Yahoo Tuning Groups Ultimate Backup tuning-math More ennealimmal notation stuff

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Unfortunately sagittal does not have an unaccented symbol pair for
> 125:126 or any other comma that would correspond to 5deg441 or
> 7deg612.

How can you propose a more or less universal scheme and be unable to
deal with one of the most important commas? You are creating something
certain people will find useless, and hence will not use. I've seen
all kinds of bizarre things given symbol pairs, so it hardly makes
sense that the commas of the first rank are not able to be notated.

Although 125:126 is common as a vanishing comma it is very
> rarely needed for notation since ratios of 125/7 with any powers of 2
> and 3 are very rarely used in tunings (according to the Scala archive).

This method of analysis clearly does not work, and the distinction
between good vanishing commas and good notation commas is not tenable.
From 126/125 = (6/5)^2/(10/7), we see some of why it is significant as
a vanishing comma, but this exact relationship also shows why it is
important notationally, especially in any septimal tuning (ennealimmal
is only one example) which emphasizes the minor third. A 10/7 can be
notated as two minor thirds, down a 126/125. This is a basic fact of
notational life, like the observation that a 5 can be notated as four
fifths, down a comma. 126/125 has exactly the same Euclidean and Hahn
size as 64/63, so dismissing it *must* be an error. There's nothing
else that close to the origin which is smaller in size, and only by
going out a touch farther, to 225/224, do we beat it. This is a
septimal analysis, but the 5 and 7 limits are crucial to get right. I
see no future in worrying about the 37 limit if you can't do a good
job with the 7 limit. And leaving aside theory, I've been complaining
of its absence for practical reasons. It is screwing things up.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > Unfortunately sagittal does not have an unaccented symbol pair for
> > 125:126 or any other comma that would correspond to 5deg441 or
> > 7deg612.
>
> How can you propose a more or less universal scheme and be unable to
> deal with one of the most important commas?

I didn't say we were unable to deal with it. I merely said we don't
have an unaccented symbol for it, and hence we don't have a single
ASCII (or UTF-8) character pair for it.

We've struggled to come up with 31 pairs of UTF-8 characters to
represent all the unaccented sagittals, plus we have 5 user-definable
pairs that can be used to represent accented symbols. You're welcome
to use one of those user-definable pairs to symbolise 125:126.

,` -+ <> [] {}

If you use more than one of these user-definable pairs, we recommend
their order of increasing size should correspond to the order of the
above list.

With only 31 unaccented symbols to assign as accidentals within the
range from 0 to 68 cents we can't do better than an average spacing of
about 2 cents between them, although they are not quite so even.

If you consider other commas within 1 cent either side of 125:126 you
find the 17-comma 4096:4131. I believe that was a comma that you
yourself put forward for a sagittal symbol in the very early days of
saggital.

And if we look at how often 17 or 1/17 with any powers of 2 and 3
occur in the Scala archive and compare it to the same for 125/7 or
7/125 with any powers of 2 and 3, in other words if you count ratios
notatable with 4096:4131 and those notatable with 125:126 (on a chain
of fifths) you find the following.

Ratios notatable with 4096:4131 318 ocurrences
Ratios notatable with 125:126 62 ocurrences

The nearest unaccented symbol on the other side of 125:126 has primary
comma role 2816:2835. It notates 35/11 or 11/35 with any powers of 2
and 3.

Ratios notatable with 2816:2835 92 ocurrences

There just happen to be 31 other commas that we found were more in
need of an unaccented symbol than 125:126. Something had to just fall
off the end of the list as far as unaccented symbols are concerned.

Sagittal tries to be as broadly applicable as possible. You're working
on something very specific and very complex, with its nominals not
being a chain of fifths, and you're requiring a single UTF-8 symbol
for every comma. You shouldn't be surprised that you have to use a few
user-definable pairs.

> You are creating something
> certain people will find useless, and hence will not use.

Undoubtedly. But we're trying our hardest to appeal to as broad a
cross-section as possible.

> I've seen
> all kinds of bizarre things given symbol pairs,

Such as?

> so it hardly makes
> sense that the commas of the first rank are not able to be notated.
>
> Although 125:126 is common as a vanishing comma it is very
> > rarely needed for notation since ratios of 125/7 with any powers of 2
> > and 3 are very rarely used in tunings (according to the Scala
archive).
>
> This method of analysis clearly does not work,

That's far from clear to me. Please propose and justify an alternative
method of determining which commas are most likely to be used to
notate ratios.

It seems to me that one must start by determining which
(octave-equivalent) ratios are most likely to be notated. Such a list
does not progress neatly by prime limits. Odd limits and product
complexity are closer. But it should come as no surprise that for
example 17/16 occurs more often in actual tunings than all of 126/125,
125/63, 125/112, 224/125 etc. combined.

> and the distinction
> between good vanishing commas and good notation commas is not tenable.

Please explain why not.

I understand that there are neat mathematical criteria for good
vanishing commas, and that is a very attractive thing. But
unfortunately when designing a notation that will have the most
general appeal we have to deal with messy human psychology. And the
best we seem able to do there is to use statistics from the only large
library of tunings that I know of. The Scala archive.

Of course I would dearly love it if you could come up with a neat
mathematical formula that gives even a rough fit to those messy
statistics. I've requested this on this list before now.

> From 126/125 = (6/5)^2/(10/7), we see some of why it is significant as
> a vanishing comma, but this exact relationship also shows why it is
> important notationally, especially in any septimal tuning (ennealimmal
> is only one example) which emphasizes the minor third. A 10/7 can be
> notated as two minor thirds, down a 126/125. This is a basic fact of
> notational life, like the observation that a 5 can be notated as four
> fifths, down a comma.

Now you're talking. You've shown there are popular ratios that would
be notated with it in your case. But it's specific to nominals in a
chain of minor thirds.

In a general notation system we really do have to cater primarily for
chain-of-fifth nominals since ninety-something percent of users will
never leave them. Like I said, 125:126 just missed out on an
unaccented symbol by the skin of its teeth. But we will certainly give
it a left-accent-only symbol. We just haven't decided which one yet.
Or have we, George? ')~| looks good to me. In UTF-8 thats .¦ for down
and '¡ for up. But I suggest either ,` or -+ as a suitable
user-definable single-character pair.

> 126/125 has exactly the same Euclidean and Hahn
> size as 64/63, so dismissing it *must* be an error. There's nothing
> else that close to the origin which is smaller in size, and only by
> going out a touch farther, to 225/224, do we beat it. This is a
> septimal analysis, but the 5 and 7 limits are crucial to get right.

But Euclidean or Hahn distance of the comma is not the only important
consideration when asessing commas for notational usefulness. Hahn
distances of the ratios most likely to be notated with those commas
are more relevant.

> I
> see no future in worrying about the 37 limit if you can't do a good
> job with the 7 limit. And leaving aside theory, I've been complaining
> of its absence for practical reasons. It is screwing things up.

No one is worrying about the 37 limit. But unfortunately we do find
that notating the 17th harmonic is a _far_ more common requirement
that notating ratios that have both 7 and 5^3 as factors.

-- Dave Keenan

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> Sagittal tries to be as broadly applicable as possible. You're working
> on something very specific and very complex, with its nominals not
> being a chain of fifths, and you're requiring a single UTF-8 symbol
> for every comma. You shouldn't be surprised that you have to use a few
> user-definable pairs.

Not that specific; while I'm concentrating on ennealimmal since I'd
like that nailed down, this is really the 7-limit itself which is at
issue here. I'd like to see microtempering worked out for the 5 and 7
limits, and it doesn't seem to be. I'm not proposing that it has to
accord with Sagittal, but it seems like a good idea to the extent
possible. Apparently it has to accord with Scala also, and I don't
even know what that entails. But it *is* clear a symbol pair for
126/125 would be useful. It seems to me the ones which came up as
1,2,3,4,5 steps of 171 are pretty basic to the septimal enterprise;
that would be 225/224, 126/125, 81/80, 64/63, 50/49-49/48.

> > You are creating something
> > certain people will find useless, and hence will not use.
>
> Undoubtedly. But we're trying our hardest to appeal to as broad a
> cross-section as possible.

You started out with the idea of notating JI, whereas I am most
interested in notating temperaments. Moreover, I'm not really
interested much in anything beyond the 13 limit, but think the 7 limit
is crucial. So it isn't clear we are going in the same direction, but
there was a desire expressed to get the temperament enterprise in
accord, as much as possible, with Sagittal. If Sagittal could supply
symbols for the stuff which is *clearly* very important for the
purposes of notating temperaments, that might help, obviously.
Otherwise, how useful will it be? It's like coming up with a set of
letters, and leaving out basic phonemes. If you do that, your letters
aren't doing the job.

> That's far from clear to me. Please propose and justify an alternative
> method of determining which commas are most likely to be used to
> notate ratios.

I've pointed out before, and now again, that commas with a low
geometric badness figure are the ones to concentrate on. In the
7-limit, that means we would want symbols for 36/35, 49/48, 126/125,
225/224. If we are going to get into the farther reaches of
complexity, then also 2401/2400 and 4375/4374. The reason for this is
that these are the comma relationships we run into soonest. The same
analysis applied to the 5 limit tells us to have symbols for 16/15 (if
we go that large), 25/24, 128/125, and 81/80; and to add 15625/15552
and 32805/32768 if we are interested in higher complexity.

