Re: A common notation for JI and ETs

[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 multi-EDO/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 A-flat. (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 prime-commas 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 one-fourth?

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 sesqui-sharps/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 half-sharp/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 tuning-math@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 multi-EDO/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 prime-limit. 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 13-comma (1024:1053) to notate
27-tET, and it looks like it would be pretty useful in 50-tET too.
There are many others under 100-tET where 4095:4096 doesn't vanish.
37-tET is another such, where I was planning to use the 13-comma for
notation. 37-tET 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 one-fourth?

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 sesqui-sharps/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 half-sharp/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 tuning-math@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 prime-limit. 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 tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@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 prime-limit. 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
/tuning-math/message/1687
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning-math@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 3-limit. Maybe I should try showing 81/80 is the smallest in the 5-limit.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@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 3-limit. Maybe I
should try showing 81/80 is the smallest in the 5-limit.
>

If it looks hard, forget it. I'm sure you've got better things to do
with your time.

--- In tuning-math@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 tuning-math@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 E-semiflat 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@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 13-comma (1024:1053) to
notate
> 27-tET, and it looks like it would be pretty useful in 50-tET too.
> There are many others under 100-tET where 4095:4096 doesn't vanish.
> 37-tET is another such, where I was planning to use the 13-comma
for
> notation. 37-tET is {1,3,5,7,13} consistent.

I spent some time wrestling with 27-ET 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:

/tuning-math/message/3768

With this it looks as if I am going to be stopping at the 17 limit,
with intervals measurable in degrees of 183-ET. 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 tuning-math@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 E-semiflat 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 tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> I spent some time wrestling with 27-ET 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 108-ET (= 9*12-ET), using a trinary notation where the 5-comma
is one step, the 7-comma 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 183-ET.

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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@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 E-semiflat 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 17-limit approach that I outline below, you will see how I
handle this. I am also outlining a 23-limit 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 11-13 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 27-ET 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 108-ET (= 9*12-ET), using a trinary notation where the 5-
comma
> is one step, the 7-comma 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 27-ET
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 27-ET 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 27-ET notation. The same could be said about
50-ET. 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 183-ET.
>
> 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 17-limit and 23-limit 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 23-limit approach) or of no more than one symbol of
any sort (accomplished in my 17-limit 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 17-limit and 23-limit 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.

*17-LIMIT APPROACH*

The 17-limit 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 17-as-flat
comma), ~14.7 cents (2deg183)
Concave right flag (vR): 2187:2176 (3^7:2^7*17, the 17-as-sharp
comma), ~8.7 cents (1deg183)

I was a bit hesitant to use two different 17-commas, 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 11-as-semisharp comma), ~53.3 cents (8deg183)
sL+xR: 35:36, ~48.8 cents, which approximates
~sL+xR: 1024:1053 (the 13-as-semisharp comma), ~48.3 cents (7deg183)
sR+xL: 26:27 (the 13-as-semiflat 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 11-as-semiflat 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 single-symbol sagittal notation it is necessary to
have alterations exceeding half an apotome, since there is no way to
notate a (two-flag) sagittal sharp/flat *less* a (two-flag)
~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.

*23-LIMIT APPROACH*

And here is the 23-limit 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 17-as-flat 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 17-as-sharp
comma), ~8.7 cents (2deg217)
Concave right flag (vR): 512:513 (the 19-as-flat comma), ~3.4 cents
(1deg217)

The difference between this and the 17-limit approach is that I have
removed the 715:729 alteration and added the 512:513 alteration,
while reassigning the 17-commas 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 11-as-semisharp comma), ~53.3 cents (10deg217)
sL+xR: 35:36, ~48.8 cents, which approximates
~sL+xR: 1024:1053 (the 13-as-semisharp 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 21-limit and
represents the building blocks of the notation as approximate
multiples of 5.5 cents.

--George

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@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 E-semiflat 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 17-limit approach that I outline below, you will see how I
> handle this.

OK. So you have gone outside of one-comma-per-prime and
one-(sub)symbol-per-prime. But you have given fair reasons for doing
so in the case of 11 and 13.

> I am also outlining a 23-limit 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 11-13 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
27-ET
> 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 27-ET 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 27-ET notation. The same could be said about
> 50-ET.

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 183-ET 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 17-limit and 23-limit 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.

> *23-LIMIT APPROACH*
>
> And here is the 23-limit arrangement, which correlates well with
217-
> ET (apotome of 21 degrees):

I don't think you can call this a 23-limit notation, since 217-ET is
not 23-limit consistent. But it is certainly 19-prime-limit.

> 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 17-as-flat
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 17-as-sharp
> comma), ~8.7 cents (2deg217)
> Concave right flag (vR): 512:513 (the 19-as-flat comma), ~3.4 cents
> (1deg217)
>
> The difference between this and the 17-limit approach is that I have
> removed the 715:729 alteration and added the 512:513 alteration,
> while reassigning the 17-commas 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 11-as-semisharp comma), ~53.3 cents (10deg217)
> sL+xR: 35:36, ~48.8 cents, which approximates
> ~sL+xR: 1024:1053 (the 13-as-semisharp 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 21-limit 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 29-limit in 282-ET if
you want both sets of 11 and 13 commas, and 31-limit in 311-ET 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 282-ET or 311-ET and so the mapping
of commas to arrows is just way too obscure.

217-ET is definitely the highest ET you can use with the above
additional constraint.

I notice that left-right confusability has gone out the window. But
maybe that's ok, if we accept that this is not a notation for
sight-reading by performers. However, it is possible to improve the
situation by making the left-right confusable pairs of symbols either
map to the same number of steps of 217-ET 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 217-ET-based scheme.
1. Swap the flags for the 17 comma and 19 comma (vL and vR)
2. Swap the flags for the 7 comma and 11-as-semisharp/5 comma. (xR and
sR)
3. Make the xL flag the 11-as-semiflat/7 comma instead of the
17-as-flat comma.

So we have
Steps Flags Commas
------------------------
1 vL 19-comma 512:513
2 vR 17-comma 2187:2176
3 vL+vR 17-as-flat-comma 4131:4096
4 sL 5-comma 80:81
5 sR 7-comma 63:64
6 xL or xR
7 vL+sR or xL+vR
8 vL+xR
9 sL+sR 13-as-semisharp comma 1024:1053
10 sL+xR 11-as-semisharp comma 32:33
11 xL+sR 11-as-semiflat comma 704:729
12 xL+xR 13-as-semiflat comma 26:27
21 apotome

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> So we have
> Steps Flags Commas
> ------------------------
> 1 vL 19-comma 512:513
> 2 vR 17-comma 2187:2176

Oops! That should have been:

Steps Flags Commas
------------------------
1 vR 19-comma 512:513
2 vL 17-comma 2187:2176
3 vL+vR 17-as-flat-comma 4131:4096
4 sL 5-comma 80:81
5 sR 7-comma 63:64
6 xL or xR
7 vL+sR or xL+vR
8 vL+xR
9 sL+sR 13-as-semisharp comma 1024:1053
10 sL+xR 11-as-semisharp comma 32:33
11 xL+sR 11-as-semiflat comma 704:729
12 xL+xR 13-as-semiflat 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > > >
>
> OK. So you have gone outside of one-comma-per-prime and
> one-(sub)symbol-per-prime. But you have given fair reasons for
doing
> so in the case of 11 and 13.

Gene had a comment about this, however:

--- In tuning-math@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 JI-to-ET, ET-to-JI, or ET-to-ET 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 high-resolution image to one of lower resolution will
be more successful than the reverse; likewise, translation of music
from JI or a large-number ET (finer resolution) to a lower-number 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 11-and-13-defining 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 23-limit 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 11-13 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
27-ET
> > is not as formidable as it seemed. ... The same could be said
about
> > 50-ET.
>
> 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 183-ET 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 quarter-commas!)

> > *23-LIMIT APPROACH*
> >
> > And here is the 23-limit arrangement, which correlates well with
> > 217-ET (apotome of 21 degrees):
>
> I don't think you can call this a 23-limit notation, since 217-ET
is
> not 23-limit consistent. But it is certainly 19-prime-limit.

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 46-EDO misses 17-limit
consistency
> for a single pair of intervals (15/13 & 26/15)! I don't think this
> precludes using it for 17-limit harmony (no EDO of lower number can
> compete with it), and I know from experience that 15-limit
harmonies
> can be successfully employed in 31-EDO without any disorientation
> whatsoever (with 13 being implied, much more successfully, in my
> opinion, than 7 is in 12-EDO), 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 19-prime
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 17-as-flat
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 17-as-sharp
comma), ~8.7 cents deg217)
> > Concave right flag (vR): 512:513 (the 19-as-flat comma), ~3.4
cents (1deg217)
> >
> > With the above used in combination, the following useful
intervals
> > are available:
> >
> > sL+sR: 32:33 (the 11-as-semisharp comma), ~53.3 cents (10deg217)
> > sL+xR: 35:36, ~48.8 cents, which approximates
> > ~sL+xR: 1024:1053 (the 13-as-semisharp 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 21-limit 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 29-limit in 282-ET if
> you want both sets of 11 and 13 commas, and 31-limit in 311-ET 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 282-ET or 311-ET 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:

> 217-ET is definitely the highest ET you can use with the above
> additional constraint.
>
> I notice that left-right 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
> sight-reading by performers.

