acceptable, the next major job before we go public is
to agree on the notation of the important ETs. I made a first pass at this in
/tuningmath/message/4188
You might address that when you have time.
I think we should use for ETs only those symbols that are necessary for
rational tunings. i.e. we should not use vv ww sv vx vs sw ws. We
should also try to make the ET complements agree with the rational
complements where possible.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> I've updated
> /tuningmath/files/Dave/Complements.bmp
> to show my latest proposal.
>
> Assuming you find it acceptable, the next major job before we go public is
> to agree on the notation of the important ETs.
What about for the important temperaments?
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4242]:
> At 22:41 13/05/02 0000, you wrote:
> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >I would like to see the complementation used in 217ET (and
available
> >for use in other ET's) compatible with the rational
complementation
> >scheme, i.e., if at all possible, all of the rational complements
> >would be valid in 217ET.
>
> You'll be pleased to know that my latest proposal has the above
property.
That's terrific!
I made an updated version of your complementation worksheet:
/tuning
math/files/secor/notation/DKComp2.xls
I removed a few nolongerrelevant rows and also added some, mostly
at the bottom. I also, added (in column K) the number of degrees
corresponding to an apotome minus column B, which will help in
selecting symbol sets for various ET's.
> However, it is not consistent with the plan B notation for 217ET
that we
> agreed on earlier. Nor is it consistent with plan A.
Since we don't have the individual flagcomplement conversion rules
anymore, there's no point in being concerned about that; lowerror
rational complements are more important. Anyone using these will
just have to memorize them. There are really only 8 pairs, since
memorizing ab <> cd also gives you cd <> ab. (Actually
there are 4 more if you count nat. <> ss, sx <> xs, ss <>
xx, and sx <> sx, but these are fairy easy to remember.)
> In particular, my
> proposal has no rational complement for ws (ws is no longer the
23'
> symbol, xv is).
This presents a problem, the only one I have found so far with your
proposal. (I'm sorry to have to bring this up, because aside from
this, I really like what you have.)
The problem is that in 217 xv is 7 degrees, whereas the 23' comma is
8, which is why we originally chose ws for its symbol. (This is not
unique to 217  the same situation also occurs in both 311 and 494,
although those don't really matter for our purposes, since we aren't
notating them.)
Now it looks as if we will need a vv symbol for the complement of
ws. (That's consistent in 217, but not 311 or 494.) It depends on
how much we want to complicate the 217 notation to make it conform to
the rational notation. Allowing wL and wL+sR to be complements in
the 217 notation makes everything much simpler in that ET, and I
think this is one place where it just might be best to apply the
guideline that the versatility (i.e., complexity) of the rational
notation should not make the simpler 217ET notation more complicated.
> 8 steps of 217ET would need to be notated as ss the 25 comma. I
don't
> have a problem with that since it involves a lower prime and still
has only
> 2 flags (it's just that they unfortunately have to be on the same
side
> because they are the same flag).
And, unfortunately, that's one more complication. I'd like to
restrict two flags on the same side to the rational notation. That
being the case, the only possibility for 8deg217 would be ws.
> Also, the 217ET 7 step symbol would need to become xv to agree
with the
> rational complement of wv. Alternatively the 3 step symbol could be
> changed to w and the 7 step symbol could remain as wx. But the
latter
> pair represent higher primes and introduce one more lateral
confusable. But
> at least it doesn't introduce 2 more like the old plan A, and I
still like
> the idea of not having a doubleflag for 3 steps, when 4, 5 and 6
are
> single flags. What do you think?
This is the sequence that I favor:
217: v w w s x s wx ws sx ss xx xs w w
s x s wx ws sx ss
Except for w <> ws and ws <> w (to avoid two flags on the
same side for 8 & 19deg217), all of these are rational complements.
In fact, except for w and w, this is the same as the plan B
notation (with that nice sequence of twoflag symbols), and now that
wR is the complement of wL+xR, your argument for its use is a very
persuasive one. Another thing that I like about it is that, in the
sequence of the first five symbols, the flags alternate from one side
to the other, which will work to good effect in your adaptive JI
example (which would need to be updated).
> > It bothers me that (the way things are at
> > present) wL doesn't have a decent 217ET complement  as you
noted,
> > the 4cent offset of wL+sR is excessive, and I want to do better
than
> > that.
>
> I believe the answer is to use ss as the complement of w in 217
ET (and
> rationally).
>
> > But (as I believe you also indicated) I would prefer not to
> > have a 3flag complement in the 217ET (or any other ET) notation.
>
> I totally agree about no 3flaggers in any ETs. What's more I don't
want
> any 3flaggers in the rational notation (including complements)!,
now that
> I know it is possible to do so with only a tiny increase in the 23'
> schisma, and larger but still modest increases in the 31 and 37'
schismas.
> Having the Reinhard property hold up to the 29 limit is good enough
for me.
>
> > I wondered whether redefining one of the flags would help us to
> > accomplish this.
>
> Well it helped, but not enough. And I don't think it is necessary.
>
> ...
>
> > Do you see any problems with this proposal?
>
> Only that it needs symbols with 3 flags. I hope I have shown that
this is
> not necessary.
Yes, you have. It was just another possibility that I wanted to
check out before wrapping this up.
> But you should go over my proposal with a finetoothed comb.
And I found only the one problem with the 23' comma.
> I've managed to fool myself into believing that various schemes
would work
> so many times only to discover later that they wouldn't, that I no
longer
> trust my own checking.
True words of wisdom, and a good reason why one person working alone
would have been hard pressed to come up with this notation.
George
Gene,
Working up sagittal notations for the most important regular temperaments
is a great idea. I assume you are volunteering to investigate this and make
some proposals. I expect you will find the table below useful, but I don't
expect you will need to use any symbols beyond the 13primelimit (only
straight or convex flags).
It seems to me that the notation for a linear temperament should be the
same as that for some large ET that represents it well. e.g meantone same
as 31ET, miracle same as 72 ET. Maybe it should be the largest compatible
ET that we can notate without using any higher prime than is approximated
by the temperament. I'd like to see them notated as a chain of generators
centered on D natural.
George,
Regarding your suggestion of redefining the w flag as 23 comma  19 comma,
while I see no benefit in doing that, it made me realise that w could be
defined as (19'19) comma, (i.e. 722:729), in the same way that x can be
defined as (11'7). i.e. using the lowest prime limit possible.
I think you suggested that x should be defined as 11'7 instead of 29
comma, but you gave the ratio as 715:729 which is 13'(115). 11'7 is
45056:45927.
Similarly xv could be defined in lowest prime terms as (11'7)+(17'17)
(1441792:1474767) instead of 23' (16384:16767), and sx could be defined as
5+7 instead of 13. Ultimately everything could be defined in terms of 5 7
11 17 19, but what would be the point?
Maybe we should not define _any_ symbol as being a _single_ comma, but
adopt the attitude that what we've done is produce a bunch of symbols each
of which may be used for a number of different commas/dieses very close
together in cents. The user can say precisely which of these she means, but
the consequences of not doing so are almost insignificant.
Someone who wants to notate a strictly rational 29limit scale, and is
willing to use multiple symbols, will define x as the 29 comma and xs as
the 13' comma, while someone notating an 11limit temperament will define
x as 11'7 and xs as apotome(5+7).
Maybe we should produce a list that shows all the possible 19limit
interpretations of each of the 20 singleshaft symbols, plus the obvious
41limit interpretations. But mostly all you want to know are
(a) the simplest interpretation, i.e. the one that involves the fewest
primes, and
(b) any interpretations that have a lower prime limit than the simplest one.

ASCII symbols

I propose we start using the following more representational ASCII versions
of our symbols in place of the "svwx" versions (although I hope that later
we can agree on a singlecharacterpersymbol version).
I suggest we use slashes /\ for straight flags, parentheses () for convex
and concave, and tildes ~ for wavy. And so that we can tell which way a
symbol is pointing when it has no straight flag, I suggest we use
exclamation marks ! for the shafts of downpointing arrows.
So here I list all 20 singleshaft upsymbols with their rational apotome
complements (not yet approved by George) and their 31limit comma names and
ratios.
Symbol Complement Comma names Ratios

// /\ natural 1:1
) (~ 19 512:513
( /) (17'17) 288:289
~ // 17 2176:2187
~( (( 17' 4096:4131
~ ~) 23 or (19'19) 729:736 or 722:729
)~ ( 19' 19456:19683
/ \ 5 80:81
) ) 7 63:64
\ / (115) 54:55
( )~ 29 or (11'7) 256:261 or 45056:45927
or (13'11+5) or 715:729
~) ~ 31 or (7+17) 243:248 or 238:243
(( ~( 23' * 729:736 *
// ~ 25 6400:6561
/) ( or (\ 13 or (5+7) 1024:1053 or 35:36
(~ ) (29+23) * 648:667 *
/\ () 11 32:33
(/ \) 31' * 31:32 *
\) (/ (7+115) 1701:1760
() /\ 11' 704:729
(\ /) 13' 26:27
* Too many lower prime interpretations and of too little interest to list.
You can see the bitmap version of the above symbols (and others which we
may not use) in
/tuningmath/files/secor/notation/Symbols3.bmp
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4242]:
>> However, it is not consistent with the plan B notation for 217ET
>that we
>> agreed on earlier. Nor is it consistent with plan A.
>
>Since we don't have the individual flagcomplement conversion rules
>anymore, there's no point in being concerned about that; lowerror
>rational complements are more important. Anyone using these will
>just have to memorize them. There are really only 8 pairs, since
>memorizing ab <> cd also gives you cd <> ab. (Actually
>there are 4 more if you count nat. <> ss, sx <> xs, ss <>
>xx, and sx <> sx, but these are fairy easy to remember.)
Agreed.
>> In particular, my
>> proposal has no rational complement for ws (ws is no longer the
>23'
>> symbol, xv is).
>
>This presents a problem, the only one I have found so far with your
>proposal. (I'm sorry to have to bring this up, because aside from
>this, I really like what you have.)
>
>The problem is that in 217 xv is 7 degrees, whereas the 23' comma is
>8, which is why we originally chose ws for its symbol. (This is not
>unique to 217  the same situation also occurs in both 311 and 494,
>although those don't really matter for our purposes, since we aren't
>notating them.)
I think the solution is easy. 217ET is not in fact 23limit consistent,
right? So if we use any possibly 23 comma symbol w xv ws, i.e. ~ ((
~\, to notate it, then we just consider them to be (19'19),
(11'7)+(17'17), (115)+17, just as we will not consider wx ~) as 31, or
x ) as 29.
So then it doesn't matter a whit to 217ET, which of those symbols is the
23' comma in the rational system. We are then free to choose xv as the 23'
symbol because it fits into a rational complement scheme with no triple
flags and low errors.
>Now it looks as if we will need a vv symbol for the complement of
>ws.
I hope I've dispensed with that.
>(That's consistent in 217, but not 311 or 494.) It depends on
>how much we want to complicate the 217 notation to make it conform to
>the rational notation.
Not at all.
>Allowing wL and wL+sR to be complements in
>the 217 notation makes everything much simpler in that ET, and I
>think this is one place where it just might be best to apply the
>guideline that the versatility (i.e., complexity) of the rational
>notation should not make the simpler 217ET notation more complicated.
Agreed.
>> 8 steps of 217ET would need to be notated as ss the 25 comma. I
>don't
>> have a problem with that since it involves a lower prime and still
>has only
>> 2 flags (it's just that they unfortunately have to be on the same
>side
>> because they are the same flag).
>
>And, unfortunately, that's one more complication. I'd like to
>restrict two flags on the same side to the rational notation. That
>being the case, the only possibility for 8deg217 would be ws.
OK.
>> Also, the 217ET 7 step symbol would need to become xv to agree
>with the
>> rational complement of wv. Alternatively the 3 step symbol could be
>> changed to w and the 7 step symbol could remain as wx. But the
>latter
>> pair represent higher primes and introduce one more lateral
>confusable. But
>> at least it doesn't introduce 2 more like the old plan A, and I
>still like
>> the idea of not having a doubleflag for 3 steps, when 4, 5 and 6
>are
>> single flags. What do you think?
>
>This is the sequence that I favor:
>
>217: v w w s x s wx ws sx ss xx xs w w
>s x s wx ws sx ss
Fine. Let's go with that, AND keep the rational system as I've described
it. There are bound to be _many_ ETs where the complements _cannot_ agree
with the rational ones.
In this case they can, but we choose not to make them so, for other
reasons. I think that in a case like this, where the symbol we pass over
actually relates to lower primes, we should tell the user that this option
exists. It may make more sense to use a double 5comma in a situation such
as a major third where the root already has a 5comma down.
I've updated
/tuningmath/files/Dave/AdaptiveJI.bmp
>Except for w <> ws and ws <> w (to avoid two flags on the
>same side for 8 & 19deg217), all of these are rational complements.
>In fact, except for w and w, this is the same as the plan B
>notation (with that nice sequence of twoflag symbols), and now that
>wR is the complement of wL+xR, your argument for its use is a very
>persuasive one. Another thing that I like about it is that, in the
>sequence of the first five symbols, the flags alternate from one side
>to the other, which will work to good effect in your adaptive JI
>example (which would need to be updated).
Neat.
>> I've managed to fool myself into believing that various schemes
>would work
>> so many times only to discover later that they wouldn't, that I no
>longer
>> trust my own checking.
>
>True words of wisdom, and a good reason why one person working alone
>would have been hard pressed to come up with this notation.
Agreed.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>
> George,
>
> Regarding your suggestion of redefining the w flag as 23 comma 
19 comma,
> while I see no benefit in doing that, it made me realise that w
could be
> defined as (19'19) comma, (i.e. 722:729), in the same way that x
can be
> defined as (11'7). i.e. using the lowest prime limit possible.
>
> I think you suggested that x should be defined as 11'7 instead of
29
> comma, but you gave the ratio as 715:729 which is 13'(115). 11'7
is
> 45056:45927.
Sorry. I forgot about the 4095:4096 schisma (which needs a
distinctive name of some sort). I meant to say that we should
redefine the xL flag as the 13'(115) comma (which is how I've been
calculating everything involving that flag up to this point), and if
that's a bit unwieldy, then we could call it the ~29 comma (if we can
figure out how to pronounce "~" (how about "quasi").
> Similarly xv could be defined in lowest prime terms as (11'7)+
(17'17)
> (1441792:1474767) instead of 23' (16384:16767), and sx could be
defined as
> 5+7 instead of 13. Ultimately everything could be defined in terms
of 5 7
> 11 17 19, but what would be the point?
>
> Maybe we should not define _any_ symbol as being a _single_ comma,
but
> adopt the attitude that what we've done is produce a bunch of
symbols each
> of which may be used for a number of different commas/dieses very
close
> together in cents. The user can say precisely which of these she
means, but
> the consequences of not doing so are almost insignificant.
That is very possibly what may need to be done in order to notate
some troublesome ET's. Allow me to quote from an earlier message of
yours (#4009):
<< When we look at the ETs where 5comma + 7comma =/= 13comma
(among those we intend to notate) we find in most cases that we only
need to use two of the 3 commas in notating the ET. e.g. In 27ET the
7comma vanishes and we use the 5 and 13 comma symbols. In 50ET the
5comma vanishes and we use the 7 and 13 comma symbols.
37ET is a case I'm not too sure about. Here we have the 5comma
being 2 steps, the 13 comma being 3 steps and the 7comma vanishing.
There is no prime comma within the 41limit that is consistently
equal to 1 step (11comma is 2 steps, same as 5comma). We could
notate 1 step as 13comma up and 5comma down, but if we insist on
single symbols, is it ok to use the 7comma symbol to mean 13comma 
5 comma? Or should we use the 19comma symbol for one step, even
though it's 1,3,pinconsistent? >>
I suggest that 37ET be notated as a subset of 111ET, with the
latter having a symbol sequence as follows:
111: w, s, s, ws, ss, xs, w, s, s, ws, ss.
However, a more difficult problem is posed by 74ET, and the idea of
having redefinable symbols may be the only way to handle situations
such as this. Should we do that, then there should probably be
standard (i.e., default) ratios for the flags, and the specific
conditions under which redefined ratios are to be used should be
identified.
> Someone who wants to notate a strictly rational 29limit scale, and
is
> willing to use multiple symbols, will define x as the 29 comma and
xs as
> the 13' comma, while someone notating an 11limit temperament will
define
> x as 11'7 and xs as apotome(5+7).
>
> Maybe we should produce a list that shows all the possible 19limit
> interpretations of each of the 20 singleshaft symbols, plus the
obvious
> 41limit interpretations. But mostly all you want to know are
> (a) the simplest interpretation, i.e. the one that involves the
fewest
> primes, and
> (b) any interpretations that have a lower prime limit than the
simplest one.
I think that we'll get more of a feel for this once we start trying
to determine symbol sequences for various ET's.
There is something else I would like to quote from one of your
earlier messages (#4188) that demonstrates a point that I would like
to make:
<< [GS:] By the way, it's been bugging me that we've yet to agree on
the
spelling of confusable vs. confusible. I finally looked up the 
able (etc.)
[DK:] Unfortunately I find "confusability" and not "confusibility"
in my Shorter Oxford. >>
I guess we should consider the English to be best authority on how to
spell English words and settle on "confusability". (Besides, even if
this is merely a difference between English vs. American usage, since
you were the one who first used the word in the present discussion,
your preference should then take precedence.)
<< [GS:] Then I think that we should decide on standard (or
preferred) sets of symbols for as many ET's as we can before doing
this [taking the notation to the main tuning list].
[DK:] What would be even better is, after doing a few very different
ones the hard way, and therefore thinking about what the issues are,
if we could simply give an algorithm for choosing the notation for
any ET. >>
I tried selecting sets of symbols (including complements) for a
number of ET's and came to the conclusion that it is not all that
obvious what is best. Among the possible objectives I identified are:
1) Consistent symbol arithmetic (a top priority);
2) A matching symbol sequence in the halfapotomes;
3) Choose flags that represent the lower prime numbers;
4) Try not to use too many different types of flags;
5) Use rational complements where possible.
In the same way that a difference of opinion occurs among experts or
authorities in the matter of English spelling (as with the
word "confusability"), a problem could result when different
composers, using the same rules and guidelines, arrive at different
sets of symbols for the same ET. Some composers won't want to use
sagittal notation if in involves puzzling with how to notate an ET
and uncertainty about the suitability of the outcome, say if, after
composing a piece in a certain ET, it turns out that others were
already using a different set of symbols.
I suspect that, in order for us to figure out how the rules should be
applied, we'll have to do all of the ET's anyway. So why not just do
as many as possible and include the symbol sequences along with the
specifications of the notation?
Notice that in doing 111 (above), I found that giving objectives 2
and 4 a higher priority than objective 5 gave me the simplest
notation.
One thing that I thought should be taken into consideration is that,
where appropriate, ET's that are subsets of others should make use of
a subset of symbols of the larger ET. This would especially be
advisable for ET's under 100 that are multiples of 12  if you learn
48ET, you have already learned half of 96ET.
I previously did symbol sets for about 20 different ET's, but that
was before the latest rational complements were determined, so I'll
have to review all of those to see what I would now do differently.
> 
> ASCII symbols
> 
> I propose we start using the following more representational ASCII
versions
> of our symbols in place of the "svwx" versions (although I hope
that later
> we can agree on a singlecharacterpersymbol version).
Okay, we'll try it and see how it works out.
George
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> It seems to me that the notation for a linear temperament should be
the
> same as that for some large ET that represents it well. e.g
meantone same
> as 31ET, miracle same as 72 ET.
hmm . . . is there going to be transparancy for cases like 76equal,
which gives you so many good lineartemperament systems within it 
can the notation show that? how about 152equal, in which most of the
lineartemperament systems use different approximations to the primes
from what 152equal as a whole would suggest?
 In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
>
> > It seems to me that the notation for a linear temperament should
be
> the
> > same as that for some large ET that represents it well. e.g
> meantone same
> > as 31ET, miracle same as 72 ET.
>
> hmm . . . is there going to be transparancy for cases like 76equal,
> which gives you so many good lineartemperament systems within it 
> can the notation show that? how about 152equal, in which most of
the
> lineartemperament systems use different approximations to the
primes
> from what 152equal as a whole would suggest?
No. The whole basis of the notation is the chain of approximate
fifths. If two temperaments available within a single ET use different
sized fifths then how could they possibly be covered by a single
notation for the ET.
You have already seen, in your adaptive JI example, how 31ET
_notation_ cannot continue to exist within 217ET, despite the fact
that 31ET exists within it. The quartercommas become explicit
instead of implicit. In exactly the same way, the 1/3 commas must
become explicit in the notation for 152ET.
The native bestfifth of 76ET is not suitable to be used a notational
fifth because, among other reasons, it is not 1,3,9consistent (i.e.
its best 4:9 is not obtained by stacking two of its best 2:3s) and I
figure folks have a right to expect C:D to be a best 4:9 when commas
for primes greater than 9 are used in the notation. So 76ET will be
notated as every second note of 152ET.
Here's my proposal for 152ET.
Steps Symbol

1 )
2 ~
3 /
4 \
5 ~)
6 (~
7 /\
11 B:C, E:F
15 #
26 A:B, C:D, D:E, F:G, G:A
Although it seems a minor problem that the 1/3 comma symbol of 152ET
is smaller in rational terms than the 1/4 comma symbol of 217ET.
We'll see what George comes up with for 152ET.
On second thoughts, here's my revised proposal for 152ET. There were
too many different flags in the previous one.
Steps Symbol

1 )
2 ~
3 /
4 \
5 /~
6 (~ or //
7 /\
11 B:C, E:F
15 #
26 A:B, C:D, D:E, F:G, G:A
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> Sorry. I forgot about the 4095:4096 schisma (which needs a
> distinctive name of some sort).
How about "the 13schisma" or the "tridecimal schisma".
> I meant to say that we should
> redefine the xL flag as the 13'(115) comma (which is how I've been
> calculating everything involving that flag up to this point), and if
> that's a bit unwieldy, then we could call it the ~29 comma (if we
can
> figure out how to pronounce "~" (how about "quasi").
"Quasi" is fine, but (11'7) is also a quasi29comma, so you can't
call (13'(115)) _the_ ~29 comma.
> I suggest that 37ET be notated as a subset of 111ET, with the
> latter having a symbol sequence as follows:
Yes. That's also what I suggested in a later message (4188).
> 111: w, s, s, ws, ss, xs, w, s, s, ws, ss.
And that's almost the notation I proposed in the same message (with
its implied complements), except that I would use xx () as the
complement of ss /\. Surely that is what you would want too, since
it represents a lower prime and is the rational complement?
> However, a more difficult problem is posed by 74ET, and the idea of
> having redefinable symbols may be the only way to handle situations
> such as this. Should we do that, then there should probably be
> standard (i.e., default) ratios for the flags, and the specific
> conditions under which redefined ratios are to be used should be
> identified.
I think 74ET is garbage.
But if someone insisted ... Due to its lack of 1,3,9 consistency and
the same going for 2*74 = 148, it would need to be notated as every
third note of 3*74 = 222ET which is itself garbage and we can't
notate it anyway. We have no 11 step symbol for it without two flags a
side.
I don't think we need to apologise for failing to notate 74ET. We can
do every ET up to 72 and many useful ones beyond.
> I think that we'll get more of a feel for this once we start trying
> to determine symbol sequences for various ET's.
Yes.
> I tried selecting sets of symbols (including complements) for a
> number of ET's and came to the conclusion that it is not all that
> obvious what is best.
I agree. We'll just have to do them all individually. I can't imagine
there being much disagreement on those using their native fifths until
we get up to 38ET. See my message 4188. The complements are implied
by them having the same sequence of flags in the second half apotome,
and the complement of /\ always being itself or ().
> Among the possible objectives I identified
are:
>
> 1) Consistent symbol arithmetic (a top priority);
>
> 2) A matching symbol sequence in the halfapotomes;
>
> 3) Choose flags that represent the lower prime numbers;
>
> 4) Try not to use too many different types of flags;
>
> 5) Use rational complements where possible.
That's an excellent list of (often conflicting) criteria.
> In the same way that a difference of opinion occurs among experts or
> authorities in the matter of English spelling (as with the
> word "confusability"), a problem could result when different
> composers, using the same rules and guidelines, arrive at different
> sets of symbols for the same ET. Some composers won't want to use
> sagittal notation if in involves puzzling with how to notate an ET
> and uncertainty about the suitability of the outcome, say if, after
> composing a piece in a certain ET, it turns out that others were
> already using a different set of symbols.
Yes. A very good point
> I suspect that, in order for us to figure out how the rules should
be
> applied, we'll have to do all of the ET's anyway. So why not just
do
> as many as possible and include the symbol sequences along with the
> specifications of the notation?
OK.
> Notice that in doing 111 (above), I found that giving objectives 2
> and 4 a higher priority than objective 5 gave me the simplest
> notation.
If you're talking about (\ as the complement of /\ then I must
disagree. In most ETs that use /\ its complement would be itself or
() so I think () should be exempt from the consideration of too many
flag types.
> One thing that I thought should be taken into consideration is that,
> where appropriate, ET's that are subsets of others should make use
of
> a subset of symbols of the larger ET. This would especially be
> advisable for ET's under 100 that are multiples of 12  if you
learn
> 48ET, you have already learned half of 96ET.
Certainly. It's only the question of how we tell "when appropriate"
that remains to be agreed. I've proposed two and only two reasons in
message 4188. You might say what you think of these.
> I previously did symbol sets for about 20 different ET's, but that
> was before the latest rational complements were determined, so I'll
> have to review all of those to see what I would now do differently.
Great!
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote
[#4272]:
>  In tuningmath@y..., "emotionaljourney22" <paul@s...> wrote:
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> >
> > > It seems to me that the notation for a linear temperament
should be the
> > > same as that for some large ET that represents it well. e.g
meantone same
> > > as 31ET, miracle same as 72 ET.
> >
> > hmm . . . is there going to be transparancy for cases like 76
equal,
> > which gives you so many good lineartemperament systems within
it 
> > can the notation show that? how about 152equal, in which most of
the
> > lineartemperament systems use different approximations to the
primes
> > from what 152equal as a whole would suggest?
>
> No. The whole basis of the notation is the chain of approximate
> fifths. ...
>
> Here's my proposal for 152ET.
>
[This has been deleted and replaced with:]
>
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote
[#4273]:
> On second thoughts, here's my revised proposal for 152ET. There
were
> too many different flags in the previous one.
>
> Steps Symbol
> 
> 1 )
> 2 ~
> 3 /
> 4 \
> 5 /~
> 6 (~ or //
> 7 /\
> 11 B:C, E:F
> 15 #
> 26 A:B, C:D, D:E, F:G, G:A
>
> Although it seems a minor problem that the 1/3 comma symbol of 152
ET
> is smaller in rational terms than the 1/4 comma symbol of 217ET.
> We'll see what George comes up with for 152ET.
Here's what I did a couple of weeks back, and after looking at the
rational complements, I would still do it this way:
Steps Symbol

1 (
2 ~
3 /
4 \
5 /~
6 /)
7 /\
8 ()
9 (\
10 ~
11 /
12 \
13 /~
14 /)
15 /\
Something that you will notice immediately is that I have used the
(17'17) comma as 1 degree. (In 152 it calculates to zero degrees
and would be unusable unless it were redefined as I have chosen to do
here.)
Dave, your solution also redefines a flag, although it is not so
obvious: since ~ is 2deg and (~ is 6deg, then ( must be 4deg.
This is so if it is calculated as the 29 comma, but it is 5deg if it
is calculated as the 715:729 comma, as I have done. (But this is not
the redefinition to which I refer.)
You didn't give symbols for 8 and 9deg, but if I assume that 8deg
would be (), so then ) would be 4deg. In 152 ) calculates to
3deg, which is where your redefinition occurs.
So the difference between our two solutions is that the flag that I
redefined is associated with a higher prime.
May I assume that you would use matching symbols for the apotome
complements? That being the case, we both chose a rational
complement for 1deg, but a 152specific complement for 2deg. I came
to the conclusion that a simple (i.e., easytoremember) sequence of
symbols is more important than using rational complements.
George
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote
[#4274]:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > Sorry. I forgot about the 4095:4096 schisma (which needs a
> > distinctive name of some sort).
>
> How about "the 13schisma" or the "tridecimal schisma".
That sounds good. We should probably propose that term on the main
tuning list, to see if anyone knows whether it has already been used
for a different schisma.
> > I meant to say that we should
> > redefine the xL flag as the 13'(115) comma (which is how I've
been
> > calculating everything involving that flag up to this point), and
if
> > that's a bit unwieldy, then we could call it the ~29 comma (if we
can
> > figure out how to pronounce "~" (how about "quasi").
>
> "Quasi" is fine, but (11'7) is also a quasi29comma, so you can't
> call (13'(115)) _the_ ~29 comma.
Since the ( flag is undoubtedly going to be used so much more often
in connection with ratios of 11 and 13  as () and (\  than for
ratios of 29, I would prefer to keep its standard definition as other
than 256:261 (the 29 comma). I would also prefer the 13'(115)
ratio to (11'7) because,
1) The numbers in the ratio are smaller (715:729 vs. 45056:45927); and
2) The 13'(115) comma (33.571 cents) is much closer in size to the
29 comma (33.487 cents) than is the (11'7) comma (33.148 cents).
> > I suggest that 37ET be notated as a subset of 111ET, with the
> > latter having a symbol sequence as follows:
>
> Yes. That's also what I suggested in a later message (4188).
>
> > 111: w, s, s, ws, ss, xs, w, s, s, ws, ss.
>
> And that's almost the notation I proposed in the same message (with
> its implied complements), except that I would use xx () as the
> complement of ss /\. Surely that is what you would want too,
since
> it represents a lower prime and is the rational complement?
I used xs (\ as 6deg111 because xx () calculates to 5deg111 and,
in addition, 26:27 is closer in size to 6deg111 than is 704:729.
However, if we think that there should be no problem in redefining
xx as 6deg111 (as it would seem to make more sense), then so be it!
> > However, a more difficult problem is posed by 74ET, and the idea
of
> > having redefinable symbols may be the only way to handle
situations
> > such as this. Should we do that, then there should probably be
> > standard (i.e., default) ratios for the flags, and the specific
> > conditions under which redefined ratios are to be used should be
> > identified.
>
> I think 74ET is garbage.
Be careful when you say something like that around here  do you
remember my "tuning scavengers" postings?
> But if someone insisted ... Due to its lack of 1,3,9 consistency
and
> the same going for 2*74 = 148, it would need to be notated as every
> third note of 3*74 = 222ET which is itself garbage and we can't
> notate it anyway. We have no 11 step symbol for it without two
flags a
> side.
>
> I don't think we need to apologise for failing to notate 74ET. We
can
> do every ET up to 72 and many useful ones beyond.
The problem is not the fault of the notation so much as the weirdness
of the division  I hesitate to call it a tonal system. Any
systematic notation is going to have problems with 74ET.
> > I think that we'll get more of a feel for this once we start
trying
> > to determine symbol sequences for various ET's.
>
> Yes.
>
> > I tried selecting sets of symbols (including complements) for a
> > number of ET's and came to the conclusion that it is not all that
> > obvious what is best.
>
> I agree.
Yes, especially since we've just had an object lesson with 111ET.
> We'll just have to do them all individually. I can't imagine
> there being much disagreement on those using their native fifths
until
> we get up to 38ET. See my message 4188. The complements are
implied
> by them having the same sequence of flags in the second half
apotome,
> and the complement of /\ always being itself or ().
We'll each work them out then and compare notes.
> > Notice that in doing 111 (above), I found that giving objectives
2
> > and 4 a higher priority than objective 5 gave me the simplest
> > notation.
>
> If you're talking about (\ as the complement of /\ then I must
> disagree. In most ETs that use /\ its complement would be itself
or
> () so I think () should be exempt from the consideration of too
many
> flag types.
Yes, I agree with that. You'll notice that this wasn't among the
reasons I gave above.
Would you also now prefer my selection of the /) symbol for 6deg111
to your choice of (~ on the grounds that it is a more commonly used
symbol, particularly in view of the probability that you might want
to use (\ instead of ) or ( for 9deg as its complement?
> > One thing that I thought should be taken into consideration is
that,
> > where appropriate, ET's that are subsets of others should make
use of
> > a subset of symbols of the larger ET. This would especially be
> > advisable for ET's under 100 that are multiples of 12  if you
learn
> > 48ET, you have already learned half of 96ET.
>
> Certainly. It's only the question of how we tell "when appropriate"
> that remains to be agreed. I've proposed two and only two reasons
in
> message 4188. You might say what you think of these.
They sound reasonable enough. Until I thought of 7ET, which seems
to be a "natural" for the 7 naturals. Of course, a simple way around
that is to put the modifying symbols from 56ET into a key signature,
a solution that would keep the manuscript clean and make everybody
happy.
> > I previously did symbol sets for about 20 different ET's, but
that
> > was before the latest rational complements were determined, so
I'll
> > have to review all of those to see what I would now do
differently.
Here's what I did a couple of weeks ago for some of the ET's (in
order of increasing complexity):
12, 19, 26: ss
17, 24, 31: ss ss
22: s s ss
36, 43: x x ss
29: wx wv ss
50: ww xs ss
34, 41: s ss s ss
27: s xs s ss
48: x ss x ss
46, 53: s ss xx s ss
58, 72: s s ss s s ss (version 1  simpler, but more
confusability)
72: s x ss x s ss (version 2  more complicated, but
less confusability)
58: s wx ss wv s ss (version 2  more complicated,
but less confusability)
96: s x s ss s x s ss (version 1  simpler, but
more confusability)
96: s x ws ss w x s ss (version 2  more
complicated, but less confusability)
94: w s ws ss xx w s ws ss
111 (37 as subset): w s s ws ss xs w s s ws
ss
140: v w s s sw sx ss xs w s s sw sx
ss
152: v w s s sw sx ss xx xs w s s sw
sx ss
171: v wv s x s ws sx ss xs wv s x s
ws sx ss
183: v wv s x s ws sx ss xx xs wv s x
s ws sx ss
181: v w wv s s wx ws sx ss xx w wv s
s wx ws sx ss
217: v w w s x s wx ws sx ss xx xs w w
s x s wx ws sx ss
I changed 217 to conform to our new standard set.
As I said, I may want to change some of these in light of the new
rational complements and to remedy (if possible) an inconsistency in
symbol arithmetic that may be lurking somewhere.
You will want to compare some of these with what you have in your
message 4188. We did not even agree on something as simple as 31ET.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote [#4283]:
>  In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote
> [#4274]:
> >  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > > I tried selecting sets of symbols (including complements) for a
> > > number of ET's and came to the conclusion that it is not all
that
> > > obvious what is best.
> >
> > I agree.
>
> Yes, especially since we've just had an object lesson with 111ET.
I meant to say 152ET.
> ...
> Would you also now prefer my selection of the /) symbol for
6deg111
> to your choice of (~ on the grounds that it is a more commonly
used
> symbol, particularly in view of the probability that you might want
> to use (\ instead of ) or ( for 9deg as its complement?
Here, again, I meant to say 6deg152.
Sorry if I caused any confusion.
George
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>Here's what I did [for 152ET] a couple of weeks back, and after looking
at the
>rational complements, I would still do it this way:
>
>Steps Symbol
>
> 1 (
> 2 ~
> 3 /
> 4 \
> 5 /~
> 6 /)
> 7 /\
> 8 ()
> 9 (\
>10 ~
>11 /
>12 \
>13 /~
>14 /)
>15 /\
>
>Something that you will notice immediately is that I have used the
>(17'17) comma as 1 degree. (In 152 it calculates to zero degrees
>and would be unusable unless it were redefined as I have chosen to do
>here.)
But redefined it as what comma? I believe a fundamental tenet of this whole
excercise, one that many people agree on, is that an accidental must never
simply represent a number of steps of the ET, but must represent a rational
comma in a manner consistent with the ETs best approximation of the primes
involved.
>Dave, your solution also redefines a flag, although it is not so
>obvious: since ~ is 2deg and (~ is 6deg, then ( must be 4deg.
>This is so if it is calculated as the 29 comma, but it is 5deg if it
>is calculated as the 715:729 comma, as I have done. (But this is not
>the redefinition to which I refer.)
That was unintentional. Thanks for spotting it. I now agree that using (~
for 6 steps is completely wrong. It corresponds to 7 steps (but should not
be used). The only possible symbols for 6 thru 9 steps are the ones you
have given. And there's no question about 3 and 4 steps either. (And
therefore also 11, 12, 14, 15).
I've (re)realised that there is no need to go beyond the 19primelimit
for notating any ETs (that we _can_ notate). So, for notating ETs, all the
symbols must be given their 19limit definitions. The fact that some of
them might also be 23, 29 or 31 commas, when used for rational scales, is
utterly irrelevant (for notating ETs). Then the only remaining ambiguity is
the one involving the 13schisma.
For notating ETs:
v is always 512:513
v is always 288:289
w is always 2176:2187
w is always 722:729
s is always 80:81
s is always 54:55
x is always 63:64
but
x can be either 45056:45927 or 715:729
In some ETs these will be the same number of steps and the choice doesn't
need to be made. But when the choice _is_ made, the meaning of (( (~ (\
and () all follow automatically from it.
I now believe that the rational complements beyond the 19limit (i.e. if
either of the pair is outside the 19limit in the rational conception) are
very unimportant for notating ETs, and would only be used as a tiebreaker
if all else fails.
Here are the valid options for 1, 2 and 5 steps of 152ET, from a 19limit
perspective.
Steps Symbol Comma Comment