> It seems to me that one must start by determining which
> (octave-equivalent) ratios are most likely to be notated. Such a list
> does not progress neatly by prime limits. Odd limits and product
> complexity are closer. But it should come as no surprise that for
> example 17/16 occurs more often in actual tunings than all of 126/125,
> 125/63, 125/112, 224/125 etc. combined.

It *never* occurs in the tunings I've been considering. It makes sense
people might stick it in a JI scale--they stick all kinds of things in
JI scales. But temperaments are creatures of reason, even if the
generators are irrational. It makes sense to have symbols for higher
limit JI, as one possible project. But to ignore or disparage
temperments, which in effect is what you are doing, does *not* make
sense, and suggests Sagittal is confining itself to a JI ghetto.

> > and the distinction
> > between good vanishing commas and good notation commas is not tenable.
>
> Please explain why not.

I did--if a comma vanishes, then examing what relationships that
implies also shows how it applies to notation.

> I understand that there are neat mathematical criteria for good
> vanishing commas, and that is a very attractive thing. But
> unfortunately when designing a notation that will have the most
> general appeal we have to deal with messy human psychology.

Then please deal with mine. I am getting more and more convinced that
I should simply write the Sagittal project off as useless for my
purposes and not a system very well able to deal with tempering, and
you are hardly making a case why I should think otherwise. But what
prevents me from doing that is that I can't figure out what the
fricking problem is with assigning symbols to the important commas of
the lower prime limits; at least through 7.

> Now you're talking. You've shown there are popular ratios that would
> be notated with it in your case. But it's specific to nominals in a
> chain of minor thirds.
>
> In a general notation system we really do have to cater primarily for
> chain-of-fifth nominals since ninety-something percent of users will
> never leave them.

However, *even if* you use chain of fifths nominals, you still want to
deal with temperaments like ennealimmal or kleismic. Or if not, then
once again Sagittal is not interesting to me, because I can't use it
for notating temperaments.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> >
> > > Unfortunately sagittal does not have an unaccented symbol pair
for
> > > 125:126 or any other comma that would correspond to 5deg441 or
> > > 7deg612.
> >
> > How can you propose a more or less universal scheme and be unable
to
> > deal with one of the most important commas?
>
> I didn't say we were unable to deal with it. I merely said we don't
> have an unaccented symbol for it, and hence we don't have a single
> ASCII (or UTF-8) character pair for it.

Yes, the problem is that, although we do have a single *symbol* for
125:126 in the Sagittal staff notation, that symbol isn't expressed
as a single *character* in Sagittal shorthand, because the symbol is
accented.

> We've struggled to come up with 31 pairs of UTF-8 characters to
> represent all the unaccented sagittals, plus we have 5 user-
definable
> pairs that can be used to represent accented symbols. You're welcome
> to use one of those user-definable pairs to symbolise 125:126.
>
> ,` -+ <> [] {}

In order to represent 125:126 (or 5deg441) with a single character,
this is the only possible solution.

> ...
> > From 126/125 = (6/5)^2/(10/7), we see some of why it is
significant as
> > a vanishing comma, but this exact relationship also shows why it
is
> > important notationally, especially in any septimal tuning
(ennealimmal
> > is only one example) which emphasizes the minor third. A 10/7 can
be
> > notated as two minor thirds, down a 126/125. This is a basic fact
of
> > notational life, like the observation that a 5 can be notated as
four
> > fifths, down a comma.
>
> Now you're talking. You've shown there are popular ratios that would
> be notated with it in your case. But it's specific to nominals in a
> chain of minor thirds.
>
> In a general notation system we really do have to cater primarily
for
> chain-of-fifth nominals since ninety-something percent of users will
> never leave them. Like I said, 125:126 just missed out on an
> unaccented symbol by the skin of its teeth. But we will certainly
give
> it a left-accent-only symbol. We just haven't decided which one yet.
> Or have we, George? ')~| looks good to me. In UTF-8 thats .¦ for
down
> and '¡ for up. But I suggest either ,` or -+ as a suitable
> user-definable single-character pair.

Dave, let me clarify something, just to make sure we're on the same
page. We have a whole menagerie of ratios represented by )~|,
ranging from 17:19k (152:153, ~11.4c) to 11:35k (2816:2835, ~11.6c)
to 143C (143:144, ~12.1c) to 49C (3^13:2^15*7^2, ~12.2c), with the
kleisma-comma boundary going smack dab through the middle of the
range. Correct me if I'm mistaken, but I think 49C is its primary
role, which gives ')~| a primary role of 243:245 (~14.2c). 125:126
is about a mina (~0.4c) smaller than that, giving ')~|. as its exact
symbol. Dropping the right accent enables you to use ')~| for both
125:126 and 243:245 (noting that Gene *would* be using the same
symbol for both). The problem with this is that 4deg441 is 22
generators distant from the original chain of ennealimal nominals, so
not a very good choice to be represented by a single character.

An alternative is to interpret |~. as 32768:33075 (2^15:3^3*5^2*7^2,
~16.1c), giving .|~. as 243:245, and .|~.. as 125:126 (hmmm, there's
something there that bothers you, but bear with me). Dropping the
right accents enables us to use .|~ for both 243:245 and 125:126.
This one (6deg441) is -16 generators distant, better, but still not
very good. See this table for comparison:

deg chars
441 dn up ratio(s) generator distance for up-symbol
--- --- --- ------------ --------------------------------
1 . ' 32768:32805 19G excessive G-distance
3 $ s 224:225 -8G
4 ¦ ¡ 49C +22G excessive G-distance
5 < > 125:126 +3G (user-defined chars)
6 z ~ 32768:33075 16G excessive G-distance
8 / \ 80:81 -5G
10 t f 63:64 +6G
13 ¿ ç 48:49, 49:50 -2G
18 n u 35:36 +1G

The schisma, at -19G, is the only character pair with an excessive
generator distance (exceeding +-8G) that Gene accepts (and that's
only because it's desirable to notate a single degree).

The 5- and 7-commas, on the other hand, are relatively close to the
original line of nominals, so single character pairs for these are
easily justified.

Gene, you were using up to 3 characters in combination to notate
441. What about doing something like this, using this character set
(no extended characters):

u n 35:36 (+1G)
\ / 80:81 (-5G)
t f 63:64 (+6G)
s $ 224:225 (-8G)
. ' 32768:32805 (-19G)

to get this symbol sequence?

deg chars
441 dn up ratio(s) generator distance for up-symbol
--- --- --- ------------ --------------------------------
1 . ' 32768:32805 19G
2 's .$ +11G
3 s $ 224:225 -8G
4 'S\ ,s/ +22G
or .s '$ -27G
5 $\ s/ 125:126 +3G
6 ss $$ 16G
7 '\ ./ diaschisma +14G
8 / \ 80:81 -5G
9 .\ '/ pyth. comma -24G
10 t f 63:64 +6G
11 .t 'f 3584:3645 -13G
12 'st .$f +17G
13 st $f 48:49, 49:50 -2G
14 .st '$f -21G
15 '\\ .// 125:128 +9G
16 \\ // 6400:6561 -10G
17 'u .n +20G
18 u n 35:36 +1G
19 .u 'n -18G

(No point in going any farther than this until I get some feedback.)

The only 3-character combinations are ones containing a schisma.
Wherever there are characters altering in opposite directions, the
smaller of the two is either a schisma or kleisma.

I avoided an additional (extended) character pair (for 48:49) by
using characters corresponding to the actual Sagittal flags:

st for ~!)
$f for ~|)

I know that Dave disapproves of double accidentals, but this isn't
Sagittal; it's a compound-character notation that uses a subset of
Sagittal shorthand characters and which is, therefore, highly
compatible with Sagittal.

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> > ,` -+ <> [] {}
>
> In order to represent 125:126 (or 5deg441) with a single character,
> this is the only possible solution.

Manuel seems to be using [] for something else, so maybe {} would be
good. I thought ,` was suggested to use for the schisma; is that
wrong? Manuel has <> down for the schisma in his 441 ennealimmal
notation.

> Dave, let me clarify something, just to make sure we're on the same
> page. We have a whole menagerie of ratios represented by )~|,
> ranging from 17:19k (152:153, ~11.4c) to 11:35k (2816:2835, ~11.6c)
> to 143C (143:144, ~12.1c) to 49C (3^13:2^15*7^2, ~12.2c), with the
> kleisma-comma boundary going smack dab through the middle of the
> range. Correct me if I'm mistaken, but I think 49C is its primary
> role, which gives ')~| a primary role of 243:245 (~14.2c). 125:126
> is about a mina (~0.4c) smaller than that, giving ')~|. as its exact
> symbol. Dropping the right accent enables you to use ')~| for both
> 125:126 and 243:245 (noting that Gene *would* be using the same
> symbol for both). The problem with this is that 4deg441 is 22
> generators distant from the original chain of ennealimal nominals, so
> not a very good choice to be represented by a single character.