I do want this to be a notation for sight-reading. 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 58-ET, which had given me problems before, now looks very good.)

> However, it is possible to improve the
> situation by making the left-right confusable pairs of symbols
either
> map to the same number of steps of 217-ET 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 217-ET-based
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 11-as-semisharp/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 11-as-semiflat/7 comma instead of the 17-as-
flat comma.

This restores a flag that was in my 17-limit 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 19-comma 512:513
> 2 vL 17-comma 2187:2176
> 3 vL+vR 17-as-flat-comma 4131:4096
> 4 sL 5-comma 80:81
> 5 sR 7-comma 63:64
> 6 xL or xR
> 7 vL+sR or xL+vR
> 8 vL+xR
> 9 sL+sR 13-as-semisharp comma 1024:1053
> 10 sL+xR 11-as-semisharp comma 32:33
> 11 xL+sR 11-as-semiflat comma 704:729
> 12 xL+xR 13-as-semiflat 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 xR-sR 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 right-hand flag for the 7-comma alteration in
the 72-ET notation. Before that all of the flags were straight.
Making the xR-sR symbol exchange would once again give the 7-comma
alteration a straight flag, which would negate the original reason
for the curved flag. The 72-ET notation could still use curved right
flags, but they would no longer symbolize the 7-comma 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 7-comma 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 7-comma with the xR (curved) flag, on the other hand,
makes a clear distinction between ratios of 5 and 7 in JI, 72-ET, 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 72-ET); 2) to distinguish 32:33 from
1024:1053 (in 46 and 53-ET, *without* lateral confusibility!); and 3)
to notate increments smaller than 80:81 (in 94-ET). 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 17-as-sharp 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 8-degree 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 tuning-math@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 quarter-commas!)

Yes. I noticed that too. But I'm not sure it matters much, since 31-ET
will of course _not_ be notated the same as every seventh note of
217-ET.

> 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 31-limit,
consistent with 311-ET, 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 23-limit, 217-ET
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 19-flag is the difference between the 11/5 flag and the 7-flag.
Probably not, since it involves a subtraction and this schisma is a
whole cent.

John's list only goes up to 23-limit. I'd like to see a list of
all the 31-limit 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 282-ET or 311-ET 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 13-as-semiflat comma in 311-ET
using only 6 flags, but if you do that, the 5 and 7 commas cannot be
single-flag. 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 58-ET, which had given me problems before, now looks very
good.)

That's great.

> > 2. Swap the flags for the 7 comma and 11-as-semisharp/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 right-hand flag for the 7-comma alteration in
> the 72-ET notation. Before that all of the flags were straight.
> Making the xR-sR symbol exchange would once again give the 7-comma
> 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
7-comma. It is most important to get the 11-limit 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 8-degree 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 single-stem saggitals used with the standard
sharp-flat 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 tuning-math@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 indigestion--what happened to 99, 118 and 140, for
starters?

> John's list only goes up to 23-limit. I'd like to see a list of
> all the 31-limit 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@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 31-limit,
> consistent with 311-ET, 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 23-limit, 217-ET
> correspondence.

Considering that the semantics of the notation have already put us
past the 19 limit, that 217 is not 23-limit 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
> 7-comma. It is most important to get the 11-limit 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 de-confuse 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 power-JI 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 8-degree 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 single-stem saggitals used with the standard
> sharp-flat 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 72-ET 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 72-ET 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 43-ET. 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 single-symbol approach or a single-symbols-
with-sharp-and-flat 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 tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> i--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> Considering that the semantics of the notation have already put us
> past the 19 limit, that 217 is not 23-limit 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 29-comma 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 11-comma 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 29-comma.

> 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 de-confuse 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 power-JI composer
> who wants a lot of prime numbers.

I totally agree.

I think we can take the 11-limit (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 13-limit 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 6/311-ET
sR 22113:22528 32.19 c 8/311-ET
xL 715:729 33.57 c 9/311-ET
xR 64:65 26.84 c 7/311-ET

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 23-comma that works so well re
no-new-flags (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 31-comma 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 17-comma (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 19-comma (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 17-comma.

What if we make the 19-comma 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 concave-left as 2176:2187 (the 17-comma) but make
concave-right 19683:19840, so that we have:

vL 2176:2187 8.73 c 3/311-ET
vR 19683:19840 13.75 c 3/311-ET

sL+vR 243:248 35.26 c 9/311-ET

By the way, I think that two straight left flags, one above the other
on the same shaft, is the best thing for two 5-commas. And do we
really need the 17'-comma, 4096:4131 (14.73 c)?

The above doesn't give us all the steps of 311-ET from 1 to 17, but I
don't think that matters. We don't need to actually be able to notate
311-ET. The gaps are 2, 5 and 15 steps (and 12 if you don't accept my
suggestion for two 5-commas).

> 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 43-ET. 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 single-symbol approach or a
single-symbols-
> with-sharp-and-flat 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 side-by
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 single-ASCII-character 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.]

5-comma 80;81

/|
/ |
| / \
|
|

7-comma 63;64
_
| \
| |
| P L
|
|

11-comma 32;33

/|\
/ | \
| ^ v
|
|

13-comma 1024;1053
_
/| \
/ | |
|
|
|

17-comma 2176;2187

|
_/|
|
|
|

19-comma 512;513

O
|
| * o
|
|

23'-comma 16384;16767 (unfortunately not 729;736)

|\
_/| \
|
|
|

29-comma 256;261
_
/ |
| |
| q d
|
|

31-comma 243;248

/|
/ |\_
|
|
|

For the down versions of these, flip them vertically (don't rotate
them 180 degrees).

The smaller 23-comma _can_ be rendered as, the unfortunately
complicated:

23-comma 729;736

O
_/O
|
|
|

Notice that lateral confusability only occurs beyond the 23-limit, and
this might be eliminated by adding blobs to the end of the curved
strokes of the 29 and 31 commas like this.

29-comma 256;261
_
/ |
b |
|
|
|

31-comma 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
sub-half-cent schismas: 4095:4096 (13-limit), 3519:3520 (23-limit
using large 23-comma).

If you use the symbols for the large 11-comma and large 13-comma as
well as the small ones, you are also assuming the vanishing of the
sub-half-cent schisma 5103:5104 (29-limit).

This is all George Secor's work (apart from the liberties I've taken
with the post 13-limit 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 17-comma (which would have been 4096:4131). That's
because it can't be done without either adding another type of
flag, disallowing the smaller 23-comma, 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 23-comma (729:736) the larger 17-comma
could have a symbol like this. [Message Index, Expand messages]

17'-comma 4096:4131

|
_/O
|
|
|

--- In tuning-math@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 twelfth-tone 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 5-flag is on the left and the 7-flag on the
right, and why the 5-flag is straight and the 7-flag curved? Why
couldn't either of these properties be switched between 5 and 7?

Here's a more complete "single-ASCII-character substitutes" proposal.

Symbol
dn up Comma Abbrev. descr. of actual symbol
-------------------------------------------------------------
\ / 5-comma 80;81 sL
L P 7-comma 63;64 xR
v ^ 11-comma 32;33 sL+sR
[ ] 11'-comma 704;729 xL+xR
{ } 13-comma 1024;1053 sL+xR
; | 13'-comma 26;27 xL+sR
j f 17-comma 2176;2187 vL
* o 19-comma 512;513 cO
w m 23-comma 729;736 vL+cO+cI
W M 23'-comma 16384;16767 vL+sR
q d 29-comma 256;261 xL
y h 31-comma 243;248 sL+vR

If it turns out we allow the large 17-comma instead of the small
23-comma then this could be:

J F 17'-comma 4096;4131 vL+cI

The abbreviated descriptions above refer to single-shaft 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 311-ET 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 311-ET 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 "single-ASCII-character substitutes" proposal.
>
>Symbol
>dn up Comma Abbrev. descr. of actual symbol
>-------------------------------------------------------------
>\ / 5-comma 80;81 sL
>L P 7-comma 63;64 xR
>v ^ 11-comma 32;33 sL+sR
>[ ] 11'-comma 704;729 xL+xR
>{ } 13-comma 1024;1053 sL+xR
>; | 13'-comma 26;27 xL+sR
>j f 17-comma 2176;2187 vL
>* o 19-comma 512;513 cO
>w m 23-comma 729;736 vL+cO+cI
>W M 23'-comma 16384;16767 vL+sR
>q d 29-comma 256;261 xL
>y h 31-comma 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Here's a more complete "single-ASCII-character substitutes" proposal.
>
> Symbol
> dn up Comma Abbrev. descr. of actual symbol
> -------------------------------------------------------------
> \ / 5-comma 80;81 sL
> L P 7-comma 63;64 xR
> v ^ 11-comma 32;33 sL+sR
> [ ] 11'-comma 704;729 xL+xR
> { } 13-comma 1024;1053 sL+xR
> ; | 13'-comma 26;27 xL+sR
> j f 17-comma 2176;2187 vL
> * o 19-comma 512;513 cO
> w m 23-comma 729;736 vL+cO+cI
> W M 23'-comma 16384;16767 vL+sR
> q d 29-comma 256;261 xL
> y h 31-comma 243;248 sL+vR