1 ) 19
1 )( 19 + (17'17)
2 ~ 17
2 ~( 17'
2 ~ 19'19
5 ( (11'7)
5 or 4 ( 13'(115) (5 steps for 1:13, 4 steps for 3:13)
5 )\ 19+(115)
5 ~) 17 + 7
5 /~ 5+(19'19)
5 (( (11'7)+(17'17)
5 or 4 (( (13'(115))+(17'17) (5 steps for 1:13, 4 steps for 3:13)
So we see that 152ET is not 1,3,13consistent.
I believe that, if for any prime p, the ET is not 1,3,pconsistent, then
commas involving that prime should not be used for notational purposes
unless there's no other option. I also think that, if possible, all
notational commas should be mutually consistent and consistent with 1,3 and
9. And if the ET can be notated by using more than one such set (unlikely),
then we should use the one with the lowest maximum error in its intervals.
So here's the information about 152ET that I find most relevant for
deciding the notation. These are its maximal consistent sets of odds in the
19limit, along with the maximum error of any interval in the set. By
maximally consistent I mean that no other 19limit odd number can be added
to a set without making it inconsistent.
{1, 13, 17} 3.7 c
{1, 3, 5, 9, 11, 15, 17} 3.7 c
{1, 3, 5, 7, 9, 11, 15, 19} 2.5 c
I haven't listed any sets that do not include 1, because
(a) I haven't computed them, and
(b) they would only be relevant if all else fails, which seems very unlikely.
The first set does not contain 3 or 9. The second set does not provide any
way of notating a single step. The third step is just right, and Goldilocks
ate it all up.
The third set works beautifully and happens to have the lowest error. It
says we shouldn't use any 13 or 17 commas, so our choice for 1, 2 and 5
steps is reduced to just these.
Steps Symbol Comma

1 ) 19
2 ~ 19'19
5 ( (11'7)
5 )\ 19+(115)
5 /~ 5+(19'19)
None of the choices for 5 introduce any new flags, but I consider the
introduction of a new flag prior to the halfapotome to be nearly as bad.
So on that basis I reject (. Also it seems like it is good to have more
equal numbers of single and doubleflag symbols.
They are both 19limit. None of them are the rational complement of ~. I
choose /~ because its size in a rational tuning is closer to 5/152 octave
than is )\.
So here's the full set for 152ET.
1 )
2 ~
3 /
4 \
5 /~
6 /)
7 /\
8 ()
9 (\ or ) ?
10 ~
11 /
12 \
13 /~
14 /)
15 /\
We now only disagree on the 1 step symbol.
>You didn't give symbols for 8 and 9deg, but if I assume that 8deg
>would be (), so then ) would be 4deg. In 152 ) calculates to
>3deg, which is where your redefinition occurs.
As I say, this wasn't an intentional redefinition, it was just dumb.
>So the difference between our two solutions is that the flag that I
>redefined is associated with a higher prime.
There is no need to redefine any, for 152ET. And only ever a need to
redefine x.
>May I assume that you would use matching symbols for the apotome
>complements?
Yes.
> That being the case, we both chose a rational
>complement for 1deg, but a 152specific complement for 2deg.
Now I choose no rational complement for either of them. It seems that the
_only_ justification for using v for 1 step is that it is the rational
complement of sx. To my mind this is not sufficient justification to
violate the definition of v as the 17 comma 2176:2187. In fact I don't
think anything could be sufficient justification for that.
>I came
>to the conclusion that a simple (i.e., easytoremember) sequence of
>symbols is more important than using rational complements.
So did I. And I came to the conclusion that 19limitcomma flagdefinitions
are more important than using rational complements.
I don't see how changing 1 step from ) to ( improves ease of remembering.
In fact with ) we have that property that you admired in 217ET, that the
flags alternate sides as you go up.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> The problem is not the fault of the notation so much as the weirdness
> of the division  I hesitate to call it a tonal system. Any
> systematic notation is going to have problems with 74ET.
Sharps and flats are a systematic notation, and since 74 is a meantone system, they would suffice.
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >Here's what I did [for 152ET] a couple of weeks back, and after
looking
> at the
> >rational complements, I would still do it this way: ...
>
> ... I've (re)realised that there is no need to go beyond the 19
primelimit
> for notating any ETs (that we _can_ notate). So, for notating ETs,
all the
> symbols must be given their 19limit definitions. ...
>
> For notating ETs:
> v is always 512:513
> v is always 288:289
> w is always 2176:2187
> w is always 722:729
> s is always 80:81
> s is always 54:55
> x is always 63:64
>
> but
>
> x can be either 45056:45927 or 715:729
>
> In some ETs these will be the same number of steps and the choice
doesn't
> need to be made. But when the choice _is_ made, the meaning of ((
(~ (\
> and () all follow automatically from it.
>
> I now believe that the rational complements beyond the 19limit
(i.e. if
> either of the pair is outside the 19limit in the rational
conception) are
> very unimportant for notating ETs, and would only be used as a tie
breaker
> if all else fails. ...
>
> So here's the full set for 152ET.
>
> 1 )
> 2 ~
> 3 /
> 4 \
> 5 /~
> 6 /)
> 7 /\
> 8 ()
> 9 (\ or ) ?
> 10 ~
> 11 /
> 12 \
> 13 /~
> 14 /)
> 15 /\
>
> We now only disagree on the 1 step symbol. ...
>
> I don't see how changing 1 step from ) to ( improves ease of
remembering.
> In fact with ) we have that property that you admired in 217ET,
that the
> flags alternate sides as you go up.
I'll give my conditional assent to what you have. I want to see how
all of this holds up once we do a lot more ET's.
I'm going to have to take some time (a week or so) away from the list
to catch up on some other things, but you can work on this, and I'll
see what you have when I get back.
George
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>> How about "the 13schisma" or the "tridecimal schisma".
>
>That sounds good. We should probably propose that term on the main
>tuning list, to see if anyone knows whether it has already been used
>for a different schisma.
Go ahead. I'm sure enough that it hasn't, that I can't be bothered.
>Since the ( flag is undoubtedly going to be used so much more often
>in connection with ratios of 11 and 13  as () and (\  than for
>ratios of 29, I would prefer to keep its standard definition as other
>than 256:261 (the 29 comma). I would also prefer the 13'(115)
>ratio to (11'7) because,
>
>1) The numbers in the ratio are smaller (715:729 vs. 45056:45927); and
>
>2) The 13'(115) comma (33.571 cents) is much closer in size to the
>29 comma (33.487 cents) than is the (11'7) comma (33.148 cents).
I say we can totally forget the 29 comma definition of ( for notating ETs.
But I think we need to decide, for every ET individually, whether x is
defined as 13'(115) or (11'7) (or both, when they are the same number of
steps).
>> > I suggest that 37ET be notated as a subset of 111ET, with the
>> > latter having a symbol sequence as follows:
>>
>> Yes. That's also what I suggested in a later message (4188).
>>
>> > 111: w, s, s, ws, ss, xs, w, s, s, ws, ss.
>>
>> And that's almost the notation I proposed in the same message (with
>> its implied complements), except that I would use xx () as the
>> complement of ss /\. Surely that is what you would want too,
>since
>> it represents a lower prime and is the rational complement?
>
>I used xs (\ as 6deg111 because xx () calculates to 5deg111 and,
>in addition, 26:27 is closer in size to 6deg111 than is 704:729.
>However, if we think that there should be no problem in redefining
>xx as 6deg111 (as it would seem to make more sense), then so be it!
(\ is only 6deg111 if you define ( as 13'(115), in which case you
should probably also use /) for 5deg111 instead of /\. In this case /)
is defined as the 13 comma, not 5+7 comma. This is something else that we
need to define on an ET by ET basis, whether ) is the 7 comma or the 135
comma. If we favout 7 over (135) in 111ET then we probably shouldn't use
any commas involving 13, and should therefore define ( as (11'7). In this
case we have /\ for 5deg111 and () for 6deg111.
>> > However, a more difficult problem is posed by 74ET, and the idea
>of
>> > having redefinable symbols may be the only way to handle
>situations
>> > such as this. Should we do that, then there should probably be
>> > standard (i.e., default) ratios for the flags, and the specific
>> > conditions under which redefined ratios are to be used should be
>> > identified.
>>
>> I think 74ET is garbage.
>
>Be careful when you say something like that around here  do you
>remember my "tuning scavengers" postings?
Yes, I remember. That's why I said it. So I'd get corrected as quickly as
possible if it _wasn't_ garbage. :) It isn't. See the topic "74EDO
challenge" on the main tuning list.
>The problem is not the fault of the notation so much as the weirdness
>of the division  I hesitate to call it a tonal system. Any
>systematic notation is going to have problems with 74ET.
Here's my proposal for notating 74ET using its native fifth (since it's a
meantone), despite the 1,3,9 inconsistency.
Steps Symbol Comma

1 )) 19+7
2 )\ 19+(115)
3 /\ 11
4 )\
5 /\
The ) flag actually has a value of 1 steps, but it never occurs alone, so
it doesn't really matter.
>Would you also now prefer my selection of the /) symbol for [6deg152]
>to your choice of (~ on the grounds that it is a more commonly used
>symbol, particularly in view of the probability that you might want
>to use (\ instead of ) or ( for 9deg as its complement?
Yes, but not on those grounds.
>> > One thing that I thought should be taken into consideration is
>that,
>> > where appropriate, ET's that are subsets of others should make
>use of
>> > a subset of symbols of the larger ET. This would especially be
>> > advisable for ET's under 100 that are multiples of 12  if you
>learn
>> > 48ET, you have already learned half of 96ET.
>>
>> Certainly. It's only the question of how we tell "when appropriate"
>> that remains to be agreed. I've proposed two and only two reasons
>in
>> message 4188. You might say what you think of these.
>
>They sound reasonable enough. Until I thought of 7ET, which seems
>to be a "natural" for the 7 naturals. Of course, a simple way around
>that is to put the modifying symbols from 56ET into a key signature,
>a solution that would keep the manuscript clean and make everybody
>happy.
A brilliant solution.
It's a pity the same thing won't work for 37ET as a subset of 111ET (or
will it?) because I know that some folks will prefer to notate it based on
its native best fifth.
The case of 74ET has shown me that my requirement of not using the native
fifth if it is 1,3,9inconsistent, unless we don't use any flags for any
prime greater than 9, may need to be relaxed in some cases.
>> > I previously did symbol sets for about 20 different ET's, but
>that
>> > was before the latest rational complements were determined, so
>I'll
>> > have to review all of those to see what I would now do
>differently.
>
>Here's what I did a couple of weeks ago for some of the ET's (in
>order of increasing complexity):
>
>12, 19, 26: ss
Agreed. My 19 and 26 were wrong.
>17, 24, 31: ss ss
17 and 24 agreed. I guess you want () for 1deg31 because it is closer in
cents than ), but I feel folks are more interested in its approximations
of 7, than 11.
>22: s s ss
I agree, but how come you didn't want ss for 1deg22? It's also arguable
that it could be s s ss, making the second halfapotome follow the
same pattern of flags as the first, but what you've got makes more sense to
me.
>36, 43: x x ss
Agreed for 36. But I wanted a singleshaft symbol for 2deg43 so it is
possible to notate it with monotonic letter names and without doubleshaft
symbols when using a notation that combines standard sharp and flat symbols
with sagittals. One could use either /\ or (\. e.g I want to be able to
notate the steps between B and C as B), B/\ or B(\, C/\ or C(\, C!).
>29: wx wv ss
Why wouldn't you use the same notation as for 22ET? There's no need to
bring in primes higher than 5.
>50: ww xs ss
For 50ET, {1, 3, 5, 7, 9, 13, 15, 17, 19} is the maximal consistent
(19limit) set containing 1,3,9. So I like xs for 2 steps (as 13'), and if
it's OK here, why not also in 43ET? But ww as 17+(19'19) is actually 1
steps of 50ET.
The only options for 1deg50, that don't involve 11 are )) as 19+7 or ~)
as 7+17 and /) as 13. /) seems the obvious choice to me.
>34, 41: s ss s ss
Agreed.
>27: s xs s ss
Why do you prefer (\ to /)?
>48: x ss x ss
In 48ET, {1, 3, 7, 9, 11} has only slightly lower errors than {1, 3, 5, 9,
11}, 10 cents versus 11 cents. Why prefer the above to the lower prime scheme
48: / /s \ /\ ?
>46, 53: s ss xx s ss
Agreed.
>58, 72: s s ss s s ss (version 1  simpler, but more
>confusability)
>72: s x ss x s ss (version 2  more complicated, but
>less confusability)
Of course, I prefer version 2 for 72ET, since I started the whole
confusability thing. It isn't significantly more complicated.
>58: s wx ss wv s ss (version 2  more complicated,
>but less confusability)
I'm inclined to go with version 1 despite the increased lateral
confusability, rather than introduce 17flags. Version 2 is a _lot_ more
complicated.
>96: s x s ss s x s ss (version 1  simpler, but
>more confusability)
>96: s x ws ss w x s ss (version 2  more
>complicated, but less confusability)
The only maximal 1,3,9consistent 19limit set for 96ET is {1, 3, 5, 9,
11, 13, 15, 17}. It is not 1,3,7consistent so the ) flag should be
defined as the 135 comma (64:65) if it's used at all. The 17 and 19 commas
vanish, so we should avoid ) ( ~ and ~. So I end up with
96: / ) /) /\ / ) /) /\
Simple _and_ non confusable.
>94: w s ws ss xx w s ws ss
Why do you prefer that to
>94: ~ / ) /\ () ~ \ ) /\
Surely we're more interested in the 7comma than the 17+(115) comma.
Also, it makes sense that / + \ = /\, but it makes the second half
apotome have a different sequence of flags to the first. Which should we
use, / or \ ?
>111 (37 as subset): w s s ws ss xs w s s ws
>ss
Dealt with above. I'd prefer () for 6deg111.
>140: v w s s sw sx ss xs w s s sw sx
>ss
>152: v w s s sw sx ss xx xs w s s sw
>sx ss
Dealt with elsewhere. I see no reason to use ( which is really zero steps,
when ) is 1 step.
>171: v wv s x s ws sx ss xs wv s x s
>ws sx ss
Why not ~ for 1 step?
>183: v wv s x s ws sx ss xx xs wv s x
>s ws sx ss
Why not use w for 1deg183, being a simpler comma than v? 17 vs. 17'17.
>181: v w wv s s wx ws sx ss xx w wv s
>s wx ws sx ss
I don't see how ) can be 5deg181 or how /\ can be 9deg181. The only
symbol that can give 9deg181 with 19limit commas is (~. Here's my proposal.
181: ( ~ ~ / /( ( (( /) (~ (\ ~ ~ / /(
( (( /) /\
>217: v w w s x s wx ws sx ss xx xs w w
>s x s wx ws sx ss
Agreed.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4297]:
> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >> How about "the 13schisma" or the "tridecimal schisma".
> >
> >That sounds good. We should probably propose that term on the
main
> >tuning list, to see if anyone knows whether it has already been
used
> >for a different schisma.
>
> Go ahead. I'm sure enough that it hasn't, that I can't be bothered.
If you're sure about that, then I'll take your word for it: 4095:4096
gets the name "tridecimal schisma."
> >Since the ( flag is undoubtedly going to be used so much more
often
> >in connection with ratios of 11 and 13  as () and (\  than
for
> >ratios of 29, I would prefer to keep its standard definition as
other
> >than 256:261 (the 29 comma). I would also prefer the 13'(115)
> >ratio to (11'7) because,
> >
> >1) The numbers in the ratio are smaller (715:729 vs. 45056:45927);
and
> >
> >2) The 13'(115) comma (33.571 cents) is much closer in size to
the
> >29 comma (33.487 cents) than is the (11'7) comma (33.148 cents).
>
> I say we can totally forget the 29 comma definition of ( for
notating ETs.
> But I think we need to decide, for every ET individually, whether
x is
> defined as 13'(115) or (11'7) (or both, when they are the same
number of
> steps).
Agreed!
> >> > I suggest that 37ET be notated as a subset of 111ET, with
the
> >> > latter having a symbol sequence as follows:
> >>
> >> Yes. That's also what I suggested in a later message (4188).
> >>
> >> > 111: w, s, s, ws, ss, xs, w, s, s, ws, ss.
> >>
> >> And that's almost the notation I proposed in the same message
(with
> >> its implied complements), except that I would use xx () as the
> >> complement of ss /\. Surely that is what you would want too,
since
> >> it represents a lower prime and is the rational complement?
> >
> >I used xs (\ as 6deg111 because xx () calculates to 5deg111
and,
> >in addition, 26:27 is closer in size to 6deg111 than is 704:729.
> >However, if we think that there should be no problem in redefining
> >xx as 6deg111 (as it would seem to make more sense), then so be
it!
>
> (\ is only 6deg111 if you define ( as 13'(115), in which case
you
> should probably also use /) for 5deg111 instead of /\. In this
case /)
> is defined as the 13 comma, not 5+7 comma. This is something else
that we
> need to define on an ET by ET basis, whether ) is the 7 comma or
the 135
> comma. If we favout 7 over (135) in 111ET then we probably
shouldn't use
> any commas involving 13, and should therefore define ( as (11'7).
In this
> case we have /\ for 5deg111 and () for 6deg111.
Yes, your notation for 111 is best and is in agreement with my latest
choice. But I arrived at () as 6deg111 by keeping ) as the 7 comma
(of 2deg) and defining ( as the 11'7 comma of 4deg. Same
difference, I guess!
It's taken me a little time to appreciate the value of your proposal
for dual roles for ( and the 19'19 role of ~. However, I believe
that a dual role should be retained for ~ also; it is quite useful
as the 23 comma for notating 135, 147, 159, 198, and 224ET
(particularly 198).
> >> > However, a more difficult problem is posed by 74ET, and the
idea of
> >> > having redefinable symbols may be the only way to handle
situations
> >> > such as this. Should we do that, then there should probably
be
> >> > standard (i.e., default) ratios for the flags, and the
specific
> >> > conditions under which redefined ratios are to be used should
be
> >> > identified.
> >>
> >> I think 74ET is garbage.
> >
> >Be careful when you say something like that around here  do you
> >remember my "tuning scavengers" postings?
>
> Yes, I remember. That's why I said it. So I'd get corrected as
quickly as
> possible if it _wasn't_ garbage. :) It isn't. See the topic "74EDO
> challenge" on the main tuning list.
>
> >The problem is not the fault of the notation so much as the
weirdness
> >of the division  I hesitate to call it a tonal system. Any
> >systematic notation is going to have problems with 74ET.
>
> Here's my proposal for notating 74ET using its native fifth (since
it's a
> meantone), despite the 1,3,9 inconsistency.
> Steps Symbol Comma
> 
> 1 )) 19+7
> 2 )\ 19+(115)
> 3 /\ 11
> 4 )\
> 5 /\
>
> The ) flag actually has a value of 1 steps, but it never occurs
alone, so
> it doesn't really matter.
While I was away, I worked on the notation for a number of ET's. I
decided to tackle 74 on my own, since it seemed to be a challenge.
The solution I came up with minimizes the use of flags with non
positive values:
74: )) /) (\ /) /\
Your solution is simpler in that it uses fewer flags and has no
lateral confusability, so it would probably be preferable on that
basis. However, I mention below that I would rather not use /\ for
anything greater than half of /\ unless absolutely necessary. On
the other hand, the 11 factor is almost exact in 74, so it would be a
shame not to represent it in the notation.
So what do you think? (I'm not going to lose any sleep over this
one.)
> >Would you also now prefer my selection of the /) symbol for
[6deg152]
> >to your choice of (~ on the grounds that it is a more commonly
used
> >symbol, particularly in view of the probability that you might
want
> >to use (\ instead of ) or ( for 9deg as its complement?
>
> Yes, but not on those grounds.
Then we agree on the following (cf. below):
152: ) ~ / \ /~ /) /\ () (\ ~ /
\ /~ /) /\
> The case of 74ET has shown me that my requirement of not using the
native
> fifth if it is 1,3,9inconsistent, unless we don't use any flags
for any
> prime greater than 9, may need to be relaxed in some cases.
Perhaps "guideline" would be a better term than "requirement."
Applying this notation to different systems is as much an art as a
science in that you need to decide which guidelines take priority
over the others to achieve the most userfriendly result.
> >> > I previously did symbol sets for about 20 different ET's, but
that
> >> > was before the latest rational complements were determined, so
I'll
> >> > have to review all of those to see what I would now do
differently.
It turns out that I did quite a few things differently this past week.
> >Here's what I did a couple of weeks ago for some of the ET's (in
> >order of increasing complexity):
> >
> >12, 19, 26: ss
>
> Agreed. My 19 and 26 were wrong.
>
> >17, 24, 31: ss ss
>
> 17 and 24 agreed. I guess you want () for 1deg31 because it is
closer in
> cents than ), but I feel folks are more interested in its
approximations
> of 7, than 11.
I think you meant /\ instead of ().
As with 17 and 24, I think it's more intuitive to use /\ (semisharp)
for half of /\ (sharp) where it's exactly half the number of
degrees. Anyone who has used the Tartini/Fokker notation already
calls an alteration of 1deg31 a semisharp or semiflat and would
expect to see this symbol used.
Besides, if there is no problem with lateral confusability, I think
that straight flags are the simplest way to go.
> >22: s s ss
>
> I agree, but how come you didn't want ss for 1deg22?
If you did that, then you wouldn't have the commaup / / and comma
down \! \!! symbols that are one of the principal features of this
notation; this is something that I would want to have in every ET in
which 80:81 does not vanish, even if that doesn't result in a
completely matched sequence of symbols in the halfapotomes. I
believe the matched sequence is more imporant once the number of
tones gets above 100, by which point / and \ are usually a
different number of degrees.
Also, with the apotome divided into fewer than 5 parts, I would want
to use /\ only when it is exactly half of /\.
In essence, what I am proposing here is that, for the lowernumbered
ET's, we should place a higher priority on the use of rational
complements than on a matching sequence of symbols. (Note that
virtually everything that we agree on below follows this principle.)
> It's also arguable
> that it could be s s ss, making the second halfapotome
follow the
> same pattern of flags as the first,
/ is 3deg22 (since \ = 0deg).
> but what you've got makes more sense to
> me.
So 22 is settled, then.
> >36, 43: x x ss
>
> Agreed for 36. But I wanted a singleshaft symbol for 2deg43 so it
is
> possible to notate it with monotonic letter names and without
doubleshaft
> symbols when using a notation that combines standard sharp and flat
symbols
> with sagittals. One could use either /\ or (\. e.g I want to be
able to
> notate the steps between B and C as B), B/\ or B(\, C/\ or C
(\, C!).
I would rather not use /\ for anything greater than half of /\
unless absolutely necessary. How about using
36, 43: ) (\ /\
for both? Since I reevaluated 72ET, I changed my mind about 36,
which hinges on how 72 is done (see below).
> >29: wx wv ss
>
> Why wouldn't you use the same notation as for 22ET? There's no
need to
> bring in primes higher than 5.
I was making it compatible with my nonconfusable version of 58,
which I no longer favor. When I discuss 58 (below), I will give
another version, which would result in this:
29: /) (\ /\
But if you prefer version 1 of 58 (with all straight flags), then we
might as well do 29 like 22ET.
> >50: ww xs ss
>
> For 50ET, {1, 3, 5, 7, 9, 13, 15, 17, 19} is the maximal consistent
> (19limit) set containing 1,3,9. So I like xs for 2 steps (as
13'), and if
> it's OK here, why not also in 43ET? But ww as 17+(19'19) is
actually 1
> steps of 50ET.
>
> The only options for 1deg50, that don't involve 11 are )) as 19+7
or ~)
> as 7+17 and /) as 13. /) seems the obvious choice to me.
This is my latest proposal:
50: /) (\ /\
So we agree!
> >34, 41: s ss s ss
>
> Agreed.
>
> >27: s xs s ss
>
> Why do you prefer (\ to /)?
2deg27 is almost 90 cents, so (\ is nearer in size than /).
Otherwise, it's a tossup.
> >48: x ss x ss
>
> In 48ET, {1, 3, 7, 9, 11} has only slightly lower errors than {1,
3, 5, 9,
> 11}, 10 cents versus 11 cents. Why prefer the above to the lower
prime scheme
> 48: / /s \ /\ ?
To make 48 compatible with 96ET (see below).
> >46, 53: s ss xx s ss
>
> Agreed.
>
> >58, 72: s s ss s s ss (version 1  simpler, but
more confusability)
> >72: s x ss x s ss (version 2  more complicated,
but less confusability)
>
> Of course, I prefer version 2 for 72ET, since I started the whole
> confusability thing. It isn't significantly more complicated.
To further confuse the issue, I now have even more options for 72ET:
72: / \ /\ / \ /\ (simplest, but most confusability)
72: / ) /\ ) \ /\ (version 2  more complicated, no
confusability, inconsistent)
72: / ) /\ ( \ /\ (version 3  simpler, no
confusability, but ( < \ )
72: / ) /\ (\ \ /\ (version 4  simple, no
confusability, consistent, harmonicoriented)
The symbol arithmetic in version 2 is inconsistent:
/\ minus ) equals 1deg72, but
/\ minus ) equals 2deg72
This is remedied in version 3, which also has a problem in that the
symbol for 4deg72 is a larger rational interval than that for 5deg72,
something I would rather not see in a division as important as 72,
although the difference between ( and \ is rather small.
This leaves me with version 4 as my choice. Notice that the first 4
symbols are, in order, the 5 comma, the 7 comma, the 11 diesis, and
the 13' diesis, all of which are the rational symbols used for a 13
limit otonal scale: C D E\! F/\ G A(!/ Bb!) or B!!!) C.
This option should also be considered in connection with our
discussion of 36 and 43 above.
> >58: s wx ss wv s ss (version 2  more
complicated, but less confusability)
>
> I'm inclined to go with version 1 despite the increased lateral
> confusability, rather than introduce 17flags. Version 2 is a _lot_
more
> complicated.
These are my latest options for both 58 and 65ET:
58, 65, 72: / \ /\ / \ /\ (simplest, but most
confusability)
58: / ~) /\ ~ \ /\ (version 2  more complicated, no
confusability)
65: / /~ /\ ~ \ /\ (version 2  more complicated, no
confusability)
58: / /) /\ (\ \ /\ (version 3  simpler, some
confusability)
Version 3 could offer 29/58 compatibility, but the straight flags of
version 1 are the simplest.
I also threw 65ET in there. Below I have a proposal for 130ET,
which results in 65 having all straight flags (as in the first
version above), so I believe I would prefer that.
> >96: s x s ss s x s ss (version 1  simpler,
but more confusability)
> >96: s x ws ss w x s ss (version 2  more
complicated, but less confusability)
>
> The only maximal 1,3,9consistent 19limit set for 96ET is {1, 3,
5, 9,
> 11, 13, 15, 17}. It is not 1,3,7consistent so the ) flag should be
> defined as the 135 comma (64:65) if it's used at all. The 17 and
19 commas
> vanish, so we should avoid ) ( ~ and ~. So I end up with
> 96: / ) /) /\ / ) /) /\
> Simple _and_ non confusable.
My latest proposal for 96 is:
96: / ) /) /\ (\ ) \ /\
As I mentioned above, I would like to see both / and \ used
whenever possible.
At least we agree on 48, if that is to be notated as a subset of 96.
> >94: w s ws ss xx w s ws ss
>
> Why do you prefer that to
>
> >94: ~ / ) /\ () ~ \ ) /\
>
> Surely we're more interested in the 7comma than the 17+(115)
comma.
>
> Also, it makes sense that / + \ = /\, but it makes the second
half
> apotome have a different sequence of flags to the first. Which
should we
> use, / or \ ?
My proposal above for a matched sequence being subordinate to having
\ and rational complements would apply here. While ~ and ~\ are
not rational complements, they are the 217ET complements  the
nearest we can get to a rational complement for 1deg94.
I calculate both ) and \ as 2deg94, so I needed something else for
3deg. The best possibilities were ( and ~\  neither one uses a
new flag. My choice was:
87, 94: ~ / ~\ /\ () ~ \ ~\ /\
The symbol sequence is fairly simple, particularly in the second half
apotome. Or is the other option:
87, 94: ~ / ( /\ () ~ \ ( /\
better? (Perhaps this is what you meant?)
> >111 (37 as subset): w s s ws ss xs w s s
ws ss
>
> Dealt with above. I'd prefer () for 6deg111.
Yes. Agreed!
> >140: v w s s sw sx ss xs w s s sw
sx ss
>
You made no comment about this one, but it's no good: /\ should be
6deg140, not 7deg (wishful thinking on my part), even though /\ is
14deg. I now propose:
140: ) ~ / )\ /~ /) (~ (\ ) ~ \ )
\ /~ /\
This is the simplest set I could come up with that uses both / and
\.
> >152: v w s s sw sx ss xx xs w s s
sw sx ss
>
> Dealt with elsewhere. I see no reason to use ( which is really
zero steps,
> when ) is 1 step.
Yes. Agreed!
> >171: v wv s x s ws sx ss xs wv s x
s ws sx ss
>
> Why not ~ for 1 step?
>
> >183: v wv s x s ws sx ss xx xs wv s
x s ws sx ss
>
> Why not use w for 1deg183, being a simpler comma than v? 17 vs.
17'17.
After reevaluating, I would keep what I had above for both 171 and
183.
The choice between ( and ~ is almost a tossup, but I found two
reasons to prefer (:
1) It is closer in size to both 1deg171 and 1deg183; and
2) It is the rational complement of /).
> >181: v w wv s s wx ws sx ss xx w wv
s s wx ws sx ss
>
> I don't see how ) can be 5deg181 or how /\ can be 9deg181.
More wishful thinking on my part that /\ should be half of /\  I
guess I was getting tired.
> The only
> symbol that can give 9deg181 with 19limit commas is (~. Here's my
proposal.
>
> 181: ( ~ ~ / /( ( (( /) (~ (\ ~
~ / /( ( (( /) /\
And here's my new proposal.
181: ( ~ ~ / /( ~) /~ /) (~ (\ ( ~ ~
\ /( ~) /~ /\
We don't agree on the symbol arithmetic in the second halfapotome.
Both / and \ are 4deg181, so /\ minus / equals /\ minus \
equals 4deg. You have / as 5deg less than /\.
My choice for 6deg ~) was on the basis of its being the rational
complement of 12deg ~; 7deg /~ logically followed as 3deg plus
4deg.
> >217: v w w s x s wx ws sx ss xx xs w
w s x s wx ws sx ss
>
> Agreed.
And here is what I came up with over the past week, prior to reading
your latest. It's easier to put the whole thing here than trying to
sort through them to figure out what wasn't covered above.
12, 19, 26: /\
17, 24, 31, 38: /\ /\
22: / \ /\
36, 43: ) ) /\ (version 1)
36, 43: ) (\ /\ (version 2)
29: ~) ~ /\
50: /) (\ /\
34, 41: / /\ \ /\
27: / (\ \ /\
48: ) /\ ) /\
55: ( /\ ( /\
39, 46, 53: / /\ () \ /\
60: / ) (\ \ /\
67: ( /) (\ /) /\
74: )) /) (\ /) /\
58, 65, 72: / \ /\ / \ /\ (simplest, but most
confusability)
72: / ) /\ ) \ /\ (version 2  more complicated, no
confusability, inconsistent)
72: / ) /\ ( \ /\ (version 3  simpler, no
confusability, but ( < \ )
72: / ) /\ (\ \ /\ (version 4  simple, no
confusability, consistent, harmonicoriented)
58: / ~) /\ ~ \ /\ (version 2  more complicated, no
confusability)
58: / /) /\ (\ \ /\ (version 3  simpler, some
confusability)
65: / /~ /\ ~ \ /\ (version 2  more complicated, no
confusability)
51: ) / /) \ ) /\
56, 63: ) / /\ () \ )\ /\
70, 77: / \ /\ () / \ /\ (simplest, but most
confusability)
77: / ) /\ () / \ /\
84: / ) /) (\ ) \ /\
68: ) / /) () ( \ /( /\
96: / ) /) /\ (\ ) \ /\
80: ) / /) /\ () (\ \ /) /\
87, 94: ~ / ~\ /\ () ~ \ ~\ /\
108: / // ) /) (\ // ) \ /\
99: ~ / /~ /) (~ (\ ~ \ /~ /\
104: ) ) )) ( /\ () ) ) )) ( /\
111 (37 as subset), 118 (59 ss.): ~ / \ ~\ /\ ()
~ / \ ~\ /\
125: ( / \ (( /\ () ( / \ (( /\
132: ~( / ) \ /) /\ / ) \ /) /\
130 (65 ss.): ~ / ) \ /) /\ (\ / )
\ /) /\
128 (64 ss.): ) ~ / ~\ ~\ /\ () ) ~ \ )\
~\ /\
135 (45 ss.): ( ~ / ( /~ /\ () ( ~ \
( /~ /\
142: ~ / ) \ /) /\ () (\ / ) \ /) /\
149: ( / /( \ /~ /) (~ (\ / /( \ /~ /\
140 (70 ss.): ) ~ / )\ /~ /) (~ (\ ) ~ \ )
\ /~ /\
147: ( ~ / \ /~ /) /\ (\ ~ / \ )
\ /~ /\
152 (76 ss.): ) ~ / \ /~ /) /\ () (\ ~ /
\ /~ /) /\
159: ( ~ / \ /~ (~ /\ () ( ~ / \ /~
(~ /\
171: ( ~( / ) \ ~\ /) /\ (\ ~( / ) \
~\ /) /\
183: ( ~( / ) \ ~\ /) /\ () (\ ~( / )
\ ~\ /) /\
181: ( ~ ~ / /( ~) /~ /) (~ (\ ( ~ ~
\ /( ~) /~ /\
193: ( ~ ~( / \ ~) ~\ /) /\ () (\ ~ ~
( / \ ~) ~\ /) /\
207: ( ~( / /( ) \ ~\ /) /\ () (\ ~( / /
( ) \ ~\ /) /\
198: ) ~ )~ / \ )\ ~) /) /\ () (\ ) ~ )
~ / \ )\ ~) /) /\
217: ( ~ ~ / ) \ ~) ~\ /) /\ () (\ ~
~ / ) \ ~) ~\ /) /\
224: ) ( ~ / ) \ /~ (( /) /\ () (\ (
~ / ) \ /~ (( /) /\
I thought that if these were listed in order of increasing
complexity, perhaps we could spot a few patterns in the arrangement
of symbols that might help to resolve differences of opinion in
instances where the best choice of symbols is not clear.
George
Dave,
I'm taking a little time to elaborate on the issue of correlating
symbols within each of two groups of systems.
36, 43, & 72