We aren't talking about 4deg441, but 5deg441. In terms of ennealimmal
generators, we have 1728/1715 = (36/35)^2/(27/25), which can be used
the same as a 126/125 in ennealimmal. Also, of course,
126/125=(6/5)^2/(10/7), if we use 6/5 instead of 36/35 as our basic
generator.

> An alternative is to interpret |~. as 32768:33075 (2^15:3^3*5^2*7^2,
> ~16.1c), giving .|~. as 243:245, and .|~.. as 125:126 (hmmm, there's
> something there that bothers you, but bear with me). Dropping the
> right accents enables us to use .|~ for both 243:245 and 125:126.
> This one (6deg441) is -16 generators distant, better, but still not
> very good. See this table for comparison:
>
> deg chars
> 441 dn up ratio(s) generator distance for up-symbol
> --- --- --- ------------ --------------------------------
> 1 . ' 32768:32805 ï¿½19G excessive G-distance

Not really; we can define ennealimmal first for 171, and then anything
larger is simply tacking on the appropriate schismas. It is a very
nice thing in ennealimma if we use 6/5 as a generator, because then it
only involves generators, not periods, up to octave equivalence. It is
equivalent in ennealimmal to the 19 comma, 32/(6/5)^19. If you have a
symbol for 648/625 handy, perhaps that could be useful also:
648/625 = (36/35)(126/125) = (6/5)^4/2, etc.

> 3 $ s 224:225 -8G
> 4 ï¿½ ï¿½ 49C +22G excessive G-distance
> 5 < > 125:126 +3G (user-defined chars)
> 6 z ~ 32768:33075 ï¿½16G excessive G-distance
> 8 / \ 80:81 -5G
> 10 t f 63:64 +6G
> 13 ï¿½ ï¿½ 48:49, 49:50 -2G
> 18 n u 35:36 +1G

If we used {} instead of <> for 126/125, and used <> for 32805/32768
as Manuel seems to be doing, and if we maybe used another pair (-+
seems to be available) for 648/625, I think we would be cooking.

> Gene, you were using up to 3 characters in combination to notate
> 441. What about doing something like this, using this character set
> (no extended characters):
>
> u n 35:36 (+1G)
> \ / 80:81 (-5G)
> t f 63:64 (+6G)
> s $ 224:225 (-8G)
> . ' 32768:32805 (-19G)
>
> to get this symbol sequence?

As I say, I think perhaps Manuel would prefer <> for the schisma, and
I think I'd like more symbol pairs. My counter-suggestion is:

32805/32768 <> 19G
225/224 s$ -8G
126/125 {} 3G
81/80 /\ -5G
64/63 tf 6G
36/35 nu 1G
648/625 -+ 4G

> deg chars
> 441 dn up ratio(s) generator distance for up-symbol
> --- --- --- ------------ --------------------------------
> 1 . ' 32768:32805 ï¿½19G
> 2 's .$ +11G
> 3 s $ 224:225 -8G
> 4 'S\ ,s/ +22G
> or .s '$ -27G
> 5 $\ s/ 125:126 +3G
> 6 ss $$ ï¿½16G
> 7 '\ ./ diaschisma +14G
> 8 / \ 80:81 -5G
> 9 .\ '/ pyth. comma -24G
> 10 t f 63:64 +6G
> 11 .t 'f 3584:3645 -13G
> 12 'st .$f +17G
> 13 st $f 48:49, 49:50 -2G
> 14 .st '$f -21G
> 15 '\\ .// 125:128 +9G
> 16 \\ // 6400:6561 -10G
> 17 'u .n +20G
> 18 u n 35:36 +1G
> 19 .u 'n -18G
>
> (No point in going any farther than this until I get some feedback.)

I don't think these multiple symbols will work for Scala, and
certainly not the ones with dots in them.

> I know that Dave disapproves of double accidentals, but this isn't
> Sagittal; it's a compound-character notation that uses a subset of
> Sagittal shorthand characters and which is, therefore, highly
> compatible with Sagittal.

But what about compatibility with Scala?

🔗Dave Keenan <d.keenan@bigpond.net.au>

I believe I have a sagittal solution for your accidental 3 ennealimmal
generators (5deg441 etc.). Since George and I readily agreed to the
use of a secondary comma role for ~| as 1024:1029 (its primary role is
to notate the 17th harmonic) maybe we can also do the same for the
alternative 17th harmonic symbol ~|( in this case.

~|( has the secondary role 243:245 and I believe that corresponds to 3
ennealimmal generators the same as 125:126 does. It also makes the
flag arithmetic work for the true sagittal symbols or the long ASCII.
The short ASCII pair for this is "h" for down and "p" for up.

I suggest not using the pair v and ^ for anything other than 32:33 or
at least only for alterations of around a quartertone.

Here are sufficient unaccented symbols, which when combined only with
schisma up or down accents, will notating up to +-24 generators of
ennealimmal (i.e. 441-ET) and therefore require no more that two ASCII
characters.

Note that in the long ASCII or the true-type symbols, all 24 symbols
are made up of only 6 components used at most 3 at a time,
'| |( ~| /| |) (| plus one ocurrence of |\ in the last symbol only.

deg ennea long short
441 gens up dn up
-----------------------------
1 -19 '| . '
2 11 |( c r
3 -8 ~| s $
5 3 ~|( h p (maybe)
8 -5 /| \ /
10 6 |) t f
12 17 (| j ?
13 -2 ~|) ¿ ç
16 -10 //| _=
18 1 /|) u n
20 12 (/| a g
22 23 (|) o @
24 -15 (|\ w m

There's only one pair that isn't ASCII and that's "¿ ç". Manuel thinks
he can make UTF-8 characters work fairly easily and will let us know
when he's done it, but in the meantime you could use - + or < > or [
] for the 2-generators pair.

If the period (dot) for schisma-down is incompatible with Scala then
please substituite the comma. But please leave the schisma-up as the
apostrophe, not the backquote. Please put the schisma symbol to the
left of the other symbols in the short ASCII for now, to avoid
confusion with schismina accents.

-- Dave Keenan

🔗Herman Miller <hmiller@IO.COM>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Gene Ward Smith wrote:
> > However, *even if* you use chain of fifths nominals, you still
want to
> > deal with temperaments like ennealimmal or kleismic. Or if not,
then
> > once again Sagittal is not interesting to me, because I can't use
it
> > for notating temperaments.
>
> I've been surprised to find so many Sagittal accidentals suitable
for
> temperaments even without accents.

Yes, it's when you use alternate spellings that the accents start
coming into play, e.g.:
B# = '/|C

> Kleismic for instance has a number of
> good sagittals in the basic set:
>
> )||( [-3, -1, 2> +4 (68.15c)
> |||( [-1, 1, 1, -1> +8 (136.30c)
> |) [6, -2, 0, -1> -15 (45.23c)
> /| [-4, 4, -1> +19 (22.92c)
>
> plus the following if you add a schisma accent:
>
> ./||\ [4, -1, -1> -11 (113.38c)
> .)X( [1, -2, 1> -7 (181.10c)
>
> So how would you notate kleismic with a chain of fifths? Something
like
> this would do:
>
> |)G \\!B \!D F !!/A )!!(C !!!)E = |)D
> |)D )||(F \!A C !!/E )!!(G !!!)B = |)A
> |)A )||(C \!E G !!/B )!!(D !)F = |)E
> |)E )||(G \!B D /|F )!!(A !)C = |)B
> |)B )||(D ||\F A /|C )!!(E !)G = |||)F
> |||)F )||(A ||\C E /|G )!!(B !)D = |||)C
> |||)C )||(E ||\G B /|D //|F !)A

Oops! Check your work. For the right half of your table I get:

Also, the equivalences in your rightmost column should read:
!!!)E = .(|\D, etc.
Eb lowered by 63:64 = D raised by 27:28

The table also shows how one arrives at ~| for the kleisma
(15552:15625), and the same symbol will also represent 224:225. Of
course, '|( would also work, but Gene wanted to avoid accented
symbols.

Herman and Dave, I've been meaning to suggest that, if you still
intend to use additional capital letters beyond G for non-diatonic
nominals, you should skip "H", because it's already used in German
notation.

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > > ,` -+ <> [] {}
> >
> > In order to represent 125:126 (or 5deg441) with a single
character,
> > this is the only possible solution.
>
> Manuel seems to be using [] for something else, so maybe {} would be
> good. I thought ,` was suggested to use for the schisma; is that
> wrong? Manuel has <> down for the schisma in his 441 ennealimmal
> notation.
>
> > Dave, let me clarify something, just to make sure we're on the
same
> > page. We have a whole menagerie of ratios represented by )~|,
> > ranging from 17:19k (152:153, ~11.4c) to 11:35k (2816:2835,
~11.6c)
> > to 143C (143:144, ~12.1c) to 49C (3^13:2^15*7^2, ~12.2c), with
the
> > kleisma-comma boundary going smack dab through the middle of the
> > range. Correct me if I'm mistaken, but I think 49C is its
primary
> > role, which gives ')~| a primary role of 243:245 (~14.2c).
125:126
> > is about a mina (~0.4c) smaller than that, giving ')~|. as its
exact
> > symbol. Dropping the right accent enables you to use ')~| for
both
> > 125:126 and 243:245 (noting that Gene *would* be using the same
> > symbol for both). The problem with this is that 4deg441 is 22
> > generators distant from the original chain of ennealimal
nominals, so
> > not a very good choice to be represented by a single character.
>
> We aren't talking about 4deg441, but 5deg441.