I don't know if anyone cares about 12-et-compatibility up to the
31-limit, 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 tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Here's a more complete "single-ASCII-character substitutes"
proposal.
> >
> > Symbol
> > dn up Comma Abbrev. descr. of actual symbol
> > -------------------------------------------------------------
> > \ / 5-comma 80;81 sL
> > L P 7-comma 63;64 xR
> > v ^ 11-comma 32;33 sL+sR
> > [ ] 11'-comma 704;729 xL+xR
> > { } 13-comma 1024;1053 sL+xR
> > ; | 13'-comma 26;27 xL+sR
> > j f 17-comma 2176;2187 vL
> > * o 19-comma 512;513 cO
> > w m 23-comma 729;736 vL+cO+cI
> > W M 23'-comma 16384;16767 vL+sR
> > q d 29-comma 256;261 xL
> > y h 31-comma 243;248 sL+vR
>
> I don't know if anyone cares about 12-et-compatibility up to the
> 31-limit, 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 31-comma 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 12-tET).

I'm also unclear how this method chooses 729/704 over 8192/8019 for
the 11-comma, and 16767/16384 over 736/729 for the 23-comma. Since
both these choices involve 3^6 versus 3^-6.

I'm guessing these have something to do with h7 or 7-ET. Could you
please explain your method in more detail?

Perhaps instead of 7-ET it would make more sense to use the C major
scale in 12-ET.

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 12-ET, although we might
favour 12-ET to the extent that all the odd numbers, up to an
odd-limit 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 12-ET, 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 odd-limit and presumably
we want the "25-comma" to simply be two 5-commas, which means two
syntonic commas (6400:6561), which means 3^8, which means our range of
allowed-exponents-of-3 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 odd-limit, or to invoke a different comma for 25,
requiring its own symbol, such as the diaschisma (2025:2048) which has
3^-4. A separate 25-comma symbol seems like a bad idea to me, unless
it is obviously made up of two 5-comma symbols, in which case it
should _be_ two 5-commas.

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 13-comma because:
(a) it has that neat sub-half-cent 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
3-exponent of 7, and that is so that we have 2187:2176 as the
preferred 17-comma. 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 19-comma of 3.4 cents and the 5-comma of 21.5 cents. I
expect the other 17-comma (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 13-comma, which would become the beloved
1024:1053. But rather than introduce the disachisma as a 25-comma I'm
inclined to allow the 3-exponents to range from -4 to +8 so that, from
C, an 8:11 is a variety of Ab (or A, using the non-preferred 13-comma)
and a 16:25 is a variety of G#. Too bad about the possibility of
simultaneous enharmonics. How do others feel about this?

--- In tuning-math@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 "single-ASCII-character substitutes"
proposal.
> >
> >Symbol
> >dn up Comma Abbrev. descr. of actual symbol
> >-------------------------------------------------------------
> >\ / 5-comma 80;81 sL
> >L P 7-comma 63;64 xR
> >v ^ 11-comma 32;33 sL+sR
> >[ ] 11'-comma 704;729 xL+xR
> >{ } 13-comma 1024;1053 sL+xR
> >; | 13'-comma 26;27 xL+sR
> >j f 17-comma 2176;2187 vL
> >* o 19-comma 512;513 cO
> >w m 23-comma 729;736 vL+cO+cI
> >W M 23'-comma 16384;16767 vL+sR
> >q d 29-comma 256;261 xL
> >y h 31-comma 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> But rather than introduce the disachisma as a 25-comma I'm
> inclined to allow the 3-exponents to range from -4 to +8 so that,
from
> C, an 8:11 is a variety of Ab (or A, using the non-preferred
13-comma)
> and a 16:25 is a variety of G#.

That should have been "... an 8:13 is a variety of Ab ...".

Dave wrote:
>v ^ 11-comma 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > But rather than introduce the disachisma as a 25-comma I'm
> > inclined to allow the 3-exponents to range from -4 to +8 so that,
> from
> > C, an 8:11 is a variety of Ab (or A, using the non-preferred
> 13-comma)
> > 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 12-equal definitely gains a lot of tonalness because
13:11 and even 13:7 are represented by their best approximations in
12-equal. and simpler chords in 12-equal simply fail to evoke the 13-
limit, under any circumstances.

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> Dave wrote:
> >v ^ 11-comma 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 arrow-heads (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 head-flags (an ordinary arrow) and the
symbol for 704:729 has convex head-flags (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 tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@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 12-equal definitely gains a lot of tonalness because
> 13:11 and even 13:7 are represented by their best approximations in
> 12-equal. and simpler chords in 12-equal 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 25-comma I'm
inclined to allow the 3-exponents to range from -4 to +8 so
that, from C, an 8:13 is a variety of Ab (or A, using the
non-preferred 13-comma) and a 16:25 is a variety of G#."

Which is one reason we have both 26:27 ; |, and 1024:1053 { } as
13-commas. Are you saying we shouldn't have 1024:1053 at all? The
notation is pythagorean-based not 12-ET 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Which is one reason we have both 26:27 ; |, and 1024:1053 { } as
> 13-commas. 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 tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Which is one reason we have both 26:27 ; |, and 1024:1053 { } as
> > 13-commas. 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 half-apotome, 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 17-commas that have been mentioned are pythagorean comma
complements, and of the two 23-commas mentioned, one is a pythagorean
comma larger than the other; similarly the two 31-commas 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 31-comma when we have
243:248 (3^5:2^3*31) 35.3 c.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > Which is one reason we have both 26:27 ; |, and 1024:1053 { }
as
> > > 13-commas. 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 half-apotome, 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > Considering that the semantics of the notation have already put
us
> > past the 19 limit, that 217 is not 23-limit 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 29-comma 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 11-comma 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 29-comma.

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 11-limit (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 13-limit 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 6/311-ET
> sR 22113:22528 32.19 c 8/311-ET
> xL 715:729 33.57 c 9/311-ET
> xR 64:65 26.84 c 7/311-ET

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-
comma-per-prime (or in this case, one-flag-per-prime) 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 comma-flags (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 23-comma that works so well re
> no-new-flags (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 F-sharp and only
one that requires a G-flat, 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 23-comma, 729:736. We
then have the alternate 23 comma (16384:16767) as a freebie.

In the process of counting the number of sharp-vs.-flat occurrences
for 23, I also kept count for some other high odd-number ratios and
came to some conclusions which I will give below.

> The 31-comma 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 17-comma (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 19-comma (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 17-comma.
>
> What if we make the 19-comma 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 23-limit 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 filled-in 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 filled-in 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 concave-left as 2176:2187 (the 17-comma) but make
> concave-right 19683:19840, so that we have:
>
> vL 2176:2187 8.73 c 3/311-ET
> vR 19683:19840 13.75 c 3/311-ET
>
> sL+vR 243:248 35.26 c 9/311-ET

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 5-commas.

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 17-as-flat 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 23-comma
(729:736) of that same number of degrees?

> The above doesn't give us all the steps of 311-ET from 1 to 17, but
I
> don't think that matters. We don't need to actually be able to
notate
> 311-ET. The gaps are 2, 5 and 15 steps (and 12 if you don't accept
my
> suggestion for two 5-commas).

I think that the gaps are unacceptable for a couple of reasons.

As I mentioned above, in the process of counting sharp-vs.-flat
occurrences for 23 in my JI heptatonic scales, I also counted the
number of sharp-vs.-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 E-flat (where 1/1 is C), there were
almost as many instances where a heptatonic scale called for a D-
sharp (using the 19-comma 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 less-traveled 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 19-limit 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 35-odd-limit
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 217-ET 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 31-ET (or train string players to
play 31-ET), then your JI can be reckoned in alterations of +/- 1 to
3 increments of 217-ET (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 12-ET,
whether for (too-coarse) 72-ET or Johnny Reinhard's (too-small) one-
cent increments. Plus you get 31-ET 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 41-ET.

So the question now becomes: Are we left with any good reason for
basing the JI notation on 311 instead of 217?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@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 twelfth-tone 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 5-flag is on the left and the 7-flag on
the
> right, and why the 5-flag is straight and the 7-flag 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 5-flag 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 72-ET); 2) to distinguish 32:33 from
1024:1053 (in 46 and 53-ET, *without* lateral confusibility!); and 3)
to notate increments smaller than 80:81 (in 94-ET). >>

Finally, the 5-comma 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 72-ET 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 72-tone notation is most unfortunate. However, I
feel no obligation to corrupt the sagittal notation in order to make
it Sims-compatible, because I believe that he is the odd-man-out 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-
to-right, which should cause them no disorientation, but I doubt that
anything like that is going to happen.