If we wish to completely correlate 36, 43, & 72, then the choice
should be clear. There are three ways to do it; only one of these
has no inconsistencies.
1) Rational complements, but ) is inconsistent in 72:
36, 43: ) ) /\
72: / ) /\ ) \ /\
If you think we can justify ) for 4deg72 on the basis of rational
complementation, I'm willing to consider it, but I think the symbol
arithmetic is sloppy. Also, if you wanted a singleshaft symbol for
2deg43, then we can forget about 3643 correlation (which may not be
all that important).
2) Mirrored flags, but ( is inconsistent in 36 & 43:
36, 43: ) ( /\
72: / ) /\ ( \ /\
This one is not my choice.
3) Use of 13limit symbols is consistent in all three:
36, 43: ) (\ /\
72: / ) /\ (\ \ /\
This is my choice: complete correlation, with a singleshaft symbol
for 2deg43.
If you are going to use only singleshaft symbols in combination with
conventional sharps and flats, I think you would still have the
option to notate something as C#!} instead of C(\, should that tone
be used in a 7 relationship.
29, 58, 87, & 94

For these it depends on how much symbol correlation is desired among
the systems.
1) Complete symbol correlation, with correlated symbols ~) and ~
being rational complements:
29: ~) ~ /\
58: / ~) /\ ~ \ /\
87, 94: ~ / ~) /\ () ~ \ ~) /\
2) Complete symbol correlation with less use of ~ flag:
29: ( ~ /\
58: / ( /\ ~ \ /\
87, 94: ~ / ( /\ () ~ \ ( /\
3) More memorable order of symbols for 87 & 94 gives almost complete
correlation and maximizes rational complementation:
29: ~) ~ /\
58: / ~) /\ ~ \ /\
87, 94: ~ / ~\ /\ () ~ \ ~\ /\
4) Using simpler flags in 29 & 58 maintains correlation only between
those two, but still maximizes rational complementation:
29: /) (\ /\
58: / /) /\ (\ \ /\
87, 94: ~ / ~\ /\ () ~ \ ~\ /\ *
5) Using only straight flags for 29 & 58 eliminates all correlation,
but still maximizes rational complementation:
29: / \ /\
58: / \ /\ / \ /\
87, 94: ~ / ~\ /\ () ~ \ ~\ /\ *
*Since 87 & 94 are not correlated with 29 & 58, they may be
considered separately.
Option 3 is my first choice in that it is least disorienting to
anyone who is going to use two or more of these systems. (Consider
that a piece in 58 or 87 might have a section entirely in 29ET.)
Regarding the use of the 17 flag: In 87 it is the best of three not
verygood choices. Its use in 58 may be appropriate, considering how
well 17 is represented in that division. It can be justified in 29
only as a subset of the other two.
I don't see any reason not to use the same symbols for 87 and 94.
Otherwise, options 4 or 5 would be okay with me.
George
Good to hear from you George.
I'm sorry I don't have time to respond to your latest posts at the
moment, but ...
Here's a spreadsheet and chart that I have found very illuminating
with regard to notating ETs. It should be selfexplanatory, except for
the mnemonic value of the markers chosen for the chart.
Red is for left flags, Green is for right. Lighter shades are the
less favoured comma interpretations. Concave are Xs, Convex are Os,
Straight are triangles, Wavy are horizontal dashes.
/tuningmath/files/Dave/ETsByBestFifth.xl
s.zip
Also, it seems you may not yet have discovered this post of mine
/tuningmath/message/4298
Regards,
 Dave Keenan
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote
[#4363]:
> Good to hear from you George.
>
> I'm sorry I don't have time to respond to your latest posts at the
> moment, but ...
>
> Here's a spreadsheet and chart that I have found very illuminating
> with regard to notating ETs. It should be selfexplanatory, except
for
> the mnemonic value of the markers chosen for the chart.
>
> Red is for left flags, Green is for right. Lighter shades are the
> less favoured comma interpretations. Concave are Xs, Convex are Os,
> Straight are triangles, Wavy are horizontal dashes.
>
> /tuning
math/files/Dave/ETsByBestFifth.xls.zip
Perhaps this could be taken a step further by taking the difference
between the ET flag values and the rational flag values to get a
deviation for each flag in each ET. Would low deviations then
identify the most suitable flags for notating those ET's?
> Also, it seems you may not yet have discovered this post of mine
> /tuningmath/message/4298
I did see it, but I figured that I had my hands full replying to your
preceding posting.
One thing that puzzles me is that you show ~) for 2deg96. In one of
your previous postings (#4297) you said the following:
<< The only maximal 1,3,9consistent 19limit set for 96ET is {1, 3,
5, 9, 11, 13, 15, 17}. It is not 1,3,7consistent so the ) flag
should be defined as the 135 comma (64:65) if it's used at all. The
17 and 19 commas vanish, so we should avoid ) ( ~ and ~. So I end
up with
96: / ) /) /\ / ) /) /\ >>
So now you would *not* avoid it?
Wait a minute! It just hit me that ~) means ~x  *alternate* x
or 135 comma  not wx. We're getting tripped up by double
meanings for single characters (for more fun, try using a symbol with
a convex flag at the end of a parenthetical statement, such as ().
(That last symbol was meant to be x, not xx.) I think that your
ASCII notation is okay; we'll just have to be more careful with it,
perhaps employing [, ], and ' in place of (, ), and ~ in certain
instances.
I have my doubts about the merits of defining ) as the 135 comma.
This would be of value only if it gives you a different number of
degrees than defining it as the 7 comma (in 96 both are 2deg).
(Otherwise, it is just an academic exercise.) Where would you use it
under these circumstances? Yes, you did use it in 111ET, but I got
the same result by defining ( as the 11'7 comma (which I discussed
in posting #4346).
Since I have said this much, I might as well address the rest of
#4298.
We do agree on the notation for 84ET.
We also agree on the symbols for 108ET as far as you take them; I
don't know exactly what you would do in the second halfapotome. I
did give a solution in posting #4346.
Our solutions for 132ET are different only for 2deg. I used /
because I wanted that for any division in which 80:81 does not
vanish; you avoided that, evidently because it didn't fit into the
curve in your diagram. I believe that I would take a cue from the
necessity to omit 120ET to establish an upper boundary for strict
adherence to the pattern that was indicated.
[DK, #4298:]
<< In general, complement symbols are a pain in the posterior, and
I'll leave it for you to wrestle with them. I'm starting to think
that the only way to make them work is to make the second half
apotome the mirror image of the first, (with the addition of a second
shaft to each symbol). >>
I need to ask what you mean by "mirror image." Which of these would
exemplify this:
/ ) /\ / ) /\
or
/ ) /\ ( \ /\ ?
I think that it would be the first one, for which I would use the
term "matching sequence" rather than "mirror image."
I sense a little frustration in your statement, perhaps because I
have seemed to be a bit capricious with the employment of the \
symbol while otherwise trying to achieve matched symbol sequences in
the halfapotomes. I tried to address that in message #4346, a
portion of which I will repeat here:
<< [DK:] > I agree, but how come you didn't want ss for 1deg22?
If you did that, then you wouldn't have the commaup / / and comma
down \! \!! symbols that are one of the principal features of this
notation; this is something that I would want to have in every ET in
which 80:81 does not vanish, even if that doesn't result in a
completely matched sequence of symbols in the halfapotomes. I
believe the matched sequence is more imporant once the number of
tones gets above 100, by which point / and \ are usually a
different number of degrees.
Also, with the apotome divided into fewer than 5 parts, I would want
to use /\ only when it is exactly half of /\.
In essence, what I am proposing here is that, for the lowernumbered
ET's, we should place a higher priority on the use of rational
complements than on a matching sequence of symbols. (Note that
virtually everything that we agree on below [i.e., 22, 50, 34, 41,
46, 53] follows this principle.) >>
In borderline cases (around 100 tones), where the symbols start to
get more numerous, but / and \ are not different numbers of
degrees, I have tried to capitalize on that equivalence by having
matching sequences for everything, *except* that the straight flag is
laterally mirrored, for example:
87, 94: ~ / ~\ /\ () ~ \ ~\ /\
99: ~ / /~ /) (~ (\ ~ \ /~ /\
128 (64 ss.): ) ~ / )\ ~\ /\ () ) ~ \ )\
~\ /\
135 (45 ss.): ( ~ / ( /~ /\ () ( ~ \
( /~ /\
(The above contains a correction of what I gave near the end of
#4346, in which my symbol for 3deg128 was given as ~\, which was in
error.)
Notice that in the notation for 87 and 94 we can use ~ for 1deg (the
simplest flag choice) and still retain some logic in the use of ~\
as the 3deg symbol as a consequence of the equivalence of / and \.
I hope that this clarifies how I have attempted to "wrestle" with
these.
George
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>It's taken me a little time to appreciate the value of your proposal
>for dual roles for ( and the 19'19 role of ~. However, I believe
>that a dual role should be retained for ~ also; it is quite useful
>as the 23 comma for notating 135, 147, 159, 198, and 224ET
>(particularly 198).
I suppose if the 23comma interpretation is confined to such large ETs it
might be ok, but I'd need to be convinced that there was no other way to do
it and the ET is actually 1,3,9,23consistent and preferably
1,3,9,...,23consistent where "..." are the other primes used for its
notation.
135, 147, 159, and 224ET are all 17limit notatable, although you'd
probably go to 19 limit for 224ET.
198ET is not 19limit notatable, but why do you feel any need to notate
it? And is it 23limit notatable anyway?
>> Here's my proposal for notating 74ET using its native fifth (since
>it's a
>> meantone), despite the 1,3,9 inconsistency.
>
>> Steps Symbol Comma
>> 
>> 1 )) 19+7
>> 2 )\ 19+(115)
>> 3 /\ 11
>> 4 )\
>> 5 /\
>>
>> The ) flag actually has a value of 1 steps, but it never occurs
>alone, so
>> it doesn't really matter.
>
>While I was away, I worked on the notation for a number of ET's. I
>decided to tackle 74 on my own, since it seemed to be a challenge.
>The solution I came up with minimizes the use of flags with non
>positive values:
>
>74: )) /) (\ /) /\
>
>Your solution is simpler in that it uses fewer flags and has no
>lateral confusability, so it would probably be preferable on that
>basis. However, I mention below that I would rather not use /\ for
>anything greater than half of /\ unless absolutely necessary. On
>the other hand, the 11 factor is almost exact in 74, so it would be a
>shame not to represent it in the notation.
I'll go with your solution, for the singleshaft symbols at least. The
lateral confusability is addressed by the large difference in widths, and
the user will just have to learn that 11 = 13' and 11' = 13.
I propose (( for 4deg74 purely because it is the mirror image of ))
(plus a shaft). More about this later.
>> >Would you also now prefer my selection of the /) symbol for
>[6deg152]
>> >to your choice of (~ on the grounds that it is a more commonly
>used
>> >symbol, particularly in view of the probability that you might
>want
>> >to use (\ instead of ) or ( for 9deg as its complement?
>>
>> Yes, but not on those grounds.
>
>Then we agree on the following (cf. below):
>
>152: ) ~ / \ /~ /) /\ () (\ ~ /
>\ /~ /) /\
Yes, for the singleshaft symbols at least.
>Perhaps "guideline" would be a better term than "requirement."
>Applying this notation to different systems is as much an art as a
>science in that you need to decide which guidelines take priority
>over the others to achieve the most userfriendly result.
You know, I won't really be happy until I have a spreadsheet that generates
the notation for every ET, based on a bunch of rules, because that's the
only way I'll be sure we're being consistent. The rules may of course end
up being very complicated, but I wouldn't want to see any rule that only
applied to a single ET.
By the way, perhaps we should use > and < instead of ) and ( to make
the concave flags more distinct from the convex ones in ASCII. What do you
think?
>> >17, 24, 31: ss ss
>>
>> 17 and 24 agreed. I guess you want () for 1deg31 because it is
>closer in
>> cents than ), but I feel folks are more interested in its
>approximations
>> of 7, than 11.
>
>I think you meant /\ instead of ().
Yes, I did. Sorry.
>As with 17 and 24, I think it's more intuitive to use /\ (semisharp)
>for half of /\ (sharp) where it's exactly half the number of
>degrees. Anyone who has used the Tartini/Fokker notation already
>calls an alteration of 1deg31 a semisharp or semiflat and would
>expect to see this symbol used.
>
>Besides, if there is no problem with lateral confusability, I think
>that straight flags are the simplest way to go.
I guess if there is already a popular notation in use for some ET, and
there is a fairly direct correspondence to that notation available in
sagittal, then we should use it.
So on that basis at least, I can tentatively agree with your proposal for
31. I reserve the right to change my mind on this on further investigation.
:)
31: /\ /\
>> >22: s s ss
>>
>> I agree, but how come you didn't want ss for 1deg22?
>
>If you did that, then you wouldn't have the commaup / / and comma
>down \! \!! symbols that are one of the principal features of this
>notation;
I understand wanting comma up and down symbols in 22ET. That's what I want
too. But what do you mean by / and \!! as comma up and down symbols?
We're proposing _not_ to have those those doubleshafters with a _left_
flag. I assume you meant the doubleshafters to have right flags. In which
case 22 is agreed.
22: / \ /\
>this is something that I would want to have in every ET in
>which 80:81 does not vanish, even if that doesn't result in a
>completely matched sequence of symbols in the halfapotomes. I
>believe the matched sequence is more imporant once the number of
>tones gets above 100, by which point / and \ are usually a
>different number of degrees.
In fact, I'm very happy that / and \, \ and / should always be
complements and to _always_ have a mirrored sequence rather than
_sometimes_ have a matched one.
>Also, with the apotome divided into fewer than 5 parts, I would want
>to use /\ only when it is exactly half of /\.
Fair enough.
>In essence, what I am proposing here is that, for the lowernumbered
>ET's, we should place a higher priority on the use of rational
>complements than on a matching sequence of symbols. (Note that
>virtually everything that we agree on below follows this principle.)
I am instead inclined to totally ignore rational complements with regard to
ETs, especially for the lower numbered ones. One reason is that I feel that
the choice of doubleshaft symbols cannot in any way be allowed to
influence the choice of singleshaft. One must first choose the best set of
single shaft symbols (ignoring complements) since some users will have no
interest in the doubleshaft symbols and should not be penalised for it.
In fact, (and I've been making gentle noises about this possibility for a
some time now), I'm willing to throw away everything we agonised over with
regard to rational complements and instead adopt a simple system that
applies automatically to all ETs and rational tunings.
I propose that the complement of ab is always ba, except that the
complement of // (natural) is /\ and the complement of /\ is /\ if it
represents the same number of steps and () if it represents a different
number of steps.
You will notice that this would require no change to the 72ET version 3
notation, nor any change to most of the smaller ETs we've agreed on such as
12, 17, 19, 22, 24, 26, 31, 46, 53.
By the way, are you collecting all those we've agreed on into one place? I
haven't been.
Why do I want to do this despite some obvious disadvantages? Because I
realised while trying to consistently notate the whole n*12ET family, that
it required us to repeat the whole somewhat arbitrary process we went thru
for rational tunings, to find complements with minimum offsets. And what's
more, that this process would have to be repeated for every such family or
small range of fifthsizes across the whole range of ETs. For example the
n*29ET family is the next largest, followed by n*17. And every such
family, or small range of fifth sizes, would have a completely different
complement mapping. The cognitive load for anyone who uses more than two
such systems would be enormous.
Now for those obvious disadvantages:
1. The second shaft does not have a fixed comma value.
This doesn't seem very important to me?
2. We lose the association of flag size with rational comma size in the
second halfapotome.
This is the biggie. It can be remedied to some degree by redesigning the
doubleshaft (and Xshaft) symbols so their concave flags are wider than
their wavy flags which are wider than their straight and convex flags.
However it will be difficult to make single flag symbols bigger than
doubleflag ones.
What other disadvantages have I omitted?
Advantages:
Simple to remember.
Covers all tunings.
Flags are more strongly associated with particular primes because the flags
don't change when the comma is complemented.
No new flag types ever need to be introduced merely to handle complements.
Doesn't require / and \ as a special case.
>I would rather not use /\ for anything greater than half of /\
>unless absolutely necessary. How about using
>
>36, 43: ) (\ /\
>
>for both? Since I reevaluated 72ET, I changed my mind about 36,
>which hinges on how 72 is done (see below).
I agree with the above for 43ET, but 36ET can be notated as every second
note of 72ET, which means I want:
36: ) ( /\
43: ) (\ /\
Do you have some argument as to why 36 and 43 should be the same? I don't
see it.
>> >29: wx wv ss
>>
>> Why wouldn't you use the same notation as for 22ET? There's no
>need to
>> bring in primes higher than 5.
>
>I was making it compatible with my nonconfusable version of 58,
>which I no longer favor. When I discuss 58 (below), I will give
>another version, which would result in this:
>
>29: /) (\ /\
>
>But if you prefer version 1 of 58 (with all straight flags), then we
>might as well do 29 like 22ET.
I now realise I will need to consider the entire n*29 family
29 58 87 116 145 174 203 232 261, before agreeing on either 29 or 58.
>> >27: s xs s ss
>>
>> Why do you prefer (\ to /)?
>
>2deg27 is almost 90 cents, so (\ is nearer in size than /).
>Otherwise, it's a tossup.
I prefer /) because it introduces only one new flag where (\ introduces
two, for singleshaftonly users.
>> >48: x ss x ss
>>
>> In 48ET, {1, 3, 7, 9, 11} has only slightly lower errors than {1,
>3, 5, 9,
>> 11}, 10 cents versus 11 cents. Why prefer the above to the lower
>prime scheme
>> 48: / /s \ /\ ?
>
>To make 48 compatible with 96ET (see below).
I now agree that 48 should be every second note from 96 and will address
all n*12ETs elsewhere.
>> >58, 72: s s ss s s ss (version 1  simpler, but
>more confusability)
>> >72: s x ss x s ss (version 2  more complicated,
>but less confusability)
>>
>> Of course, I prefer version 2 for 72ET, since I started the whole
>> confusability thing. It isn't significantly more complicated.
>
>To further confuse the issue, I now have even more options for 72ET:
>
>72: / \ /\ / \ /\ (simplest, but most confusability)
>72: / ) /\ ) \ /\ (version 2  more complicated, no
>confusability, inconsistent)
>72: / ) /\ ( \ /\ (version 3  simpler, no
>confusability, but ( < \ )
>72: / ) /\ (\ \ /\ (version 4  simple, no
>confusability, consistent, harmonicoriented)
>
>The symbol arithmetic in version 2 is inconsistent:
>
>/\ minus ) equals 1deg72, but
>/\ minus ) equals 2deg72
>
>This is remedied in version 3, which also has a problem in that the
>symbol for 4deg72 is a larger rational interval than that for 5deg72,
>something I would rather not see in a division as important as 72,
>although the difference between ( and \ is rather small.
>
>This leaves me with version 4 as my choice. Notice that the first 4
>symbols are, in order, the 5 comma, the 7 comma, the 11 diesis, and
>the 13' diesis, all of which are the rational symbols used for a 13
>limit otonal scale: C D E\! F/\ G A(!/ Bb!) or B!!!) C.
>
>This option should also be considered in connection with our
>discussion of 36 and 43 above.
I can only agree to your version 3.
72: / ) /\ ( \ /\
I prefer the above to version 4, with (\ as 4 steps, because I think that
in any given ET, ( and ) flags should either both be 13based or both be
7based, so that () is always 11' (whether it is used or not). Otherwise
we have 3 different possible values for (), the largest and smallest of
which differ by 0.84 cents in rational tuning.
It's bad enough that folks have to know whether the convex flags refer to 7
or 13 in ETs where the tridecimal schisma doesn't vanish. I wouldn't want
them to have to worry about the two convex flags _independently_.
Also, 72ET is not terribly good at the 13limit, the error hikes from 3.9
cents at the 11limit to 7.2 cents at the 13limit, and in any case folks
can learn that the 7comma symbol doubles as the 13comma symbol in 72ET,
just as they must learn that the 11comma symbol doubles as the 7comma
symbol in 31ET.
>> >58: s wx ss wv s ss (version 2  more
>complicated, but less confusability)
>>
>> I'm inclined to go with version 1 despite the increased lateral
>> confusability, rather than introduce 17flags. Version 2 is a _lot_
>more
>> complicated.
>
>These are my latest options for both 58 and 65ET:
>
>58, 65, 72: / \ /\ / \ /\ (simplest, but most
>confusability)
>58: / ~) /\ ~ \ /\ (version 2  more complicated, no
>confusability)
>65: / /~ /\ ~ \ /\ (version 2  more complicated, no
>confusability)
>58: / /) /\ (\ \ /\ (version 3  simpler, some
>confusability)
>
>Version 3 could offer 29/58 compatibility, but the straight flags of
>version 1 are the simplest.
I don't see any need for 29 to be every second of 58, but I do want to look
at the whole huge family first.
>I also threw 65ET in there. Below I have a proposal for 130ET,
>which results in 65 having all straight flags (as in the first
>version above), so I believe I would prefer that.
>
>> >96: s x s ss s x s ss (version 1  simpler,
>but more confusability)
>> >96: s x ws ss w x s ss (version 2  more
>complicated, but less confusability)
>>
>> The only maximal 1,3,9consistent 19limit set for 96ET is {1, 3,
>5, 9,
>> 11, 13, 15, 17}. It is not 1,3,7consistent so the ) flag should be
>> defined as the 135 comma (64:65) if it's used at all. The 17 and
>19 commas
>> vanish, so we should avoid ) ( ~ and ~. So I end up with
>> 96: / ) /) /\ / ) /) /\
>> Simple _and_ non confusable.
>
>My latest proposal for 96 is:
>
>96: / ) /) /\ (\ ) \ /\
>
>As I mentioned above, I would like to see both / and \ used
>whenever possible.
>
>At least we agree on 48, if that is to be notated as a subset of 96.
I changed my mind on 96, as you will have seen in other posts, but might
end up changing it back.
>> >94: w s ws ss xx w s ws ss
>>
>> Why do you prefer that to
>>
>> >94: ~ / ) /\ () ~ \ ) /\
>>
>> Surely we're more interested in the 7comma than the 17+(115)
>comma.
>>
>> Also, it makes sense that / + \ = /\, but it makes the second
>half
>> apotome have a different sequence of flags to the first. Which
>should we
>> use, / or \ ?
>
>My proposal above for a matched sequence being subordinate to having
>\ and rational complements would apply here. While ~ and ~\ are
>not rational complements, they are the 217ET complements  the
>nearest we can get to a rational complement for 1deg94.
>
>I calculate both ) and \ as 2deg94, so I needed something else for
>3deg. The best possibilities were ( and ~\  neither one uses a
>new flag. My choice was:
>
>87, 94: ~ / ~\ /\ () ~ \ ~\ /\
>
>The symbol sequence is fairly simple, particularly in the second half
>apotome. Or is the other option:
>
>87, 94: ~ / ( /\ () ~ \ ( /\
>
>better? (Perhaps this is what you meant?)
Yes. That's what I meant. Sorry.
I now want one of
94: ~ / ( /\ () ) \ ~ /\
94: ~ / ~\ /\ () /~ \ ~ /\
and need to look at 94 188 282 to decide.
>> >111 (37 as subset): w s s ws ss xs w s s
>ws ss
>>
>> Dealt with above. I'd prefer () for 6deg111.
>
>Yes. Agreed!
Only now I want the fullymirrored halfapotomes:
111 (37): ~ / \ ~\ /\ () /~ / \ ~ /\
>140: ) ~ / )\ /~ /) (~ (\ ) ~ \ )
>\ /~ /\
>
>This is the simplest set I could come up with that uses both / and
>\.
I'll leave the second halfapotome out of it for now. It seems we have 4
options:
140: ) ~ / )\ ( /) (~ 6 flags
140: ) ~ / )) ( /) (~ 5 flags
140: ) ~ / )\ /~ /) (~ 6 flags monotonic flags per symb
140: ) ~ / )) /~ /) (~ 5 flags monotonic flags per symb
I prefer the last one, and with mirror complements it would be
140: ) ~ / )) /~ /) (~ (\ ~\ (( \ ~ ( /\
Note that with mirror complements, (\ is the same as (\.
>> >152: v w s s sw sx ss xx xs w s s
>sw sx ss
>>
>> Dealt with elsewhere. I see no reason to use ( which is really
>zero steps,
>> when ) is 1 step.
>
>Yes. Agreed!
With mirror complements we have:
152: ) ~ / \ /~ /) /\ () (\ ~ / \ ~ ( /\
>> >171: v wv s x s ws sx ss xs wv s x
>s ws sx ss
>>
>> Why not ~ for 1 step?
>>
>> >183: v wv s x s ws sx ss xx xs wv s
>x s ws sx ss
>>
>> Why not use w for 1deg183, being a simpler comma than v? 17 vs.
>17'17.
>
>After reevaluating, I would keep what I had above for both 171 and
>183.
>
>The choice between ( and ~ is almost a tossup, but I found two
>reasons to prefer (:
>
>1) It is closer in size to both 1deg171 and 1deg183; and
>
>2) It is the rational complement of /).
I'll buy 1), but no longer care about 2). So I agree with the above, as far
as the singleshaft symbols.
>> >181: v w wv s s wx ws sx ss xx w wv
>s s wx ws sx ss
>>
>> I don't see how ) can be 5deg181 or how /\ can be 9deg181.
>
>More wishful thinking on my part that /\ should be half of /\  I
>guess I was getting tired.
>
>> The only
>> symbol that can give 9deg181 with 19limit commas is (~. Here's my
>proposal.
>>
>> 181: ( ~ ~ / /( ( (( /) (~ (\ ~
>~ / /( ( (( /) /\
>
>And here's my new proposal.
>
>181: ( ~ ~ / /( ~) /~ /) (~ (\ ( ~ ~
>\ /( ~) /~ /\
>
>We don't agree on the symbol arithmetic in the second halfapotome.
>Both / and \ are 4deg181, so /\ minus / equals /\ minus \
>equals 4deg. You have / as 5deg less than /\.
>
>My choice for 6deg ~) was on the basis of its being the rational
>complement of 12deg ~; 7deg /~ logically followed as 3deg plus
>4deg.
It is still unclear to me what's best for 181, but you will realise that
rational complements may no longer be of any relevance to me.
>> >217: v w w s x s wx ws sx ss xx xs w
>w s x s wx ws sx ss
>>
>> Agreed.
Except for the mirror complement thingy that we need to thrash out now.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4405]:
> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >It's taken me a little time to appreciate the value of your
proposal
> >for dual roles for ( and the 19'19 role of ~. However, I
believe
> >that a dual role should be retained for ~ also; it is quite
useful
> >as the 23 comma for notating 135, 147, 159, 198, and 224ET
> >(particularly 198).
>
> I suppose if the 23comma interpretation is confined to such large
ETs it
> might be ok, but I'd need to be convinced that there was no other
way to do
> it and the ET is actually 1,3,9,23consistent and preferably
> 1,3,9,...,23consistent where "..." are the other primes used for
its
> notation.
>
> 135, 147, 159, and 224ET are all 17limit notatable, although you'd
> probably go to 19 limit for 224ET.
For that you would need to use a twoflag symbol ~( for 2deg of 135,
147, and 159 and for 3deg224. I thought that you might prefer a
singleflag symbol ~ for a small interval such as that, as with 217.
However, if you think that we should use something such as the
following,
135 (45 ss.): ~ ~( / ( (( /\ () ~ ~( \ (
(( /\
147: ~ ~( / \ (( /) /\ (\ ~( / \ )
\ /) /\ or
147: ~ ~( / \ ~\ /) /\ (\ ~( / \
~\ /) /\
159: ~ ~( / \ ~\ (~ /\ () ~ ~( / \ ~\
(~ /\
then I would have no problem with it.
> 198ET is not 19limit notatable, but why do you feel any need to
notate
> it?
I thought that we were notating everything that we possibly could.
Who knows what the tuning scavengers might want to use?
> And is it 23limit notatable anyway?
Only the 17 factor is so far removed from everything else that it
would need to be avoided in the notation. (As it turns out I had to
use ~) for 7deg.) The 19 factor is actually more accurately
represented that most everything else, and it's inconsistent only
with respect to 15. When I did the notation, this is what I got:
198: ) ~ )~ / \ )\ ~) /) /\ () (\ ) ~ )
~ / \ )\ ~) /) /\
Aside from the mirroring issue (which I will adress below), do you
have any problem with this?
> >> Here's my proposal for notating 74ET using its native fifth
(since it's a
> >> meantone), despite the 1,3,9 inconsistency.
> >
> >> Steps Symbol Comma
> >> 
> >> 1 )) 19+7
> >> 2 )\ 19+(115)
> >> 3 /\ 11
> >> 4 )\
> >> 5 /\
> >>
> >> The ) flag actually has a value of 1 steps, but it never
occurs alone, so
> >> it doesn't really matter.
> >
> >While I was away, I worked on the notation for a number of ET's.
I
> >decided to tackle 74 on my own, since it seemed to be a
challenge.
> >The solution I came up with minimizes the use of flags with non
> >positive values:
> >
> >74: )) /) (\ /) /\
> >
> >Your solution is simpler in that it uses fewer flags and has no
> >lateral confusability, so it would probably be preferable on that
> >basis. However, I mention below that I would rather not use /\
for
> >anything greater than half of /\ unless absolutely necessary.
On
> >the other hand, the 11 factor is almost exact in 74, so it would
be a
> >shame not to represent it in the notation.
>
> I'll go with your solution, for the singleshaft symbols at least.
The
> lateral confusability is addressed by the large difference in
widths, and
> the user will just have to learn that 11 = 13' and 11' = 13.
>
> I propose (( for 4deg74 purely because it is the mirror image of )
)
> (plus a shaft). More about this later.
>
> >> >Would you also now prefer my selection of the /) symbol for
[6deg152]
> >> >to your choice of (~ on the grounds that it is a more commonly
used
> >> >symbol, particularly in view of the probability that you might
want
> >> >to use (\ instead of ) or ( for 9deg as its complement?
> >>
> >> Yes, but not on those grounds.
> >
> >Then we agree on the following (cf. below):
> >
> >152: ) ~ / \ /~ /) /\ () (\ ~ /
\ /~ /) /\
>
> Yes, for the singleshaft symbols at least.
>
> >Perhaps "guideline" would be a better term than "requirement."
> >Applying this notation to different systems is as much an art as a
> >science in that you need to decide which guidelines take priority
> >over the others to achieve the most userfriendly result.
>
> You know, I won't really be happy until I have a spreadsheet that
generates
> the notation for every ET, based on a bunch of rules, because
that's the
> only way I'll be sure we're being consistent. The rules may of
course end
> up being very complicated, but I wouldn't want to see any rule that
only
> applied to a single ET.
That sounds like a worthy goal. I just wonder how long it would take
us to agree on all of the rules and the hierarchy.
> By the way, perhaps we should use > and < instead of ) and ( to
make
> the concave flags more distinct from the convex ones in ASCII. What
do you
> think?
I think that we would have a problem with that in email when the
lines wrap and an extra > character gets inserted right in the middle
of a symbol. (Yes, Yahoo does break lines that way, and I try to
clean them up before sending my replies  there were several
instances where I did it in this message  and using > to represent
a flag would make that more difficult.)
> >> >17, 24, 31: ss ss
> >>
> >> 17 and 24 agreed. I guess you want () for 1deg31 because it is
closer in
> >> cents than ), but I feel folks are more interested in its
approximations
> >> of 7, than 11.
> >
> >I think you meant /\ instead of ().
>
> Yes, I did. Sorry.
>
> >As with 17 and 24, I think it's more intuitive to use /\
(semisharp)
> >for half of /\ (sharp) where it's exactly half the number of
> >degrees. Anyone who has used the Tartini/Fokker notation already
> >calls an alteration of 1deg31 a semisharp or semiflat and would
> >expect to see this symbol used.
> >
> >Besides, if there is no problem with lateral confusability, I
think
> >that straight flags are the simplest way to go.
>
> I guess if there is already a popular notation in use for some ET,
and
> there is a fairly direct correspondence to that notation available
in
> sagittal, then we should use it.
>
> So on that basis at least, I can tentatively agree with your
proposal for
> 31. I reserve the right to change my mind on this on further
investigation.
> :)
>
> 31: /\ /\
>
> >> >22: s s ss
> >>
> >> I agree, but how come you didn't want ss for 1deg22?
> >
> >If you did that, then you wouldn't have the commaup / / and
comma
> >down \! \!! symbols that are one of the principal features of this
> >notation;
>
> I understand wanting comma up and down symbols in 22ET. That's
what I want
> too. But what do you mean by / and \!! as comma up and down
symbols?
> We're proposing _not_ to have those those doubleshafters with a
_left_
> flag. I assume you meant the doubleshafters to have right flags.
In which
> case 22 is agreed.
>
> 22: / \ /\
Sorry. I got laterally confused. I meant \ and !!/.
> >this is something that I would want to have in every ET in
> >which 80:81 does not vanish, even if that doesn't result in a
> >completely matched sequence of symbols in the halfapotomes. I
> >believe the matched sequence is more imporant once the number of
> >tones gets above 100, by which point / and \ are usually a
> >different number of degrees.
>
> In fact, I'm very happy that / and \, \ and / should always be
> complements and to _always_ have a mirrored sequence rather than
> _sometimes_ have a matched one.
As you said above, more about this below.
> >Also, with the apotome divided into fewer than 5 parts, I would
want
> >to use /\ only when it is exactly half of /\.
>
> Fair enough.
Conversely, should we not also specify that if /\ *can* be used for
exactly half of /\, then it *should* be used. (I would consider
this the primary justification for its use as 1deg31.) This would be
one of the rules for deriving ET notations in a spreadsheet that I
would like to see.
> >In essence, what I am proposing here is that, for the lower
numbered
> >ET's, we should place a higher priority on the use of rational
> >complements than on a matching sequence of symbols. (Note that
> >virtually everything that we agree on below follows this
principle.)
>
> I am instead inclined to totally ignore rational complements with
regard to
> ETs, especially for the lower numbered ones. One reason is that I
feel that
> the choice of doubleshaft symbols cannot in any way be allowed to
> influence the choice of singleshaft. One must first choose the
best set of
> single shaft symbols (ignoring complements) since some users will
have no
> interest in the doubleshaft symbols and should not be penalised
for it.
>
> In fact, (and I've been making gentle noises about this possibility
for a
> some time now), I'm willing to throw away everything we agonised
over with
> regard to rational complements and instead adopt a simple system
that
> applies automatically to all ETs and rational tunings.
>
> I propose that the complement of ab is always ba, except that the
> complement of // (natural) is /\ and the complement of /\
is /\ if it
> represents the same number of steps and () if it represents a
different
> number of steps.
>
> You will notice that this would require no change to the 72ET
version 3
> notation, nor any change to most of the smaller ETs we've agreed on
such as
> 12, 17, 19, 22, 24, 26, 31, 46, 53.
>
> By the way, are you collecting all those we've agreed on into one
place? I
> haven't been.
Yes, in two places.
> Why do I want to do this despite some obvious disadvantages?
Because I
> realised while trying to consistently notate the whole n*12ET
family, that
> it required us to repeat the whole somewhat arbitrary process we
went thru
> for rational tunings, to find complements with minimum offsets. And
what's
> more, that this process would have to be repeated for every such
family or
> small range of fifthsizes across the whole range of ETs. For
example the
> n*29ET family is the next largest, followed by n*17. And every such
> family, or small range of fifth sizes, would have a completely
different
> complement mapping. The cognitive load for anyone who uses more
than two
> such systems would be enormous.
>
> Now for those obvious disadvantages:
>
> 1. The second shaft does not have a fixed comma value.
>
> This doesn't seem very important to me?
>
> 2. We lose the association of flag size with rational comma size in
the
> second halfapotome.
>
> This is the biggie. It can be remedied to some degree by
redesigning the
> doubleshaft (and Xshaft) symbols so their concave flags are wider
than
> their wavy flags which are wider than their straight and convex
flags.
> However it will be difficult to make single flag symbols bigger than
> doubleflag ones.
>
> What other disadvantages have I omitted?
One very big one that I will state below, when you give a couple of
examples.
> Advantages:
>
> Simple to remember.
> Covers all tunings.
> Flags are more strongly associated with particular primes because
the flags
> don't change when the comma is complemented.
> No new flag types ever need to be introduced merely to handle
complements.
> Doesn't require / and \ as a special case.
>
> >I would rather not use /\ for anything greater than half of /\
> >unless absolutely necessary. How about using
> >
> >36, 43: ) (\ /\
> >
> >for both? Since I reevaluated 72ET, I changed my mind about 36,
> >which hinges on how 72 is done (see below).
>
> I agree with the above for 43ET, but 36ET can be notated as every
second
> note of 72ET, which means I want:
> 36: ) ( /\
> 43: ) (\ /\
>
> Do you have some argument as to why 36 and 43 should be the same? I
don't
> see it.
I have no particular reason, other than notating these two the same
way might make them easier to remember.
Then we can agree on what you have above for 36 and 43, inasmuch as I
will also be agreeing with you on 72, below.
> >> >29: wx wv ss
> >>
> >> Why wouldn't you use the same notation as for 22ET? There's no
need to
> >> bring in primes higher than 5.
> >
> >I was making it compatible with my nonconfusable version of 58,
> >which I no longer favor. When I discuss 58 (below), I will give
> >another version, which would result in this:
> >
> >29: /) (\ /\
> >
> >But if you prefer version 1 of 58 (with all straight flags), then
we
> >might as well do 29 like 22ET.
>
> I now realise I will need to consider the entire n*29 family
> 29 58 87 116 145 174 203 232 261, before agreeing on either 29 or
58.
Why bother with anything above 145?
> >> >27: s xs s ss
> >>
> >> Why do you prefer (\ to /)?
> >
> >2deg27 is almost 90 cents, so (\ is nearer in size than /).
> >Otherwise, it's a tossup.
>
> I prefer /) because it introduces only one new flag where (\
introduces
> two, for singleshaftonly users.
As if this were already too many flags? Okay, let's use /); it's
not that big a deal, anyway.
> >> >48: x ss x ss
> >>
> >> In 48ET, {1, 3, 7, 9, 11} has only slightly lower errors than
{1, 3, 5, 9,
> >> 11}, 10 cents versus 11 cents. Why prefer the above to the lower
prime scheme
> >> 48: / /s \ /\ ?
> >
> >To make 48 compatible with 96ET (see below).
>
> I now agree that 48 should be every second note from 96 and will
address
> all n*12ETs elsewhere.
>
> > ... To further confuse the issue, I now have even more options
for 72ET:
> >
> >72: / \ /\ / \ /\ (simplest, but most
confusability)
> >72: / ) /\ ) \ /\ (version 2  more complicated,
no confusability, inconsistent)
> >72: / ) /\ ( \ /\ (version 3  simpler, no
confusability, but ( < \ )
> >72: / ) /\ (\ \ /\ (version 4  simple, no
confusability, consistent, harmonicoriented)
> >
> ... I can only agree to your version 3.
>
> 72: / ) /\ ( \ /\
>
> I prefer the above to version 4, with (\ as 4 steps, because I
think that
> in any given ET, ( and ) flags should either both be 13based or
both be
> 7based, so that () is always 11' (whether it is used or not).
Otherwise
> we have 3 different possible values for (), the largest and
smallest of
> which differ by 0.84 cents in rational tuning.
>
> It's bad enough that folks have to know whether the convex flags
refer to 7
> or 13 in ETs where the tridecimal schisma doesn't vanish. I
wouldn't want
> them to have to worry about the two convex flags _independently_.
That makes sense. However, I would look at it a little differently.
Adhering more closely to the onecommaperprime ideal, I would
consider 715:729 the preferred ratio for (, which gives an exact
26:27 for (\, which I consider the principal 13 diesis (even if it
exceeds a halfapotome by a few cents) that modifies a natural note
(downward) to give 13/8. We already have \ as 54:55, which gives an
exact 11 diesis of 32:33 that modifies a natural note (upward) to
give 11/8. So we would have one comma each for 11 and 13, not two
commas for 11.
As long as the tridecimal schisma vanishes, then () will be /\
minus /\, and we can leave well enough alone. In cases where it
doesn't vanish, then we can redefine *one* of the two commas: either
( as the 11'7 comma or ) as the 135 comma. (To avoid confusion,
I have used your nomenclature, which calls 1024:1053 the 13 diesis,
although I would prefer to label it the 13' diesis, with 26:27 as the
13 diesis, in which case the redefined ) would then be labeled the
13'5 comma.)
Anyway, however one chooses to look at it, we are in agreement on how
to apply the symbols in the notation.
> Also, 72ET is not terribly good at the 13limit, the error hikes
from 3.9
> cents at the 11limit to 7.2 cents at the 13limit, and in any case
folks
> can learn that the 7comma symbol doubles as the 13comma symbol in
72ET,
> just as they must learn that the 11comma symbol doubles as the 7
comma
> symbol in 31ET.
I will agree to use version 3, then, with 36ET being notated as a
subset.
One thing that I like about this version is that it closely resembles
the sagittal notation as I originally presented it, with all straight
flags, except that convex flags replace straight flags in the tones
neighboring the halfapotome to eliminate lateral confusability. For
the Miracle family of ET's, you will recall that I was advocating the
possibility of reading uncomplicated music in 72ET directly into 31
and 41ET by a mental "roundingoff" process. The 72ET symbols with
straight flags can usually be read directly, and the convex flags
would readily identify the symbols that would need to be "rounded
off" to (straightflag) 31 and 41ET symbols. (Of course, this would
work only if we use /\ as 1deg31, which would be another reason for
doing 31 that way.)
> >> >58: s wx ss wv s ss (version 2  more
> >complicated, but less confusability)
> >>
> >> I'm inclined to go with version 1 despite the increased lateral
> >> confusability, rather than introduce 17flags. Version 2 is a
_lot_ more
> >> complicated.
> >
> >These are my latest options for both 58 and 65ET:
> >
> >58, 65, 72: / \ /\ / \ /\ (simplest, but most
confusability)
> >58: / ~) /\ ~ \ /\ (version 2  more complicated,
no confusability)
> >65: / /~ /\ ~ \ /\ (version 2  more complicated,
no confusability)
> >58: / /) /\ (\ \ /\ (version 3  simpler, some
confusability)
> >
> >Version 3 could offer 29/58 compatibility, but the straight flags
of
> >version 1 are the simplest.
>
> I don't see any need for 29 to be every second of 58, but I do want
to look
> at the whole huge family first.
My subsequent posting #4354 also addressed this in more detail. I
will have to take a closer look at 116 and 145. I made a comment on
those above 145 below.
> >I also threw 65ET in there. Below I have a proposal for 130ET,
> >which results in 65 having all straight flags (as in the first
> >version above), so I believe I would prefer that.
> >
> >> >96: s x s ss s x s ss (version 1 
simpler, but more confusability)
> >> >96: s x ws ss w x s ss (version 2  more
complicated, but less confusability)
> >>
> >> The only maximal 1,3,9consistent 19limit set for 96ET is {1,
3, 5, 9,
> >> 11, 13, 15, 17}. It is not 1,3,7consistent so the ) flag
should be
> >> defined as the 135 comma (64:65) if it's used at all. The 17
and 19 commas
> >> vanish, so we should avoid ) ( ~ and ~. So I end up with
> >> 96: / ) /) /\ / ) /) /\
> >> Simple _and_ non confusable.
> >
> >My latest proposal for 96 is:
> >
> >96: / ) /) /\ (\ ) \ /\
> >
> >As I mentioned above, I would like to see both / and \ used
> >whenever possible.
> >
> >At least we agree on 48, if that is to be notated as a subset of
96.
>
> I changed my mind on 96, as you will have seen in other posts, but
might
> end up changing it back.
>
> >> >94: w s ws ss xx w s ws ss
> >>
> >> Why do you prefer that to
> >>
> >> >94: ~ / ) /\ () ~ \ ) /\
> >>
> >> Surely we're more interested in the 7comma than the 17+(115)
> >comma.
> >>
> >> Also, it makes sense that / + \ = /\, but it makes the
second half
> >> apotome have a different sequence of flags to the first. Which
should we
> >> use, / or \ ?
> >
> >My proposal above for a matched sequence being subordinate to
having
> >\ and rational complements would apply here. While ~ and ~\
are
> >not rational complements, they are the 217ET complements  the
> >nearest we can get to a rational complement for 1deg94.
> >
> >I calculate both ) and \ as 2deg94, so I needed something else
for
> >3deg. The best possibilities were ( and ~\  neither one uses
a
> >new flag. My choice was:
> >
> >87, 94: ~ / ~\ /\ () ~ \ ~\ /\
> >
> >The symbol sequence is fairly simple, particularly in the second
half
> >apotome. Or is the other option:
> >
> >87, 94: ~ / ( /\ () ~ \ ( /\
> >
> >better? (Perhaps this is what you meant?)
>
> Yes. That's what I meant. Sorry.
>
> I now want one of
>
> 94: ~ / ( /\ () ) \ ~ /\
> 94: ~ / ~\ /\ () /~ \ ~ /\
>
> and need to look at 94 188 282 to decide.
Very well! But I don't think that 188 has much going for it, and 282
is going to have gaps.
Otherwise, if we think it advisable to keep the same sequence of
symbols for both 87 and 94 (for ease of learning), then our choice
may be influenced by what we do for multiples of 29.
> >> >111 (37 as subset): w s s ws ss xs w s s
ws ss
> >>
> >> Dealt with above. I'd prefer () for 6deg111.
> >
> >Yes. Agreed!
>
> Only now I want the fullymirrored halfapotomes:
>
> 111 (37): ~ / \ ~\ /\ () /~ / \ ~ /\
Hmm. That makes /~ is smaller than either / or ~. Let me
give this a little thought.
> >140: ) ~ / )\ /~ /) (~ (\ ) ~ \ )
\ /~ /\
> >
> >This is the simplest set I could come up with that uses both /
and
> >\.
>
> I'll leave the second halfapotome out of it for now. It seems we
have 4
> options:
> 140: ) ~ / )\ ( /) (~ 6 flags
> 140: ) ~ / )) ( /) (~ 5 flags
> 140: ) ~ / )\ /~ /) (~ 6 flags monotonic flags per
symb
> 140: ) ~ / )) /~ /) (~ 5 flags monotonic flags per
symb
>
> I prefer the last one, and with mirror complements it would be
>
> 140: ) ~ / )) /~ /) (~ (\ ~\ (( \ ~ 
( /\
>
> Note that with mirror complements, (\ is the same as (\.
In 70ET )\ is 2deg, whereas )) is 1deg, so I prefer the former.
However, with the mirrored symbols ~\ is a smaller interval than
either ~ or \ and (( is smaller than (. This goes counter to
what I would expect.
> >> >152: v w s s sw sx ss xx xs w s s
sw sx ss
> >>
> >> Dealt with elsewhere. I see no reason to use ( which is really
zero steps,
> >> when ) is 1 step.
> >
> >Yes. Agreed!
>
> With mirror complements we have:
> 152: ) ~ / \ /~ /) /\ () (\ ~ / \ ~ 
( /\
I think you intended ~\ for 10deg, which gives:
152: ) ~ / \ /~ /) /\ () (\ ~\ / \ ~ 
( /\
So ~\ is smaller than either \ or ~. I need to think about
this a little more.
> >> >171: v wv s x s ws sx ss xs wv s x
s ws sx ss
> >>
> >> Why not ~ for 1 step?
> >>
> >> >183: v wv s x s ws sx ss xx xs wv s
x s ws sx ss
> >>
> >> Why not use w for 1deg183, being a simpler comma than v? 17
vs. 17'17.
> >
> >After reevaluating, I would keep what I had above for both 171
and
> >183.
> >
> >The choice between ( and ~ is almost a tossup, but I found two
> >reasons to prefer (:
> >
> >1) It is closer in size to both 1deg171 and 1deg183; and
> >
> >2) It is the rational complement of /).
>
> I'll buy 1), but no longer care about 2). So I agree with the
above, as far
> as the singleshaft symbols.
>
> >> >181: v w wv s s wx ws sx ss xx w wv
s s wx ws sx ss
> >>
> >> I don't see how ) can be 5deg181 or how /\ can be 9deg181.
> >
> >More wishful thinking on my part that /\ should be half of /\ 
I
> >guess I was getting tired.
> >
> >> The only
> >> symbol that can give 9deg181 with 19limit commas is (~. Here's
my proposal.
> >>
> >> 181: ( ~ ~ / /( ( (( /) (~ (\ ~
~ / /( ( (( /) /\
> >
> >And here's my new proposal.
> >
> >181: ( ~ ~ / /( ~) /~ /) (~ (\ ( ~ ~
\ /( ~) /~ /\
> >
> >We don't agree on the symbol arithmetic in the second half
apotome.
> >Both / and \ are 4deg181, so /\ minus / equals /\ minus \
> >equals 4deg. You have / as 5deg less than /\.
> >
> >My choice for 6deg ~) was on the basis of its being the rational