I was talking about 4deg being )~| (which is represented by a single
character) and thus 5deg being ')~| -- which would have to be
represented by two characters, one for )~| and another character for
the accent mark. But you don't seem to like having 125:126 shorthand
as two characters, so we've been trying to find a way around that.

The way things work out, some ratios, such as the 5-comma have no
accent marks, while others, such as the diaschisma and pythagorean
comma do, because they differ from the unaccented ratio by a 5-
schisma. Multiples of the 5-comma 80:81 have no accent marks, e.g.,
80^3:81^3, which is represented by (|\, as are 26:27 and 8192:8505.
However, 27:28 is a 5-schisma less than that last one, so it's
notated .(|\ -- with an accent mark. Ratios with accented symbols
aren't necessarily unimportant -- we started out notating the most
important ratios without accents marks, and this is how it happened
to work out.

> In terms of ennealimmal
> generators, we have 1728/1715 = (36/35)^2/(27/25), which can be used
> the same as a 126/125 in ennealimmal. Also, of course,
> 126/125=(6/5)^2/(10/7), if we use 6/5 instead of 36/35 as our basic
> generator.
>
> > An alternative is to interpret |~. as 32768:33075
(2^15:3^3*5^2*7^2,
> > ~16.1c), giving .|~. as 243:245, and .|~.. as 125:126 (hmmm,
there's
> > something there that bothers you, but bear with me). Dropping
the
> > right accents enables us to use .|~ for both 243:245 and
125:126.
> > This one (6deg441) is -16 generators distant, better, but still
not
> > very good. See this table for comparison:
> >
> > deg chars
> > 441 dn up ratio(s) generator distance for up-symbol
> > --- --- --- ------------ --------------------------------
> > 1 . ' 32768:32805 -19G excessive G-distance
>
> Not really; we can define ennealimmal first for 171, and then
anything
> larger is simply tacking on the appropriate schismas. It is a very
> nice thing in ennealimma if we use 6/5 as a generator, because then
it
> only involves generators, not periods, up to octave equivalence. It
is
> equivalent in ennealimmal to the 19 comma, 32/(6/5)^19. If you have
a
> symbol for 648/625 handy, perhaps that could be useful also:
> 648/625 = (36/35)(126/125) = (6/5)^4/2, etc.

625:648 would be a mina (short for "schismina", ~0.4c) less than
27:28, or .(|\. Since the mina vanishes, you can drop the right
accent mark from that and use the same symbol as for 27:28, .(|\,
which is .m in Sagittal shorthand. Gene, there's no easy way around
it -- since we've decided to use accent marks for schismas, and since
you have intervals differing by a schisma, some of them will have to
have accent marks. Otherwise, if there were *no* symbols at all with
accent marks, then we'd need more unaccented symbols of some sort and
more character pairs for those, which would be much more complicated
than what we now have. People are already telling us that we have
too many symbols, and we've already run out of characters for the
ones we now have, so that's clearly out of the question.

I found a way around notating 224:225 with an accent as '|( by using
a 17-kleisma symbol, ~|, and I see that Dave's just come up with a
proposal to notate 125:126 with a 17-comma symbol that may work for
you (which I'll reply to shortly). But if you also want an
unaccented symbol for 625:648, then we would probably have to look at
|\\ (11:19L diesis, 2^15*11:3^9*19, ~63.8c), which is so far out that
we've never used it to notate anything (and have in fact considered
dropping it from our symbol set).

A much better solution would be for you to stop thinking at the 5-
schisma-characters in the shorthand as separate characters and start
thinking of them as a belonging to the adjacent character to the
right, unless, of course, there is no character to the right (in
which case you would simply read it as a 5-schisma).

> ...
> > Gene, you were using up to 3 characters in combination to notate
> > 441. What about doing something like this, using this character
set
> > (no extended characters):
> >
> > u n 35:36 (+1G)
> > \ / 80:81 (-5G)
> > t f 63:64 (+6G)
> > s $ 224:225 (-8G)
> > . ' 32768:32805 (-19G)
> >
> > to get this symbol sequence?
>
> As I say, I think perhaps Manuel would prefer <> for the schisma,
and
> I think I'd like more symbol pairs. My counter-suggestion is:
>
> 32805/32768 <> 19G
> 225/224 s$ -8G
> 126/125 {} 3G
> 81/80 /\ -5G
> 64/63 tf 6G
> 36/35 nu 1G
> 648/625 -+ 4G

More symbol pairs are fine, but there are limitations on how many
character pairs are available. I was rather hoping that one's
imagination would not be so limited that it would be impossible to
merge character pairs into a single mental unit so that they could be
interpreted as representing single symbols, such as these:

dn up ratio
-- -- -----
.c 'r 224:225
'z .~ 125:126, 243:245
.a 'g 392:405
'm .w 27:28, 625:648

Please give this some thought -- please?

> ...
> But what about compatibility with Scala?

What about it? Is "Scala" a notation that's locked into <> for the
schisma, or what? We already suggested replacing the period "." with
a comma "," and I also suggested putting the accidental to the left
of the capital letter nominal. Won't either of these work for you?

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> I believe I have a sagittal solution for your accidental 3
ennealimmal
> generators (5deg441 etc.). Since George and I readily agreed to the
> use of a secondary comma role for ~| as 1024:1029 (its primary role
is
> to notate the 17th harmonic) maybe we can also do the same for the
> alternative 17th harmonic symbol ~|( in this case.
>
> ~|( has the secondary role 243:245 and I believe that corresponds
to 3
> ennealimmal generators the same as 125:126 does. It also makes the
> flag arithmetic work for the true sagittal symbols or the long
ASCII.
> The short ASCII pair for this is "h" for down and "p" for up.

Hmmm, ~| is 3deg441 and |( is 2deg, so ~|( can be taken as 5deg, and
since we're not anywhere near the 17 limit it wouldn't be interpreted
as the 17 comma, so I guess it's possible to do that.

> I suggest not using the pair v and ^ for anything other than 32:33
or
> at least only for alterations of around a quartertone.

Yes, please!

> Here are sufficient unaccented symbols, which when combined only
with
> schisma up or down accents, will notating up to +-24 generators of
> ennealimmal (i.e. 441-ET) and therefore require no more that two
ASCII
> characters.
>
> Note that in the long ASCII or the true-type symbols, all 24 symbols
> are made up of only 6 components used at most 3 at a time,
> '| |( ~| /| |) (| plus one ocurrence of |\ in the last symbol
only.
>
> deg ennea long short
> 441 gens up dn up
> -----------------------------
> 1 -19 '| . '
> 2 11 |( c r
> 3 -8 ~| s $
> 5 3 ~|( h p (maybe)
> 8 -5 /| \ /
> 10 6 |) t f
> 12 17 (| j ?
> 13 -2 ~|) ¿ ç
> 16 -10 //| _=
> 18 1 /|) u n
> 20 12 (/| a g
> 22 23 (|) o @
> 24 -15 (|\ w m
>
> There's only one pair that isn't ASCII and that's "¿ ç". Manuel
thinks
> he can make UTF-8 characters work fairly easily and will let us know
> when he's done it, but in the meantime you could use - + or < > or
[
> ] for the 2-generators pair.
> ...
> If the period (dot) for schisma-down is incompatible with Scala then
> please substituite the comma. But please leave the schisma-up as the
> apostrophe, not the backquote. Please put the schisma symbol to the
> left of the other symbols in the short ASCII for now, to avoid
> confusion with schismina accents.

Yes, please!

--George

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> Herman and Dave, I've been meaning to suggest that, if you still
> intend to use additional capital letters beyond G for non-diatonic
> nominals, you should skip "H", because it's already used in German
> notation.

And should we also skip "B" because it's already used in German
notation (to mean what the rest of us call "Bb")? ;-)

The full-alphabet thing will probably only play a minor role, however
Germans have had to cope with B meaning something different to others
for quite some time. I'm sure they are smart enough to cope with H
meaning something different too (down-A = semiflat-A) in what is
obviously a completely new notation system. I think HIJKLMN as a
parallel diatonic and O as the zero-point a half-octave from the
central D, are too good to give up.

By the way George, I don't really like the idea of reversing the usual
order of nominal and accidentals in English text. It is normal to
spell English words as they sound from left to right.

So if you pronounce it as "sharp F comma-up" as I (and I assume
Herman) would do for this compound-nominal LT notation, then I think
it should be spelled "#F/". If you pronounce something as "comma-up F
sharp" as Bosanquet may have done then you should spell it "/F#". But
if, as is standard for diatonic-based notations (on this list at
least), you pronounce it as "F sharp comma-up" then I think you should
spell it as "F#/".

It is the order on the staff that is anomalous. You might as well try
to convince people to change that, although I think you'd be wasting
your time.