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> > > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > >
> > > > Which is one reason we have both 26:27 ; |, and 1024:1053 { }
> as
> > > > 13-commas. 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 half-apotome, 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 217-ET-based
notation) 13 different symbols less than half an apotome in size from
which to select.

Only the two largest of these have dedicated apotome-complement
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 tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@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 half-apotome, 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 tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@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 half-apotome, 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 tuning-math@y..., "annelizkeenan" <annelizkeenan@y...> wrote:

>
> -- Dave Keenan

who's anneliz and does she like to annelize things as much as you
do? :)

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@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.
:-)

In-Reply-To: <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 31-ET (or train string players to
> play 31-ET), then your JI can be reckoned in alterations of +/- 1 to
> 3 increments of 217-ET (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 12-ET,
> whether for (too-coarse) 72-ET or Johnny Reinhard's (too-small) one-
> cent increments. Plus you get 31-ET 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 41-ET.

31-ET for the 31-limit, great!

In fact, a temperament with a period of 1/31 octaves is already my top
inconsistent 21-limit 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 12-equal in the 5-limit, which many put up with. But if the 21-limit
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
21-limit 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 31-limit, 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)

217-equal 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 31-equal fifths as you like, and they're
good enough in another context. Again, there isn't enough (any!) history
of tempered 31-limit music to pronounce on the importance of these
differences.

Graham

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@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 311-ET was 31-limit-unique until Graham
corrected me. What's the smallest ET that has a 31-limit-unique mapping,
even if it's not consistent? Could it be 624-ET. How about a 35-limit
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 11-limit
notation then they shouldn't have to know anything about xL also being
715:729 (13-limit).

> As I mentioned above, in the process of counting sharp-vs.-flat
> occurrences for 23 in my JI heptatonic scales, I also counted the
> number of sharp-vs.-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
11-comma or a 13-comma 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 217-ET.

However I do not wish to base a JI (rational) notation on _any_ temperament
that has errors larger than 0.5 c. For me, 217-ET and 311-ET 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
1200-ET is good enough as an ET-based JI notation.

What is the first ET above 1200 that is 35-limit consistent? 35-limit unique?

But I am more interested in a 31-limit (or 35-limit) temperament (not equal
or linear or even planar, but maybe 6D or 7D or 8D) with as many
sub-half-cent schismas as possible, vanishing. Hence the challenge which
no-one'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 217-ET, 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

In-Reply-To: <3.0.6.32.20020404214429.00b1bb90@uq.net.au>
David C Keenan wrote:

> What is the first ET above 1200 that is 35-limit consistent? 35-limit
> unique?

Above 1200? 1600. The only ones below that are 311 and 388. If I can
remember how I did uniqueness for the 11-limit, 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 1600-equal is unique.

Graham

I double-checked the following and found it to be in error:

--- In tuning-math@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 E-flat (where 1/1 is C), there were
> almost as many instances where a heptatonic scale called for a D-
> sharp (using the 19-comma 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 less-traveled 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 19-comma 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 311-ET was 31-limit-unique until Graham
>corrected me. What's the smallest ET that has a 31-limit-unique mapping,
>even if it's not consistent? Could it be 624-ET.

Yes, 624 to 633-tET are all 35-limit unique.
1600-tET is 37-limit consistent and 55-limit unique.

Manuel

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <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 31-ET (or train string players
to
> > play 31-ET), then your JI can be reckoned in alterations of +/- 1
to
> > 3 increments of 217-ET (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 12-ET,
> > whether for (too-coarse) 72-ET or Johnny Reinhard's (too-small)
one-
> > cent increments. Plus you get 31-ET 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 41-ET.

monzo was going to make midi mahler renditions in 217-equal some time
ago, when we were discussing how 217 is the et that supports adaptive
5-limit ji similar to vicentino's second tuning. 152-equal 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 tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> --- In tuning-math@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 311-ET was 31-limit-unique 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
> 11-comma or a 13-comma 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 53-ET this is not the case, and you
either need to have both 11-commas 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 apotome-complements,
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
217-ET.

This amounts, then, to a 19-limit-unique-&-consistent, polyphonic-
readable sagittal notation with non-unique 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, 217-ET and 311-ET 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
> 1200-ET is good enough as an ET-based JI notation.
>
> What is the first ET above 1200 that is 35-limit consistent? 35-
limit unique?

Graham suggests 1600-ET in his message #3947. It looks like a good
choice, inasmuch as:

1) It is 37-limit consistent;

2) The largest error for any 37-limit 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 31-schisma 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 31-limit (or 35-limit) temperament
(not equal
> or linear or even planar, but maybe 6D or 7D or 8D) with as many
> sub-half-cent schismas as possible, vanishing. Hence the challenge
which
> no-one'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 217-ET, 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 72-ET 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 217-ET:

There is a question that needs to be asked: are we notating JI or are
we notating 217-ET? 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
3-cent 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 fine-tuning (by
ear) you would be entitled to contemplate 3-cent 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 217-ET performance.

Putting this another way: If I write a piece for 13-limit 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 12-ET 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 tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> > In all the lower ETs where either an
> > 11-comma or a 13-comma 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 53-ET this is not the case, and you
> either need to have both 11-commas 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.
39-ET is 1,3,13-inconsistent 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 apotome-complements,
> 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 sub-half-cent 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
> 217-ET.
>
> This amounts, then, to a 19-limit-unique-&-consistent, polyphonic-
> readable sagittal notation with non-unique 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 217-ET or 311-ET have to do with it.
217-ET 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 217-ET.

What do you mean by "polyphonic-readable"? 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, 217-ET and 311-ET 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
> > 1200-ET is good enough as an ET-based JI notation.
> >
> > What is the first ET above 1200 that is 35-limit consistent? 35-
> limit unique?
>
> Graham suggests 1600-ET in his message #3947. It looks like a good
> choice, inasmuch as:
>
> 1) It is 37-limit consistent;
>
> 2) The largest error for any 37-limit 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 31-schisma 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 31-limit (or 35-limit) temperament
> (not equal
> > or linear or even planar, but maybe 6D or 7D or 8D) with as many
> > sub-half-cent schismas as possible, vanishing. Hence the challenge
> which
> > no-one'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
31-limit 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 1600-ET. In other words, any schisma that
vanishes in 1600-ET is acceptable to be built into the notation, but
certainly not all the schismas that vanish in 217-ET (or 311-ET or
388-ET).

So in that sense only, I could say that the notation is "based on"
1600-ET, but there is certainly no desire to notate every degree of
1600-ET 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 1600-ET at all, and it is fine that 217-ET 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 217-ET, 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 72-ET 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 217-ET:
>
> There is a question that needs to be asked: are we notating JI or
are
> we notating 217-ET? 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
> 3-cent errors a bit irrelevant.

OK. Good. So I wish you'd stop talking about it being "based on" or
"going with" 217-ET, 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 fine-tuning (by
> ear) you would be entitled to contemplate 3-cent 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 217-ET performance.

Yes, so 217-ET 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 13-limit 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 12-ET 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 217-ET is no different from showing how it maps to 22-ET. The
_definition_ of the symbols is in terms of the commas and schismas.

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and
> 20735:20736 (but not the 31-schisma 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > 3) It conflates all three of my schismas: 4095:4096, 3519:3520,
and
> > 20735:20736 (but not the 31-schisma 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 31-schisma. What's wrong with 253935:253952
(3^5*5*11*19 : 2^13*31) 0.12 cents. Consistent with 311-ET 388-ET
1600-ET, but not 217-ET.

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 217-ET doesn't matter because
the notation is not "based on" 217-ET and the 31 comma is not needed
in order to notate 217-ET.

George,

Here's another pass at a full set of 31-limit 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.]

5-comma 80:81

/|
/ |
| \ /
|
|

7-comma 63:64
_
| \
| |
| L P
|
|

11-comma 32:33

/|\
/ | \
| v ^
|
|

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

17-comma 2176:2187

|
_/|
| j f
|
|

19-comma 512:513
_
(_)
|
| o *
|
|

23-comma 729:736

|
|\_
| w m
|
|

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

31-comma 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:

19-comma 512:513
_
(_)
|
|
|
|

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

or 31-comma 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 37-schisma that vanishes in 1600-ET (and
388-ET) that lets us make a symbol for a 37-comma, 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 17-flag and the 29-flag. 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 11-5
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 217-ET
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 1600-ET 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 37-limit 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.]

5-comma 80:81

/|
/ |
| \ /
|
|

7-comma 63:64
_
| \
| |
| L P
|
|

11-comma 32:33

/|\
/ | \
| v ^
|
|

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

17-comma 2176:2187

|
|\_
| j f
|
|

19-comma 512:513
_
(_)
|
| o *
|
|

23-comma 729:736

|
_/|
| w m
|
|

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

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

37-comma 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 11-5 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> > > --- In tuning-math@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 217-ET.
> >
> > This amounts, then, to a 19-limit-unique-&-consistent, polyphonic-
> > readable sagittal notation with non-unique 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 217-ET or 311-ET have to do with
it.
> 217-ET 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 217-ET.
>
> What do you mean by "polyphonic-readable"? As opposed to what?