> >complement of 12deg ~; 7deg /~ logically followed as 3deg plus
> >4deg.
>
> It is still unclear to me what's best for 181, but you will realise
that
> rational complements may no longer be of any relevance to me.
>
> >> >217: v w w s x s wx ws sx ss xx xs
w w s x s wx ws sx ss
> >>
> >> Agreed.
>
> Except for the mirror complement thingy that we need to thrash out
now.
In effect, mirroring gives the flags negative values, with the zero
point being the apotome, which itself is notated as an
exception, /\ ,when its proper mirror should be . For the
simpler ET's that use no concave or wavy flags, I don't see much of a
problem, since the symbol arithmetic usually works in spite of the
mirroring. But as soon as you introduce concave or wavy flags,
particularly in twoflag symbols, the symbol arithmetic goes crazy.
After all that we went through figuring out the rational complements,
I can't see replacing that with something in which the order of
symbols in the second halfapotome makes very little sense? All
to "fix" a problem involving notquitematched symbols / and \ in
a few ET's? I say: "forget it."
If we can get mirroring in the lowernumbered ET's by means of the
complementation that we already worked out, then that's a commendable
goal. But please, let's not dump the concept of consistent symbol
arithmetic in the process.
If you feel that the best choice of singleshaft symbols is in some
instances compromised by the need to have doubleshaft complements,
then I'll work with you to address that problem.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> I thought that we were notating everything that we possibly could.
> Who knows what the tuning scavengers might want to use?
Good ideaI recently found some interesting uses for the 108et.
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote [#4412]:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote
[#4405]:
> > > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > >140: ) ~ / )\ /~ /) (~ (\ ) ~ \ )
\ /~ /\
> > >
> > >This is the simplest set I could come up with that uses both /
and \.
> >
> > I'll leave the second halfapotome out of it for now. It seems we
have 4
> > options:
> > 140: ) ~ / )\ ( /) (~ 6 flags
> > 140: ) ~ / )) ( /) (~ 5 flags
> > 140: ) ~ / )\ /~ /) (~ 6 flags monotonic flags
per symb
> > 140: ) ~ / )) /~ /) (~ 5 flags monotonic flags
per symb
> >
> > I prefer the last one, and with mirror complements it would be
> >
> > 140: ) ~ / )) /~ /) (~ (\ ~\ (( \ ~
( /\
> >
> > Note that with mirror complements, (\ is the same as (\.
>
> In 70ET )\ is 2deg, whereas )) is 1deg, so I prefer the former.
Let me correct myself: in 70ET ) will be the 135 comma of 2deg,
so ( will then be the 13'7 comma of 2deg. So )) must also be
2deg70.
My reason for choosing ~ for 2deg140 is that the flag is also used
for 7deg (where this is no other choice), which serves to limit the
total number of flags. However, this flag (taken as either a 19'19
or a 23 flag) is not really a very good choice for either 1deg70 or
2deg140; the 17 flag ~ would be better for both. In 70ET both 5
and 7 are awful, and whatever inconsistency we find for 17 is only
with respect to 5 and 7. Taken separately, 17 is one of the best
factors in 70ET, and it's also better than either 19 or 23 in 140ET.
Using the 17 flag, I got the following easytoremember sequence of
symbols:
140 (70 ss.): ) ~ / )) ~) /) (~ (\ ) ~ \ )
) ~) /\
The two disadvantages with this are that it uses 7 flags and has two
pairs of laterally confusable symbols. So I would be more inclined
to go with one of yours.
Of the four options you gave, I agree with your choice of the last
one (5 flags monotonic flags per symb), but with nonmirroring double
shaft symbols:
140 (70 ss.): ) ~ / )) /~ /) (~ (\ ) ~ \ )
) /~ /\ 5 flags monotonic flags per symb
George
Hi George,
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote
> > In fact, (and I've been making gentle noises about this
possibility
> for a
> > some time now), I'm willing to throw away everything we agonised
> over with
> > regard to rational complements and instead adopt a simple system
> that
> > applies automatically to all ETs and rational tunings.
> >
> > I propose that the complement of ab is always ba, except that
the
> > complement of // (natural) is /\ and the complement of /\
> is /\ if it
> > represents the same number of steps and () if it represents a
> different
> > number of steps.
> >
...
> > Why do I want to do this despite some obvious disadvantages?
> Because I
> > realised while trying to consistently notate the whole n*12ET
> family, that
> > it required us to repeat the whole somewhat arbitrary process we
> went thru
> > for rational tunings, to find complements with minimum offsets.
And
> what's
> > more, that this process would have to be repeated for every such
> family or
> > small range of fifthsizes across the whole range of ETs. For
> example the
> > n*29ET family is the next largest, followed by n*17. And every
such
> > family, or small range of fifth sizes, would have a completely
> different
> > complement mapping. The cognitive load for anyone who uses more
> than two
> > such systems would be enormous.
> >
> > Now for those obvious disadvantages:
> >
> > 1. The second shaft does not have a fixed comma value.
> >
> > This doesn't seem very important to me?
> >
> > 2. We lose the association of flag size with rational comma size
in
> the
> > second halfapotome.
> >
> > This is the biggie. It can be remedied to some degree by
> redesigning the
> > doubleshaft (and Xshaft) symbols so their concave flags are
wider
> than
> > their wavy flags which are wider than their straight and convex
> flags.
> > However it will be difficult to make single flag symbols bigger
than
> > doubleflag ones.
> >
> > What other disadvantages have I omitted?
>
> One very big one that I will state below, when you give a couple of
> examples.
>
> > Advantages:
> >
> > Simple to remember.
> > Covers all tunings.
> > Flags are more strongly associated with particular primes because
> the flags
> > don't change when the comma is complemented.
> > No new flag types ever need to be introduced merely to handle
> complements.
> > Doesn't require / and \ as a special case.
..
> > Except for the mirror complement thingy that we need to thrash out
> now.
>
> In effect, mirroring gives the flags negative values, with the zero
> point being the apotome, which itself is notated as an
> exception, /\ ,when its proper mirror should be . For the
> simpler ET's that use no concave or wavy flags, I don't see much of
a
> problem, since the symbol arithmetic usually works in spite of the
> mirroring. But as soon as you introduce concave or wavy flags,
> particularly in twoflag symbols, the symbol arithmetic goes crazy.
As you imply above, it's just different arithmetic, not necessarily crazy.
> If we can get mirroring in the lowernumbered ET's by means of the
> complementation that we already worked out, then that's a
commendable
> goal. But please, let's not dump the concept of consistent symbol
> arithmetic in the process.
So do we agree to always use mirrorred complements when they agree with
both kinds of arithmetic? as in 72ET version 3. But I haven't yet given up
on the idea of using them everywhere.
> If you feel that the best choice of singleshaft symbols is in some
> instances compromised by the need to have doubleshaft complements,
> then I'll work with you to address that problem.
OK. Thanks.
> After all that we went through figuring out the rational
complements,
I think we agreed that the effort expended in designing something should
never be a consideration in whether to abandon it if something better is
possible.
> I can't see replacing that with something in which the order of
> symbols in the second halfapotome makes very little sense?
But it makes perfect sense. Just different sense to what you have become
used to over the past months. It is always an exact mirror image of the
first halfapotome. The same pairs of symbols are _always_ complements of
each other, in _every_ tuning. No exceptions.
We should try to think like someone coming to sagittal notation for the
first time. Or maybe we should actually ask a few people in that boat (e.g.
Ted Mook). Would they rather have complements that were a complete
nobrainer (just flip horizontally and add a stroke [except for !//
(natural) and /\ ]), or would they rather have to learn the order of twice
as many symbols for each new tuning. Clearly, under the current system the
second halfapotome cannot easily be derived from the first, or we wouldn't
be presenting various options and arguing over their merits.
> All
> to "fix" a problem involving notquitematched symbols / and \ in
> a few ET's? I say: "forget it."
This is not the only reason for the mirror proposal. See the list of
advantages I gave above for mirrored apotomes and read their converse as
_disadvantages_ of the current system.
It's just that, so long as I thought the halfapotomes were _always_ going
to match (except for natural), and therefore folks only needed to learn the
first halfapotome of any new tuning, then the mirror solution was, for me,
hovering just below the the threshold of being considered.
So far I have identified 4 solutions to be considered for the second apotome.
1. Always matched, except that !// has various pseudomatches [e.g. () or
(\ ] when /\ is not an exact halfapotome.
2. Always mirrored, except !// is always the pseudomirror of /\, and
/\ is always the pseudomirror of () when /\ is not an exact halfapotome.
3. Mostly matched. As for 1, except that / and \ are mirrored instead of
matched when matching disagrees with an arithmetic that says that / + \
= / + \ = /\
4. Mostly mirrored. As for 2, except that mirroring is replaced by matching
if the mirroring does not agree with an arithmetic where the second shaft
represents the addition of an 11' comma.
I thought we had agreed on 1, but it seems you actually had in mind 3, and
presumably vice versa. Now, pending more argument from you (or anyone
else), my order of preference is 1,2,4,3.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> Hi George,
>
> In fact, (and I've been making gentle noises about this possibility
for a
> some time now), I'm willing to throw away everything we agonised
over with
> regard to rational complements and instead adopt a simple system
that
> applies automatically to all ETs and rational tunings.
>
> I propose that the complement of ab is always ba, except that the
> complement of // (natural) is /\ and the complement of /\
is /\ if it
> represents the same number of steps and () if it represents a
different
> number of steps.
>
> ...
I agree that the biggest disadvantage of the rational complements is
that they are not easy to remember. And the second biggest
disadvantage is that not all of the singleshaft symbols have
rational complements. However, I don't like the symbol arithmetic
being completely different in the doubleshaft symbols; it is counter
intuitive, especially in the fact that more flags make a smaller
alteration, e.g., /~ is a smaller alteration than either / or
~, which occurs in your porposal for 111ET.
After carefully considering your mirroring proposal, I am making a
counterproposal for the determination of apotome complements that
also eliminates both of these biggest disadvantages of the rational
complements. This will look familiar, except that it has one added
clause (to cover wavy flags):
<< For a symbol consisting of:
1) a left flag (or blank)
2) a single (or triple) stem, and
3) a right flag (or blank):
4) convert the single stem to a double (or triple to an X);
5) replace the left and right flags with their opposites according to
the following:
a) a straight flag is the opposite of a blank (and vice versa);
b) a convex flag is the opposite of a concave flag (and vice versa);
c) a wavy flag is its own opposite.
This preserves most of the symbol arithmetic without encountering
either of the two disadvantages you gave for mirrored complements.
It also retains most of the advantages of your mirroring proposal.
> > Advantages:
> >
> > Simple to remember  check!
> > Covers all tunings.  check!
> > Flags are more strongly associated with particular primes because
the flags don't change when the comma is complemented; and
> > No new flag types ever need to be introduced merely to handle
complements.
For straight flags used alone  check! (they just change sides;
otherwise, used in combination, they just disappear)
For convex right flag used alone  check! Observe that ) has as
complement /(, which is the virtual equivalent of ), which may
then be used as the complement  we discussed this previously.
For wavy flags  check! (these completely retain their identities)
For convex left flag used in combination with straight or convex
right flag  check! (these don't require doubleshaft complements)
This covers the situation for most of the lowernumbered ET's, which
should keep the simple things simple.
Convex flags are not used with concave or wavy flags (nor are concave
flags generally used at all) until the notation starts getting more
complicated, and I don't think that the complementation is going to
result in too many new flags in those situations (we would have to
try this to see).
> > Doesn't require / and \ as a special case.  check!
Plus there are the following additional advantages:
Doesn't require /\ to be an exception.
Retains most of the symbol arithmetic used in singleshaft symbols.
The exceptions are with the left convex and left concave flags.
After looking at several ET's in which these are used, I believe that
problems with these can be easily avoided in most cases, especially
in the lowernumbered ET's.
Let me know what you think about this (notsonew) idea.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> I agree that the biggest disadvantage of the rational complements is
> that they are not easy to remember. And the second biggest
> disadvantage is that not all of the singleshaft symbols have
> rational complements. However, I don't like the symbol arithmetic
> being completely different in the doubleshaft symbols; it is
counter
> intuitive, especially in the fact that more flags make a smaller
> alteration, e.g., /~ is a smaller alteration than either / or
> ~, which occurs in your porposal for 111ET.
Yes. More flags being a smaller alteration, is a serious problem with
mirrored complements.
> After carefully considering your mirroring proposal, I am making a
> counterproposal for the determination of apotome complements that
> also eliminates both of these biggest disadvantages of the rational
> complements. This will look familiar, except that it has one added
> clause (to cover wavy flags):
>
> << For a symbol consisting of:
> 1) a left flag (or blank)
> 2) a single (or triple) stem, and
> 3) a right flag (or blank):
> 4) convert the single stem to a double (or triple to an X);
> 5) replace the left and right flags with their opposites according
to
> the following:
> a) a straight flag is the opposite of a blank (and vice versa);
> b) a convex flag is the opposite of a concave flag (and vice
versa);
> c) a wavy flag is its own opposite.
Wavy being its own opposite isn't new either. I think I proposed that
back when we were still deciding what the wavy's would mean.
> This preserves most of the symbol arithmetic without encountering
> either of the two disadvantages you gave for mirrored complements.
>
> It also retains most of the advantages of your mirroring proposal.
>
> > > Advantages:
> > >
> > > Simple to remember  check!
>
> > > Covers all tunings.  check!
>
> > > Flags are more strongly associated with particular primes
because
> the flags don't change when the comma is complemented; and
> > > No new flag types ever need to be introduced merely to handle
> complements.
>
> For straight flags used alone  check! (they just change sides;
> otherwise, used in combination, they just disappear)
> For convex right flag used alone  check! Observe that ) has as
> complement /(, which is the virtual equivalent of ), which may
> then be used as the complement  we discussed this previously.
>
> For wavy flags  check! (these completely retain their identities)
>
> For convex left flag used in combination with straight or convex
> right flag  check! (these don't require doubleshaft complements)
>
> This covers the situation for most of the lowernumbered ET's, which
> should keep the simple things simple.
>
> Convex flags are not used with concave or wavy flags (nor are
concave
> flags generally used at all) until the notation starts getting more
> complicated, and I don't think that the complementation is going to
> result in too many new flags in those situations (we would have to
> try this to see).
>
> > > Doesn't require / and \ as a special case.  check!
>
> Plus there are the following additional advantages:
>
> Doesn't require /\ to be an exception.
>
> Retains most of the symbol arithmetic used in singleshaft symbols.
> The exceptions are with the left convex and left concave flags.
> After looking at several ET's in which these are used, I believe
that
> problems with these can be easily avoided in most cases, especially
> in the lowernumbered ET's.
>
> Let me know what you think about this (notsonew) idea.
Now that we've exhaustively (exhaustingly?) considered the
alternatives, I think it looks absolutely brilliant!!!!
If you have time, could you repost your latest proposals, including
any whose first halfapotome we've agreed on, using these complements?
I've been working on what I call horizontal consistency in the first
halfapotome. I believe it is more important than vertical
consistency. Vertical consistency is between ETs where fifthsize is
the same but number of steps per apotome in one is a multiple of the
other, e.g. 48ET and 96ET. Horizontal consistency is between ETs
that have the same number of steps per apotome, but have slightly
different fifth sizes. ETs with same stepsperapotome and adjacent
fifth sizes, always differ by 7, e.g. 41ET, 48ET, 55ET.
Here are the proposals that have come from that investigation so far.
I've added complements as per the above.
To be notated as subsets of larger ETs:
2,3,4,5,6,7,8,9,10,11,13,14,15,16,18,20,21,23,25,28,30,33,35,40,47.
1 step per apotome
12,19,26: /\
2 steps per apotome
17,24,31,38: /\ /\
45,52: /) /\ [13comma]
3 steps per apotome
22,29: / \ /\
36: ) ) /\
43,50,57,64: /) (\ /\ [13commas]
4 steps per apotome
27: / /) \ /\ [13comma]
34,41,(48?): / /\ \ /\
(48?),55: ~) /\ ~( /\
62: ) /\ (\ /\ [13commas]
69,76: ) ?? (\ /\ [13comma]
5 steps per apotome
32: ) /\ () (\ /\
39,46,53: / /\ () \ /\
60: / ) ) \ /\
67,74: ~) /) (\ ~( /\
81,88: )) /) (\ (( /\ [13commas]
6 steps per apotome
37,44,51: ) / /) \ (\ /\ [13commas]
or
37,44,51: ) )) /) (( ) /\ [13commas]
58: / \ /\ / \ /\
or
58: / ) /\ ) \ /\ [13comma]
65,72,79: / ) /\ ) \ /\
86,93,100: )) ) )\ (\ (( /\ [13commas]
or
86,100: )( ) )\ (\ () /\ [13commas]
93: ( ) )\ (\ /) /\ [13commas]
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote
[#4434]:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >  In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> [GS:]
> > After carefully considering your mirroring proposal, I am making
a
> > counterproposal for the determination of apotome complements
that
> > also eliminates both of these biggest disadvantages of the
rational
> > complements. This will look familiar, except that it has one
added
> > clause (to cover wavy flags):
> >
> > For a symbol consisting of:
> > 1) a left flag (or blank)
> > 2) a single (or triple) stem, and
> > 3) a right flag (or blank):
> > 4) convert the single stem to a double (or triple to an X);
> > 5) replace the left and right flags with their opposites
according to
> > the following:
> > a) a straight flag is the opposite of a blank (and vice versa);
> > b) a convex flag is the opposite of a concave flag (and vice
versa);
> > c) a wavy flag is its own opposite.
>
> Wavy being its own opposite isn't new either. I think I proposed
that
> back when we were still deciding what the wavy's would mean.
That's correct, except that we decided that ) and ( wouldn't have
any complements. In determining the 217ET notation, each time we
avoided ) and didn't use ( for any singleflag symbol smaller than
a halfapotome. That enabled us to arrive at a consistent set of
complements for 217.
> > This preserves most of the symbol arithmetic without encountering
> > either of the two disadvantages you gave for mirrored complements.
> >
> >
> > ... Let me know what you think about this (notsonew) idea.
>
> Now that we've exhaustively (exhaustingly?) considered the
> alternatives, I think it looks absolutely brilliant!!!!
(Whew, that's a relief!)
> If you have time, could you repost your latest proposals, including
> any whose first halfapotome we've agreed on, using these
complements?
For the ones you haven't covered below, I'll do that in a another
message. We'll also have to discuss 217 again  you'll recall that
the way we did it was slightly different before we worked on the
rational complements, and I think we'll need to get that settled
(again) before tackling the other divisions above 100.
> I've been working on what I call horizontal consistency in the
first
> halfapotome. I believe it is more important than vertical
> consistency. Vertical consistency is between ETs where fifthsize
is
> the same but number of steps per apotome in one is a multiple of
the
> other, e.g. 48ET and 96ET. Horizontal consistency is between ETs
> that have the same number of steps per apotome, but have slightly
> different fifth sizes. ETs with same stepsperapotome and adjacent
> fifth sizes, always differ by 7, e.g. 41ET, 48ET, 55ET.
With largenumbered ET's, we might also wish to have some commonality
of symbol usage even among those with different numbers of steps per
apotome. For example, the difference between the 171 and 183
notations will probably involve only the addition of the () symbol
for the latter. And the symbols for 183 might be a subset of those
for 217.
> Here are the proposals that have come from that investigation so
far.
> I've added complements as per the above.
>
> To be notated as subsets of larger ETs:
> 2,3,4,5,6,7,8,9,10,11,13,14,15,16,18,20,21,23,25,28,30,33,35,40,47.
Okay.
> 1 step per apotome
> 12,19,26: /\
Agreed.
> 2 steps per apotome
> 17,24,31,38: /\ /\
Agreed.
> 45,52: /) /\ [13comma]
Okay, that works.
> 3 steps per apotome
> 22,29: / \ /\
Okay. I don't think that there would be a problem having a
difference between the notation for 29 and that for every other
degree of 58. The latter gives \ / /\ (version 1) or )
) /\ (version 2), either of which is easy enough to comprehend,
e.g., in a portion of a piece in 58ET using only every other tone.
> 36: ) ) /\
Agreed!
> 43,50,57,64: /) (\ /\ [13commas]
Agreed! (Yikes, this is too easy!)
> 4 steps per apotome
> 27: / /) \ /\ [13comma]
Okay.
> 34,41,(48?): / /\ \ /\
For 34 & 41, agreed! In 48, the 5 factor is more than 45 percent of
a degree false, so there would not be a strong reason to do 48 this
way. While I would prefer this to what might be done for 55 (below),
I have yet another preference (following).
> (48?),55: ~) /\ ~( /\
I think that 48 and 55 have sufficiently different properties that
there would be no reason to insist on doing them alike. Since I
would do 96 this way:
96: / ) /) /\ (\ ) \ /\
I wouldn't see any problem with doing 48 as a subset of 96,
particularly since 7 and 11 are among the best factors represented in
48:
48: ) /\ ) /\
Now 55 is a real problem, because nothing is really very good for
1deg. The only single flags that will work are ( (17'17) or ( (as
the 29 comma), and the only primes that are 1,3,5,nconsistent are
17, 23, and 29.
If I wanted to minimize the number of flags, I could do it by
introducing only one new flag:
55: ~\ /\ ~ /\
so that 1deg55 is represented by the larger version of the 23' comma
symbol. Or doing it another way would introduce only two new flags:
55: ~~ /\ ~~ /\
The latter has for 1deg the 17+23 symbol, and its actual size (~25.3
cents) is fairly close to 1deg55 (~21.8 cents). Besides, the symbols
are very easy to remember. So this would be my choice.
What was your reason for choosing ~)?
> 62: ) /\ (\ /\ [13commas]
Considering that 7 is so well represented in this division, I would
hesitate to use ) in the notation if it isn't being used as the 7
comma. In fact, I don't think I would want to use ) for a symbol
unless it *did* represent the 7 comma (lest the notation be
misleading), although I would allow its use it in combination with
other flags. So I would prefer this:
62: /) /\ (\ /\ [13commas]
> 69,76: ) ?? (\ /\ [13comma]
Again, I wouldn't use ) by itself defined as a 13comma symbol, but
would choose /) instead:
69,76: /) )\ (\ /\ [13comma]
For 2deg of either 69 or 76, )\ is about the right size.
> 5 steps per apotome
> 32: ) /\ () (\ /\
Very good! The 19 comma is small, but its usage is quite vailid,
considering how accurately 19 is represented. (This is one division
I hadn't looked at before.)
> 39,46,53: / /\ () \ /\
Agreed!
> 60: / ) ) \ /\
I notice that 13 is much better represented than 7, so I would prefer
this (in which the JI symbols also more closely approximate the ET
intervals):
60: / /) (\ \ /\
> 67,74: ~) /) (\ ~( /\
I'm certainly in agreement with the 2deg and 3deg symbols, and if you
must do both ET's alike, then what you have for 1deg would be the
only choice (apart from ( as the 29 comma). We both previously
chose )) for 1deg74 (see message #4412), presumably because it's the
smallest symbol that will work, and I chose ( for 1deg67 (in #4346),
which would give this:
67: ( /) (\ /) /\
74: )) /) (\ (( /\
So what do you prefer?
> 81,88: )) /) (\ (( /\ [13commas]
This is exactly what I have for 74, above. Should we do 67 as I did
it above and do 74, 81, and 88 alike?
On the other hand, why wouldn't 88 be done as a subset of 176?
It is with some surprise that I find that ( is 1deg in both 67 and
81, so 81 could also be done the same way as I have for 67, above.
> 6 steps per apotome
> 37,44,51: ) / /) \ (\ /\ [13commas]
> or
> 37,44,51: ) )) /) (( ) /\ [13commas]
For 51 I had something a bit simpler (using lower primes):
51: ) / /) \ ) /\
> 58: / \ /\ / \ /\
> or
> 58: / ) /\ ) \ /\ [13comma]
I think I would avoid your version2  this is another instance where
it's too easy to be misled into thinking that ) is the 7 comma. If
we wanted to avoid the confusability of all straight flags, we could
try:
58: / /) /\ (\ \ /\
Here ) would be kept as the 7 comma and ( would be the 11'7 comma
of 2deg58. However, I think that it would be too easy to forget
that /) and (\ aren't representing ratios of 13. So I think that
the safest choice is version 1  all straight flags.
> 65,72,79: / ) /\ ) \ /\
Agreed! (After all we went through before about 72, this one is now
almost a nobrainer!)
> 86,93,100: )) ) )\ (\ (( /\ [13commas]
> or
> 86,100: )( ) )\ (\ () /\ [13commas]
> 93: ( ) )\ (\ /) /\ [13commas]
I would do 93ET and 100ET as subsets of 186ET and 200ET,
respectively.
For 86, I wouldn't use ) by itself as anything other than the 7
comma, as explained above, but would use convex flags for symbols
that are actual ratios of 13. So this is how I would do it:
86: ~~ /) (~ (\ ~~ /\ [13commas and 23comma]
The two best primes are 13 and 23, so there is some basis for
defining ~ as the 23 flag. In any event, I believe that (~ can be
a strong candidate for half an apotome if neither /\ nor /) nor (\
can be used.
George
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4405]:
>
> I am instead inclined to totally ignore rational complements with
regard to
> ETs, especially for the lower numbered ones. One reason is that I
feel that
> the choice of doubleshaft symbols cannot in any way be allowed to
> influence the choice of singleshaft. One must first choose the
best set of
> single shaft symbols (ignoring complements) since some users will
have no
> interest in the doubleshaft symbols and should not be penalised
for it.
>
> In fact, (and I've been making gentle noises about this possibility
for a
> some time now), I'm willing to throw away everything we agonised
over with
> regard to rational complements and instead adopt a simple system
that
> applies automatically to all ETs and rational tunings.
I am replying to an earlier message with a different proposal, now
that I have had the experience of actually trying to do a
considerable number of ET's (both small and large) following both
your mirroring proposal and my counterproposal. This experience can
be summarized in the following observations:
1) I find that all of the ET's below 100 on which we agree (per
message #4443) can be done, without exception, in exactly the same
way using the rational complementation scheme that we just abandoned.
2) The ET's on which we did not agree fall into two categories:
a) Those that are fairly simple to do (but have more than one
possible way), on which we have not yet formally agreed; all of these
can also be done in exactly the same way using the rational
complementation scheme.
b) Those that are more difficult or not so obvious; these are the
lessknown, littleused, and justplainweird ET's, for which the
choice of symbols amounts to "we do the best we can."
3) I tried some of the easier ET's above 100 and found that at least
half of them ended up with unavoidable inconsistencies in symbol
arithmetic, and those that had a matching sequence of symbols in the
halfapotomes were a rarity. I consider this a rather high price to
pay for an easytoremember complementation scheme.
4) When I previously did these ET's above 100 within the rational
complementation scheme, I was able to do all of these with completely
consistent symbol arithmetic and most of them with either a matching
sequence of symbols or rational complementation  and sometimes both.
5) Our chief problem with the rational complements is that they are
not very easy to remember. However, when you consider that this
statement applies *only* to commas above the 13 limit, I don't think
that this is a major drawback. There are only 8 pairs of rational
complements to remember, and nearly half of them can be formulated
into rules (represented symbolically as):
[ / <=> b ,  <=>  , \ <=> b ]
/ <=> \
\ <=> /
nat. <=> /\
[ b <=> b ,  <=>  , ) <=> ) ]
) <=> )
[ ( <=> ) ,  <=>  , ~ <=> b ]
) <=> (~
)~ <=> (
( <=> )~
(~ <=> )
I would therefore recommend going back to the rational
complementation system and doing the ET's that way as well.
Or, if you like, we could do them both ways and then decide.
I would be agreeable to doing all of the ET's (with the rational
complementation scheme) using the symbols that we agreed on in
message #4443.
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> I would therefore recommend going back to the rational
> complementation system and doing the ET's that way as well.
Agreed. Provided we _always_ use rational complements, whether this
results in matching halfapotomes or not.
> Or, if you like, we could do them both ways and then decide.
No need.
> I would be agreeable to doing all of the ET's (with the rational
> complementation scheme) using the symbols that we agreed on in
> message #4443.
OK.
I will respond to your suggestions for the remaining ones of 6 or less
steps per apotome when I get more time. Then move on to
7 steps per apotome
42,49,56,63,70,77,84,91,98,105
8 steps per apotome
54,61,68,75,82,89,96,103,110,117
9 steps per apotome
59,66,73,80,87,94,101,108,115,122,129
10 steps per apotome
71,78,85,92,99,106,113,120,127,134,141
etc.
I think we can do some with 23 steps per apotome, maybe even 25.
 In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > I would therefore recommend going back to the rational
> > complementation system and doing the ET's that way as well.
>
> Agreed. Provided we _always_ use rational complements, whether this
> results in matching halfapotomes or not.
In other words, you would prefer having this:
152 (76 ss.): ) ~ / \ ~) /) /\ () (\ ~ /
\ ~) (~ /\
instead of this:
152 (76 ss.): ) ~ / \ /~ /) /\ () (\ ~ /
\ /~ /) /\
even if it isn't as easy to remember.
I suggest that you try some more ET's before insisting on rational
complements across the board. In addition to less memorable symbol
sequences, strict rational complementation will also result in some
bad symbol arithmetic in instances where the complement symbols are
not consistent in some ET's. I will accept some symbol arithmetic
inconsistency (e.g., with ) in 72ET), if it isn't too
disorienting, but I think that users will need all the help they can
get to keep the symbols straight in the larger ET's, and too many
flags and bad symbol arithmetic aren't going to help.
> > I would be agreeable to doing all of the ET's (with the rational
> > complementation scheme) using the symbols that we agreed on in
> > message #4443.
>
> OK.
I erroneously stated that everything that we last agreed on (using
what I would call "inverse complements") would stay the same.
However, there is one exception. This:
32: ) /\ () (\ /\ (DK  inverse complements
would become this:
32: ) /\ () (~ /\ (rational complements)
To this I am agreeable.
> I will respond to your suggestions for the remaining ones of 6 or
less
> steps per apotome when I get more time. ...
My time will also be rather limited for at least the next several
days, so I will not be working on this a great deal.
George
At 01:03 18/06/02 0000, you wrote:
> In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>> > I would therefore recommend going back to the rational
>> > complementation system and doing the ET's that way as well.
>>
>> Agreed. Provided we _always_ use rational complements, whether this
>> results in matching halfapotomes or not.
>
>In other words, you would prefer having this:
>
>152 (76 ss.):
) ~ / \ ~) /) /\ ()
(\ ~ / \ ~) (~ /\
>
>instead of this:
>
>152 (76 ss.):
) ~ / \ /~ /) /\ ()
(\ ~ / \ /~ /) /\
>
>even if it isn't as easy to remember.
OK. I think you've got me there. :) Remember I said I thought we shouldn't let complements cause us to choose an inferior set of singleshaft symbols, because some people won't use the purely saggital complements. I think we both agree that /~ is a better choice for 5deg152 than ~) since it introduces fewer new flags and puts the ET value closer to the rational value.
I don't think we have defined a rational complement for /~ because it isn't needed for rational tunings. But if we look at complements consistent with 494ET (as all the rational complements are) the only complement for /~ is ~(. So we end up with
152 (76 ss.):
) ~ / \ /~ /) /\ ()
(\ ~( / \ ~) /) /\
But this is bad because the flag sequence is different in the two halfapotomes _and_ ~( = 10deg152 is inconsistent _and_ too many flag types. So you're right. I don't want to use strict rational complements for this, particularly with its importance in representing 1/3 commas. I'd rather have
>152 (76 ss.):
) ~ / \ /~ /) /\ ()
(\ ~ / \ /~ /) /\
I note that 76ET can also be notated using its native fifth, as you give (and I agree) below.
>I suggest that you try some more ET's before insisting on rational
>complements across the board. In addition to less memorable symbol
>sequences, strict rational complementation will also result in some
>bad symbol arithmetic in instances where the complement symbols are
>not consistent in some ET's. I will accept some symbol arithmetic
>inconsistency (e.g., with ) in 72ET), if it isn't too
>disorienting, but I think that users will need all the help they can
>get to keep the symbols straight in the larger ET's, and too many
>flags and bad symbol arithmetic aren't going to help.
Point taken.
>> > I would be agreeable to doing all of the ET's (with the rational
>> > complementation scheme) using the symbols that we agreed on in
>> > message #4443.
>>
>> OK.
>
>I erroneously stated that everything that we last agreed on (using
>what I would call "inverse complements") would stay the same.
>However, there is one exception. This:
>
>32: ) /\ () (\ /\ (DK  inverse complements
>
>would become this:
>
>32: ) /\ () (~ /\ (rational complements)
>
>To this I am agreeable.
That's ok with me too.
Now to start on the others with 6 or less steps per apotome.
I won't necessarily include the doubleshaft symbols from here on. You should assume they correspond to the rational complements.
We are really having problems with 1deg48 aren't we?
You wrote:
>I think that 48 and 55 have sufficiently different properties that
>there would be no reason to insist on doing them alike. Since I
>would do 96 this way:
>
>96: / ) /) /\ (\ ) \ /\
>
>I wouldn't see any problem with doing 48 as a subset of 96,
>particularly since 7 and 11 are among the best factors represented in
>48:
>
>48: ) /\ ) /\
We agree 48 should be every second step of 96, but we haven't agreed on 96 yet.
I agree 48 doesn't _need_ to be the same as either 41 or 55, but it would be good to minimise the number of different notations for all the scales with 4 steps to the apotome.
Both ~) and ~\ are consistently 1 degree of 48, 55 and 62ET, but of these only ~) is also 2 degrees of 96ET.
That's one reason why I favour ~).
But lets forget 55 and 62 for now. You propose to use ) which is certainly correct as the 7comma for both 48 and 96ET. Why would I want to add the ~ 17flag to it when this is zero steps?
One problem is that we're already using ) as one degree of 36ET and 2 degrees of 72ET. People will naturally attach the meaning of 1/3 semitone to it in this application, and may find it confusing if 48 and 96ET use it for 1/4 semitone.
This opens a whole other can of worms regarding notation relative to 12ET. Lots of people would like to notate their tunings (even those which are not n*12ETs) as deviations from 12ET, rather than as deviations from a chain of the tuning's own native fifths (or it may have none).
Since people are going to try to do it anyway, shouldn't we look at standardising a consistent way of doing it? Some time ago I investigated this in depth and I now offer a first pass at a spreadsheet that does it semiautomatically. And, you guessed it, it requires 1deg48 and 2deg96 to be ~).
/tuningmath/files/Dave/Notating12ETDeviations.xls.zip
If you examine the formulae in this spreadsheet you will see that the principle is that each symbol is given, in a lookup table, a range of cents deviations that it covers. In general the ranges overlap, but there is a strict order of precedence to resolve the cases where more than one symbol could notate the same degree. Determining the ranges was quite tedious, but the main requirement is to ensure that the symbols actually agree with their comma values, given 12ET fifths. e.g. the changover between one symbol and the next, at the same precedence level, occurs at the point equidistant from their two comma values relative to a chain of 12ET fifths.
But how did I choose which symbols to use in the first place? It's so long ago I've almost forgotten, but the basic idea was for example, to look at all the n*12ETs that contained a 25c step and find which symbol corresponded to 25 cents in all of them, and so on.
Here's what it gives for all the n*12ETs whose best fifth is the 12ET fifth. The dots indicate degrees that cannot be notated.
12:
24: /\
36: )
48: ~) /\
60: / \
72: / ) /\
84: / ) /)
96: / ~) \ /\
108: / /( ) /)
120: / ( ) \ /\
132: ~( / ) \ /)
144: ~( / ~) ) /) /\
156: ~( / ~) ) \ /)
168: ~( / /( ) \ /) /\
180: ~( / ( ~) ) \ /)
192: ~( / ( ~) ) \ /) /\
204: ~( / ( ~) ) \ () /\
216: ~( / ( /( ~) ) \ /) /\
228: ~( ( / /( ~) ) \ /) () /\
240: ~( ( / ( ~) ) ~\ \ /) /\
252: ~( ( / ( ~) ) ~\ \ /) () /\
264: ~( ( / ( /( ~) ) \ . /) /\
276: ~( ( / ( /( ~) ) ~\ \ /) () /\
288: ~( ( / . ( ~) ) ~\ \ /) . ()
300: ~( ( / . ( ~) ) ~\ \ /) . () /\
Notice that this scheme only uses 6 types of flag since it doesn't go beyond 17limit. Of course one has to get used to the fact that ~ is negative (5.0 cents).
Notice that 276ET is the largest that can be fully notated, and that 12,24,36,72 are as previously agreed. We haven't agreed on 60ET yet, but the proposal above is different from what either of us suggested recently.
Notice that 144ET has bad flag arithmetic, since / and ) [7 flag] are 2 and 4 steps respectively and thereby agree with 72ET, but /) is 5 steps and must be interpreted as the 13 flag. If we are not willing to do this, then we must accept that 144ET cannot be fully notated in a manner consistent with 72ET, simply because we don't have a separate symbol for the 13comma, and the 13schisma doesn't vanish.
>Now 55 is a real problem, because nothing is really very good for
>1deg. The only single flags that will work are ( (17'17) or ( (as
>the 29 comma), and the only primes that are 1,3,5,nconsistent are
>17, 23, and 29.
>
>If I wanted to minimize the number of flags, I could do it by
>introducing only one new flag:
>
>55: ~\ /\ ~ /\
>
>so that 1deg55 is represented by the larger version of the 23' comma
>symbol. Or doing it another way would introduce only two new flags:
>
>55: ~~ /\ ~~ /\
>
>The latter has for 1deg the 17+23 symbol, and its actual size (~25.3
>cents) is fairly close to 1deg55 (~21.8 cents). Besides, the symbols
>are very easy to remember. So this would be my choice.
I would not use a 23 comma to notate this when it can be done in 17limit. Luckily ~\ works for 1 step as the 17+(115) comma (which also agrees with 2 steps of 110ET). So I go for your first (min flags) suggestion:
55: ~\ /\
>What was your reason for choosing ~)?
Probably only because I could make it agree with 48ET.
>> 62: ) /\ (\ /\ [13commas]
>
>Considering that 7 is so well represented in this division, I would
>hesitate to use ) in the notation if it isn't being used as the 7
>comma. In fact, I don't think I would want to use ) for a symbol
>unless it *did* represent the 7 comma (lest the notation be
>misleading), although I would allow its use it in combination with
>other flags.
Good point.
> So I would prefer this:
>
>62: /) /\ (\ /\ [13commas]
Agreed.
>> 69,76: ) ?? (\ /\ [13comma]
>
>Again, I wouldn't use ) by itself defined as a 13comma symbol, but
>would choose /) instead:
>
>69,76: /) )\ (\ /\ [13commas]
>
>For 2deg of either 69 or 76, )\ is about the right size.
Agreed.
I note that 62, 69 and 76 are all 1,3,9inconsistent and might also be notated as subsets of 2x or 3x ETs.
We should take a look at the n*19ET family now that it is complete.
19: /\
38: /\ /\
57: /) (\ /\ [13commas]
76: /) )\ (\ /\ [13commas]
>> 60: / ) ) \ /\
>
>I notice that 13 is much better represented than 7, so I would prefer
>this (in which the JI symbols also more closely approximate the ET
>intervals):
>
>60: / /) (\ \ /\
As described above, this would not work in with the other n*12ETs. My current proposal uses neither 7 nor 13 comma symbols.
60: / \ / \ /\
>> 67,74: ~) /) (\ ~( /\
>
>I'm certainly in agreement with the 2deg and 3deg symbols, and if you
>must do both ET's alike, then what you have for 1deg would be the
>only choice (apart from ( as the 29 comma). We both previously
>chose )) for 1deg74 (see message #4412), presumably because it's the
>smallest symbol that will work, and I chose ( for 1deg67 (in #4346),
>which would give this:
>
>67: ( /) (\ /) /\
>74: )) /) (\ (( /\
>
>So what do you prefer?
I prefer yours, but I'm uncertain about the complement used for 4 steps of 74.
>> 81,88: )) /) (\ (( /\ [13commas]
>
>This is exactly what I have for 74, above. Should we do 67 as I did
>it above and do 74, 81, and 88 alike?
Yes.
>On the other hand, why wouldn't 88 be done as a subset of 176?
I have a reason to do both 81 and 88 as subsets, apart from the fact that they are 1,3,9inconsistent. When using their native fifths they need a single shaft symbol for 4 steps and none is available.
>It is with some surprise that I find that ( is 1deg in both 67 and
>81, so 81 could also be done the same way as I have for 67, above.
Better to do it the same as 74 and 88 (or as a subset).
>> 6 steps per apotome
>> 37,44,51: ) / /) \ (\ /\ [13commas]
>> or
>> 37,44,51: ) )) /) (( ) /\ [13commas]
>
>For 51 I had something a bit simpler (using lower primes):
So are you agreeing to one of these for 37 and 44? Presumably not the second one because of ) not being the 7comma. And with rational complements?
>51: ) / /) \ ) /\
OK.
>> 58: / \ /\ / \ /\
>> or
>> 58: / ) /\ ) \ /\ [13comma]
>
>I think I would avoid your version2  this is another instance where
>it's too easy to be misled into thinking that ) is the 7 comma. If
>we wanted to avoid the confusability of all straight flags, we could
>try:
>
>58: / /) /\ (\ \ /\
>
>Here ) would be kept as the 7 comma and ( would be the 11'7 comma
>of 2deg58. However, I think that it would be too easy to forget
>that /) and (\ aren't representing ratios of 13. So I think that
>the safest choice is version 1  all straight flags.
Agreed:
58: / \ /\ / \ /\
>> 86,93,100: )) ) )\ (\ (( /\ [13commas]
>> or
>> 86,100: )( ) )\ (\ () /\ [13commas]
>> 93: ( ) )\ (\ /) /\ [13commas]
>
>I would do 93ET and 100ET as subsets of 186ET and 200ET,
>respectively.
I can agree to that for 100ET since there is no singleshaft symbol for 5 steps, but it is of course 2*50, and 93 is 3*31, so the fifth sizes are quite acceptable.
>For 86, I wouldn't use ) by itself as anything other than the 7
>comma, as explained above,
I totally agree we should avoid this in all cases.
> but would use convex flags for symbols
>that are actual ratios of 13. So this is how I would do it:
>
>86: ~~ /) (~ (\ ~~ /\ [13commas and 23comma]
>
>The two best primes are 13 and 23, so there is some basis for
>defining ~ as the 23 flag. In any event, I believe that (~ can be
>a strong candidate for half an apotome if neither /\ nor /) nor (\
>can be used.
I have no argument about the even steps (they agree with 43 and 50ET). But again I don't see the need to use a 23comma. We have already used )\ for a halfapotome in the case of 69 and 76ETs. It works here too. 86ET is 1,3,7,13,19consistent. So why not:
86,93,100: )) /) )\ (\ ?? /\ [13commas]
We can now consider the 31ET family.
31: /\ /\
62: /) /\ (\ /\ [13commas]
93: )) /) )\ (\ ?? /\ [13commas]
and compare it to the 19ET family
19: /\
38: /\ /\
57: /) (\ /\ [13commas]
76: /) )\ (\ /\ [13commas]
Whew!
With that I must sadly inform you that I will not be able to contribute to this discussion again for quite some time. I need to get seriously involved in an electronic design project for some months now. The trouble is I'm a tuning theory addict. I can't have just a little.
George, I strongly encourage you to present what we've agreed upon so far, to the wider community for comment.
Regards,
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4532]:
> [a lengthy reply] ...
>
> Whew!
>
> With that I must sadly inform you that I will not be able to
contribute to this discussion again for quite some time. I need to
get seriously involved in an electronic design project for some
months now. The trouble is I'm a tuning theory addict. I can't have
just a little.
>
> George, I strongly encourage you to present what we've agreed upon
so far, to the wider community for comment.
>
> Regards,
>  Dave Keenan
Dave,
Thank you for your latest comments and ideas.
It will take me some time to digest and thoughtfully consider all of
what you discussed. I have also been busy with other things for the
past few weeks and will not be looking at this in detail for at least
a few more, at which time I will be able to review all of this with a
fresher perspective.
So I expect that it will be at least a month before I present
anything about what we have accomplished. And once that's started, I
imagine that it's going to take a while to cover, given that there
will probably be a lot of questions.
So let's both enjoy our summer break.
Best regards,
George
Note: Dave Keenan has kindly agreed to work with me (offlist) on
the notation project again for a short time to deal with the latest
modifications that I am proposing. (Will there ever be an end to
this? I think there's light at the end of the tunnel.) Otherwise, I
expect that he will continue to take time off from the Tuning List.
We will be posting our correspondence here to maintain a complete
record of how the notation is being developed.
I have a long reply to his last message, and I will post this in
installments. GS
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4532]:
> At 01:03 18/06/02 0000, you wrote:
> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> > In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
> >> > I would therefore recommend going back to the rational
> >> > complementation system and doing the ET's that way as well.
> >>
> >> Agreed. Provided we _always_ use rational complements, whether
this
> >> results in matching halfapotomes or not.
> >
> >In other words, you would prefer having this:
> >
> >152 (76 ss.): ) ~ / \ ~) /) /\ () (\
~ / \ ~) (~ /\
> >
> >instead of this:
> >
> >152 (76 ss.): ) ~ / \ /~ /) /\ () (\
~ / \ /~ /) /\
> >
> >even if it isn't as easy to remember.
>
> OK. I think you've got me there. :) Remember I said I thought we
shouldn't let complements cause us to choose an inferior set of
singleshaft symbols, because some people won't use the purely
saggital complements. I think we both agree that /~ is a better
choice for 5deg152 than ~) since it introduces fewer new flags and
puts the ET value closer to the rational value.
New Rational Complements Â– Part 1