I'm not against allowing the reverse order in text as representing a
direct transfer from staff notation, as I understand the value of
being able to take in the whole collection as a gestalt, but it is
clear that you must still allow the standard order, particularly since
it is assumed by software such as Scala and FTS.

Therefore your proposal does not constitute a solution to the
syntactic or lexical issues I raised regarding strings of multiple
sagittals with (a) both left and right accents and (b) accents versus
schisma and schismina symbols in short ASCII.

I understand your solution to both of these was simply to prohibit
left-accented symbols (including schisma symbols) except as the
leftmost symbol of a string, and to prohibit right-accented symbols
(including schismina symbols) except as the rightmost symbol of a string.

Robert has reported his solution which is a little less draconian. He
assumes that all accents are left accents unless they are on the right
of the rightmost symbol. In other words, as he parses a string from
left to right he assumes each accent character belongs with the
non-accent character immediately to its right unless there is no such
character, in which case it belongs with the non-accent character to
its left.

I note that this parser could be written so as to allow that if a
sequence of two accent characters occurs between two non-accent
characters they could be apportioned one to the left and one to the
right. Similarly if three occur together then two would be right
accents and one left.

Robert hopes that spaces can be avoided in multi-symbol strings. I'm
not sure how he handles cases in long ASCII such as /|)|) which could
be parsed as either /| )|) or /|) |). A rule saying that flags belong
with the shaft to the left unless this already has two flags (the
second choice above), would seem reasonable, provided that spaces
_can_ be used to obtain the first result above when required.

Gene quite reasonably wanted the form /' to be interpreted as /| '|
since he wanted them to be considered separate symbols ordered by
decreasing size from left to right. However Robert's parser would
quite reasonable interpret /' as the single right-accented symbol /|'
with a quite different meaning.

It is better if Gene violates the usual descending order and puts the
schisma to the left since Robert's parser will at least interpret it
as the single symbol '/| which has the same total result as '| /| or
very close to it. But note that for any unaccented symbol X, '| X will
not necessarily have _exactly_ the same total comma interpretation as 'X .

I note that if we used comm and backquote ,` for right accents and
period and apostrophe .' for left accents and insisted that even in
short ASCII the full-symbols for shisma and schismina must appear as
'| .| and |` !, (i.e. with two characters per symbol), that would
solve _all_ of these lexical problems, including the one where Scala
used the dot as a separator for octave numbers.

But it would unfortunately not give Gene a single ASCII character for
the schisma symbol and he would be forced to use one of the user
definable pairs as he would for _any_ accented symbol. Unfortunately
the set of available user-definable pairs would now be one pair
smaller since it would not then include comma and backquote , ` .
However he would then be free to put the chosen schisma symbols as the
rightmost in a string of symbols, without ambiguity.

This is my favourite solution as it has no special cases and allows
strings of arbitrarily accented symbols without spaces, without
ambiguity. We still need spaces for cases like /|)|) but I don't see
any need to worry about that since if one is using long ASCII one must
accept its lack of compactness.

It also doesn't solve the accent-assignment problem for the True-type
symbols as we are currently using the same characters there for both
right and left accents, but again spaces can be used to disambiguate.

-- Dave

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Not that specific; while I'm concentrating on ennealimmal since I'd
> like that nailed down, this is really the 7-limit itself which is at
> issue here. I'd like to see microtempering worked out for the 5 and 7
> limits, and it doesn't seem to be.

Relative to 7 nominals in a chain of fifths, sagittal has the 7 limit
well taken care of. But it should be no surprise that 5-schisma
accents must be used once the sum of the absolute powers of 5 and 7
exceeds 3 (as it does in 125:126).

> I'm not proposing that it has to
> accord with Sagittal, but it seems like a good idea to the extent
> possible. Apparently it has to accord with Scala also, and I don't
> even know what that entails. But it *is* clear a symbol pair for
> 126/125 would be useful.

Yes. And we have a symbol pair for it. It's just that it happens to be
an accented symbol and therefore not representable in the short ASCII
aa a single character without appropriating one of the user-definable
pairs for the purpose. We don't find this unreasonable.

Given that there are only 31 single-character pairs available in UTF-8
(36 if we were to leave none to be user-definable), we would be very
interested in your list of the 31 commas most deserving of single
character pairs. Not just for ennealimmal, but weighted towards
notating the most commonly used pitches relative to the most commonly
used sets of nominals in the most commonly used temperaments (or JI).

> You started out with the idea of notating JI,

That's simply not true. Surely you remember the title of the Sagittal
development thread as it appeared on this list for over a year: "A
Common notation for JI and ETs", and by implication everything in between.

> whereas I am most
> interested in notating temperaments.

Certainly we are using comma definitions for the symbols, to give them
a commonality of meaning across both JI and various temperaments. But
I understood you were in favour of that approach too.

> Moreover, I'm not really
> interested much in anything beyond the 13 limit, but think the 7 limit
> is crucial.

George and I personally agree with you there. However there are others
who need a way to notate harmonics to at least the 19th, and we want
to cater for them as well.

Dave:
> > It seems to me that one must start by determining which
> > (octave-equivalent) ratios are most likely to be notated. Such a list
> > does not progress neatly by prime limits. Odd limits and product
> > complexity are closer. But it should come as no surprise that for
> > example 17/16 occurs more often in actual tunings than all of 126/125,
> > 125/63, 125/112, 224/125 etc. combined.

Gene:
> It *never* occurs in the tunings I've been considering. It makes sense
> people might stick it in a JI scale--they stick all kinds of things in
> JI scales. But temperaments are creatures of reason, even if the
> generators are irrational. It makes sense to have symbols for higher
> limit JI, as one possible project. But to ignore or disparage
> temperments, which in effect is what you are doing, does *not* make
> sense, and suggests Sagittal is confining itself to a JI ghetto.

I think you are mainly having a problem with it because you are
insisting on restricting yourself to the small subset of symbols which
have a single-character representation in ASCII.

The _real_ sagittal symbols are the much larger (but logically
constructed from a few components) set of graphical or true-type
symbols, which obviously will sometimes require more than one
character to represent in ASCII.

Perhaps you should revisit some of the material on the sagittal
website to better understand its philosophy.

-- Dave Keenan

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Herman Miller <hmiller@IO.COM>

George D. Secor wrote:
> Oops! Check your work. For the right half of your table I get:
> > F !!/A )!!(C \!!!)E !!!)G
> C !!/E )!!(G \!!!)B !!!)D
> G !!/B )!!(D \!)F !!!)A
> D /|F )!!(A \!)C !!!)E !!!(G = .!!)F ~!A
> A /|C )!!(E \!)G !!!)B !!!(D = .!!)C ~!E
> E /|G )!!(B \!)D !)F !!!(A = .!!)G ~!B
> B /|D //|F \!)A !)C !!!(E = .!!)D

Which version of kleismic are you using? In the Xenharmonikon article at http://dkeenan.com/sagittal/Sagittal.pdf, the |) symbol is listed as representing 63:64. In the version of kleismic with the map [<1, 0, 1, 2|, <0, 6, 5, 3|], 64/63 works out to <6, -15] (6 octaves up, 15 minor thirds down). /|) would be +4 generators if it represents 36/35, or -15 if it represents 250/243. But 36/35 is listed first, so that would be the primary role of /|) in a 7-limit temperament. In 5-limit kleismic, /|) would be suitable for -15 generators, but then what would a 7-limit symbol like |) be used for?

It's possible that you're thinking of the [<1, 0, 1, -3|, <0, 6, 5, 22|] temperament, which has been called "catakleismic" or "complex kleismic". I was assuming that Gene was referring to the simpler one, which is "keemun" (a kind of black tea) in Paul's paper.

> Also, the equivalences in your rightmost column should read:
> !!!)E = .(|\D, etc.
> Eb lowered by 63:64 = D raised by 27:28

28/27 and 64/63 are equivalent in keemun temperament (but not in catakleismic).

> The table also shows how one arrives at ~| for the kleisma > (15552:15625), and the same symbol will also represent 224:225. Of > course, '|( would also work, but Gene wanted to avoid accented > symbols.
> > Herman and Dave, I've been meaning to suggest that, if you still > intend to use additional capital letters beyond G for non-diatonic > nominals, you should skip "H", because it's already used in German > notation.