As opposed to polyphonic-confusible or polyphonic-difficult-or-slow-
to-read. 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, 217-ET and 311-ET
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
> > > 1200-ET is good enough as an ET-based 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 217-ET 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 super-fine correction in intonation by ear. I should
emphasize that those intervals in which you are most likely to be
able to hear 2-cent errors are the 5-limit consonances, none of which
have an error greater than 1 cent in 217-ET.

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 217-ET? 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
> > 3-cent errors a bit irrelevant.
>
> OK. Good. So I wish you'd stop talking about it being "based on" or
> "going with" 217-ET, or any other ET with larger than 0.5 cent
errors.

How about a compromise in which we "go with" both 217 and 1600-ET (37-
limit), with a specific set of symbols for 217 and a superset for
1600? (This might also make it possible to notate 311-ET using the
full set of symbols.) I am suggesting this in light of your
observation:

> Yes, so 217-ET 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 217-ET);

2) Introduce new symbols that I have no idea how to incorporate into
a single-symbol notation (this makes it difficult to do something
that I was previously able to do with 217-ET); and

3) Employ semantics inconsistent with 217-ET, as in the following:

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > > 3) It conflates all three of my schismas: 4095:4096, 3519:3520,
and
> > > 20735:20736 (but not the 31-schisma 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 31-schisma. What's wrong with 253935:253952
> (3^5*5*11*19 : 2^13*31) 0.12 cents. Consistent with 311-ET 388-ET
> 1600-ET, but not 217-ET.
>
> 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 217-ET doesn't matter
because
> the notation is not "based on" 217-ET and the 31 comma is not
needed
> in order to notate 217-ET.

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 31-schisma (above). During the two weeks or so that I
spent leading up to my 17-limit (183-tone) and 23-limit (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.

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 217-ET 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 upward-pointing arrow) to notate
the 31-comma 243:248, using the schisma 353935:253952. Also, the
alternate 37-comma 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 23-comma, 729:736. This would
then allow us to use newL+xR+vR to notate the 37-comma 36:37, using
the schisma 6992:6993. (Thus, the symbols for the two 37-commas both
contain a combination of a convex and concave flag on the same side,
which is most appropriate!)

--- In tuning-math@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 37-limit 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 37-limit 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
no-more-than-one-new-comma-per prime guideline throughout. Also,
there are some divisions between 100 and 217 that the 217-notation
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 tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Sure. But I don't understand what 217-ET or 311-ET have to do with
> it.
> > 217-ET 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 217-ET.
> >
> > What do you mean by "polyphonic-readable"? As opposed to what?
>
> As opposed to polyphonic-confusible or polyphonic-difficult-or-slow-
> to-read. 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" 1600-ET.
1600-ET 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 19-prime-limit
rational pitches which may be _more_ than 3 cents apart.
e.g. 19/14 and 34/25 are the same in 217-ET, 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 217-ET 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 217-ET 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 super-fine correction in intonation by ear. I should
> emphasize that those intervals in which you are most likely to be
> able to hear 2-cent errors are the 5-limit consonances, none of
which
> have an error greater than 1 cent in 217-ET.

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 half-step of 217-ET.

For example, I consider 72-ET to be a 7-limit microtemperament, but not a
9-limit or higher one. 217-ET is therefore the smallest ET that is a
21-limit microtemperament, and if that cutoff were bumped to 2.9 cents it
would be a 35-limit microtemperament (max 37-limit error is 4.6 cents).
311-ET is a 45-limit microtemperament and has no error greater than 1.9
cents in the 41-limit. So 311-ET is way more than we need from this point
of view, and 217-ET is just right.

By the way, 1600-ET gets us to the 45-limit 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" 217-ET, or any other ET with larger than 0.5 cent
> errors.
>
> How about a compromise in which we "go with" both 217 and 1600-ET
(37-
> limit), with a specific set of symbols for 217 and a superset for
> 1600? (This might also make it possible to notate 311-ET using the
> full set of symbols.)

OK. Except I'd probably prefer to put it this way:

The notation is based on pythagorean A-G,#,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
half-apotome (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 half-arrowhead 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 11-5 flag.

Because we use 1600-ET 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 19-limit to notate most ETs of
interest.

217-ET 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). 217-ET
has no error greater than 2.9 cents in the 35-limit, and so provided that
such errors are acceptable, we can use it to notate up to 35-limit 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 double-flat to a double-sharp using single symbols.

> I am suggesting this in light of your
> observation:
>
> > Yes, so 217-ET 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 217-ET);

217-ET only needs 19-limit, correct? I don't understand why you consider
that changing the 19-flag 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 11-limit. 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 19-limit (and thereby greatly reducing it in 217-ET).

> 2) Introduce new symbols that I have no idea how to incorporate into
> a single-symbol notation (this makes it difficult to do something
> that I was previously able to do with 217-ET); and

I think this is the big one, but I have a proposed solution. Later.

> 3) Employ semantics inconsistent with 217-ET, ...

I don't see this as a problem because I don't think that anything employing
those semantics is required in order to notate 217-ET

> 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 31-schisma (above). During the two weeks or so that I
> spent leading up to my 17-limit (183-tone) and 23-limit (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 217-ET 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 upward-pointing arrow) to notate
> the 31-comma 243:248, using the schisma 353935:253952. Also, the
> alternate 37-comma 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
lower-prime 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 23-comma, 729:736.

Beginning and ending with a new 23-flag. 7 flags is enough.

> This would
> then allow us to use newL+xR+vR to notate the 37-comma 36:37, using
> the schisma 6992:6993. (Thus, the symbols for the two 37-commas
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 tuning-math@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 37-limit 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 37-limit 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 217-ET
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 8-degree 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 double-shafted 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 half-apotome is simply repeated in the second half-apotome (and all
other half-apotomes). 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 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 217-ET.

This also solves your problem number 2 above.

Objections?

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > How about a compromise in which we "go with" both 217 and 1600-ET
> > (37-limit), with a specific set of symbols for 217 and a superset
for
> > 1600? (This might also make it possible to notate 311-ET using
the
> > full set of symbols.)
>
> OK. Except I'd probably prefer to put it this way:
>
> The notation is based on pythagorean A-G,#,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
> half-apotome (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 half-arrowhead 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
11-5 flag.
>
> Because we use 1600-ET 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 19-limit to notate most
ETs of
> interest.
>
> 217-ET 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).
217-ET
> has no error greater than 2.9 cents in the 35-limit, and so
provided that
> such errors are acceptable, we can use it to notate up to 35-limit
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 double-flat to a double-sharp 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 11-comma 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 11-5 comma that is used in achieving it, just as the 13-diesis is
also the (approximate) sum of two commas.

> > I am suggesting this in light of your
> > observation:
> >
> > > Yes, so 217-ET 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 217-ET);
>
> 217-ET only needs 19-limit, correct? I don't understand why you
consider
> that changing the 19-flag 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 11-limit. 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 19-limit (and thereby greatly reducing it in
217-ET).

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 single-symbol notation (this makes it difficult to do something
> > that I was previously able to do with 217-ET); 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 217-ET, ...
>
> I don't see this as a problem because I don't think that anything
employing
> those semantics is required in order to notate 217-ET

I had the impression that the 23-flag used in combination with
something else defined another prime inconsistenly in 217, but that
one (for the 37-comma 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 31-schisma (above). During the two weeks or so that
I
> > spent leading up to my 17-limit (183-tone) and 23-limit (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 217-ET 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 upward-pointing arrow) to
notate
> > the 31-comma 243:248, using the schisma 353935:253952. Also, the
> > alternate 37-comma 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
> lower-prime 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 13-diesis.
Taking C as 1/1, to get 13/8 I want to lower A (27/16) by a semiflat
(26:27) instead of raising A-flat 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 41-comma, 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 23-comma, 729:736.
>
> Beginning and ending with a new 23-flag. 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 37-comma 36:37,
using
> > the schisma 6992:6993. (Thus, the symbols for the two 37-commas
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 tuning-math@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 37-limit 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 37-limit 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 217-ET
(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 19-limit notation, with a *newL* (different-looking *left*) flag
for the 23 comma being foreign to all three: the 19-limit, 217-ET,
and the single-symbol notation.

Otherwise, I would need to have a way to incorporate the new flag
into the single-symbol notation, which will be discussed next.

> Now to the problems that occur when you try to make this work for
217-ET
> 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 8-degree 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 double-shafted 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 5-comma and 7-comma flags convex, which re-
introduces the problem of lateral confusibility, not only between
ratios of 5 and 7, but also for the two 11-dieses, which I think is a
more serious issue. (In addition, a curved 5-flag would not have a
constant slope, thereby obscuring the comma-up meaning.)

Another inconsistency is that vL||sR 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: vL||sR does not have to be
used, inasmuch as it is the same number of degrees as sL||. (And
vL||xR can also be avoided, being almost the same size as xL||vR.)