Now that I've talked you into this, I'm going to have to try to talk
you out of it (to some extent) because of something that I have come
to realize over these past few weeks. There's nothing like some time
off to create a new perspective: I have come back to this as if I
were a JI composer new to the notation who is asking the
question, "How would I notate a 15limit tonality diamond?" And now
that I've taken a fresh look at the notation, I came up with some
ideas on how to improve a few things.
First of all, here is how I was able to notate all of the 15odd
limit consonances taking C as 1/1. (Don't bother to look through all
of this now; I'll be referring to many of these below, so this
listing is just given for reference.)
1/1 = C 2/1 = C
3/2 = G 4/3 = F
5/3 = A\! 6/5 = Eb/ or E!!/
5/4 = E\! 8/5 = Ab/ or A!!/
7/4 = Bb!) or B!!!) 8/7 = D)
7/5 = Gb!( or G!!!( 10/7 = F#( or F(
7/6 = Eb!) or E!!!) 12/7 = A)
9/5 = Bb/ or B!!/ 10/9 = D\!
9/7 = E) 14/9 = Ab!) or A!!!)
9/8 = D 16/9 = Bb or B\!!/
11/6 = B(!) 12/11 = D\!/
11/7 = G#(! or G)~ 14/11 = Fb( or F)~
11/8 = F/\ 16/11 = G\!/
11/9 = E(!) 18/11 = A\!/
11/10 = D\!~ 20/11 = Bb/~ or B~!!(
13/7 = B\!~ 14/13 = Db/~ or D~!!(
13/8 = A(!/ 16/13 = E\!)
13/9 = G(!/ 18/13 = F(\
13/10 = F\\! 20/13 = G//
13/11 = Eb!( or E!!!( 22/13 = A(
13/12 = D(!/ 24/13 = B\)
15/8 = B\! 16/15 = Db/ or D!!/
15/11 = F/~ 22/15 = G/!~
15/13 = D// 26/15 = Bb\\! or B\\!!!
15/14 = C#( or C( 28/15 = Cb!( or C!!!(
In determining the notation for all of the 15oddlimit consonances I
found that the symbols of the sagittal notation fall into three
groups: 1) those that are very useful for the 15 limit, 2) those that
are useful only for primes above 13, and 3) those for which I haven't
yet found a use. (Symbols not having rational complements should be
in the third category, which is not the case at present.) This is
relevant to your very next statement:
> I don't think we have defined a rational complement for /~ because
it isn't needed for rational tunings.
On the contrary, I found that /~ is in fact quite useful for
rational tunings (see above table of ratios), but its lack of a
rational complement is a problem. To remedy this, I propose ~( as
its rational complement. With C as 1/1, the following ratios would
then use these two symbols (which also appear in the table of ratios
above):
11/10 = D\!~
20/11 = Bb/~ or B~!!(
15/11 = F/~
13/7 = B\!~
14/13 = Db/~ or D~!!(
In effect, /~ functions not only as the 5+23 comma (~38.051c), but
also as the 11'5 comma (~38.906c) and the 13'7 comma (~38.073c)
This would replace (( <> ~( as rational complements. I found
that (( is not needed for any rational intervals in the 15odd
limit, so this has no adverse consequences. (However, it leaves the
23' comma without a rational complement; I will address that problem
below.) The new pair of complements that I am proposing also has a
lower offset (0.49 cents) than the old (1.03 cents), so, apart from
the 23' comma, I can't think of a single reason not to do this.
The reverse pair of complements, ~( <> /~, would be used for the
following ratios of 17:
17/16 = Db~( or D\!!~
17/12 = Gb~( or G\!!~
17/9 = Cb~( or C\!!~
32/17 = B~!(
24/17 = F#~!( or F/~
18/17 = C#~!( or C/~
All of this is going to affect how we will want to notate not only
152, but also other ET's, including 217. (More about this later.)
> But if we look at complements consistent with 494ET (as all the
rational complements are) the only complement for /~ is ~(. So we
end up with
>
> 152 (76 ss.): ) ~ / \ /~ /) /\ () (\ ~
( / \ ~) /) /\
>
> But this is bad because the flag sequence is different in the two
halfapotomes _and_ ~( = 10deg152 is inconsistent _and_ too many
flag types. So you're right. I don't want to use strict rational
complements for this, particularly with its importance in
representing 1/3 commas. I'd rather have
>
> > 152 (76 ss.): ) ~ / \ /~ /) /\ () (\
~ / \ /~ /) /\
I don't follow the part about ~( = 10deg152 being inconsistent:
The 17' comma ~( is 2deg, and the apotome (15deg) minus the
unidecimal diesis (7deg) is () = 8deg, so () + ~( = ~( = 10deg.
So I would replace ~, the 23comma, with ~(, the 17' comma, which
gives:
152 (76 ss.): ) ~( / \ /~ /) /\ () (\ ~( /
\ /~ /) /\ (RC w/ 14deg AC)
This not only uses a symbol ~( that corresponds to a lower prime
symbol for 2deg, but also uses a rational symbol ~( that has
meaning for certain ratios of 11 and 13, as also will /~. The 14deg
symbol /) is not the rational complement of 1deg ), but its offset
(~1.12 cents) is small enough that it would have qualified as a
rational complement (RC) if we had no other choice. I'll call this
an alternate complement (AC)  one that may be used for notating an
ET in the absence of a RC consistent in that ET, but which is not
used for rational notation.
The principle that I am advancing here is that there is another goal
or rule that should take precedence over that of an easytomemorize
symbol sequence  symbols which are used to represent JI consonances
should be used in preference to those that can be expressed only as
sums of commaflags. These are the symbols that will be used for JI
most frequently, and they will therefore (through repeated use)
become *the most familiar* ones. And these are the symbols that
should have first priority in the assignment of rational
complements. This is why I want to eliminate (( in the rational
complement scheme  it is the (13'(115))+(17'17) comma or, if you
prefer, the (11'7)+(17'17) comma, neither of which is simple enough
to indicate that it would ever be used to notate a rational interval;
and none of the 15limit consonances (relative to C=1/1) require it.
This will be continued, following a short digression about 76ET.
> I note that 76ET can also be notated using its native fifth, as
you give (and I agree) below.
In the process of looking over what we discussed regarding 76 (in
connection with 62 and 69 a bit later in your message #4532), I
noticed that it was given above as a subset of 152. I then noticed
how bad the 5limit is in 76 and wondered why it was being considered
on its own.
I then reviewed our correspondence. In response to a question from
Paul about 76ET, you told him this (in message #4272):
<< The native bestfifth of 76ET is not suitable to be used a
notational fifth because, among other reasons, it is not 1,3,9
consistent (i.e. its best 4:9 is not obtained by stacking two of its
best 2:3s) and I figure folks have a right to expect C:D to be a best
4:9 when commas for primes greater than 9 are used in the notation.
So 76ET will be notated as every second note of 152ET. >>
It gets even worse than this: not only is 3 over 45 percent of a
degree false, but 5 deviates even more.
Your next mention of 76ET was in message #4434, in which you
treated the divisions of the apotome systematically:
4 steps per apotome ...
69,76: ) ?? (\ /\ [13comma]
From that point we had 76 listed both as a subset of 152 and with
69. So after looking at all this, which will it be? (I would prefer
it as the subset.)
(To be continued.)
At 11:52 13/08/02 0700, George Secor wrote:
>From: George Secor, 8/13/2002 (tuningmath #4577)
>Subject: A common notation for JI and ETs
>
>Note: Dave Keenan has kindly agreed to work with me (offlist) on the
>notation project again for a short time to deal with the latest
>modifications that I am proposing. (Will there ever be an end to this?
> I think there's light at the end of the tunnel.) Otherwise, I expect
>that he will continue to take time off from the Tuning List. We will
>be posting our correspondence here to maintain a complete record of how
>the notation is being developed.
That's right. I am not reading any lists. Only CCing my replies to George, to tuningmath.
>I have a long reply to his last message, and I will post this in
>installments. GS
>
> In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4532]:
>> At 01:03 18/06/02 0000, you wrote:
>> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>> > In tuningmath@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>> >>  In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
>> >> > I would therefore recommend going back to the rational
>> >> > complementation system and doing the ET's that way as well.
>> >>
>> >> Agreed. Provided we _always_ use rational complements, whether
>this
>> >> results in matching halfapotomes or not.
>> >
>> >In other words, you would prefer having this:
>> >
>> >152 (76 ss.): ) ~ / \ ~) /) /\ () (\ ~
>/ \ ~) (~ /\
>> >
>> >instead of this:
>> >
>> >152 (76 ss.): ) ~ / \ /~ /) /\ () (\ ~
>/ \ /~ /) /\
> > >
>> >even if it isn't as easy to remember.
>>
>> OK. I think you've got me there. :) Remember I said I thought we
>shouldn't let complements cause us to choose an inferior set of
>singleshaft symbols, because some people won't use the purely saggital
>complements. I think we both agree that /~ is a better choice for
>5deg152 than ~) since it introduces fewer new flags and puts the ET
>value closer to the rational value.
>
>New Rational Complements Â– Part 1
>
>
>Now that I've talked you into this, I'm going to have to try to talk
>you out of it (to some extent) because of something that I have come to
>realize over these past few weeks. There's nothing like some time off
>to create a new perspective: I have come back to this as if I were a JI
>composer new to the notation who is asking the question, "How would I
>notate a 15limit tonality diamond?"
An excellent question. I think I posed a similar one earlier, but only considering the 11limit diamond.
>And now that I've taken a fresh
>look at the notation, I came up with some ideas on how to improve a few
>things.
>
>First of all, here is how I was able to notate all of the 15oddlimit
>consonances taking C as 1/1. (Don't bother to look through all of this
>now; I'll be referring to many of these below, so this listing is just
>given for reference.)
...
That's marvellous, except of course it looks like gobbledygook when up to 5 ASCII symbols are being used to represent a single sagittal symbol. How big is the biggest schisma involved?
>> I don't think we have defined a rational complement for /~ because
>it isn't needed for rational tunings.
>
>On the contrary, I found that /~ is in fact quite useful for rational
>tunings (see above table of ratios), but its lack of a rational
>complement is a problem. To remedy this, I propose ~( as its
>rational complement.
Fair enough, and yes, that would seem the obvious complement.
>With C as 1/1, the following ratios would then
>use these two symbols (which also appear in the table of ratios above):
>
>11/10 = D\!~
>20/11 = Bb/~ or B~!!(
>15/11 = F/~
>13/7 = B\!~
>14/13 = Db/~ or D~!!(
>
>In effect, /~ functions not only as the 5+23 comma (~38.051c), but
>also as the 11'5 comma (~38.906c) and the 13'7 comma (~38.073c)
OK, so a 0.86 c schisma. I can certainly live with that for such obscure ratios.
>This would replace (( <> ~( as rational complements. I found that
>(( is not needed for any rational intervals in the 15odd limit, so
>this has no adverse consequences. (However, it leaves the 23' comma
>without a rational complement; I will address that problem below.) The
>new pair of complements that I am proposing also has a lower offset
>(0.49 cents) than the old (1.03 cents), so, apart from the 23' comma,
>I can't think of a single reason not to do this.
Me neither. Apart from the 23' comma.
We could resurrect ~), with two left flags, as the complement of the 23' comma. It isn't like a lot of people really care about ratios of 23 anyway. We already made a good looking bitmap for ~) with the wavy and the concave making a loop.
>The reverse pair of complements, ~( <> /~, would be used for the
>following ratios of 17:
>
> 17/16 = Db~( or D\!!~
> 17/12 = Gb~( or G\!!~
> 17/9 = Cb~( or C\!!~
> 32/17 = B~!(
> 24/17 = F#~!( or F/~
> 18/17 = C#~!( or C/~
>
>All of this is going to affect how we will want to notate not only 152,
>but also other ET's, including 217. (More about this later.)
If rational complements don't have to be consistent with 217ET any more, how about making rational complements consistent with 665ET, as proposed earlier?
>> But if we look at complements consistent with 494ET (as all the
>rational complements are) the only complement for /~ is ~(. So we
>end up with
>>
>> 152 (76 ss.): ) ~ / \ /~ /) /\ () (\ ~(
>/ \ ~) /) /\
>>
>> But this is bad because the flag sequence is different in the two
>halfapotomes _and_ ~( = 10deg152 is inconsistent _and_ too many flag
>types. So you're right. I don't want to use strict rational complements
>for this, particularly with its importance in representing 1/3 commas.
>I'd rather have
>>
>> > 152 (76 ss.): ) ~ / \ /~ /) /\ () (\ ~
>/ \ /~ /) /\
>
>I don't follow the part about ~( = 10deg152 being inconsistent: The
>17' comma ~( is 2deg, and the apotome (15deg) minus the unidecimal
>diesis (7deg) is () = 8deg, so () + ~( = ~( = 10deg.
My mistake. Sorry.
>So I would replace ~, the 23comma, with ~(, the 17' comma,
Well of course I think of ~ as 19'19 when notating ETs.
>which gives:
>
>152 (76 ss.): ) ~( / \ /~ /) /\ () (\ ~( / \
> /~ /) /\ (RC w/ 14deg AC)
Unfortunately this gives up a desirable property: Monotonicity of flagspersymbol with scale degree.
>This not only uses a symbol ~( that corresponds to a lower prime
>symbol for 2deg, but also uses a rational symbol ~( that has meaning
>for certain ratios of 11 and 13, as also will /~. The 14deg symbol
>/) is not the rational complement of 1deg ), but its offset (~1.12
>cents) is small enough that it would have qualified as a rational
>complement (RC) if we had no other choice. I'll call this an alternate
>complement (AC)  one that may be used for notating an ET in the
>absence of a RC consistent in that ET, but which is not used for
>rational notation.
Fair enough.
>The principle that I am advancing here is that there is another goal or
>rule that should take precedence over that of an easytomemorize
>symbol sequence  symbols which are used to represent JI consonances
>should be used in preference to those that can be expressed only as
>sums of commaflags. These are the symbols that will be used for JI
>most frequently, and they will therefore (through repeated use) become
>*the most familiar* ones.
But many people using ETs couldn't care less about JI, so why should rational approximations take precedence over mnemonics, particularly if they only involve ratios as uncommon as 5:11 and 7:13?
>And these are the symbols that should have
>first priority in the assignment of rational complements.
Yes. I can accept that.
> This is why
>I want to eliminate (( in the rational complement scheme  it is the
>(13'(115))+(17'17) comma or, if you prefer, the (11'7)+(17'17)
>comma, neither of which is simple enough to indicate that it would ever
>be used to notate a rational interval; and none of the 15limit
>consonances (relative to C=1/1) require it.
I'll wait and see where this leads. By the way, I assume we agree that many of those 15limit "consonances" are not consonant at all, and are not even Just, being indistinguishable from the intervals in their vicinity, except if they are a subset of a very large otonality or with the most contrived timbre.
>This will be continued, following a short digression about 76ET.
>
>> I note that 76ET can also be notated using its native fifth, as you
>give (and I agree) below.
>
>In the process of looking over what we discussed regarding 76 (in
>connection with 62 and 69 a bit later in your message #4532), I noticed
>that it was given above as a subset of 152. I then noticed how bad the
>5limit is in 76 and wondered why it was being considered on its own.
>
>I then reviewed our correspondence. In response to a question from
>Paul about 76ET, you told him this (in message #4272):
>
><< The native bestfifth of 76ET is not suitable to be used a
>notational fifth because, among other reasons, it is not
>1,3,9consistent (i.e. its best 4:9 is not obtained by stacking two of
>its best 2:3s) and I figure folks have a right to expect C:D to be a
>best 4:9 when commas for primes greater than 9 are used in the
>notation. So 76ET will be notated as every second note of 152ET. >>
>
>It gets even worse than this: not only is 3 over 45 percent of a degree
>false, but 5 deviates even more.
>
>Your next mention of 76ET was in message #4434, in which you treated
>the divisions of the apotome systematically:
>
>4 steps per apotome ...
>69,76: ) ?? (\ /\ [13comma]
>
>>From that point we had 76 listed both as a subset of 152 and with 69.
>So after looking at all this, which will it be? (I would prefer it as
>the subset.)
If we are proposing a _single_ standard way of notating every ET then 76 should be as a subset of 152ET. However I think there are several such ETs where some composers may have very good reasons for wanting to notate them based on their native best fifth, (for example because the 76ET native fifth is the 19ET fifth), and we should attempt to standardise those too. So I say give both, but favour the 152ET subset.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> At 11:52 13/08/02 0700, George Secor wrote:
> >From: George Secor, 8/13/2002 (tuningmath #4577)
> >Subject: A common notation for JI and ETs
> >
> > ... And now that I've taken a fresh
> >look at the notation, I came up with some ideas on how to improve
a few
> >things.
> >
> >First of all, here is how I was able to notate all of the 15odd
limit
> >consonances taking C as 1/1. (Don't bother to look through all of
this
> >now; I'll be referring to many of these below, so this listing is
just
> >given for reference.)
> ...
>
> That's marvellous, except of course it looks like gobbledygook when
up to 5 ASCII symbols are being used to represent a single sagittal
symbol. How big is the biggest schisma involved?
As best I can remember there was nothing significantly more than 1
cent. (I'll be reporting most of them as I go along.) I would like
to have the 217tone symbols used for JI only as a last resort, which
enables us to keep the 15 limit notation systemindependent.
> >> I don't think we have defined a rational complement for /~
because
> >it isn't needed for rational tunings.
> >
> >On the contrary, I found that /~ is in fact quite useful for
rational
> >tunings (see above table of ratios), but its lack of a rational
> >complement is a problem. To remedy this, I propose ~( as its
> >rational complement.
>
> Fair enough, and yes, that would seem the obvious complement.
>
> >With C as 1/1, the following ratios would then
> >use these two symbols (which also appear in the table of ratios
above):
> >
> >11/10 = D\!~
> >20/11 = Bb/~ or B~!!(
> >15/11 = F/~
> >13/7 = B\!~
> >14/13 = Db/~ or D~!!(
> >
> >In effect, /~ functions not only as the 5+23 comma (~38.051c), but
> >also as the 11'5 comma (~38.906c) and the 13'7 comma (~38.073c)
>
> OK, so a 0.86 c schisma. I can certainly live with that for such
obscure ratios.
>
> >This would replace (( <> ~( as rational complements. I found
that
> >(( is not needed for any rational intervals in the 15odd limit,
so
> >this has no adverse consequences. (However, it leaves the 23'
comma
> >without a rational complement; I will address that problem
below.) The
> >new pair of complements that I am proposing also has a lower offset
> >(0.49 cents) than the old (1.03 cents), so, apart from the 23'
comma,
> >I can't think of a single reason not to do this.
>
> Me neither. Apart from the 23' comma.
>
> We could resurrect ~), with two left flags, as the complement of
the 23' comma. It isn't like a lot of people really care about ratios
of 23 anyway. We already made a good looking bitmap for ~) with the
wavy and the concave making a loop.
I'll be addressing this later.
> >The reverse pair of complements, ~( <> /~, would be used for
the
> >following ratios of 17:
> >
> > 17/16 = Db~( or D\!!~
> > 17/12 = Gb~( or G\!!~
> > 17/9 = Cb~( or C\!!~
> > 32/17 = B~!(
> > 24/17 = F#~!( or F/~
> > 18/17 = C#~!( or C/~
> >
> >All of this is going to affect how we will want to notate not only
152,
> >but also other ETs, including 217. (More about this later.)
>
> If rational complements don't have to be consistent with 217ET any
more, how about making rational complements consistent with 665ET,
as proposed earlier?
And I'll answer this one at the same time as the previous, because I
believe they're related.
> >... The principle that I am advancing here is that there is
another goal or
> >rule that should take precedence over that of an easytomemorize
> >symbol sequence  symbols which are used to represent JI
consonances
> >should be used in preference to those that can be expressed only as
> >sums of commaflags. These are the symbols that will be used for
JI
> >most frequently, and they will therefore (through repeated use)
become
> >*the most familiar* ones.
>
> But many people using ETs couldn't care less about JI, so why
should rational approximations take precedence over mnemonics,
particularly if they only involve ratios as uncommon as 5:11 and 7:13?
I think you meant 15:11, because I was going to remark: Are there
really so few who would venture beyond the 7limit? (But you are
still going to encounter 15:11 in the 11 limit.) My experience is
that two things take place the longer you are into microtonality:
1) If you use temperaments, you tend to prefer systems with less
error in the intervals than you did at first; and
2) You are able to accept (or find use for) a higher harmonic limit.
When I performed some of Ben Johnston's music in the mid '70s he was
composing in 5limit JI, but he didn't stop there. Given enough
time, I think that you're going to find 15limit ratios becoming more
and more common.
Anyway, my objective is to *minimize* the total number of symbols
that are likely to be encountered by performers, who are likely to
be involved with *both* JI and ETs. Fewer symbols would, in turn,
decrease the possibility of misreading or confusing them and would
also make the process of memorization easier. Remember, this is
supposed to be a *common* notation for JI and ETs, and having symbols
in the ETs that are even more uncommon (i.e., both different and
infrequent) than 15limit JI ones (which is what occurs much of the
time when you try to minimize the wavy and concave flags) tends to do
the opposite.
Let's pass judgment on this after we've looked at how this works out
with a number of ETs.
> >And these are the symbols that should have
> >first priority in the assignment of rational complements.
>
> Yes. I can accept that.
>
> > This is why
> >I want to eliminate (( in the rational complement scheme  it is
the
> >(13'(115))+(17'17) comma or, if you prefer, the (11'7)+(17'17)
> >comma, neither of which is simple enough to indicate that it would
ever
> >be used to notate a rational interval; and none of the 15limit
> >consonances (relative to C=1/1) require it.
>
> I'll wait and see where this leads. By the way, I assume we agree
that many of those 15limit "consonances" are not consonant at all,
and are not even Just, being indistinguishable from the intervals in
their vicinity, except if they are a subset of a very large otonality
or with the most contrived timbre.
Try mistuning an 11:13:15 triad  you will hear the combinational
tones beat against one another. As I understand it, this is the
essence of JI.
> >This will be continued, following a short digression about 76ET.
> > ...
> If we are proposing a _single_ standard way of notating every ET
then 76 should be as a subset of 152ET. However I think there are
several such ETs where some composers may have very good reasons for
wanting to notate them based on their native best fifth, (for example
because the 76ET native fifth is the 19ET fifth), and we should
attempt to standardise those too. So I say give both, but favour the
152ET subset.
Okay, that makes sense!
George
(This is a continuation of message #4577.)
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote [#4532]:
> At 01:03 18/06/02 0000, you wrote:
> > In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote:
New Rational Complements Â– Part 2