I thought of that, but German notation is incompatible in any case, since German B = the rest of the world's Bb.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
> wrote:
> > Gene Ward Smith wrote:
> > > However, *even if* you use chain of fifths nominals, you still
> want to
> > > deal with temperaments like ennealimmal or kleismic. Or if not,
> then
> > > once again Sagittal is not interesting to me, because I can't use
> it
> > > for notating temperaments.
> >
> > I've been surprised to find so many Sagittal accidentals suitable
> for
> > temperaments even without accents.
>
> Yes, it's when you use alternate spellings that the accents start
> coming into play, e.g.:
> B# = '/|C
>
> > Kleismic for instance has a number of
> > good sagittals in the basic set:
> >
> > )||( [-3, -1, 2> +4 (68.15c)
> > |||( [-1, 1, 1, -1> +8 (136.30c)
> > |) [6, -2, 0, -1> -15 (45.23c)
> > /| [-4, 4, -1> +19 (22.92c)
> >
> > plus the following if you add a schisma accent:
> >
> > ./||\ [4, -1, -1> -11 (113.38c)
> > .)X( [1, -2, 1> -7 (181.10c)
> >
> > So how would you notate kleismic with a chain of fifths? Something
> like
> > this would do:
> >
> > |)G \\!B \!D F !!/A )!!(C !!!)E = |)D
> > |)D )||(F \!A C !!/E )!!(G !!!)B = |)A
> > |)A )||(C \!E G !!/B )!!(D !)F = |)E
> > |)E )||(G \!B D /|F )!!(A !)C = |)B
> > |)B )||(D ||\F A /|C )!!(E !)G = |||)F
> > |||)F )||(A ||\C E /|G )!!(B !)D = |||)C
> > |||)C )||(E ||\G B /|D //|F !)A
>
> Oops! Check your work. For the right half of your table I get:
>
> F !!/A )!!(C \!!!)E !!!)G
> C !!/E )!!(G \!!!)B !!!)D
> G !!/B )!!(D \!)F !!!)A
> D /|F )!!(A \!)C !!!)E !!!(G = .!!)F ~!A
> A /|C )!!(E \!)G !!!)B !!!(D = .!!)C ~!E
> E /|G )!!(B \!)D !)F !!!(A = .!!)G ~!B
> B /|D //|F \!)A !)C !!!(E = .!!)D

You've got me confused here George.

The nominals are in a chain of fifths. In kleismic a fifth is 6
generators. So what we need for accidentals are the 3 most common
commas which when multiplied by the generator mapping <0 6, 5 3] and
reduced modulo 6 (to the range +-3) give results of 1, 2 and 3
generators. Herman's suggestion looks ideal.

OK. I see yours is correct for the <0 6, 5 22] mapping, as Herman
suggested.

-- Dave

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > Herman and Dave, I've been meaning to suggest that, if you still
> > intend to use additional capital letters beyond G for non-
diatonic
> > nominals, you should skip "H", because it's already used in
German
> > notation.
>
> And should we also skip "B" because it's already used in German
> notation (to mean what the rest of us call "Bb")? ;-)
>
> The full-alphabet thing will probably only play a minor role,
however
> Germans have had to cope with B meaning something different to
others
> for quite some time. I'm sure they are smart enough to cope with H
> meaning something different too (down-A = semiflat-A) in what is
> obviously a completely new notation system. I think HIJKLMN as a
> parallel diatonic and O as the zero-point a half-octave from the
> central D, are too good to give up.

Yes, B is already one point of confusion, but I was suggesting that H
be skipped over to avoid doubling the potential for confusion. But
as long as you have a "too-good-to-give-up" situation, then I would
say let it be.

> By the way George, I don't really like the idea of reversing the
usual
> order of nominal and accidentals in English text. It is normal to
> spell English words as they sound from left to right.
>
> So if you pronounce it as "sharp F comma-up" as I (and I assume
> Herman) would do for this compound-nominal LT notation, then I think
> it should be spelled "#F/". If you pronounce something as "comma-up
F
> sharp" as Bosanquet may have done then you should spell it "/F#".
But
> if, as is standard for diatonic-based notations (on this list at
> least), you pronounce it as "F sharp comma-up" then I think you
should
> spell it as "F#/".

If this were some sort of spectral notation and the F# occurred in
different colors, say black and red, we would quite naturally call
them "black F-sharp" and "red F-sharp" rather than "F-sharp black",
etc. Then what's wrong with saying "comma-up F-sharp" or "5-up F-
sharp", if it were written that way? The only reason we haven't
already been doing that is that we haven't been writing them that
way, and the only reason we haven't been writing them that way is
that we haven't had a good enough reason to do that -- but now we
do. Sagittal is still very new, and it's not too late to think
through these possibilities.

> It is the order on the staff that is anomalous. You might as well
try
> to convince people to change that, although I think you'd be wasting
> your time.

True. So instead why not change the order in the way we *say* them
to correspond to the order in which we *see* them, so as to minimize
the anomaly? We would allow sharps and flats on the staff to remain
an exception, since that's been in practice for so long.

> I'm not against allowing the reverse order in text as representing a
> direct transfer from staff notation, as I understand the value of
> being able to take in the whole collection as a gestalt, but it is
> clear that you must still allow the standard order, particularly
since
> it is assumed by software such as Scala and FTS.

Has the Sagittal implementation in Scala and FTS already been set in
stone for so long that it can't be changed?

> Therefore your proposal does not constitute a solution to the
> syntactic or lexical issues I raised regarding strings of multiple
> sagittals with (a) both left and right accents and (b) accents
versus
> schisma and schismina symbols in short ASCII.
>
> I understand your solution to both of these was simply to prohibit
> left-accented symbols (including schisma symbols) except as the
> leftmost symbol of a string, and to prohibit right-accented symbols
> (including schismina symbols) except as the rightmost symbol of a
string.

I meant as the rightmost symbol of the accidental string, meaning
that the nominal would follow to the right of that.

> Robert has reported his solution which is a little less draconian.
He
> assumes that all accents are left accents unless they are on the
right
> of the rightmost symbol. In other words, as he parses a string from
> left to right he assumes each accent character belongs with the
> non-accent character immediately to its right unless there is no
such
> character, in which case it belongs with the non-accent character to
> its left.
>
> I note that this parser could be written so as to allow that if a
> sequence of two accent characters occurs between two non-accent
> characters they could be apportioned one to the left and one to the
> right. Similarly if three occur together then two would be right
> accents and one left.

I think that having two accented characters next to each other to
modify a nominal is a mess, but ...

> Robert hopes that spaces can be avoided in multi-symbol strings. I'm
> not sure how he handles cases in long ASCII such as /|)|) which
could
> be parsed as either /| )|) or /|) |). A rule saying that flags
belong
> with the shaft to the left unless this already has two flags (the
> second choice above), would seem reasonable, provided that spaces
> _can_ be used to obtain the first result above when required.

If Robert's going to be doing anything like that, then I think he
needs a delimiting character.

> But none of these solutions would allow one to distinguish
> '| /| from |' /| from '/| or
> /| |' from /| '| from /|' or indeed
> '| from |'
> when these are encoded in short ASCII without spaces, since the
first
> three all appear as '/ and the next three as /' and the last two
as '.
>
> Gene quite reasonably wanted the form /' to be interpreted as /| '|
> since he wanted them to be considered separate symbols ordered by
> decreasing size from left to right. However Robert's parser would
> quite reasonable interpret /' as the single right-accented
symbol /|'
> with a quite different meaning.
>
> It is better if Gene violates the usual descending order and puts
the
> schisma to the left since Robert's parser will at least interpret it
> as the single symbol '/| which has the same total result as '| /| or
> very close to it. But note that for any unaccented symbol X, '| X
will
> not necessarily have _exactly_ the same total comma interpretation
as 'X .

It would be better if Gene could dispense with multiple accidentals
altogether and consider the ' and . (or ' and ,) characters
(representing a schisma) to be the first character of two-character
entities that represent a single symbol.

> I note that if we used comm and backquote ,` for right accents and
> period and apostrophe .' for left accents and insisted that even in
> short ASCII the full-symbols for shisma and schismina must appear as
> '| .| and |` !, (i.e. with two characters per symbol), that would
> solve _all_ of these lexical problems, including the one where Scala
> used the dot as a separator for octave numbers.
>
> But it would unfortunately not give Gene a single ASCII character
for
> the schisma symbol and he would be forced to use one of the user
> definable pairs as he would for _any_ accented symbol. Unfortunately
> the set of available user-definable pairs would now be one pair
> smaller since it would not then include comma and backquote , ` .
> However he would then be free to put the chosen schisma symbols as
the
> rightmost in a string of symbols, without ambiguity.

If you put a ' or . (without any other accidental) to the *left* of
the nominal, then it would default to interpretation as a 5-schisma,
and you would then use |' or !. to indicate a schismina alteration
(which I imagine would be very rare).

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> George D. Secor wrote:
> > Oops! Check your work. For the right half of your table I get:
> >
> > F !!/A )!!(C \!!!)E !!!)G
> > C !!/E )!!(G \!!!)B !!!)D
> > G !!/B )!!(D \!)F !!!)A
> > D /|F )!!(A \!)C !!!)E !!!(G = .!!)F ~!A
> > A /|C )!!(E \!)G !!!)B !!!(D = .!!)C ~!E
> > E /|G )!!(B \!)D !)F !!!(A = .!!)G ~!B
> > B /|D //|F \!)A !)C !!!(E = .!!)D
>
> Which version of kleismic are you using? In the Xenharmonikon
article at
> http://dkeenan.com/sagittal/Sagittal.pdf, the |)
> symbol is listed as representing 63:64. In the version of kleismic
with
> the map [<1, 0, 1, 2|, <0, 6, 5, 3|], 64/63 works out to <6, -15]
(6
> octaves up, 15 minor thirds down). /|) would be +4 generators if it
> represents 36/35, or -15 if it represents 250/243. But 36/35 is
listed
> first, so that would be the primary role of /|) in a 7-limit
> temperament. In 5-limit kleismic, /|) would be suitable for -15
> generators, but then what would a 7-limit symbol like |) be used
for?