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 half-apotome is simply repeated in the second half-apotome
(and all
> other half-apotomes). 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 72-ET 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 half-apotome (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 217-ET.

I would want xL in 217 anyway, since it does handle ratios of 29.
After all, this is supposed to allow 35-limit (nonunique) notation,
and it would be better not to have a new flag appearing out of the
blue, just for 29.

Now regarding 72-ET, you will recall that I said this earlier:

<< Using curved flags in the 72-ET 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 43-ET. 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 72-ET and 43-ET. 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 full-octave diagram for
217 as well, just so we have a better idea of how everything comes
out.

--George

--- In tuning-math@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 full-octave 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 adaptive-ji version
of the chord progression

Cmajor -> A minor -> D minor -> G major -> C major

in 217-equal. 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 217-ET 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 11-5 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
19-odd-limit-unique, using single symbols spanning from double-flat to
double-sharp (or even just from flat to sharp), and use subsets of it to
notate lower ETs, and extend it to uniquely notate 19-or-higher-prime-limit
RTs (rational tunings),
and an approach
(b) that seeks to notate 19-or-higher-prime-limit 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 double-flat to double-sharp (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 _right-angle_ flag. It indicates direction simply by being
close to one end of the shaft. Since none of our arrows have
"tail-feathers" 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 ASCII-graphics, 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 ASCII-graphics, 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 tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@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 full-octave 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 adaptive-ji version
> of the chord progression
>
> Cmajor -> A minor -> D minor -> G major -> C major
>
> in 217-equal. 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 tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> > --- In tuning-math@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 single-symbol notation (this makes it difficult to do
something
> > > that I was previously able to do with 217-ET); 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 half-apotome is simply repeated in the second half-apotome
(and all
> > other half-apotomes). 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 72-ET 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 half-apotome (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 half-apotome repeated
in the second half-apotome.

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
183-ET, and at that time the vL and vR symbols were both being used
for 17-commas. 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 217-ET set"). This standard set of 217-ET 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
symbol-ratio 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
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 217-ET.
>
[gs:]
> I would want xL in 217 anyway, since it does handle ratios of 29.
> After all, this is supposed to allow 35-limit (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 apotome-complement symbol of one that has an xL
or vR flag. (These are among the "non-standard" 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 "non-standard" complement symbol that doesn't
occur in the expected order.

[gs:]
> Now regarding 72-ET, you will recall that I said this earlier:
>
> << Using curved flags in the 72-ET 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 43-ET. 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 72-ET and 43-ET. 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 72-ET (but not 43-ET), 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
counter-productive if the true complement were used everywhere else.
But in any case (as you note), there would be no concave or small
flag in 72-ET.

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 tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@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 full-octave 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 adaptive-ji version
> of the chord progression
>
> Cmajor -> A minor -> D minor -> G major -> C major
>
> in 217-equal. 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 19-comma (512:513, the
single degree of 217), then I would be happy to make a figure
illustrating this on a real staff.

--George

--- In tuning-math@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 217-ET 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 11-5 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
> 19-odd-limit-unique, using single symbols spanning from double-flat
to
> double-sharp (or even just from flat to sharp), and use subsets of
it to
> notate lower ETs, and extend it to uniquely notate 19-or-higher-
prime-limit
> RTs (rational tunings),
> and an approach
> (b) that seeks to notate 19-or-higher-prime-limit 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 double-flat to double-sharp (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 apotome-complement 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 tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> > --- In tuning-math@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 full-octave
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 adaptive-ji
version
> > of the chord progression
> >
> > Cmajor -> A minor -> D minor -> G major -> C major
> >
> > in 217-equal. 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 tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@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 1600-ET).

_19__
_____17_____
__________23__________
______________5______________
_________________7__________________
___________________11-5___________________
_____________________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
1600-ET, 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___________________________
____________________________11-5___________________________
_____________________________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________________________________
_______________________________11-5_________________________________
________________________________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
small-concave 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 right-angle 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 right-angle-ness can happen in the other direction too. We
actually have 5 types of flag. From smallest to largest they are:

concave right-angle
concave quarter-circle
straight
convex quarter-circle
convex right-angle

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
right-angle 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 apotome-complement 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 tuning-math@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 11-comma 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 11-5 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.

> 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
> > lower-prime 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 13-diesis.
> Taking C as 1/1, to get 13/8 I want to lower A (27/16) by a semiflat
> (26:27) instead of raising A-flat 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 (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
(Pythagorean-12).

> 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 41-comma, 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 217-ET, 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 11-5) 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 7-limit and from 72-ET.

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 19-limit and from 217-ET.

The problem is of course the lateral confusability of the symbols for 1
step and 2 steps of 217-ET. 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 217-ET 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 v|n
4 s|
5 s|n (confusable alternative is |x)
6 x| (confusable alternative is |s)
7 v|x or x|n
8 v|s or ss|
9 s|x
10 s|s
11 x|x
12 ||n (confusable alternative is x|s)

The worst thing about this is not using the 7-comma |x for 5 steps. The
schisma that says 7 comma = 5 comma + 19 comma, only works in 217-ET, not
1600-ET.

However, if we do this (changing only 5 and 6)

1 |n
2 v|
3 v|n
4 s|
5 |x
6 |s
7 v|x or x|n
8 v|s or ss|
9 s|x
10 s|s
11 x|x
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 v|v
4 s|
5 |x
6 |s
7 v|x or x|v
8 v|s or ss|
9 s|x
10 s|s
11 x|x
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 v|v
4 s|
5 |x
6 x|
7 v|x or x|v
8 v|s or ss|
9 s|x
10 s|s
11 x|x
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
x||x instead of s||s?

> Then with 217-ET
> (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 19-limit notation, with a *newL* (different-looking *left*) flag
> for the 23 comma being foreign to all three: the 19-limit, 217-ET,
> and the single-symbol notation.

Well yes, minimising the number of flag-types 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 7-limit 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 single-symbol 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 35-limit (nonunique) notation,
> and it would be better not to have a new flag appearing out of the
> blue, just for 29.

When dealing with 217-ET (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 217-ET set"). This standard set of 217-ET 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
> symbol-ratio 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 apotome-complement symbol of one that has an
xL
> or vR flag.

Yes. I was forgetting about the 11 step symbol x|x. The 12 step one can be
replaced with a double-shaft symbol if need be, but you're right, there's
no way to do 217-ET without the 29 flag although we should probably call it
the 13'-7 flag in this context, except when we describe the non-unique
35-limit possibilities of 217-ET.

> (These are among the "non-standard" 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 "non-standard" 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 217-ET? s||s 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 arrow-like 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 right-angle
@@
@@
@@

@@@@
@@ @@ 5
@@ @@
@@ convex quadrant
@@
@@
@@

@@
@@@@@@ 4
@@ @@
@@ concavoconvex
@@
@@
@@

@@
@@@@ 3
@@ @@
@@ straight
@@
@@
@@

@@
@@ 2
@@@@@@
@@ concave quadrant
@@
@@
@@

@@
@@@@ 1 (the minimum possible)
@@
@@ small concave right-angle
@@
@@
@@

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 right-angle
@@
@@
@@
@@
@@

@@@@
@@ @@ 9
@@ @@
@@ @@ convex quadrant
@@
@@
@@
@@
@@

@@
@@@@@@ 8
@@ @@
@@ @@ concavoconvex
@@
@@
@@
@@
@@

@@
@@@@ 6
@@ @@
@@ @@ straight
@@
@@
@@
@@
@@

@@
@@ 5
@@@@@@
@@ @@ alternative concavoconvex
@@
@@
@@
@@
@@

@@
@@ 4
@@@@
@@@@ @@ concave quadrant
@@
@@
@@
@@
@@
@@

@@
@@ 2
@@@@@@
@@ small concave right-angle
@@
@@
@@
@@
@@

@@
@@ @@
@@ @@@@
@@@@@@@@
@@@@ @@@@
@@@@@@@@
@@@@ @@
@@ @@
@@

@@
@@
@@
@@
@@@@
@@ @@
@@ @@
@@@@
@@

I figure, by measuring some sharps and flats on scores, that at full size
the symbols will be 11 (computer-screen) 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 right-angle
@@ @@
@@
@@
@@
@@
@@
@@

@@@@@@
@@ @@ 17
@@ @@
@@ @@ convex quadrant
@@ @@
@@
@@
@@
@@
@@
@@

@@@@
@@ @@ 14
@@ @@
@@ @@ alternative convex quadrant
@@ @@
@@
@@
@@
@@
@@
@@

@@
@@ 11
@@@@@@@@
@@ @@ concavoconvex
@@ @@
@@
@@
@@
@@
@@
@@

@@
@@@@ 10
@@ @@
@@ @@ straight
@@ @@
@@
@@
@@
@@
@@
@@

@@
@@ 7
@@@@
@@ @@ concave quadrant
@@@@ @@
@@
@@
@@
@@
@@
@@
@@

@@
@@ 5
@@
@@@@ alternative concave quadrant
@@@@@@ @@
@@
@@
@@
@@
@@
@@
@@

@@
@@ 3
@@
@@@@@@@@ small concave right-angle
@@
@@
@@
@@
@@
@@
@@

@@
@@ 2
@@@@@@
@@ alternative small concave right-angle
@@
@@
@@
@@
@@
@@
@@

@@
@@ @@
@@ @@@@
@@@@@@@@
@@@@ @@
@@ @@
@@ @@@@
@@@@@@@@
@@@@ @@
@@ @@
@@

@@
@@
@@
@@
@@
@@@@@@
@@ @@
@@ @@
@@ @@
@@@@
@@

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
right-angle.