Since I proposed changing a pair of rational complements above, I
would like to reexamine the subject of rational complements a bit
further.
One flag combination that is very useful is 5+5, //:
25/16 = G#\\! or G~
13/10 = F\\!
26/15 = Bb\\! or B\\!!!
20/13 = G//
15/13 = D//
17/14 = Eb// or E~!!
and // is useful for 17 complements:
17/16 = C#~! or C//
17/12 = F#~! or F//
32/17 = Cb~ or C//!!
24/17 = Gb~ or G//!!
So we see that // functions not only as the 5+5 comma (6400:6561,
~43.013c) but also as the 13'5 comma (39:40, ~43.831c) and the 17'+7
comma (448:459, ~41.995c).
Until recently I had a prejudice against //, because it has two
flags on the same side. But now that I see that other symbols of
this sort haven't popped up all over the place, and since its
rational complement ~ is simple and useful, I would like to include
it in the standard 217 notation instead of ~\ (which is only the 11
5+17 comma, and which is not needed for any 15limit consonances).
For reference, here is the 217 standard notation as it presently
stands:
217: ( ~ ~ / ) \ ~) ~\ /) /\ () (\ ~
~ / ) \ ~) ~\ /) /\ (present)
Making this change would give us:
217: ( ~ ~ / ) \ ~) // /) /\ () (\ ~
~ / ) \ ~) // /) /\ (all RCs)
So we would now have true rational complements throughout.
However, there is a second change that I wish to propose. It
incorporates the change of rational complements from (( <> ~(
to /~ <> ~( that I also proposed above. For 7deg we now have
~), which is used for the following ratios, but for nothing in the
15limit:
17/10 = Bbb~) or Bx~
17/15 = Ebb~) or Ex~
(For this ascii notation I have used x instead of X to specify a
*downward* alteration of pitch, as we have already done with !
instead of . I hope the presence the wavy flag in combination with
it is enough to indicate it is not being used here to indicate a
double sharp. Otherwise, would a capital Y be a suitable alternative?)
The proposed replacement standard symbol /~ for 7deg217 is used for
11/10, 14/13, and 15/11 (plus their inversions).
In order to maintain rational complements and a matching symbol
sequence throughout, the symbols for 3, 14, and 18deg217 would also
need to be changed, which would give this for the standard 217 set:
217: ( ~ ~( / ) \ /~ // /) /\ () (\ ~ ~
( / ) \ /~ // /) /\ (new RCs)
The 3deg symbol changes from the 23 comma (or 19'19 comma, if you
prefer) to the 17' comma. This is a more complicated symbol, but it
symbolizes a lower prime number, making it more likely to be used.
(Besides, it has mnemonic appeal.)
My goal is to minimize the differences between the 217ET notation
and the rational notation (while maintaining a matched symbol
sequence), with the lowest primes (i.e., the 17 limit) being
favored. This would make the transition from purely rational symbols
to 217ET standard symbols as painless as possible in instances where
the composer has run out of rational symbols and has no other choice
but to use 217 symbols to indicate rational intervals.
With this set of symbols there are only two intervals (including
inversions) in the 15odd limit that, relative to C=1/1, require
symbols outside of the standard 217 set for the rational notation:
11/7 = G#(! or G)~
14/11 = Fb( or F)~
This uses ( as the 11'7 comma (45056:45927, ~33.148c), which is
already defined as part of the notation.
Curiously, these could easily be incorporated into the 217 notation
by replacing \ and / with ( with )~, respectively:
217mapped JI: ( ~ ~( / ) ( /~ // /) /\ ()
(\ ~ ~( )~ ) \ /~ // /) /\
No 15odd limit consonances require either \ or / (they are not
needed until 19/14 and 38/21 are encountered), so no 15limit symbols
are lost in the process. This completely minimizes the differences
between the 217ET and the rational notation.
Rational complementation is maintained, but the matching sequence of
symbols is lost (an important consideration with this many symbols),
not to mention losing half of the easytoremember straightflag
symbols. Also, while the lateral confusability between the straight
flag symbols has been eliminated, that has been replaced by laterally
mirrored 14deg and 15deg symbols.
So I think it would be best to retain the straight flags in the
standard 217 set, but to have in mind the ( and )~ symbols as
supplementary rational complements. A composer would have the option
to use ( and )~ to clarify the harmonic function of the tones
which they represent for either 217ET or JI mapped to 217. The same
could be said for the rational symbols for ratios of 19 and 23,
should one want to use a higher harmonic limit. (These would be less
used, lessfamiliar symbols that would be rarely be needed below the
19 limit.)
With these changes in the standard 217 notation, it would be
necessary to memorize only 8 rational complement pairs (half of which
use only straight and convex flags, and half of which are singles,
not pairs) to notate all of the 15limit consonances and a majority
of the ratios of 17 in JI:
5 and 115 commas: / <> \ and \ <> /
7 comma: ) <> )
11 diesis: /\ <> ()
13 diesis: /) <> (\
75 comma or 1113 comma: ( <> /)
17 comma and 25 comma: ~ <> // and // <> ~
17' comma and 11'5 or 13'7 comma: ~( <> /~ and /~ <> ~(
19' comma and 11'7 comma: )~ <> ( and ( <> )~
(The last pair of RCs are the supplementary symbols that are not part
of the standard 217ET set.)
With these symbols you have more than enough symbols to notate a 15
limit tonality diamond (with 49 distinct tones in the octave).
Notice that I identified ( as something other than the 17'17
comma. This is because it is used for the following rational
intervals:
7/5 = Gb!( or G!!!(
10/7 = F#( or F(
13/11 = Eb!( or E!!!(
22/13 = A(
15/14 = C#( or C(
28/15 = Cb!( or C!!!(
Thus ( can assume the role of either the 17'17 comma (288:289,
~6.001c), the 75 comma (5103:5120, ~5.758c), or the 1113 comma
(351:352, ~4.925c). However, there are a limited number of ETs in
which it can function as all three commas (159, 171, 183, 217, 311,
400, 494, and 653) or at least as both the 75 and 1113 commas (130
and 142).
(To be continued.)
At 11:27 14/08/02 0700, George Secor wrote:
>Until recently I had a prejudice against //, because it has two flags
>on the same side. But now that I see that other symbols of this sort
>haven't popped up all over the place, and since its rational complement
>~ is simple and useful, I would like to include it in the standard
>217 notation instead of ~\ (which is only the 115+17 comma, and which
>is not needed for any 15limit consonances).
That's fine by me. I totally approve of making more use of //, but it should only be used in an ET if it is valid as the double 5comma.
>For reference, here is the 217 standard notation as it presently
>stands:
>
>217: ( ~ ~ / ) \ ~) ~\ /) /\ () (\ ~ ~
>/ ) \ ~) ~\ /) /\ (present)
>
>Making this change would give us:
>
>217: ( ~ ~ / ) \ ~) // /) /\ () (\ ~ ~
>/ ) \ ~) // /) /\ (all RCs)
>
>So we would now have true rational complements throughout.
>
>However, there is a second change that I wish to propose. It
>incorporates the change of rational complements from (( <> ~( to
>/~ <> ~( that I also proposed above. For 7deg we now have ~),
>which is used for the following ratios, but for nothing in the
>15limit:
>
>17/10 = Bbb~) or Bx~
> 17/15 = Ebb~) or Ex~
>
>(For this ascii notation I have used x instead of X to specify a
>*downward* alteration of pitch, as we have already done with ! instead
>of . I hope the presence the wavy flag in combination with it is
>enough to indicate it is not being used here to indicate a double
>sharp. Otherwise, would a capital Y be a suitable alternative?)
Little x for downward is fine with me.
>The proposed replacement standard symbol /~ for 7deg217 is used for
>11/10, 14/13, and 15/11 (plus their inversions).
>
>In order to maintain rational complements and a matching symbol
>sequence throughout, the symbols for 3, 14, and 18deg217 would also
>need to be changed, which would give this for the standard 217 set:
>
>217: ( ~ ~( / ) \ /~ // /) /\ () (\ ~ ~(
>/ ) \ /~ // /) /\ (new RCs)
>
>The 3deg symbol changes from the 23 comma (or 19'19 comma, if you
>prefer) to the 17' comma. This is a more complicated symbol, but it
>symbolizes a lower prime number, making it more likely to be used.
>(Besides, it has mnemonic appeal.)
Yes I suppose I can give up monotonic flagspersymbol, but if you don't want to know about JI or don't care about 11/10, 14/13, or 15/11, then that /~ now seems to come out of nowhere. Why suddenly introduce the right wavy flag. At least ~) introduces no new flags.
>My goal is to minimize the differences between the 217ET notation and
>the rational notation (while maintaining a matched symbol sequence),
>with the lowest primes (i.e., the 17 limit) being favored.
That's fine so long as it is the 217ET notation that gets compromised, not the rational.
>This would
>make the transition from purely rational symbols to 217ET standard
>symbols as painless as possible in instances where the composer has run
>out of rational symbols and has no other choice but to use 217 symbols
>to indicate rational intervals.
I don't understand why there would be no choice but 217ET. Is 217ET really the best ET that we can fully notate? What about 282ET? It's 29limit consistent. I've never really understood the deference to 217ET.
...
>So I think it would be best to retain the straight flags in the
>standard 217 set,
Agreed.
> but to have in mind the ( and )~ symbols as
>supplementary rational complements. A composer would have the option
>to use ( and )~ to clarify the harmonic function of the tones which
>they represent for either 217ET or JI mapped to 217. The same could
>be said for the rational symbols for ratios of 19 and 23, should one
>want to use a higher harmonic limit. (These would be lessused,
>lessfamiliar symbols that would be rarely be needed below the 19
>limit.)
>
>With these changes in the standard 217 notation, it would be necessary
>to memorize only 8 rational complement pairs (half of which use only
>straight and convex flags, and half of which are singles, not pairs) to
>notate all of the 15limit consonances and a majority of the ratios of
>17 in JI:
>
>5 and 115 commas: / <> \ and \ <> /
>7 comma: ) <> )
>11 diesis: /\ <> ()
>13 diesis: /) <> (\
>75 comma or 1113 comma: ( <> /)
>17 comma and 25 comma: ~ <> // and // <> ~
>17' comma and 11'5 or 13'7 comma: ~( <> /~ and /~ <> ~(
>19' comma and 11'7 comma: )~ <> ( and ( <> )~
>
>(The last pair of RCs are the supplementary symbols that are not part
>of the standard 217ET set.)
>
>With these symbols you have more than enough symbols to notate a
>15limit tonality diamond (with 49 distinct tones in the octave).
Good work. I'd like to see that listed in pitch order.
>Notice that I identified ( as something other than the 17'17 comma.
>This is because it is used for the following rational intervals:
>
>7/5 = Gb!( or G!!!(
>10/7 = F#( or F(
>13/11 = Eb!( or E!!!(
>22/13 = A(
>15/14 = C#( or C(
>28/15 = Cb!( or C!!!(
>
>Thus ( can assume the role of either the 17'17 comma (288:289,
>~6.001c), the 75 comma (5103:5120, ~5.758c), or the 1113 comma
>(351:352, ~4.925c). However, there are a limited number of ETs in
>which it can function as all three commas (159, 171, 183, 217, 311,
>400, 494, and 653) or at least as both the 75 and 1113 commas (130
>and 142).
Hmm. It is certainly arguable that we should favour the interpretation of ( as the 75 comma when notating ETs. What's the smallest ET that would be affected by this?
Is ) still to be interpreted as the 19 comma and what is to be its complement?
Is there a lower prime interpretation of ~ now too?
It seems to me that what we are discussing here is unlikely to impact on many ETs below 100. Is that the case?
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> At 11:27 14/08/02 0700, George Secor wrote:
> >Until recently I had a prejudice against //, because it has two
flags
> >on the same side. But now that I see that other symbols of this
sort
> >haven't popped up all over the place, and since its rational
complement
> >~ is simple and useful, I would like to include it in the
standard
> >217 notation instead of ~\ (which is only the 115+17 comma, and
which
> >is not needed for any 15limit consonances).
>
> That's fine by me. I totally approve of making more use of //, but
it should only be used in an ET if it is valid as the double 5comma.
Yes, a mandatory test for the use of this symbol in an ET is that the
ET be 1,5,25 consistent.
> >For reference, here is the 217 standard notation as it presently
> >stands:
> >
> >217: ( ~ ~ / ) \ ~) ~\ /) /\ () (\ ~
~ / ) \ ~) ~\ /) /\ (present)
> >
> >Making this change would give us:
> >
> >217: ( ~ ~ / ) \ ~) // /) /\ () (\ ~
~ / ) \ ~) // /) /\ (all RCs)
> >
> >So we would now have true rational complements throughout.
> >
> >However, there is a second change that I wish to propose. It
> >incorporates the change of rational complements from (( <> ~(
to
> >/~ <> ~( that I also proposed above. For 7deg we now have
~),
> >which is used for the following ratios, but for nothing in the
> >15limit:
> >
> >17/10 = Bbb~) or Bx~
> > 17/15 = Ebb~) or Ex~
> >
> >(For this ascii notation I have used x instead of X to specify a
> >*downward* alteration of pitch, as we have already done with !
instead
> >of . I hope the presence the wavy flag in combination with it is
> >enough to indicate it is not being used here to indicate a double
> >sharp. Otherwise, would a capital Y be a suitable alternative?)
>
> Little x for downward is fine with me.
>
> >The proposed replacement standard symbol /~ for 7deg217 is used
for
> >11/10, 14/13, and 15/11 (plus their inversions).
> >
> >In order to maintain rational complements and a matching symbol
> >sequence throughout, the symbols for 3, 14, and 18deg217 would also
> >need to be changed, which would give this for the standard 217 set:
> >
> >217: ( ~ ~( / ) \ /~ // /) /\ () (\ ~
~( / ) \ /~ // /) /\ (new RCs)
> >
> >The 3deg symbol changes from the 23 comma (or 19'19 comma, if you
> >prefer) to the 17' comma. This is a more complicated symbol, but
it
> >symbolizes a lower prime number, making it more likely to be used.
> >(Besides, it has mnemonic appeal.)
>
> Yes I suppose I can give up monotonic flagspersymbol, but if you
don't want to know about JI or don't care about 11/10, 14/13, or
15/11, then that /~ now seems to come out of nowhere. Why suddenly
introduce the right wavy flag. At least ~) introduces no new flags.
Three reasons:
1) As I said above, /~ is used for 3 15limit ratios (not including
inversions), while ~) is used for only one ratio of 17. Hence /~
will have a wider use.
2) Those who don't care about 11/10 _et al_ will probably be using
tempered versions of these ratios in one way or another if /~ occurs
in the particular ET they are using. Use of the same symbol in
*both* JI and the ET exploits the *commonality* of the symbols for
both applications.
3) As I said below, I am now placing a higher priority on minimizing
the number of the most commonly used *symbols* than on minimizing the
number of *flags* used for an ET. This "most commonly used" set of
symbols was summarized in the 8 sets of rational complements that I
listed at the end of my last message.
> >My goal is to minimize the differences between the 217ET notation
and
> >the rational notation (while maintaining a matched symbol
sequence),
> >with the lowest primes (i.e., the 17 limit) being favored.
>
> That's fine so long as it is the 217ET notation that gets
compromised, not the rational.
>
> >This would
> >make the transition from purely rational symbols to 217ET standard
> >symbols as painless as possible in instances where the composer
has run
> >out of rational symbols and has no other choice but to use 217
symbols
> >to indicate rational intervals.
>
> I don't understand why there would be no choice but 217ET. Is 217
ET really the best ET that we can fully notate? What about 282ET?
It's 29limit consistent. I've never really understood the deference
to 217ET.
I never considered 282 before, but I do see some problems with it:
1) 11 is almost 1.9 cents in error, and 13 is over 2 cents; these
errors approach the maximum possible error for the system. (This is
the same sort of problem that we have with 13 in 72ET.)
2) The ) flag is not the same number of degrees for the 7 and 135
commas (which is by itself reason enough to reject 282), nor is (
the same number of degrees for the 11'7 and 13'(115) commas.
3) The following rational complements for the 15limit symbols are
not consistent in 282:
)~ <> ( 19' comma
( <> /) as 75 comma or 1113 comma (but 17'17 is okay)
~ <> // 17 comma
) <> ) 7 comma
// <> ~ 25 comma
( <> )~ 11'7 comma
And besides this, there are others that are inconsistent, such as:
~ <> ~) as both the 19'19 and 23 comma
What makes 217 so useful is that *everything* is consistent to the 19
limit, and, except for 23, to the 29 limit. And I think that the
problems with 23 can be managed, considering how rarely it is likely
to be used. You have to have a way to accommodate the electronic JI
composer who might want to modulate all over the place, and a
consistent ET mapping for JI intervals is the only way to do it with
a finite number of symbols; this is where 217 really delivers the
goods!
> ...
> >So I think it would be best to retain the straight flags in the
> >standard 217 set,
>
> Agreed.
>
> > but to have in mind the ( and )~ symbols as
> >supplementary rational complements. A composer would have the
option
> >to use ( and )~ to clarify the harmonic function of the tones
which
> >they represent for either 217ET or JI mapped to 217. The same
could
> >be said for the rational symbols for ratios of 19 and 23, should
one
> >want to use a higher harmonic limit. (These would be lessused,
> >lessfamiliar symbols that would be rarely be needed below the 19
> >limit.)
> >
> >With these changes in the standard 217 notation, it would be
necessary
> >to memorize only 8 rational complement pairs (half of which use
only
> >straight and convex flags, and half of which are singles, not
pairs) to
> >notate all of the 15limit consonances and a majority of the
ratios of
> >17 in JI:
> >
> >5 and 115 commas: / <> \ and \ <> /
> >7 comma: ) <> )
> >11 diesis: /\ <> ()
> >13 diesis: /) <> (\
> >75 comma or 1113 comma: ( <> /)
> >17 comma and 25 comma: ~ <> // and // <> ~
> >17' comma and 11'5 or 13'7 comma: ~( <> /~ and /~ <> ~
(
> >19' comma and 11'7 comma: )~ <> ( and ( <> )~
> >
> >(The last pair of RCs are the supplementary symbols that are not
part
> >of the standard 217ET set.)
> >
> >With these symbols you have more than enough symbols to notate a
> >15limit tonality diamond (with 49 distinct tones in the octave).
>
> Good work. I'd like to see that listed in pitch order.
At first I thought you meant listing the symbols like this:
Symbol set used for 15limit JI

)~ <> ( 19' comma (not in standard 217 set)
( <> /) 75 comma or 1113 comma
~ <> // 17 comma
~( <> /~ 17' comma
/ <> \ 5 comma
) <> ) 7 comma
\ <> / 115 comma
( <> )~ 11'7 comma (not in standard 217 set)
// <> ~ 25 comma
/~ <> ~( 11'5 or 13'7 comma
/\ <> () 11 diesis
/) <> (\ 13 diesis
But now I think you meant listing the ratios like this:
Sagittal Notation for 15limit JI