In the 5 limit |) would represent 3^9:2^5*5^4, ~27.660c; technically
that one should have a right up-accent added.

> It's possible that you're thinking of the [<1, 0, 1, -3|, <0, 6, 5,
22|]
> temperament, which has been called "catakleismic" or "complex
kleismic".
> I was assuming that Gene was referring to the simpler one, which is
> "keemun" (a kind of black tea) in Paul's paper.

I was under the mistaken impression that this was somehow related to
notating ennealimal, so I was simply taking a string of just minor
thirds and dispensing with the right-accents.

> > Also, the equivalences in your rightmost column should read:
> > !!!)E = .(|\D, etc.
> > Eb lowered by 63:64 = D raised by 27:28
>
> 28/27 and 64/63 are equivalent in keemun temperament (but not in
> catakleismic).

Okay, I've been reading through some of this stuff too quickly and
didn't realize what you were doing. Sorry -- my apologies!

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> A much better solution would be for you to stop thinking at the 5-
> schisma-characters in the shorthand as separate characters and
start
> thinking of them as a belonging to the adjacent character to the
> right, unless, of course, there is no character to the right (in
> which case you would simply read it as a 5-schisma).
> ...
> More symbol pairs are fine, but there are limitations on how many
> character pairs are available. I was rather hoping that one's
> imagination would not be so limited that it would be impossible to
> merge character pairs into a single mental unit so that they could
be
> interpreted as representing single symbols, such as these:
>
> dn up ratio
> -- -- -----
> .c 'r 224:225
> 'z .~ 125:126, 243:245
> .a 'g 392:405
> 'm .w 27:28, 625:648
>
> Please give this some thought -- please?

This is how the symbol sequence would look (as accidentals to 9
ennealimal nominals):

deg chars
441 dn up ratio(s) generators (up-symbol)
--- -- -- ----------- ----------------------
1 . ' 32768:32805 19G
2 c r 5103:5120 +11G
3 'c .r 224:225 -8G
4 ¦ ¡ 49 comma +22G (3^13:2^15*7^2)
5 'z .~ 243:245 +3G (my preference)
or '¦ .¡ and 125:126
6 z ~ 32768:33075 16G
7 '\ ./ diaschisma +14G
8 \ / 80:81 -5G
9 .\ '/ pyth. comma -24G
10 t f 63:64 +6G
11 .t 'f 3584:3645 -13G
12 '¿ .ç +17G
13 ¿ ç 48:49, 49:50 -2G
14 .¿ 'ç -21G
15 '_ .= 125:128 +9G
16 _ = 6400:6561 -10G
17 'u .n +20G
18 u n 35:36 +1G
19 .u 'n -18G
20 a g 3969:4096 +12G
21 .a 'g 392:405 -7G
22 ¤ ° 49L +23G (2^23:3^11*7^2)
23 'w .m 27:28 +4G
24 w m 8192:8505 -15G
25 .w 'm -34G

--George

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> Has the Sagittal implementation in Scala and FTS already been set in
> stone for so long that it can't be changed?

No, but their implementation of notation systems in general, has been
set in stone for a long time, and I certainly don't want Sagittal to
be the odd one out in this regard.

It's one thing to have staff-like order as an alternative, but quite
unacceptable for it to be the only game in town.

Nominals and accidentals have been spoken, and hence written in text,
in the order "B flat" (earlier "B molle") ever since they have existed.

Can we please keep discussions of the parsing of strings of multiple
accented-sagittals, separate from considerations of
(a) compound nominals, and
(b) staff-like-ordering of accidentals in text.

The same problems of multiple sagittals exist whether they are to the
left or the right of the nominal and whether they are adjacent to or
not adjacent to conventional sharps and flats and whether they are
written in order of decreasing or increasing size.

> I meant as the rightmost symbol of the accidental string,

Yes. I assumed that. I was only considering the accidental string, not
the nominal.

> meaning
> that the nominal would follow to the right of that.

I find it irrelevant which side of the nominal the string of sagittal
accidentals is on.

> I think that having two accented characters next to each other to
> modify a nominal is a mess, but ...

Well so do I. But sagittal isn't only for us.

> > Robert hopes that spaces can be avoided in multi-symbol strings. I'm
> > not sure how he handles cases in long ASCII such as /|)|) which
> could
> > be parsed as either /| )|) or /|) |). A rule saying that flags
> belong
> > with the shaft to the left unless this already has two flags (the
> > second choice above), would seem reasonable, provided that spaces
> > _can_ be used to obtain the first result above when required.
>
> If Robert's going to be doing anything like that, then I think he
> needs a delimiting character.

Right. But do you agree it is OK to omit the delimiter when there's no
ambiguity, and to use the rule I described above when there otherwise
would be ambiguity. In other words, if you write it without delimiters
you deserve what you get (but at least you get something).

Robert has a strong preference for minimising error messages and
instead making a valid interpretation of a string if one is at all
possible.

> > But none of these solutions would allow one to distinguish
> > '| /| from |' /| from '/| or
> > /| |' from /| '| from /|' or indeed
> > '| from |'
> > when these are encoded in short ASCII without spaces, since the
> first
> > three all appear as '/ and the next three as /' and the last two
> as '.
> >
> > Gene quite reasonably wanted the form /' to be interpreted as /| '|
> > since he wanted them to be considered separate symbols ordered by
> > decreasing size from left to right. However Robert's parser would
> > quite reasonable interpret /' as the single right-accented
> symbol /|'
> > with a quite different meaning.
> >
> > It is better if Gene violates the usual descending order and puts
> the
> > schisma to the left since Robert's parser will at least interpret it
> > as the single symbol '/| which has the same total result as '| /| or
> > very close to it. But note that for any unaccented symbol X, '| X
> will
> > not necessarily have _exactly_ the same total comma interpretation
> as 'X .
>
> It would be better if Gene could dispense with multiple accidentals
> altogether and consider the ' and . (or ' and ,) characters
> (representing a schisma) to be the first character of two-character
> entities that represent a single symbol.

Sure. But even if he did there will still be others who want strings
of multiple sagittals. And we've agreed they will occur in FTS with
its comma-shifting. So we can't avoid dealing with it.

> > I note that if we used comma and backquote ,` for right accents and
> > period and apostrophe .' for left accents and insisted that even in
> > short ASCII the full-symbols for shisma and schismina must appear as
> > '| .| and |` !, (i.e. with two characters per symbol), that would
> > solve _all_ of these lexical problems, including the one where Scala
> > used the dot as a separator for octave numbers.

I've now confirmed that Scala does not have a problem with trailing
dots (even multiple trailing dots) in accidental strings, even in
combination with a dot-delimited octave number. Manuel fixed that when
he first implemented sagittal. So we can stop worrying about that one
at least.

But using comma and backquote for right accents would still have the
advantage of allowing arbitrary strings of accented sagittals in short
form with no spaces and no ambiguity.

Don't you think this would be better than a bunch of tricky rules
about which symbol an accent belongs to when it has cores on both
sides of it?

> >
> > But it would unfortunately not give Gene a single ASCII character
> for
> > the schisma symbol and he would be forced to use one of the user
> > definable pairs as he would for _any_ accented symbol. Unfortunately
> > the set of available user-definable pairs would now be one pair
> > smaller since it would not then include comma and backquote , ` .
> > However he would then be free to put the chosen schisma symbols as
> the
> > rightmost in a string of symbols, without ambiguity.
>
> If you put a ' or . (without any other accidental) to the *left* of
> the nominal, then it would default to interpretation as a 5-schisma,
> and you would then use |' or !. to indicate a schismina alteration
> (which I imagine would be very rare).

I don't see that it makes any difference whether the accidental string
is to the left or the right of the nominal.

However I agree that if an accidental string consists only of . or '
then it should be interpreted as equivalent to the full symbol .! or '| .

Likewise an accidental string consisting only of , or ` could be
interpreted as the full symbol !, or |` . Similarly if it consists
only of ,, or `` it would be !,, or |`` .

So what we've agreed on here so far is that if you want to indicate a
separate symbol for the 5-schisma in a short-form string with other
sagittals then you must show it as .! or '| and so these two-character
symbols must be allowed in the short ASCII form. Similarly the
(schis)mina and double-mina two and three-character symbols must be
allowed.

So we do not have the property that if it contains ! or | then it must
be in the long ASCII form. But I can live with that. We never really
had this property anyway, since we previously used the long-form
symbols in the short-form when a symbol had no single ASCII character
allocated to it. Now every unaccented symbol has a single 8-bit
character from the ISO-LATIN-1 set. Manuel has successfully
implemented these in Scala.

If we used only , and ` for right accents then a . or ' as the
rightmost character in a multi-accidental string could also be
interpreted as a separate schisma symbol.

-- Dave

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> ...
> Can we please keep discussions of the parsing of strings of multiple
> accented-sagittals, separate from considerations of
> (a) compound nominals, and
> (b) staff-like-ordering of accidentals in text.

Okay.