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 flag-type 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
True-type 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.

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 sub-29-limit situations. Of course it's actually the 11'-7 flag.
There is no 13'-7 flag.

Here are some possible accidentals as bitmaps at computer-screen
resolution, shown on the staff, both line-centred and space-centred.

@@
@@@@@@@@@@@@@@@@@@@@
@@ @@@@
@@@@@@@@
@@@@ @@
@@@@@@@@@@@@@@@@@@@@ sharp
@@ @@@@
@@@@@@@@
@@@@ @@
@@@@@@@@@@@@@@@@@@@@
@@

@@@@@@@@@@@@@@@@@@@@
@@
@@ @@
@@ @@@@
@@@@@@@@@@@@@@@@@@@@
@@@@ @@
@@ @@ sharp
@@ @@@@
@@@@@@@@@@@@@@@@@@@@
@@@@ @@
@@ @@
@@
@@@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@@@
@@ @@
@@ double sharp
@@ @@
@@@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@@@

@@ @@
@@ @@
@@@@@@@@@@@@@@@@@@@@ double sharp
@@ @@
@@ @@

@@@@@@@@@@@@@@@@@@@@

@@
@@@@@@@@@@@@@@@@@@@@
@@ @@
@@@@@@@@
@@ @@
@@@@@@@@@@@@@@@@@@@@ natural
@@ @@
@@@@@@@@
@@ @@
@@@@@@@@@@@@@@@@@@@@
@@

@@@@@@@@@@@@@@@@@@@@
@@
@@
@@ @@
@@@@@@@@@@@@@@@@@@@@
@@ @@
@@ @@ natural
@@ @@
@@@@@@@@@@@@@@@@@@@@
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@@@

@@
@@
@@@@@@@@@@@@@@@@
@@
@@@@@@
@@ @@
@@@@@@@@@@@@@@@@ flat
@@ @@
@@@@
@@
@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@
@@ @@
@@ @@ flat
@@ @@
@@@@@@@@@@@@@@@@
@@

@@@@@@@@@@@@@@@@@@
@@@@@@@@@@
@@ @@
@@ @@
@@@@@@@@@@@@@@@@@@ convex right-angle
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@

@@@@@@@@@@@@@@@@@@

@@@@@@@@@@
@@@@@@@@@@@@@@@@@@
@@ @@
@@ @@ convex right-angle
@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@
@@@@@@
@@ @@
@@ @@
@@@@@@@@@@@@@@@@@@ convex quadrant
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@

@@@@@@@@@@@@@@@@@@

@@@@@@
@@@@@@@@@@@@@@@@@@
@@ @@
@@ @@ convex quadrant
@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@
@@
@@@@
@@ @@
@@@@@@@@@@@@@@@@@@ straight
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@

@@@@@@@@@@@@@@@@@@

@@
@@@@@@@@@@@@@@@@@@
@@ @@
@@ @@ straight
@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@
@@
@@
@@@@@@
@@@@@@@@@@@@@@@@@@ narrow concavoconvex
@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@

@@@@@@@@@@@@@@@@@@

@@
@@@@@@@@@@@@@@@@@@
@@@@@@
@@ @@ narrow concavoconvex
@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@@ narrow concave quadrant
@@@@ @@
@@
@@
@@@@@@@@@@@@@@@@@@
@@
@@

@@@@@@@@@@@@@@@@@@

@@
@@@@@@@@@@@@@@@@@@
@@
@@@@ narrow concave quadrant
@@@@ @@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@

@@
@@@@@@@@@@@@@@@@@@ small concave right-angle
@@@@@@
@@
@@
@@@@@@@@@@@@@@@@@@
@@

@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@
@@
@@ small concave right-angle
@@@@@@
@@@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@@@

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
right-hand down-pointing and space-centred.

@@@@@@@@@@@@@@@@
@@
@@
@@
@@@@@@@@@@@@@@@@
@@ @@
@@ @@ 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 (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
(Pythagorean-12)."

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 1600-ET.

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 tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#3994]:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > --- In tuning-math@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 1600-ET).
>
> _19__
> _____17_____
> __________23__________
> ______________5______________
> _________________7__________________
> ___________________11-5___________________
> _____________________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
> 1600-ET, 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 23-comma 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) Modifying-12-things/modulo-1600; vs.

b) modifying-7-things/modulo-217.

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 19-comma; and

B) Using the new flag for the 23-comma,

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 tuning-math/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 tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#3995]:
> --- In tuning-math@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 11-comma 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 11-5 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 co-author 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 off-list.)

> > 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
> > > lower-prime 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 sharp-vs.-
flat (or in this case natural-vs.-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
> (Pythagorean-12).

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 41-comma, 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 217-ET, 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 17-limit-in-183 and 23-limit-in-
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
17-comma if the inequality doesn't vanish in that ET.

> On another matter: Can you tell me why the apotome symbol should
not be
> x||x instead of s||s?

The simplest ET notations (17, 22, 24, 31, 41, which require only 5
and 11-comma 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 s|s in each of these (except 22, where s|s
isn't used). Curved flags appear in the notation only when they are
necessary or helpful, etc., etc.

> When dealing with 217-ET (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 13-diesis, 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 217-ET set"). This standard set of 217-ET
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
> > symbol-ratio 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 19-comma. Its apotome
complement symbol ( s||x ) 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 217-ET? s||s type notation will do fine.

Assuming Plan B for the moment, in which the 19-comma is vR, the
standard symbol set is:

degs symbol ratio
---- ------ -----
1 |v 512:513
2 v| 2176:2187
3 v|v 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 v|x 238:243
8 v|s 4352:4455, ~16384:16767
9 s|x 35:36, ~1024:1053
10 s|s 32:33
11 x|x 704:729, ~5005:5184
12 x|s 26:27

The apotome complements:
13 v|| 2048:2187 less 4352:4455
14 v||v 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 v||x 2048:2187 less 1114112:1121931
19 v||x 2048:2187 less 2176:2187
20 s||x 2048:2187 less 512:513
21 s||s 2048:2187 (the apotome)

The non-standard combinations:
5' s|v 40960:41553 (has ||v as complement)
6' x| 715:729, ~256:261 (used separately as 29-comma)
7' x|v 366080:373977 (has v||v as complement)

The only inconsistent apotome complement:
15' x|| 2048:2187 less 256:261 (used separately as /29-comma)

--- In tuning-math@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 full-page 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 low-budget, 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 single-pixel 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, x|||x, 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
> True-type 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 tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

> To further complicate this, we're each looking at this with a
> different primary objective in mind:
>
> a) Modifying-12-things/modulo-1600; vs.
>
> b) modifying-7-things/modulo-217.
>
> 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
(11-5)
17
19
23
29

Now I list the effect of all the notationally-useful 1600-ET 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 1600-ET was
what I was looking for. We can forget my 31-limit challenge now.

symb lft-flgs rt-flgs
------------------------
5 = 5
7 = 7
11 = 5 + (11-5)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (11-5)
17 = 17
17' = [none except 19 + 19 + 19 + 19]
19 = 19
19' = 23 + 19
23 = 23 [also 17 + 19 + 19]
23' = 17 + (11-5)
29 = 29
31 = 19 + (11-5)
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 217-ET. Also the one for 31'.

The following are valid in 217-ET, but not 1600-ET. Note that 217-ET only
gives unique mappings up to the 19-odd-limit. (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 tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > To further complicate this, we're each looking at this with a
> > different primary objective in mind:
> >
> > a) Modifying-12-things/modulo-1600; vs.
> >
> > b) modifying-7-things/modulo-217.
> >
> > 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
19-limit since beyond that is not required for 217-ET or any other ETs
that we plan to notate.

In fact I'm comitted to first ensuring that the 19-limit and 217-ET
notation is as good as it can be, ignoring any higher-limits or higher
ETs.

Our current disagreement re the 19 comma symbol is not a chromatic vs.
diatonic thing or a 217-ET vs. 1600-ET thing or a 19-limit vs. higher
limits thing. It is a question of whether it is better (at the
19-limit and in 217-ET) 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 same-style 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
one-symbol-for-one-comma-per-prime use of the symbols. In this way of
using it, you get a single-shaft 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
double-flag symbols are generally larger than the single flag.

1600-ET 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 5-comma and a 7-comma
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 13-comma and you have introduced an approximation.

In rational tunings this approximation is kept to inaudible levels by
basing it on 1600-ET. 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 5-comma + 7-comma =/= 13-comma (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 27-ET the 7-comma
vanishes and we use the 5 and 13 comma symbols. In 50-ET the 5-comma
vanishes and we use the 7 and 13 comma symbols.