1/1 = C
16/15 = Db/ or D!!/
15/14 = C#( or C(
14/13 = Db/~ or D~!!(
13/12 = D(!/
12/11 = D\!/
11/10 = D\!~
10/9 = D\!
9/8 = D
8/7 = D)
15/13 = D//
7/6 = Eb!) or E!!!)
13/11 = Eb!( or E!!!(
6/5 = Eb/ or E!!/
11/9 = E(!)
16/13 = E\!)
5/4 = E\!
14/11 = Fb( or F)~
9/7 = E)
13/10 = F\\!
4/3 = F
15/11 = F/~
11/8 = F/\
18/13 = F(\
7/5 = Gb!( or G!!!(
10/7 = F#( or F(
13/9 = G(!/
16/11 = G\!/
22/15 = G/!~
3/2 = G
20/13 = G//
14/9 = Ab!) or A!!!)
11/7 = G#(! or G)~
8/5 = Ab/ or A!!/
13/8 = A(!/
18/11 = A\!/
5/3 = A\!
22/13 = A(
12/7 = A)
26/15 = Bb\\! or B\\!!!
7/4 = Bb!) or B!!!)
16/9 = Bb or B\!!/
9/5 = Bb/ or B!!/
20/11 = Bb/~ or B~!!(
11/6 = B(!)
24/13 = B\)
13/7 = B\!~
28/15 = Cb!( or C!!!(
15/8 = B\!
2/1 = C
> >Notice that I identified ( as something other than the 17'17
comma.
> >This is because it is used for the following rational intervals:
> >
> >7/5 = Gb!( or G!!!(
> >10/7 = F#( or F(
> >13/11 = Eb!( or E!!!(
> >22/13 = A(
> >15/14 = C#( or C(
> >28/15 = Cb!( or C!!!(
> >
> >Thus ( can assume the role of either the 17'17 comma (288:289,
> >~6.001c), the 75 comma (5103:5120, ~5.758c), or the 1113 comma
> >(351:352, ~4.925c). However, there are a limited number of ETs in
> >which it can function as all three commas (159, 171, 183, 217, 311,
> >400, 494, and 653) or at least as both the 75 and 1113 commas
(130
> >and 142).
>
> Hmm. It is certainly arguable that we should favour the
interpretation of ( as the 75 comma when notating ETs. What's the
smallest ET that would be affected by this?
It's hard to say what is the smallest ET in which they differ
consistently. We're looking at three different commas: 75, 1113,
and 17'17, and we're dealing with all of the primes in the 17
limit. All three commas are the same number of degrees (without
vanishing) in 19, 43, 159, 171, 183, 217, 311, 400, 494, and 653.
All three are a different number of degrees in 612 and 1600. 75 and
1113 are like degrees but differ from 17'17 in 26, 60, 72, 84, 96,
130, 142, 176, 224, 270, 282, and 364. The 75 and 1113 agreement
is important for rational complementation, because it is the number
of degrees in the 1113 comma that determines whether ( <> /) is
consistent, whereas the 17'17 agreement is important only if the
notation for a given ET also uses both the 17 and 17' symbols. The
ETs above 100 that I looked at in which these don't agree are 108,
118, 120, 125, 132, 144, 147, 149, 152, 193, 207, 388, 525, 612, 742,
and 1600; many of these won't even need the ( symbol, and 193 is
probably the most important one in which ( would be used that is not
consistent with the 75 comma (although the 19 comma could be used
instead with /) as its alternate complement; another alternate
complement is already required for the 17 comma of 2deg, so this
doesn't harm a notation that might otherwise have all rational
complements and a matching sequence).
> Is ) still to be interpreted as the 19 comma and what is to be its
complement?
Yes, and its complement is still (~. I don't see any lowerprime
interpretations of it without going into rational complements, where
we have only one: 11/7 = G)~. This is greater than G()
(2187/1408) by 15309:15488, ~20.125c (vs. the 19' comma, 19456:19683,
~20.082c). But this is for )~, so we must subtract ~ from this,
but what comma would ~ be? Since your next question has a positive
answer (and since I did that one first I can peek at the answer),
I'll use the 11limit comma 99:100, which gives 42525:42592
(3^5*5^2*7:2^5*11^3, ~2.725c) as the 11limit interpretation of ).
This is meaningful only if you are using rational complements, i.e.,
singlesymbol notation.
> Is there a lower prime interpretation of ~ now too?
Hmm, good question! Yes, using /~ as the 11'5 comma for 11/10
would make that symbol 44:45, so ~ would be 99:100, ~17.399 cents.
And using /~ as the 13'7 comma for 13/7 would make /~ 1664:1701,
so ~ would be 104:105, ~16.567 cents. ^
> It seems to me that what we are discussing here is unlikely to
impact on many ETs below 100. Is that the case?
Yes, I think that this will affect mostly the weird and difficult
ones. We have been able to do the simpler ones using only straight
and convexright flags, which have remained unchanged.
George
At 12:31 15/08/02 0700, George Secor wrote:
>> That's fine by me. I totally approve of making more use of //, but
>it should only be used in an ET if it is valid as the double 5comma.
>
>Yes, a mandatory test for the use of this symbol in an ET is that the
>ET be 1,5,25 consistent.
That's a little more strict that what I had in mind, but I guess it's a good idea. I'd be inclined to allow it to represent two 5commas whether that gives the best 25 or not.
>> >217: ( ~ ~( / ) \ /~ // /) /\ () (\ ~
>~( / ) \ /~ // /) /\ (new RCs)
>> >
>> >The 3deg symbol changes from the 23 comma (or 19'19 comma, if you
>> >prefer) to the 17' comma. This is a more complicated symbol, but it
>> >symbolizes a lower prime number, making it more likely to be used.
>> >(Besides, it has mnemonic appeal.)
>>
>> Yes I suppose I can give up monotonic flagspersymbol, but if you
>don't want to know about JI or don't care about 11/10, 14/13, or 15/11,
>then that /~ now seems to come out of nowhere. Why suddenly introduce
>the right wavy flag. At least ~) introduces no new flags.
>
>Three reasons:
>
>1) As I said above, /~ is used for 3 15limit ratios (not including
>inversions), while ~) is used for only one ratio of 17. Hence /~
>will have a wider use.
This seems a little circular. If we did not limit ET notations to using only those symbols used for 15limit JI, but instead tried to minimise the number of different flags each uses (as we have been until recently), then ~) may well have wider use than /~, purely due to the number of ETs it is used in. So I don't buy this one.
>2) Those who don't care about 11/10 _et al_ will probably be using
>tempered versions of these ratios in one way or another if /~ occurs
>in the particular ET they are using. Use of the same symbol in *both*
>JI and the ET exploits the *commonality* of the symbols for both
>applications.
Yes, I agree that is the whole point of our "common notation". However I'm not convinced that there will be many times when somone uses an approximate 11:15 or 13:14 _as_ an approximate just interval when the lower note is a natural (or has only # or b). But in the case of a 5:11 I guess it's more likely. So I find this reason to be marginally valid.
>3) As I said below, I am now placing a higher priority on minimizing
>the number of the most commonly used *symbols* than on minimizing the
>number of *flags* used for an ET. This "most commonly used" set of
>symbols was summarized in the 8 sets of rational complements that I
>listed at the end of my last message.
On examing these in more detail I find that I don't understand at all why you chose /~ as the appropriate symbol for the 11'5 comma, 44:45 (and the 13'7 comma). (( seems the obvious choice to me, since ( is the 11'7 comma and ( is the 75 comma and ( + ( = (( . (11'7)+(75) = 11'5. and (11'7)+(13'11')=(13'7).
If (( is the symbol for the 11'5 comma (or we could more usefully call it the 11/5 comma) then you don't need to change any rational complements from what we had (the 494ETconsistent ones) and what's more we don't need to introduce any more flags into 217ET when (( is used for 7 steps.
>> I don't understand why there would be no choice but 217ET. Is 217ET
>really the best ET that we can fully notate? What about 282ET? It's
>29limit consistent. I've never really understood the deference to
>217ET.
>
>I never considered 282 before, but I do see some problems with it:
>
>1) 11 is almost 1.9 cents in error, and 13 is over 2 cents; these
>errors approach the maximum possible error for the system. (This is
>the same sort of problem that we have with 13 in 72ET.)
You're only looking at the primes themselves. What about the ratios between them. 217ET has a 2.8 cent error in its 7:11 whereas 282ET never gets worse than that 2.0 cents in the 1:13.
>2) The ) flag is not the same number of degrees for the 7 and 135
>commas (which is by itself reason enough to reject 282), nor is ( the
>same number of degrees for the 11'7 and 13'(115) commas.
Reason enough to reject 282ET as what? Reject it as a good way of having a fully notatable closed system that approximates 29limit JI? I seriously disagree. It just means that we should use ( and ) with their non13 meanings in 282ET.
>3) The following rational complements for the 15limit symbols are not
>consistent in 282:
>
>)~ <> ( 19' comma
> ( <> /) as 75 comma or 1113 comma (but 17�17 is okay)
>~ <> // 17 comma
> ) <> ) 7 comma
>// <> ~ 25 comma
> ( <> )~ 11'7 comma
>
>And besides this, there are others that are inconsistent, such as:
>
> ~ <> ~) as both the 19�19 and 23 comma
All this means is that maybe we should consider making our rational complements consistent with 282ET rather than 217ET.
>What makes 217 so useful is that *everything* is consistent to the 19
>limit, and, except for 23, to the 29 limit.
I don't know what you mean by *everything* here. Isn't 282ET consistent to the 29limit with no exceptions?
>And I think that the
>problems with 23 can be managed, considering how rarely it is likely to
>be used. You have to have a way to accommodate the electronic JI
>composer who might want to modulate all over the place, and a
>consistent ET mapping for JI intervals is the only way to do it with a
>finite number of symbols; this is where 217 really delivers the goods!
I still fail to see why 217 is better than 282, except that various choices we have made along the way, regarding the symbols, have been biased toward 217.
>> >With these symbols you have more than enough symbols to notate a
>> >15limit tonality diamond (with 49 distinct tones in the octave).
>>
>> Good work. I'd like to see that listed in pitch order.
>
>At first I thought you meant listing the symbols like this:
>
>Symbol set used for 15limit JI
>
> )~ <> ( 19' comma (not in standard 217 set)
> ( <> /) 75 comma or 1113 comma
> ~ <> // 17 comma
> ~( <> /~ 17' comma
> / <> \ 5 comma
> ) <> ) 7 comma
> \ <> / 115 comma
> ( <> )~ 11'7 comma (not in standard 217 set)
>// <> ~ 25 comma
> /~ <> ~( 11'5 or 13'7 comma
> /\ <> () 11 diesis
> /) <> (\ 13 diesis
No. Although that's interesting too.
>But now I think you meant listing the ratios like this:
>
>Sagittal Notation for 15limit JI
>
...
Yes that was it, but now I realise there are only 6 that are independent and that we haven't already agreed on. Here they are in oder of decreasing importance:
1/1 = C
7/5 = Gb!( or G!!!(
11/5 = D\!~
11/7 = G#(! or G)~
13/5 = F\\!
13/7 = B\!~
13/11 = Eb!( or E!!!(
But I think they should be:
1/1 = C
7/5 = Gb!( or G!!!(
11/5 = D(!(
11/7 = G#(! or G)~
13/5 = F\\!
13/7 = B(!(
13/11 = Eb!( or E!!!(
>> Hmm. It is certainly arguable that we should favour the
>interpretation of ( as the 75 comma when notating ETs. What's the
>smallest ET that would be affected by this?
>
>It's hard to say what is the smallest ET in which they differ
>consistently.
I mean: What's the smallest one we've agreed on that uses (, where the 75 comma interpretation of it would be a different number of steps from what we've used it for.
>> Is ) still to be interpreted as the 19 comma and what is to be its
>complement?
>
>Yes, and its complement is still (~. I don't see any lowerprime
>interpretations of it without going into rational complements, where we
>have only one: 11/7 = G)~. This is greater than G() (2187/1408) by
>15309:15488, ~20.125c (vs. the 19' comma, 19456:19683, ~20.082c). But
>this is for )~, so we must subtract ~ from this, but what comma would
>~ be? Since your next question has a positive answer (and since I did
>that one first I can peek at the answer), I'll use the 11limit comma
>99:100, which gives 42525:42592 (3^5*5^2*7:2^5*11^3, ~2.725c) as the
>11limit interpretation of ).
>
>This is meaningful only if you are using rational complements, i.e.,
>singlesymbol notation.
No. Going via complements isn't what I had in mind. Does ) want to be used as a comma for any of 17/5, 17/7, 17/11, 17/13?
>> Is there a lower prime interpretation of ~ now too?
>
>Hmm, good question! Yes, using /~ as the 11'5 comma for 11/10 would
>make that symbol 44:45, so ~ would be 99:100, ~17.399 cents. And
>using /~ as the 13'7 comma for 13/7 would make /~ 1664:1701, so ~
>would be 104:105, ~16.567 cents.
OK. But this is not so, if we adopt (( as the 7/5comma symbol.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 David C Keenan <d.keenan@uq.net.au> wrote:
> At 12:31 15/08/02 0700, George Secor wrote:
> >> That's fine by me. I totally approve of making more use of //,
but
> >it should only be used in an ET if it is valid as the double 5
comma.
> >
> >Yes, a mandatory test for the use of this symbol in an ET is that
the
> >ET be 1,5,25 consistent.
>
> That's a little more strict that what I had in mind, but I guess
it's
> a good idea. I'd be inclined to allow it to represent two 5commas
> whether that gives the best 25 or not.
In working out a spreadsheet to automatically assign the symbols for
ETs, one of the criteria I am using is to select ones that eliminate
(or at least minimize) the inconsistencies. This can get not only
complicated, but tricky.
> >> >217: ( ~ ~( / ) \ /~ // /) /\ () (\
~ ~( / ) \ /~ // /) /\ (new RCs)
> >> >
> >> >The 3deg symbol changes from the 23 comma (or 19'19 comma, if
you
> >> >prefer) to the 17' comma. This is a more complicated symbol,
but it
> >> >symbolizes a lower prime number, making it more likely to be
used.
> >> >(Besides, it has mnemonic appeal.)
> >>
> >> Yes I suppose I can give up monotonic flagspersymbol, but if
you
> >don't want to know about JI or don't care about 11/10, 14/13, or
15/11,
> >then that /~ now seems to come out of nowhere. Why suddenly
introduce
> >the right wavy flag. At least ~) introduces no new flags.
> >
> >Three reasons:
> >
> >1) As I said above, /~ is used for 3 15limit ratios (not
including
> >inversions), while ~) is used for only one ratio of 17.
Hence /~
> >will have a wider use.
>
> This seems a little circular. If we did not limit ET notations to
> using only those symbols used for 15limit JI, but instead tried to
> minimise the number of different flags each uses (as we have been
> until recently), then ~) may well have wider use than /~, purely
> due to the number of ETs it is used in. So I don't buy this one.
>
> >2) Those who don't care about 11/10 _et al_ will probably be using
> >tempered versions of these ratios in one way or another if /~
occurs
> >in the particular ET they are using. Use of the same symbol in
*both*
> >JI and the ET exploits the *commonality* of the symbols for both
> >applications.
>
> Yes, I agree that is the whole point of our "common notation".
> However I'm not convinced that there will be many times when somone
> uses an approximate 11:15 or 13:14 _as_ an approximate just interval
> when the lower note is a natural (or has only # or b). But in the
> case of a 5:11 I guess it's more likely. So I find this reason to be
> marginally valid.
>
> >3) As I said below, I am now placing a higher priority on
minimizing
> >the number of the most commonly used *symbols* than on minimizing
the
> >number of *flags* used for an ET. This "most commonly used" set of
> >symbols was summarized in the 8 sets of rational complements that I
> >listed at the end of my last message.
>
> On examing these in more detail I find that I don't understand at
all
> why you chose /~ as the appropriate symbol for the 11'5 comma,
> 44:45 (and the 13'7 comma). (( seems the obvious choice to me,
> since ( is the 11'7 comma and ( is the 75 comma and ( + ( = (
(.
> (11'7)+(75) = 11'5. and (11'7)+(13'11')=(13'7).
Oops, you're right! I've been using ( as the 13'(115) comma all
along in computing these ratios, and it looks like I'm going to have
to redo a few things on account of our recent lowerprime symbol
definitions. Glad you caught this! Of course, the 13'7 comma is
1664:1701, about 0.833 cents smaller. So (( will definitely have to
be among the 217 standard symbols, and it's back to the drawing
board! (Really, I'm very delighted that you found this, because it's
going to make things a lot easier.)
So it looks like this will be the 217 standard set:
217: ( ~ ~( / ) \ (( // /) /\ () (\ ~ ~
( / ) \ (( // /) /\ (new RCs)
> If (( is the symbol for the 11'5 comma (or we could more usefully
> call it the 11/5 comma) then you don't need to change any rational
> complements from what we had (the 494ETconsistent ones) and what's
> more we don't need to introduce any more flags into 217ET when ((
> is used for 7 steps.
>
> >> I don't understand why there would be no choice but 217ET. Is
217ET
> >really the best ET that we can fully notate? What about 282ET?
It's
> >29limit consistent. I've never really understood the deference to
> >217ET.
> >
> >I never considered 282 before, but I do see some problems with it:
> >
> >1) 11 is almost 1.9 cents in error, and 13 is over 2 cents; these
> >errors approach the maximum possible error for the system. (This
is
> >the same sort of problem that we have with 13 in 72ET.)
>
> You're only looking at the primes themselves. What about the ratios
> between them. 217ET has a 2.8 cent error in its 7:11 whereas 282ET
> never gets worse than that 2.0 cents in the 1:13.
>
> >2) The ) flag is not the same number of degrees for the 7 and 135
> >commas (which is by itself reason enough to reject 282), nor is (
the
> >same number of degrees for the 11'7 and 13'(115) commas.
>
> Reason enough to reject 282ET as what? Reject it as a good way of
> having a fully notatable closed system that approximates 29limit
JI?
> I seriously disagree. It just means that we should use ( and )
with
> their non13 meanings in 282ET.
I guess I didn't get my point across. I want to be able to use a
largenumbered ET (217 or 282 or whatever) to notate *JI* when there
are no suitable rational symbols that will do the job. If ( or )
don't have 13 meanings in 282, then there cannot be a good transition
between the rational notation and the largeET notation  symbols
would have to be converted from one to the other should a JI
composition suddenly require 282ET symbols. This problem is
minimized with 217, because even the nonstandard symbols such as )
and ( can be kept, because they are all the correct number of
degrees.
> >3) The following rational complements for the 15limit symbols are
not
> >consistent in 282:
> >
> >)~ <> ( 19' comma
> > ( <> /) as 75 comma or 1113 comma (but 17'17 is okay)
> >~ <> // 17 comma
> > ) <> ) 7 comma
> >// <> ~ 25 comma
> > ( <> )~ 11'7 comma
> >
> >And besides this, there are others that are inconsistent, such as:
> >
> > ~ <> ~) as both the 19'19 and 23 comma
>
> All this means is that maybe we should consider making our rational
> complements consistent with 282ET rather than 217ET.
>
> >What makes 217 so useful is that *everything* is consistent to the
19
> >limit, and, except for 23, to the 29 limit.
>
> I don't know what you mean by *everything* here. Isn't 282ET
> consistent to the 29limit with no exceptions?
It isn't consistent with the schismas that are essential to the
rational notation:
1) The 5 comma / (5deg) plus the 7 comma ) (6deg) doesn't equal the
13 comma /) (12deg); this is the 4095:4096 schisma, ~0.423c. So you
can't notate ratios of 7 that are consistent with ratios of 13 in 282.
2) The 17'17 comma (2deg) doesn't equal the 75 (1deg), or put
another way, ) <> /(; this is the 163840:163863 schisma, ~0.243c.
So you can't notate ratios of 17 that are consistent with ratios of 7
and 13 in 282.
Or should we discard these and start over  I think I would then be
entitled to say that you have either a 288bias or an anti217 bias.
> >And I think that the
> >problems with 23 can be managed, considering how rarely it is
likely to
> >be used. You have to have a way to accommodate the electronic JI
> >composer who might want to modulate all over the place, and a
> >consistent ET mapping for JI intervals is the only way to do it
with a
> >finite number of symbols; this is where 217 really delivers the
> goods!
>
> I still fail to see why 217 is better than 282, except that various
> choices we have made along the way, regarding the symbols, have been
> biased toward 217.
My latest solution for the 23' comma is actually biased more toward
low error and 494 than it is toward 217, as you'll see in the
continuation of my reply to your message #4543.
> ...
> >> Hmm. It is certainly arguable that we should favour the
> >interpretation of ( as the 75 comma when notating ETs. What's the
> >smallest ET that would be affected by this?
> >
> >It's hard to say what is the smallest ET in which they differ
> >consistently.
>
> I mean: What's the smallest one we've agreed on that uses (, where
> the 75 comma interpretation of it would be a different number of
> steps from what we've used it for.
Our latest agreement has been on mostly ETs below 100, and I don't
think any of those even used (. The largernumbered ones were still
subject to review at the time you took your break, so they are still
open to review. I said that 193 would be affected, but the 19 comma )
 can be used instead of ( for 1deg, so that's no problem.
> >> Is ) still to be interpreted as the 19 comma and what is to be
its
> >complement?
> >
> >Yes, and its complement is still (~. I don't see any lowerprime
> >interpretations of it without going into rational complements,
where we
> >have only one: 11/7 = G)~. This is greater than G()
(2187/1408) by
> >15309:15488, ~20.125c (vs. the 19' comma, 19456:19683, ~20.082c).
But
> >this is for )~, so we must subtract ~ from this, but what comma
would
> >~ be? Since your next question has a positive answer (and since
I did
> >that one first I can peek at the answer), I'll use the 11limit
comma
> >99:100, which gives 42525:42592 (3^5*5^2*7:2^5*11^3, ~2.725c) as
the
> >11limit interpretation of ).
> >
> >This is meaningful only if you are using rational complements,
i.e.,
> >singlesymbol notation.
>
> No. Going via complements isn't what I had in mind. Does ) want to
> be used as a comma for any of 17/5, 17/7, 17/11, 17/13?
No.
> >> Is there a lower prime interpretation of ~ now too?
> >
> >Hmm, good question! Yes, using /~ as the 11'5 comma for 11/10
would
> >make that symbol 44:45, so ~ would be 99:100, ~17.399 cents. And
> >using /~ as the 13'7 comma for 13/7 would make /~ 1664:1701, so
~
> >would be 104:105, ~16.567 cents.
>
> OK. But this is not so, if we adopt (( as the 7/5comma symbol.
True (except that you meant the 11/5 comma, but I would prefer
calling it the 11'5 comma for now).
I'm going to have to take some time to figure out how everything
works out with this change (even if it is a change back for me),
since I've been using /~ for the past couple of weeks. But there is
no question that we should use ((.
George
(This is a continuation of my message #4580, which is in reply to
Dave Keenan's message #4543.)
New Rational Complements Â– Part 3

You previously mentioned that all of the rational complements are
consistent with 494ET (as they are also with 217ET). I would like
to define another pair of supplementary rational complements; we
didn't need these before, but they just might be useful when we're
doing some of the more obscure ETs. They're consistent in both 217
and 494, and the offset is 0.49 cents. They are:
~~ <> /( and
/( <> ~~
There are at least a couple of ratios that these can be used to
notate:
19/10 = Cb~~ or C\!(
19/15 = Fb~~ or F\!(
Also, we might want to allow both /( and ~~ as their own alternate
complements in certain instances:
/( <> /(
~~ <> ~~
This is just in case we need them. I would really not want to use
these unless it were a last resort. (After all, I want to keep the
number of symbols to a minimum.)
New Rational Complements Â– Part 4

Now for what may be the most controversial issue  actually, at the
last minute I came up with a very noncontroversial solution to the
whole thing (almost a nobrainer), but I'll leave what I had here;
just don't reply to any of it until you get to the end  I would
like to propose a definition of yet another supplementary pair of
rational complements:
)( <> ~\ and
~\ <> )(
Both of these are symbols that formerly lacked rational complements.
This is being done so that ~\, which I am now proposing to be the
23' comma instead of ((, may have a rational complement.
The reason that we did not previously use ~\ as the 23' comma is
that it lacked a rational complement. Using ~\ for this purpose has
the advantage of making the 23' comma consistent in the majority of
the best largenumbered ETs, including 152, 171, 217, 224, 270, 311,
494 (yes, 494 too!!!), and 612, *none* of which use (( consistently
as the 23' comma. (This is one more thing that would make a
transition between rational notation and 217 notation for JI as easy
and consistent  seamless might be a good word  as possible.)
Another advantage relates to the Reinhard property: The accuracy for
((, 1441792:1474767, ~39.149c, as the 23' comma, 16384:16767,
~40.004c, is contingent on the definition of ( as the 13'(115)
comma (715:729) or as the 29 comma (256:261). But if ( is defined
as the 11'7 comma (45056:45927), then the schisma is 2023:2024,
~0.856 cents, which is larger than what we have with ~\, 4352:4455,
~40.496c, for a schisma of 3519:3520, ~0.492c. Using ~\ makes the
schisma independent of the size of (.
There are a couple of possible objections to this:
1) The rational complementation offset is ~3.40 cents, which is
relatively large. (This would apply only to the singlesymbol
notation.) I don't think this is much of a problem, because the
complement symbols are *defined* as rational intervals, not as the
sum of their component stems and flags. We wanted to keep the
offsets low in order to minimize the inconsistencies, but consider
the alternative: when we had (( as the 23' comma we had an
inconsistency for the symbol itself in both 217 and 494; this new
proposal eliminates that.
2) The rational complement being proposed is consistent in 217, but
not in 494. I checked consistency for a number of the better ETs in
this general neighborhood; most of those under 300 are consistent,
and all of those above 300 are inconsistent, so it's definitely
related to the offset. (Again, this would apply only to the single
symbol notation, and the inconsistency occurs mostly in systems that
we are not even going to notate.)
Is it all that important to have all of the rational complements
consistent with 494? If it is, then I just got an idea for what may
be an even better solution, one that you suggested, but with a twist:
<< We could resurrect ~), with two left flags, as the complement
of the 23' comma. It isn't like a lot of people really care about
ratios of 23 anyway. We already made a good looking bitmap for ~)
with the wavy and the concave making a loop. >>
You were intending ~) to be the complement of ((, which has the
following consequences:
1) The complement has an offset of 1.59c with xL as the 13'(115)
comma, which increases to 2.02 cents if you make xL the 11'7 comma.
2) The complement is inconsistent in 494, but consistent in 217.
3) And as I said above, the 23' comma itself is inconsistent in both
217 and 494.
But if we were to make ~) the rational complement of ~\, then:
1) The offset would be 0.67c, independent of the xL flag.
2) The complement would be consistent in 494, but inconsistent in 217.
3) And as I said above, the 23' comma itself would be consistent in
both 217 and 494.
As for the inconsistency of the complement in 217, the ~) symbol
could either be replaced with the standard ~ symbol or else with )
( to specially designate the 23' complement. Thus only one obscure
complementary symbol would have to be changed in going from the
strict rational to the 217 quasirational version.
The foregoing was written before you pointed out that (( is the true
11'5 and 13'7 comma in your latest message. In light of this, I
would still assign ~\ as the 23' comma, while making (( a standard
symbol with rational complement ~(, thereby eliminating /~ from
the picture. (I was also using /~ for 17/11 as Ab\!~ or A\!!!~, but
I'll see how well (( works later.) One thing I am very happy about
is that the lateral confusability between /~ and ~\ is eliminated
if one of those two symbols is eliminated.
So what do you think?
George
 In tuningmath@y..., "gdsecor" <gdsecor@y...> wrote [#4586]:
> ... The foregoing was written before you pointed out that (( is
the true
> 11'5 and 13'7 comma in your latest message. In light of this, I
> would still assign ~\ as the 23' comma, while making (( a
standard
> symbol with rational complement ~(, thereby eliminating /~ from
> the picture. (I was also using /~ for 17/11 as Ab\!~ or A\!!!~,
but
> I'll see how well (( works later.)
I checked how well (( would work for 17/11. The commas we now have
for (( are:
11'5 comma (44:45, ~38.906c)
13'7 comma (1664:1701, ~38.073c)
The comma for 17/11 is intermediate in size, so it works just fine:
1117' comma (1377:1408, ~38.543c)
So with 1/1 as C, 17/11 will be Ab(!( or A(!!!(.
George
Sorry for the long delay in replying.
At 11:56 AM 8/17/2002 +0000, George Secor wrote:
>In working out a spreadsheet to automatically assign the symbols for
>ETs, one of the criteria I am using is to select ones that eliminate
>(or at least minimize) the inconsistencies. This can get not only
>complicated, but tricky.
Indeed. Good on you for doing this!
>So it looks like this will be the 217 standard set:
>
>217: ( ~ ~( / ) \ (( // /) /\ () (\ ~ ~(
>/ ) \ (( // /) /\ (new RCs)
Looks OK to me.
> > Reason enough to reject 282ET as what? Reject it as a good way of
> > having a fully notatable closed system that approximates 29limit JI?
> > I seriously disagree. It just means that we should use ( and ) with
> > their non13 meanings in 282ET.
>
>I guess I didn't get my point across. I want to be able to use a
>largenumbered ET (217 or 282 or whatever) to notate *JI* when there
>are no suitable rational symbols that will do the job. If ( or )
>don't have 13 meanings in 282, then there cannot be a good transition
>between the rational notation and the largeET notation  symbols
>would have to be converted from one to the other should a JI
>composition suddenly require 282ET symbols. This problem is minimized
>with 217, because even the nonstandard symbols such as ) and ( can
>be kept, because they are all the correct number of degrees.
I see what you mean.
> > >3) The following rational complements for the 15limit symbols are
>not
> > >consistent in 282:
> > >
> > >)~ <> ( 19' comma
> > > ( <> /) as 75 comma or 1113 comma (but 17Â’17 is okay)
> > >~ <> // 17 comma
> > > ) <> ) 7 comma
> > >// <> ~ 25 comma
> > > ( <> )~ 11'7 comma
> > >
> > >And besides this, there are others that are inconsistent, such as:
> > >
> > > ~ <> ~) as both the 19Â’19 and 23 comma
> >
> > All this means is that maybe we should consider making our rational
> > complements consistent with 282ET rather than 217ET.
> >
> > >What makes 217 so useful is that *everything* is consistent to the
>19
> > >limit, and, except for 23, to the 29 limit.
> >
> > I don't know what you mean by *everything* here. Isn't 282ET
> > consistent to the 29limit with no exceptions?
>
>It isn't consistent with the schismas that are essential to the
>rational notation:
>
>1) The 5 comma / (5deg) plus the 7 comma ) (6deg) doesn't equal the
>13 comma /) (12deg); this is the 4095:4096 schisma, ~0.423c. So you
>can't notate ratios of 7 that are consistent with ratios of 13 in 282.
>
>2) The 17'17 comma (2deg) doesn't equal the 75 (1deg), or put
>another way, ) <> /(; this is the 163840:163863 schisma, ~0.243c. So
>you can't notate ratios of 17 that are consistent with ratios of 7 and
>13 in 282.
>
>Or should we discard these and start over  I think I would then be
>entitled to say that you have either a 288bias or an anti217 bias.
OK. I understand now. Yes we definitely have a 217ET bias (or rather a
bias toward systems whose fifth is close to that of 217ET, like 494) in
the sense that we are only using schismas that vanish (I think we've been
overloading or overusing the term "consistent") in 217ET. And it may well
be possible to start completely from scratch and build a different system
where we only use subcent (or subhalfcent) schimas that vanish in
282ET. Then we'd have a 282ET bias (not anti 217ET). But then the 282ET
fifth _is_ closer to the precise 2:3 that the system is supposedly based on.
This is a daunting prospect, having come this far with the current system.
But wouldn't it be terrible if there was a _better_ system waiting to be
discovered, based on 282ET schismas, and we passed it over? Perhaps you
can come up with a simple argument as to why this is not possible, short of
a complete investigation?
> > I mean: What's the smallest one we've agreed on that uses (, where
> > the 75 comma interpretation of it would be a different number of
> > steps from what we've used it for.
>
>Our latest agreement has been on mostly ETs below 100, and I don't
>think any of those even used (. The largernumbered ones were still
>subject to review at the time you took your break, so they are still
>open to review.
We agreed on ( for 1deg67 which is wrong (or at least not
1,3,5,7consistently right) if ( is the 75 comma. I also proposed it for
93ET (3*31) but we didn't agree on a notation for that.
> > OK. But this is not so, if we adopt (( as the 7/5comma symbol.
>
>True (except that you meant the 11/5 comma, but I would prefer calling
>it the 11'5 comma for now).
Yes I did mean the 11/5 comma, and yes I will continue to call it the 11'5
comma.
At 08:17 AM 8/20/2002 +0000, George Secor wrote:
>(This is a continuation of my message #4580, which is in reply to Dave
>Keenan's message #4543.)
>
>New Rational Complements Part 3
>
>
>You previously mentioned that all of the rational complements are
>consistent with 494ET (as they are also with 217ET). I would like to
>define another pair of supplementary rational complements; we didn't
>need these before, but they just might be useful when we're doing some
>of the more obscure ETs. They're consistent in both 217 and 494, and
>the offset is 0.49 cents. They are:
>
>~~ <> /( and
>/( <> ~~
I have no objection to these at this stage.
>There are at least a couple of ratios that these can be used to notate:
>
>19/10 = Cb~~ or C\!(
>19/15 = Fb~~ or F\!(
You could more generally just say that it can notate 19/5. We know that
adding any number of factors of 2 or 3 doesn't change the saggital
accidental required.
>Also, we might want to allow both /( and ~~ as their own alternate
>complements in certain instances:
>
>/( <> /(
>~~ <> ~~
>
>This is just in case we need them. I would really not want to use
>these unless it were a last resort. (After all, I want to keep the
>number of symbols to a minimum.)
Definitely last resort.
>New Rational Complements Part 4
>
>
>Now for what may be the most controversial issue  actually, at the
>last minute I came up with a very noncontroversial solution to the
>whole thing (almost a nobrainer), but I'll leave what I had here; just
>don't reply to any of it until you get to the end  I would like to
>propose a definition of yet another supplementary pair of rational
>complements:
>
>)( <> ~\ and
>~\ <> )(
>
>Both of these are symbols that formerly lacked rational complements.
>This is being done so that ~\, which I am now proposing to be the 23'
>comma instead of ((, may have a rational complement.
>
>The reason that we did not previously use ~\ as the 23' comma is that
>it lacked a rational complement. Using ~\ for this purpose has the
>advantage of making the 23' comma consistent in the majority of the
>best largenumbered ETs, including 152, 171, 217, 224, 270, 311, 494
>(yes, 494 too!!!), and 612, *none* of which use (( consistently as the
>23' comma. (This is one more thing that would make a transition
>between rational notation and 217 notation for JI as easy and
>consistent  seamless might be a good word  as possible.)
>
>Another advantage relates to the Reinhard property: The accuracy for
>((, 1441792:1474767, ~39.149c, as the 23' comma, 16384:16767,
>~40.004c, is contingent on the definition of ( as the 13'(115) comma
>(715:729) or as the 29 comma (256:261). But if ( is defined as the
>11'7 comma (45056:45927), then the schisma is 2023:2024, ~0.856 cents,
>which is larger than what we have with ~\, 4352:4455, ~40.496c, for a
>schisma of 3519:3520, ~0.492c. Using ~\ makes the schisma independent
>of the size of (.
>
>There are a couple of possible objections to this:
>
>1) The rational complementation offset is ~3.40 cents, which is
>relatively large. (This would apply only to the singlesymbol
>notation.) I don't think this is much of a problem, because the
>complement symbols are *defined* as rational intervals, not as the sum
>of their component stems and flags. We wanted to keep the offsets low
>in order to minimize the inconsistencies, but consider the alternative:
>when we had (( as the 23' comma we had an inconsistency for the symbol
>itself in both 217 and 494; this new proposal eliminates that.
I really don't think I could have accepted a 3.4c offset.
>2) The rational complement being proposed is consistent in 217, but not
>in 494. I checked consistency for a number of the better ETs in this
>general neighborhood; most of those under 300 are consistent, and all
>of those above 300 are inconsistent, so it's definitely related to the
>offset. (Again, this would apply only to the singlesymbol notation,
>and the inconsistency occurs mostly in systems that we are not even
>going to notate.)
>
>Is it all that important to have all of the rational complements
>consistent with 494?
No. But to minimise offsets I think it needs to be consistent with _some_
similarly high numbered ET. 653ET was a favourite of mine for this purpose
at one time.
> If it is, then I just got an idea for what may be
>an even better solution, one that you suggested, but with a twist:
>
><< We could resurrect ~), with two left flags, as the complement of
>the 23' comma. It isn't like a lot of people really care about ratios
>of 23 anyway. We already made a good looking bitmap for ~) with the
>wavy and the concave making a loop. >>
>
>You were intending ~) to be the complement of ((, which has the
>following consequences:
>
>1) The complement has an offset of 1.59c with xL as the 13'(115)
>comma, which increases to 2.02 cents if you make xL the 11'7 comma.
>
>2) The complement is inconsistent in 494, but consistent in 217.
>
>3) And as I said above, the 23' comma itself is inconsistent in both
>217 and 494.
>
>But if we were to make ~) the rational complement of ~\, then:
>
>1) The offset would be 0.67c, independent of the xL flag.
>
>2) The complement would be consistent in 494, but inconsistent in 217.
>
>3) And as I said above, the 23' comma itself would be consistent in
>both 217 and 494.
>
>As for the inconsistency of the complement in 217, the ~) symbol
>could either be replaced with the standard ~ symbol or else with )(
>to specially designate the 23' complement. Thus only one obscure
>complementary symbol would have to be changed in going from the strict
>rational to the 217 quasirational version.
>
>The foregoing was written before you pointed out that (( is the true
>11'5 and 13'7 comma in your latest message. In light of this, I
>would still assign ~\ as the 23' comma, while making (( a standard
>symbol with rational complement ~(, thereby eliminating /~ from the
>picture. (I was also using /~ for 17/11 as Ab\!~ or A\!!!~, but I'll
>see how well (( works later.) One thing I am very happy about is that
>the lateral confusability between /~ and ~\ is eliminated if one of
>those two symbols is eliminated.
>
>So what do you think?
I think I'm confused, and I think I would have preferred you to spare me
the foregoing and just given me the "almost nobrainer".
So I think what you want to know is, do I think it is OK to have ~\ as the
23' comma with a rational complement of ~), and (( as the 11'5 and
13'7 commas with rational complement ~(. And I've already agreed to ~~
as the 5+19 comma with complement /(.
Well ~( already was the complement of (( because we needed (( as the
complement of ~( which is the 17' comma. So that's no problem.
And I also have no problem with ~) as the complement of ~\ since the
offset is so low and it interleaves nicely between the existing
complements. Given this option I must totally reject )( as a possible
rational complement for ~\ . Now the remaining question is whether I can
accept ~\ as the 23' comma. The answer is yes.
But the whole 282ET schisma question still haunts me.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 David C Keenan <d.keenan@uq.net.au> wrote:
> Sorry for the long delay in replying.
No problem  take your time.
> >... So it looks like this will be the 217 standard set:
> >
> >217: ( ~ ~( / ) \ (( // /) /\ () (\ ~
~( / ) \ (( // /) /\ (new RCs)
>
> Looks OK to me.
Good! This is point of agreement #1.
> > > >... The following rational complements for the 15limit
symbols are not
> > > >consistent in 282:
> > > >
> > > >)~ <> ( 19' comma
> > > > ( <> /) as 75 comma or 1113 comma (but 17'17 is
okay)
> > > >~ <> // 17 comma
> > > > ) <> ) 7 comma
> > > >// <> ~ 25 comma
> > > > ( <> )~ 11'7 comma
> > > >
> > > >And besides this, there are others that are inconsistent, such
as:
> > > >
> > > > ~ <> ~) as both the 19'19 and 23 comma
> > >
> > > All this means is that maybe we should consider making our
rational
> > > complements consistent with 282ET rather than 217ET.
> > >
> > > >What makes 217 so useful is that *everything* is consistent to
the 19
> > > >limit, and, except for 23, to the 29 limit.
> > >
> > > I don't know what you mean by *everything* here. Isn't 282ET
> > > consistent to the 29limit with no exceptions?
> >
> >It isn't consistent with the schismas that are essential to the
> >rational notation:
> >
> >1) The 5 comma / (5deg) plus the 7 comma ) (6deg) doesn't equal
the
> >13 comma /) (12deg); this is the 4095:4096 schisma, ~0.423c. So
you
> >can't notate ratios of 7 that are consistent with ratios of 13 in
282.
> >
> >2) The 17'17 comma (2deg) doesn't equal the 75 (1deg), or put
> >another way, ) <> /(; this is the 163840:163863 schisma,
~0.243c. So
> >you can't notate ratios of 17 that are consistent with ratios of 7
and
> >13 in 282.
> >
> >Or should we discard these and start over  I think I would then
be
> >entitled to say that you have either a 288bias or an anti217
bias.
>
> OK. I understand now. Yes we definitely have a 217ET bias (or
rather a
> bias toward systems whose fifth is close to that of 217ET, like
494) in
> the sense that we are only using schismas that vanish (I think
we've been
> overloading or overusing the term "consistent") in 217ET. And it
may well
> be possible to start completely from scratch and build a different
system
> where we only use subcent (or subhalfcent) schimas that vanish
in
> 282ET. Then we'd have a 282ET bias (not anti 217ET). But then
the 282ET
> fifth _is_ closer to the precise 2:3 that the system is supposedly
based on.
But then the fifth of 494 is closer to an exact 2:3 than that of 282:
217: ~702.304c  0.349c or 0.063deg wide
288: ~702.128c  0.173c or 0.041deg wide
494: ~702.024c  0.069c or 0.029deg wide
2:3 ~701.955c
Yet 494 uses the virtually the same schismas as 217.
> This is a daunting prospect, having come this far with the current
system.
> But wouldn't it be terrible if there was a _better_ system waiting
to be
> discovered, based on 282ET schismas, and we passed it over?
Perhaps you
> can come up with a simple argument as to why this is not possible,
short of
> a complete investigation?
To answer this, let me begin by quoting from prior correspondence:
[gs]
<< >I never considered 282 before, but I do see some problems with
it:
>
>1) 11 is almost 1.9 cents in error, and 13 is over 2 cents; these
>errors approach the maximum possible error for the system. (This is
>the same sort of problem that we have with 13 in 72ET.)
[dk]
You're only looking at the primes themselves. What about the ratios
between them. 217ET has a 2.8 cent error in its 7:11 whereas 282ET
never gets worse than that 2.0 cents in the 1:13. >>
This is only because the error in either one can never exceed half a
system degree, which for 282 is ~2.127 cents, but in 217 is ~2.765c.
So whatever advantage 282 has is only because it divides the octave
into more parts, which would be an advantage in itself. But there is
more than this to take into consideration  something that will
demonstrate that it is better to have the error of the primes
distributed in both directions rather than in a single direction,
given that primelimit consistency is maintained in each case. For
situations involving schisma consistency, sometimes the error of two
primes will accumulate rather than cancel, so that large
unidirectional errors added together exceed 1/2 degree, resulting in
an inconsistency.
Both 72 and 282 are consistent to at least the 17 limit (as are 217
and 494). Since the tridecimal schisma (4095:4096, ~0.423c) vanishes
in our notation but in neither ET, we cannot notate *both* ratios of
7 and 13 consistently in either one. I found this schisma at least a
week before I considered 217 as a basis for mapping out the symbols,
so we can't say that its selection was 217biased; indeed it vanishes
in a majority of the best ETs above 100.
The fact that it doesn't vanish in either 72 or 282 is a consequence
of the relatively large error for 13 (approaching the maximum) that I
referred to above. Since the functional 13 diesis (1024:1053) is
computed as the number of degrees (rounded) in the best fifth times
4, less the number of degrees in 3 octaves, plus the number of
degrees (rounded) for 8:13, we can calculate the number of degrees
for each of four divisions as follows:
Interval deg72 deg282 deg217 deg494
    
fifth (2:3) 42.117 164.959 126.937 288.971
rounded 42 165 127 289
times 4 168 660 508 1156
less 3 octaves 216 846 651 1482
equals 48 186 143 326
plus 8:13 rounded 50 198 152 346
equals 13 diesis 2 12 9 20
We then calculate the number of degrees in the 5+7 comma for each:
Interval deg72 deg282 deg217 deg494
    
5 comma 80:81 1 5 4 9
7 comma 63:64 2 6 5 11
5+7 diesis 35:36 3 11 9 20
and compare these with the actual (as opposed to functional) number
of degrees for 1024:1053, the ratio of the 13 diesis:
Interval deg72 deg282 deg217 deg494
    
actual 13 diesis 2.901 11.362 8.743 19.903
rounded 3 11 9 20
for which we find complete agreement in all four divisions, as
opposed to the functional values calculated above:
Interval deg72 deg282 deg217 deg494
    
funct'l 13 diesis 2 12 9 20
We see that there is indeed an inconsistency in both the 72 and 282
divisions in that the number of degrees in the functional 13 diesis
does not agree with the number of degrees for the actual interval;
this inconsistency exists apart from the tridecimal schisma, but it
happens to cause this schisma not to vanish. This is due principally
to the excessive relative error in the representation of 13:
Interval deg72 deg282 deg217 deg494
    
actual 8:13 50.432 197.524 151.995 346.017
8:13 rounded 50 198 152 346
error in degrees 0.432 0.476 0.005 0.017
I don't think that we would want to devise a system of notation in
order to work around an inconsistency such as this, because I expect
that we would then have some problems notating those ETs in which the
tridecimal schisma *does* vanish. Our goal should be to make the
smallest schismas vanish.
As for what schismas do vanish in 282, maybe Gene would best be able
to answer that. I thought that it was most productive to start with
rational intervals, find the most useful schismas that can vanish,
and then look for ETs that are consistent with those schismas.
Working backwards by starting with a largenumber ET and then finding
the schismas that vanish in that ET is something that I don't have
much experience with, and I have a feeling that we're not going to
find anything better in 282 that will be useful in devising a
notation that offers a better economy of symbols.
> >Our latest agreement has been on mostly ETs below 100, and I don't
> >think any of those even used (. The largernumbered ones were
still
> >subject to review at the time you took your break, so they are
still
> >open to review.
>
> We agreed on ( for 1deg67 which is wrong (or at least not
> 1,3,5,7consistently right) if ( is the 75 comma. I also proposed
it for
> 93ET (3*31) but we didn't agree on a notation for that.
It is valid as 1deg67 for the 1113 comma, but I would prefer not to
use ( here (or elsewhere) unless it were valid for *both* the 75
and 1113 commas. In addition, if *both* the 17 ~ and 17' ~(
symbols were to occur in an ET notation, then it would also have to
be valid as the 17'17 comma in order to maintain consistent symbol
arithmetic. (At least that's the ideal I'm shooting for.)
Anyway, after looking at 67 again, I don't see any clear choice for
1deg among several possibilities. I would prefer to do the easier
ETs first (again) and in the process establish a hierarchy of rules
for choosing the symbols. As we attempt to do increasingly difficult
ones, we should get a better perspective on how to handle problems
such as this one.
I'll be discussing these issues in more detail in my next message,
when I will again address the hows and whys of notating some of the
less difficult ETs.
> >... You previously mentioned that all of the rational complements
are
> >consistent with 494ET (as they are also with 217ET). I would
like to
> >define another pair of supplementary rational complements; we
didn't
> >need these before, but they just might be useful when we're doing
some
> >of the more obscure ETs. They're consistent in both 217 and 494,
and
> >the offset is 0.49 cents. They are:
> >
> >~~ <> /( and
> >/( <> ~~
>
> I have no objection to these at this stage.
Good! This is point of agreement #2.
> >New Rational Complements Part 4
> >
> >
> >Now for what may be the most controversial issue  actually, at
the
> >last minute I came up with a very noncontroversial solution to the
> >whole thing (almost a nobrainer), but I'll leave what I had here;
just
> >don't reply to any of it until you get to the end  I would like
to
> >propose a definition of yet another supplementary pair of rational
> >complements:
> >
> >)( <> ~\ and
> >~\ <> )(
> > ...
>
> I think I'm confused, and I think I would have preferred you to
spare me
> the foregoing and just given me the "almost nobrainer".
I was just trying to compare it with the alternatives, because until
I did that, I didn't realize how much of a nobrainer it was.
> So I think what you want to know is, do I think it is OK to have
~\ as the
> 23' comma with a rational complement of ~), and (( as the 11'5
and
> 13'7 commas with rational complement ~(. And I've already agreed
to ~~
> as the 5+19 comma with complement /(.
>
> Well ~( already was the complement of (( because we needed ((
as the
> complement of ~( which is the 17' comma. So that's no problem.
>
> And I also have no problem with ~) as the complement of ~\ since
the
> offset is so low and it interleaves nicely between the existing
> complements. Given this option I must totally reject )( as a
possible
> rational complement for ~\ .
I meant )( to be only a 217specific alternative complement; ~)
would be the true rational complement.
> Now the remaining question is whether I can
> accept ~\ as the 23' comma. The answer is yes.
Good! Then this is point of agreement #3.
> But the whole 282ET schisma question still haunts me.
Did I deal with it above adequately?
George
(This is a continuation of my message #4586, which is in reply to
Dave Keenan's message #4543.)
Summary of Additional Rational Complements

In addition to the seven 217 standard symbol RC pairs and the
supplementary pair of RCs I listed previously, there are then four
additional pairs of supplementary symbols in my proposal. These
rational complements are used for some of the ratios of 17, 19, and
23:
19 comma and (11'7)+(19'19) comma: ) <> (~ and (~ <> )
23 comma and 7+17 comma or 5+17' comma: ~ <> ~) and ~) <> ~
17+19 comma and 115+17 comma or 23' comma: ~) <> ~\ and ~\ <
> ~)
17+23 comma and 5+(1717') comma: ~~ <> /( and /( <> ~~
Here is how the ratios from 17 through the 21 limit are notated:
17/9 = B~! 18/17 = Db~ or D//!!
or Cb~( or C(!!( C#~!( or C((
17/10 = Bbb~) or Bx~ 20/17 = D#~!) or D~
17/11 = Ab(!( or A(!!!( 22/17 = E((
17/12 = F#~! or F// 24/17 = Gb~ or G//!!
or Gb~( or G(!!( F#~!( or F((
17/13 = 17/13 = F(! 26/17 = G(
17/14 = Eb// or E~!! 28/17 = A\\!
17/15 = Ebb~) or Ex~ 30/17 = A#~!) or A~
17/16 = C#~! or C// 32/17 = Cb~ or C//!!
or Db~( or D(!!( B~!(
19/10 = Cb~~ or C\!( 20/19 = C#~!~ or C/(
or B or Db or D\/
19/11 = Bb(!~ or B(!!!~ 22/19 = D(~
19/12 = Ab) or A(!!~ 24/19 = E)
or G#)!~ or G(!! or Dx(! or D)X~
19/13 = G\\! 26/19 = F//
19/14 = F)) 28/19 = G)!)
(no rational complement defined; use / as alternate complement)
19/15 = Fb~~ or F\!( 30/19 = G#~!~ or G/(
or E or Ab or A\/
19/16 = Eb) or E(!!~ 32/19 = A)
or D#)!~ or D(!! or Gx(! or G)X~
19/17 = D~)! 34/19 = Bb~) or B~!!/
19/18 = Db) or D(!!~ 36/19 = B)
or C#)!~ or C(!! or Ax(! or A)X~
21/11 = Cb( or C)!!~ 22/21 = C#(! or C)~
21/13 = Ab(( or A~!!( 26/21 = E(!(
21/16 = F!) 32/21 = G)
21/17 = E\\! 34/21 = Ab// or A~!!
21/19 = )!) 38/21 = Bb)) or B\!!
21/20 = Db!( or D!!!( 40/21 = B(
We will have to prepare a comprehensive listing of these in some
form. It would be nice if we could have a spreadsheet in which you
could input a letterplussymbol(s) for a tone and a ratio up or down
for a second tone, and letterplussymbol options for the second tone
would be displayed in both single and doublesymbol versions.
(Something like this would be useful for ETs as well.)
Notation of ETs

Since we would want to see how well the proposals I have made for
modifying the RCs would work for various ETs, following are some that
I have tried.
First, for reference I am listing symbol sequences for some of the
ETs on which we have most recently agreed. These are the ones that
will not change as a result of the latest proposals.
12, 19, 26: /\ (RC)
17, 24, 31, 38: /\ /\ (RC)
45: /) /\ (RC)
22, 29: / \ /\ (RC)
36: ) ) /\ (RC & MS)
43, 50, 57, 64: /) (\ /\ (RC)
27: / /) \ /\ (RC)
34, 41: / /\ \ /\ (RC)
62: /) /\ (\ /\ (RC)
39, 46, 53: / /\ () \ /\ (RC)
51: ) / /) \ ) /\ (RC)
65, 72, 79: / ) /\ ) \ /\ (RC; ISA ) 65,72,79)
58: / \ /\ / \ /\ (RC & MS)
84: / ) /) (\ ) \ /\ (RC)
RC = rational complementation
AC = alternate complementation
MS = matching symbol sequence
MM = most memorable sequence
ISA = inconsistent symbol arithmetic
Some of the conditions that I needed to get the above symbols in my
spreadsheetunderconstruction are:
1) The 5 comma and 7 comma must each be less than 2/5 apotome.
2) The 11 diesis must be greater than 1/3 apotome.
3) The 11 diesis must be less than the 11' diesis if both symbols are
used.
This avoids results such as:
17: / /\
36: () /\ /\
27: /\ /) () /\
60: / ) ) \ /\
In the meantime I have discovered that a couple of those that we did
agree on have properties that now persuade me either to question or
reject outright the symbol sequences:
52: /) /\ (RC)
32: ) /\ () (~ /\ (RC)
for the following reasons.
The 13 comma /) is not valid as 1deg52. Instead I propose the half
apotome symbol of last resort that can usually be made to work when
nothing else will:
52a: (~ /\ [(117)+23 comma] (RC)
However, after doing 69, 76, 86, 93, and 100 (see below), where )\
is quite useful for the halfapotome, I thought that this might also
be a possibility:
52b: )\ /\ (RC)
With 32 the best we could do for 1deg was the 19 comma, which is
quite a bit smaller than 1deg52, 37.5 cents. We have subsequently
defined (( as the 11'5 comma (~38.9 cents), which would give us
this:
32: (( /\ () ~( /\ [11'5 comma] (RC)
~( would be 3deg32 with ( as the 75 comma, but since ~ is not
being used as the 17 comma (or as anything else, for that matter),
there is no inconsistency in symbol arithmetic, and the symbol can be
justified as 4deg simply on the basis of its being the rational
complement of ((, which is the same thing we did before for (~.
I have a question. In doing the symbol selection spreadsheet, my
logic gives this for both 36 and 43:
43: ) ) /\
but it gives 50, 57, and 64 with the 13 commas (as we agreed on
above), because the 7 comma ) is not 1deg for those systems. You
said that you wanted 2deg43 (~55.8c) to be a singleshaft symbol (\,
but I don't know what sort of test to introduce to give this result
for 43 without giving the 13 diesis precedence over the 7 comma for
other ETs in which both are valid (such as 36). Why is it so
important to have (\ as 2deg43?
Now for some of the larger ETs that we need to reconsider.
With most of those over 100 tones, I have found that matching symbol
sequences or, where that's not possible, most memorable symbol
selections will be the most important factor. Most memorable means
minimizing the number of flags while keeping the symbol arithmetic
consistent.
I have found that the easiest ETs between 100 and 217 can be done
with both matching symbol sequences and rational complementation:
118a (59 ss.): ~ / \ // /\ () ~ /
\ // /\ (RC & MS)
130 (65 ss.): ( / ) \ /) /\ (\ / )
\ /) /\ (RC & MS  75 comma)
142 (71 ss): ( / ) \ /) /\ () (\ / )
\ /) /\ (RC & MS  75 comma)
176a (88 ss.): ( ~ / ) \ ~) /) /\ () (\
~ / ) \ ~) /) /\ (RC & MS)
183: ( ~( / ) \ (( /) /\ () (\ ~( / )
\ /~ /) /\ (RC & MS)
217: ( ~ ~( / ) \ (( // /) /\ () (\ ~ ~
( / ) \ (( // /) /\ (RC & MS)
Even if you're not interested in the singlesymbol notation, the
rational complementation ideal still has validity in these larger
divisions, because in each halfapotome pairs of symbols are also
related as unidecimaldiesis complements such as ~ = /\  // and 
( = /\  /).
In all of the above, in every case where a particular symbol is used,
it is valid as all of the comma aliases for which it is called upon:
( is valid as both the 75 (or 7/5) and 1113 (or 13/11) comma in
all of these divisions (and also as the 17'17 comma in 183 & 217)
and is consistent as the rational complement of /).
~ is valid as the 17 comma and // as the both the 5+5 comma (1,5,25
consistent) and 13'5 (or 13/5) comma, and the two symbols are
consistent with their rational complements.
~( is valid as the 17' comma and (( as both the 11'5 (or 11/5) and
13'7 (or 13/7) commas and also the 1117' (or 17/11) comma, and the
two symbols are consistent with their rational complements.
~ is valid as the 23 comma and ~) as both the 7+17 and 5+17' (or
17/5) commas, and the two are consistent with their rational
complements. (If the ~ flag is used alone, I want to use it only if
it is valid as the 23 comma; I would use it as the 19'19 comma only
if it is being used in combination so that it would not be
misinterpreted as the 23 comma.)
So everything works perfectly for these select halfdozen ETs above
100.
For 176 ~ was not used for 2deg for two reasons: 1) // is not
consistent as its rational complement, which would have also
necessitated // for 6deg to match flags in the halfapotomes; 2)
also, // is compromised because 176 is not 1,5,25 consistent. I
chose not to use ~( for 2deg because ( was used as the 75 comma
for 1deg, but this is not valid as the 17'17 comma, which usage
would be required for ~( as 2deg. So it is by process of
elimination that I arrived at ~ for 2deg176, which is not bad,
because 23 is represented much better than 17 in this ET; the only
problem is that ~) is not valid as the 5+17' (or 17/5) comma.
However, if you don't like this many flags for 176, I have another
solution below.
This next one was not quite perfect:
125: ~( / \ (( /\ () ~( / \ (( /\ (RC &
MS)
~( is valid as the 17' comma but (( is valid as the 11'5 (or 11/5)
and 1117' (or 17/11) commas, and the two symbols are consistent as
rational complements. However, (( is not valid as the 13'7 (or
13/7) comma, so 13 usage must be excluded from the notation, which is
not inappropriate, since 13 is not well represented in this ET. I
don't think we can complain about the number of flags in this one.
The next easiest ETs can be done with matching sequences and mostly
rational complementation with a little bit of alternate
complementation:
171: ( ~( / ) \ ~\ /) /\ (\ ~( / ) \
~\ /) /\ (MS; 10,14deg AC)
I'll spare you the details, other than to say that ~\, which is
valid here as the 23' comma, serves nicely to keep the number of
flags down.
Here's the other solution for 176, which keeps the flags to a minimum:
176b (88 ss.): ( ~ / ) \ ~) /) /\ () (\
~ / ) \ ~) /) /\ (MS; 11,15deg AC)
I really don't know if I prefer this to version a; 23 is much better
represented than 17 in 176, which would justify using ~ for 2deg.
For 152 I have three solutions:
152a (76 ss.): ) ~( / \ (( /) /\ () (\ ~( /
\ (( /) /\ (MS; 14deg AC)
152b (76 ss.): ) ~ / \ ~) /) /\ () (\ ~ /
\ ~) /) /\ (MS; 14deg AC)
152c (76 ss.): ) ~ / \ ~) /) /\ () (\ ~ /
\ ~) /) /\ (MS; 10,13,14deg AC)
In version a, (( as 6deg152 is valid as the 11'5 (or 11/5) and 11
17' (or 17/11) commas, but not the 13'7 (or 13/7) comma. The
replacements in version b result in higher primes and more flags;
here ~) is valid as both the 7+17 and 5+17 (or 17/5) commas.
Version c uses the simplest matching symbols, and I am inclined to go
with that. (I have reached the conclusion that if a set of symbols
isn't close to flawless with rational complements, then we should
just go for the most memorable set, with matching symbols in the half
apotomes where possible.)
For some of the more difficult ETs above 100 I have matching
sequences and consistent symbol complementation using as few flags as
possible. In order to avoid using invalid 13limit symbol
indications (such as (( not being valid for both the 11'5 and 13'7
commas), I found that ~) and ~\ come in very handy, particularly
because they introduce no new flags in instances where either the 17
or 17' comma is also being used.
111 (37 ss): ~ / \ ~\ /\ () ~ / \ ~\ /\
(MS)
144: ~( / )) \ /) /\ (\ / )) \ /) /\
(MS)
193: ) ~ ~( / \ ~) ~\ /) /\ () (\ ~ ~
( / \ ~) ~\ /) /\ (MS)
207: ( ~( / /( ( \ ~\ /) /\ () (\ ~( / /
( ( \ ~\ /) /\ (MS)
224: ) ~ )~ / ) \ /~ // /) /\ () (\ ~ )
~ / ) \ /~ // /) /\ (MS  19 commas)
The symbol set for 111 is one that we previously agreed upon; I
didn't use // for 111 because it is not 1,5,25 consistent, whereas
118 and 125 are.
The ) flag in 144 is the 135 comma, so )) is 3deg. I'm
considering this without regard to a comprehensive multiplesof12
plan for the time being.
For the most difficult ETs (RC, AC, and MS not possible), it's a
matter of doing them any way you can to find the most memorable
selection of symbols:
128b (64 ss.): ) ~( / (( ~\ /\ () ) ~( \ (
( ~\ /\ (MM)
135a (45 ss.): ~ ~ / ( /~ /\ () ~ ~ \
( /~ /\ (MM)
135b (45 ss.): ~ ~( / ( /~ /\ () ~ ~( \
( /~ /\ (MM)
140 (70 ss.): ) ~( / )\ ~\ /) (~ (\ ) ~( \ )
\ ~\ /\ (MM)
181a: ( ~ ~ / /( ~) /~ /) (~ (\ ( ~ ~
\ /( ~) /~ /\ (MM)
181b: ( ~ ~( / /( ( /~ /) (~ (\ ( ~ ~(
\ /( ( /~ /\ (MM)
I have another version for 128 below, which uses rational
complementation without matching symbols in the halfapotomes.
I have two options for 135, with no choice other than /~ for 5deg.
Take your pick.
70 is given above as a subset of 140, but I found that it works
better on its own:
70: / \ /\ () / \ /\ (RC & MS)
77: / ) /\ () ) \ /\ (RC; ISA5deg)
While 77 can be done like 70, my spreadsheet selects ) in preference
to \ to eliminate lateral confusability. I have come to the
conclusion that there is not much point in trying to notate ETs 7
tones apart alike if one of them can be done a better way (hence 111,
118, and 125 were all done differently above).
For ETs below 100 (for which matching sequences are often not
possible) and for those above 100 for which / and \ are the same
number of degrees (and therefore for which matching sequences are not
possible), I think that rational complementation is the most
important principle, including those that we already agreed upon
(which I summarized above). Even for the doublesymbol notation
(where rational complementation is of little concern), some of the
flags in the second apotome will also occur in the first apotome,
where symbols are paired as unidecimaldiesis complements.
Here are some more of the ET notations that I am proposing:
68: \ / /\ /) () \ / /\ (RC & MS) if we
permit /\ < /)
80: ) / (~ /\ () ) \ (~ /\ [13'(115)+23 =
1119 diesis] (RC)
87a: ~ / ~) /\ () ~ \ ~) /\ (RC)
94a: ~( / (( /\ () ~( \ (( /\ (RC)
99a: ~ / ~) /) (~ (\ ~ \ ~) /\ (RC)
108a: / // ) /) (\ ) ~ \ /\ (RC; ~ as RC)
108b: / (( ) /) (\ ) ~( \ /\ (RC)
104a: ) ) / ( /\ () )~ \ ) (~ /\ [~ as
23 comma] (RC)
104b: ) ) / (~ /\ () ) \ ) (~ /\ [~ as
23 comma] (RC)
128a (64 ss.): ) ~( / (( (~ /\ () ) ~( \ (
( (~ /\ [~ as 23 comma] (RC)
The familiar flags can be used for 68 if we allow some the symbols to
be used in an unusual order. (Also see 51 above, which we previously
agreed on.)
I think that if the apotome is 10 or more degrees and if the symbols
in the halfapotomes *can* be made to match, then they *should* be
made to match, even if they are not all strict rational complements 
it is easier to remember them that way.
I want to make special mention of 94, since it is such an important
division. For 3deg94 (( is valid as the 11'5, 13'7, and 1117'
commas, and I chose ~( for 1deg as its unidecimaldiesis complement;
this pairing works perfectly, much better than ~ and //, which is
plagued with inconsistencies. I would like to use these rational
complement pairs whenever they work this well, but perhaps you would
prefer another option (to follow).
One problem I find with 108 version a is that // is not valid as the
13'5 comma, which is significant, since we are using the 13 commas
in the notation. In version b I have written off 11 for excessive
error and used (( as the 13'7 comma (even though it is invalid as
the 11'7 comma) and have used )~ as its rational complement.
For some divisions under 100 (with matching sequences not possible),
we would have to decide whether we prefer strict rational complement
sets to a "most memorable" selection of symbols, such as these
(including ones for 67 and 81):
87b, 94b: ~ / ~\ /\ () ~ \ ~\ /\ (MM)
87c, 94c: ~ / /~ /\ () ~ \ /~ /\ (MM)
87d, 94d: ~ / /~ /\ () ~ \ ~\ /\ (MM)
99b: ~ / ~\ /) (~ (\ ~ \ ~\ /\ (MM)
99c: ~ / /~ /) (~ (\ ~ \ /~ /\ (MM)
An important principle that I have employed in arriving at these
symbols sets is that the more difficult or obscure ETs should not
dictate what the notation for the easier and morelikelytobeused
ETs should be. (This is a corollary to the principle that the more
difficult things should not make the simpler things more difficult.)
So I have considered each division on its own without attempting to
use the same symbols in ETs differing by 7.
I have shown sets for both 108 and 144, which I will discuss further
when I reply to your proposal for the symbol sets for multiples of
12, where the two principles mentioned in the previous paragraph will
be relevant.
George
(To be continued.)
Just a quick reply to one question.
At 11:35 AM 29/08/2002 0700, George Secor wrote:
>I have a question. In doing the symbol selection spreadsheet, my logic
>gives this for both 36 and 43:
>
>43: ) ) /\
>
>but it gives 50, 57, and 64 with the 13 commas (as we agreed on above),
>because the 7 comma ) is not 1deg for those systems. You said that
>you wanted 2deg43 (~55.8c) to be a singleshaft symbol (\, but I don't
>know what sort of test to introduce to give this result for 43 without
>giving the 13 diesis precedence over the 7 comma for other ETs in which
>both are valid (such as 36). Why is it so important to have (\ as
>2deg43?
What seems important to me, is to be able to notate any ET using only singleshaft symbols in combination with # and b.
In that case, the largest number of steps to need a singleshaft symbol in an ET is given by
=TRUNC(MAX(steps_in_tone, steps_in_diatonic semitone)/2)
in some cases the largest number of steps will be catered for by the # or b itself.
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
 In tuningmath@y..., David C Keenan <d.keenan@u...> wrote:
> Just a quick reply to one question.
>
> At 11:35 AM 29/08/2002 0700, George Secor wrote:
> >I have a question. In doing the symbol selection spreadsheet, my
logic
> >gives this for both 36 and 43:
> >
> >43: ) ) /\
> >
> >but it gives 50, 57, and 64 with the 13 commas (as we agreed on
above),
> >because the 7 comma ) is not 1deg for those systems. You said
that
> >you wanted 2deg43 (~55.8c) to be a singleshaft symbol (\, but I
don't
> >know what sort of test to introduce to give this result for 43
without
> >giving the 13 diesis precedence over the 7 comma for other ETs in
which
> >both are valid (such as 36). Why is it so important to have (\ as
> >2deg43?
>
> What seems important to me, is to be able to notate any ET using
only
> singleshaft symbols in combination with # and b.
>
> In that case, the largest number of steps to need a singleshaft
symbol in
> an ET is given by
> =TRUNC(MAX(steps_in_tone, steps_in_diatonic semitone)/2)
> in some cases the largest number of steps will be catered for by
the # or b
> itself.
>  Dave Keenan
I don't understand this at all. For 43, steps_in_tone=7 and
diatonic_semitone=4, for which your formula gives 3. Did you mean
TRUNC(MAX(steps_in_tone/2, steps_in_diatonic_semitone)/2), for which
your formula gives 2? (However, I found that doesn't work either,
because it gives 1 for 27, 34, and 41ET, but we want 2.) TRUNC
(steps_in_apotome/2), which gives 1, is what I think it should be; we
can still notate 43 with singleshaft symbols using only ):
0 1 2 3 4 5 6 7
C C) C#!) C# C#) Cx!) Cx
Dbb Dbb) Db!) Db D!) D
This is how it would be with the 13comma symbols:
C C/) C(\ C# C#/) C#(\ Cx
Dbb Db(!\ Db/!) Db(!\ D/!) D
I don't recall that we previously objected to having a 7 comma alter
in the opposite direction in combination with a sharp or flat.
So I am at a loss as to what to do.
George
At 12:47 PM 30/08/2002 0700, George Secor wrote:
> > In that case, the largest number of steps to need a singleshaft
>symbol in
> > an ET is given by
> > =TRUNC(MAX(steps_in_tone, steps_in_diatonic semitone)/2)
> > in some cases the largest number of steps will be catered for by the
># or b
> > itself.
> >  Dave Keenan
>
>I don't understand this at all. For 43, steps_in_tone=7 and
>diatonic_semitone=4, for which your formula gives 3. Did you mean
>TRUNC(MAX(steps_in_tone/2, steps_in_diatonic_semitone)/2), for which
>your formula gives 2? (However, I found that doesn't work either,
>because it gives 1 for 27, 34, and 41ET, but we want 2.)
>TRUNC(steps_in_apotome/2), which gives 1, is what I think it should be;
>we can still notate 43 with singleshaft symbols using only ):
>
>0 1 2 3 4 5 6 7
>
>C C) C#!) C# C#) Cx!) Cx
> Dbb Dbb) Db!) Db D!) D
>
>This is how it would be with the 13comma symbols:
>
>C C/) C(\ C# C#/) C#(\ Cx
> Dbb Db(!\ Db/!) Db(!\ D/!) D
>
>I don't recall that we previously objected to having a 7 comma alter in
>the opposite direction in combination with a sharp or flat.
>
>So I am at a loss as to what to do.
Sorry George,
I screwed up. You nearly got it. What I meant to say was
=TRUNC(MAX(steps_in_apotome, steps_in_Pythagorean_limma)/2)
apotome = 2187:2048
Pythagorean limma = 243:256
(i.e. the Pythagorean versions of the chromatic and diatonic semitones)
and sure, it doesn't matter if you put the dividebytwos before the MAX. And there's certainly no objection to having a 7 comma alter in
the opposite direction in combination with a sharp or flat.
By the way, you left out the Db) in your first example and the Db in your second.
The way of thinking that will favour using saggitals in combination with # and b, is one that thinks of C# as a single symbol, and would rather not have to accept Db as being a different pitch. In this person's mind there are not 7 but 12 basic symbols which are to be modified by the saggitals. For example, when the key is nominally C or Am then the 12 symbols are Eb Bb F C G D A E B F# C# G#
So it could be:
0 1 2 3 4 5 6 7
C C) C#!) C# C#) C#(\
D(!/ D!) D
So you see it's the 4 step _limma_ (between C# and D) that causes the problem here. Similarly:
0 1 2 3 4
B B) B(\
C(!/ C!) C
 Dave Keenan
Brisbane, Australia
http://dkeenan.com
I wrote:
"The way of thinking that will favour using saggitals in combination with # and b, is one that thinks of C# as a single symbol, and would rather not have to accept Db as being a different pitch. In this person's mind there are not 7 but 12 basic symbols which are to be modified by the saggitals. For example, when the key is nominally C or Am then the 12 symbols are Eb Bb F C G D A E B F# C# G#"
I should have said "_One_ way of thinking that will favour using sagittals in combination with # and b ...", since some folks will prefer it even though they don't prescribe to this way of thinking. However I think that many trained musicians, who have never before had to deal with tunings other than 12ET, will think this way, in particular keyboard players and players of other fixed pitch instruments where all 12 equallyspaced pitches are almost equally playable. I became convinced of this through discussions with Paul Erlich and Joseph Pehrson.
It's clear that you and I have trouble seeing things from this perspective, immersed as we have been, in tuning theory, for many years.
I realised after sending the previous message that I have not followed it consistently either. A person who does not want to see C# and Db as different pitches (and therefore should use only one of them at a time to avoid inconsistencies) will need a single shaft symbol for TRUNC(steps_in_Pythagorean_limma/2) even if this is the same as
steps_in_apotome and could therefore be symbolised by # or b, e.g in 19ET, 26ET, 38ET and 45ET.
I certainly wouldn't expect you to _replace_ /\ and \!!/ with single shaft symbols in these (the extreme meantones), but I do feel that we must provide singleshaft _alternatives_ for them, when used with a chainoftwelvefifths basis (as opposed to a chainofsevenfifths). The same goes for 2deg43, with an alternative to ).
(\ is a sensible alternative for 1deg19 and 1deg26, but 2deg38 presents a problem. I can find no consistent candidate below the 23 limit, but it seems like we should use (\ on the basis that 2deg38 is the same as 1deg19.
) is 2deg45 but it doesn't seem wise to use this symbol for something that large and again I fall back on (\. Neither 38 nor 45 are 1,3,13consistent, but a 2 step shift does at least give the best 3:13 in both cases.
A single shaft alternative for ) as 2deg43 is no problem. It's fine to use both ) as 1deg43 and (\ as 2deg43, since the 13schisma vanishes.
2deg50 is already the singleshaft (\ as standard.
(\ also works for 3deg62, 3deg67, 3deg69, 3deg74, 4deg86, 4deg91.
But I can't see any possibility of meaningful singleshaft alternatives for:
3deg52, 3deg57, 3deg64, 4deg76, 4deg81, 4deg88, 4deg93 etc., so I'm prepared to give up on them. These ETs are all 1,3,9inconsistent and will be better notated as subsets anyway.
Here's a proposed rule:
if TRUNC(steps_in_Pythagorean_limma/2) > TRUNC(steps_in_apotome/2) then
the alternative singleshaft symbol for
d