37-ET is a case I'm not too sure about. Here we have the 5-comma being
2 steps, the 13 comma being 3 steps and the 7-comma vanishing. There
is no prime comma within the 41-limit that is consistently equal to 1
step (11-comma is 2 steps, same as 5-comma). We could notate 1 step as
13-comma up and 5-comma down, but if we insist on single symbols, is
it ok to use the 7-comma symbol to mean 13-comma - 5 comma? Or should
we use the 19-comma symbol for one step, even though it's
1,3,p-inconsistent?

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 tuning-math@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 non-prime 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 41-limit 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 tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

These commas don't seem to be h217-unique--was there some 217 mapping which you were looking at, or do I have the wrong set, or the wrong idea?

>--- In tuning-math@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 23-comma 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) Modifying-12-things/modulo-1600; vs.
>
>b) modifying-7-things/modulo-217.
>
>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 19-comma; and
>
>B) Using the new flag for the 23-comma,
>
>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 tuning-math/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 co-author 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 off-list.)

Thanks. Yes I'd be delighted to coauthor it with you. Do you mean that we
should take all further discussion of the notation off-list, or just
discussions re the article?

>Now that you mention it, it does have more to do with the sharp-vs.-
>flat (or in this case natural-vs.-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 17-limit-in-183 and 23-limit-in-
>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
>17-comma 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 217-ET).

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
>> x||x instead of s||s?
>
>The simplest ET notations (17, 22, 24, 31, 41, which require only 5
>and 11-comma 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 s|s in each of these (except 22, where s|s
>isn't used). Curved flags appear in the notation only when they are
>necessary or helpful, etc., etc.

I'm convinced.

>> When dealing with 217-ET (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 13-diesis, 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'-(11-5)) flag. Yes 26:27 is
certainly the primary 13-diesis 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 19-comma. Its apotome
>complement symbol ( s||x ) 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 19-comma as a concave flag.

>Assuming Plan B for the moment, in which the 19-comma is vR, the
>standard symbol set is:
>
>degs symbol ratio
>---- ------ -----
> 1 |v 512:513
> 2 v| 2176:2187
> 3 v|v 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 v|x 238:243
> 8 v|s 4352:4455, ~16384:16767
> 9 s|x 35:36, ~1024:1053
> 10 s|s 32:33
> 11 x|x 704:729, ~5005:5184
> 12 x|s 26:27
>
>The apotome complements:
> 13 v|| 2048:2187 less 4352:4455
> 14 v||v 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 v||x 2048:2187 less 1114112:1121931
> 19 v||x 2048:2187 less 2176:2187
> 20 s||x 2048:2187 less 512:513
> 21 s||s 2048:2187 (the apotome)
>
>The non-standard combinations:
> 5' s|v 40960:41553 (has ||v as complement)
> 6' x| 715:729, ~256:261 (used separately as 29-comma)
> 7' x|v 366080:373977 (has v||v as complement)
>
>The only inconsistent apotome complement:
> 15' x|| 2048:2187 less 256:261 (used separately as /29-comma)

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 v||s, not v||x. 5' should say " has ||x as complement".

So we have plan B):

1 |v +19
2 v| 17+
3 v|v 17+19
4 s| 5+
5 |x +7
6 |s +(11-5)
7 v|x 17+7
8 v|s 17+(11-5)
9 s|x 5+7
10 s|s 5+(11-5)
11 x|x (11'-7)+7
12 x|s (11'-7)+5 ~= (13'-(11-5))+(11-5)
13 v||
14 v||v
15 s||
16 ||x
17 ||s
18 v||x
19 v||s
20 s||x
21 s||s

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 217-ET) 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 half-apotome 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 left-hand concavoconvex flag
(the 17 flag) to be narrower than the right-hand one (the 23 flag), but
will use "c" for both. Here's option C1)

1 |v +19
2 c| 17+
3 c|v 17+19
4 s| 5+
5 |x +7
6 |s +(11-5)
7 c|x 17+7
8 c|s 17+(11-5)
9 s|x 5+7
10 s|s 5+(11-5)
11 x|x (11'-7)+7
12 x|s (11'-7)+5 ~= (13'-(11-5))+(11-5)
13 c||
14 c||v
15 s||
16 ||x
17 ||s
18 c||x
19 c||s
20 s||x
21 s||s

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 23-flag 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 +(11-5)
7 c|x 17+7
8 c|s 17+(11-5)
9 s|x 5+7
10 s|s 5+(11-5)
11 x|x (11'-7)+7
12 x|s (11'-7)+5 ~= (13'-(11-5))+(11-5)
13 c||
14 ||c
15 s||
16 ||x
17 ||s
18 c||x
19 c||s
20 s||x
21 s||s

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 23-flag wider than the 17-flag, 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
larger-in-cents 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+(11-5)
symbol (s|s) 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 217-ET I like the fact that it
doesn't give a double-flag symbol for something as small as 3 steps.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@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 non-prime
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 7-ET 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 41-limit 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 5-ET
one disagrees on 13 and 17, and the 7-ET one disagrees on
17,23,31,37,41, about something.

> These commas don't seem to be h217-unique--was there some 217
> mapping which you were looking at, or do I have the wrong set, or >
the wrong idea?

No. I understand that 217-ET is only unique up to the 19-limit. But I
understand they are all unique in 1600-ET. 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 1600-ET
that are 41-limit unique, or 37 limit unique, or even 31-limit unique.

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@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 17-limit-in-183 and 23-limit-in-
> 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
> 17-comma if the inequality doesn't vanish in that ET.

But I thought about it this afternoon and found one for the 19-as-
sharp comma, 19456:19683. The schisma is 531392:531441
(2^6*19^2*23:3^12, ~0.160 cents). This occurs when the 19-as-sharp
comma is approximated by adding the main 19-comma (512:513) to the 23-
comma (729:736). It vanishes in both 217 and 1600-ET, but not in 311-
ET.

--George

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > Here are the corresponding 5 and 7 et mappings of the non-prime
> 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 7-ET 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 41-limit JI.

> > These can be modifed to give the 41-limit 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 discussion--that 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 1600-ET
> that are 41-limit unique, or 37 limit unique, or even 31-limit unique.

I'll run a search for "standard" ones, at any rate, and report the results.

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.

They show respectively single-symbol and multi-symbol (i.e. with
conventional sharps and flats) notation of all the steps of 217-ET
from a double-flat to a double-sharp.

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 +(11-5)
7 w|x 17+7
8 w|s 17+(11-5)
9 s|x 5+7
10 s|s 5+(11-5)
11 x|x (11'-7)+7
12 ||v [single] x|s [multi] (11'-7)+5 ~= (13'-(11-5))+(11-5)
13 w||
14 ||w
15 s||
16 ||x
17 ||s
18 w||x
19 w||s
20 s||x
21 s||s

The flag apotome-complementation 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 tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@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 17-limit-in-183 and 23-limit-in-
> 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
> 17-comma if the inequality doesn't vanish in that ET.
>
> But I thought about it this afternoon and found one for the 19-as-
> sharp comma, 19456:19683. The schisma is 531392:531441
> (2^6*19^2*23:3^12, ~0.160 cents). This occurs when the 19-as-sharp
> comma is approximated by adding the main 19-comma (512:513) to the
23-
> comma (729:736). It vanishes in both 217 and 1600-ET, but not in
311-
> ET.

Judging from your message #4008, you found this one also:

symb lft-flgs rt-flgs
------------------------
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 17-commas 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 tuning-math@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: up-arrows 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 straight-flag
symbols should look before doing any more with the rest of them.

--George

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> Judging from your message #4008, you found this one also:
>
> symb lft-flgs rt-flgs
> ------------------------
> 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 17-commas 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 1600-ET).

They might give us a 41-comma symbol.

They might fall near the middle of a big gap in flag-comma 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 41-comma 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 217-ET,
253-ET and 311-ET. 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
(11-5) 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
(11-5)
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 1600-ET, with a possible allocation to
left and right flags.

Symbol Left Right
for flags flags
------------------------------
5 = 5
7 = 7
11 = 5 + (11-5)
11' = 29 + 7
13 = 5 + 7
13' = 29 + (11-5)
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 + (11-5)
29 = 29
31 = 19 + (11-5) [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 (11-5)
Wavy | 17 23
Concave | (17'-17) 19

Want to propose an alternative?

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning-math@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 2-3 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 double-shaft
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
double-flat 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.
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
space-centered, 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: up-arrows 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 half-way 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 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. 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 tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > Judging from your message #4008, you found this one also:
> >
> > symb lft-flgs rt-flgs
> > ------------------------
> > 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 17-as-flat scale function (hence 4096:4131) is musically
more useful (Margo Schulter uses the 14:17:21 triad) than the 17-as-
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 (11-5)
> Wavy | 17 23
> Concave | (17'-17) 19
>
> Want to propose an alternative?

Maybe -- and maybe not. We shall see.

--George