> ...
> > > Robert hopes that spaces can be avoided in multi-symbol
strings. I'm
> > > not sure how he handles cases in long ASCII such as /|)|) which
could
> > > be parsed as either /| )|) or /|) |). A rule saying that flags
belong
> > > with the shaft to the left unless this already has two flags
(the
> > > second choice above), would seem reasonable, provided that
spaces
> > > _can_ be used to obtain the first result above when required.
> >
> > If Robert's going to be doing anything like that, then I think he
> > needs a delimiting character.
>
> Right. But do you agree it is OK to omit the delimiter when there's
no
> ambiguity, and to use the rule I described above when there
otherwise
> would be ambiguity. In other words, if you write it without
delimiters
> you deserve what you get (but at least you get something).

Yeah, I guess so.

> ...
> But using comma and backquote for right accents would still have the
> advantage of allowing arbitrary strings of accented sagittals in
short
> form with no spaces and no ambiguity.
>
> Don't you think this would be better than a bunch of tricky rules
> about which symbol an accent belongs to when it has cores on both
> sides of it?
> ...
> If we used only , and ` for right accents then a . or ' as the
> rightmost character in a multi-accidental string could also be
> interpreted as a separate schisma symbol.

Okay, you've convinced me! We would have the following then:

ASCII shorthand
long down up Comma
----- ---- -- -----------------------
|' , ` schismina, 4095:4096 & 4374:4375, 3^9*5^2*7^2:2^20*23
|'' ,, `` double-schismina, 2079:2080 & 255879:256000
'| . ' 5-schisma, 32768:32805

It appears that 4374:4375 (I think this has been called the ragisma)
might be as important as 4095:4096 as the value of the schismina,
particularly when you're below the 13 limit. I also threw in that
secondary 23-limit ratio that represents 243:245 as .|~. (that last
period being a right-accent).

I would still allow that the long version of the schismina could be
written as either |' !. or |` !, (the former in order to correspond
to the actual symbol) and that the ` , pair must then be used in
instances where ambiguity might occur in a character string.

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
>
> > This is how the symbol sequence would look (as accidentals to 9
> > ennealimal nominals):
>
> It's hard for me to respond to this without knowing what Manuel
thinks
> about it, since I am wanting a Scala compatible notation.

I've had a chance to look at the EL441 implementation in the latest
Scala (v. 2.21m), and I found a few problems and issues with it --
both in notational semantics (6 occurrences) and in conflicts with
Sagittal (1.5 occurrences).

I'm still trying to figure out exactly what "Scala-compatible" means,
because I'm not very familiar with the E-notations in Scala. My
understanding is that they're based on Paul Rapoport's symbols, which
are all 5-limit and which agree with Sagittal shorthand in having \ /
as the 5-comma character pair. The two disagree as to which pair
should be used for the 5-schisma, but since Sagittal shorthand does
not use the < > pair, at least there is no direct conflict.

I recall that Paul Rapoport had separate symbols for certain ratios
that could otherwise be notated as a combination of 5-commas and the
5-schisma, and on looking thru the Scala sequence, I found the
following:

Scala Sagit
v ^ '\ ./ diaschisma
( ) '_ .= meantone diesis

The character pair v ^ for the diaschisma directly conflicts with
Sagittal shorthand, since that's already used for the 11M-diesis,
32:33. While the meantone-diesis parentheses are not used in
Sagittal shorthand, they do have a meaning in its long form that's in
conflict, so that amounts to 1.5 instances of conflict.

In EL441 I didn't see any microtonal accidentals modifying the same
notehead altering in opposite directions. Allowing this to occur in
the case of the schisma (as we have done in Sagittal) would allow two
character pairs to be used for other ratios. It will also be evident
from the table below that this will also be essential in order to
avoid faulty semantics.

I noticed the pair L 7 for the 7-comma, evidently an addition to
Paul's symbol set. Again this doesn't conflict with Sagittal
shorthand, since those characters aren't used. It's only a matter of
Sagittal using different characters "t" and "f" for these (as with
the 5-schisma).

The s $ pair is one that's been appropriated from Sagittal (at my
suggestion), so it's completely compatible. It's not the usual way
that Sagittal would notate the ratios for 2 and 3deg441, and since
the semantics for s< $> for 4deg is not valid anyway, there doesn't
appear to be any advantage in using that in preference to the c r
pair we have in Sagittal. (I've given alternatives for all of the
other places where the s $ pair is presently used.)

Here's a comparison of the symbol sequences, which will identify the
6 semantical problems:

deg Sagit EL441
441 dn up dn up ratio(s) generators (up-symbol)
--- -- -- --- --- ------------ -----------------------
1 . ' < > 32768:32805 19G
2 c r 5103:5120 +11G
<< >> -38G WRONG #G's; need s> $<
3 'c .r s $ 224:225 -8G
4 ¦ ¡ 49 comma +22G
s< $> -27G WRONG #G's
5 'z .~ 243:245 +3G
or '¦ .¡ and 125:126 +3G
s<< $>> -46G WRONG #G's; need \$ /s
6 z ~ 32768:33075 16G
7 '\ ./ v ^ diaschisma +14G could use \> /<
8 \ / \ / 80:81 -5G
9 .\ '/ \< /> pyth. comma -24G
10 t f L 7 63:64 +6G
11 .t 'f L< 7> 3584:3645 -13G
12 '¿ .ç +17G
L<< 7>> -32G WRONG #G's; need Ls> 7$<
13 ¿ ç Ls 7$ 48:49, 49:50 -2G \z /~ would also work
14 .¿ 'ç -21G
vv ^^ +28G WRONG #G's; want Ls< 7$>
15 '_ .= ( ) 125:128 +9G could use \\> //<
16 _ = \\ // 6400:6561 -10G
17 'u .n +20G
\\< //> -29G WRONG #G's
18 u n (s )$ 35:36 +1G L\ 7/ would also work
19 .u 'n \\s //$ -18G L\> 7/< would also work
20 a g [ ] 3969:4096 +12G
21 .a 'g [< ]> 392:405 -7G
22 ¤ ° (v )^ 49L +23G
23 'w .m (\ )/ 27:28 +4G
24 w m \\\ /// 8192:8505 -15G
25 .w 'm -34G WRONG # G'S
(L )7 +15G

The semantical problem is that the symbol arithmetic must be correct
not only in the number of scale degrees, but also in the number of
generators indicated by the character components. If this is to be a
true ennealimal notation, then it must be valid not only for 441-ET,
but also 270, 171, 99, etc., and that will be the case only if you
don't go beyond +-24 generators.

There's one instance (25deg) where the Sagittal syntax doesn't give
the right number of generators (which leaves something to be
addressed), but this is more than half the distance to the next
higher nominal, so it is not necessarily a fatal error.

--George

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > Don't you think this would be better than a bunch of tricky rules
> > about which symbol an accent belongs to when it has cores on both
> > sides of it?
> > ...
> > If we used only , and ` for right accents then a . or ' as the
> > rightmost character in a multi-accidental string could also be
> > interpreted as a separate schisma symbol.
>
> Okay, you've convinced me! We would have the following then:
>
> ASCII shorthand
> long down up Comma
> ----- ---- -- -----------------------
> |' , ` schismina, 4095:4096 & 4374:4375, 3^9*5^2*7^2:2^20*23
> |'' ,, `` double-schismina, 2079:2080 & 255879:256000
> '| . ' 5-schisma, 32768:32805

Yikes!

I may have convinced you, but I've just realised it may well be out of
the question for FTS and Scala since they use commas as delimiters.
I've asked them and will let you know, (if they don't Cc you).

> It appears that 4374:4375 (I think this has been called the ragisma)

Yes.

> might be as important as 4095:4096 as the value of the schismina,
> particularly when you're below the 13 limit.

Certainly.

> I also threw in that
> secondary 23-limit ratio that represents 243:245 as .|~. (that last
> period being a right-accent).

Can't argue with that.

> I would still allow that the long version of the schismina could be
> written as either |' !. or |` !, (the former in order to correspond
> to the actual symbol) and that the ` , pair must then be used in
> instances where ambiguity might occur in a character string.

Assuming we can actually use the ASCII comma as an accidental, then
I'd prefer to say that the , ` pair must be used in all cases except
where . ' can be interpreted in no other way than as a right accent.

This is subtly different in allowing n. to be parsed as /|) .!

-- Dave

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> I'm still trying to figure out exactly what "Scala-compatible" means,
> because I'm not very familiar with the E-notations in Scala.

I'm afraid I don't know what it means either. My working definition is
anything Manuel is willing to install as a Scala notation.

> The semantical problem is that the symbol arithmetic must be correct
> not only in the number of scale degrees, but also in the number of
> generators indicated by the character components. If this is to be a
> true ennealimal notation, then it must be valid not only for 441-ET,
> but also 270, 171, 99, etc., and that will be the case only if you
> don't go beyond +-24 generators.

I don't see why you need to go that far. My present theory as to how
it should work is that -9 to 9 generators are notated first, giving a
notation for 171. The 99 and 72 would be strict subsets of that. 270,
441 and 612 would be notated purely by using the schisma.

> There's one instance (25deg) where the Sagittal syntax doesn't give
> the right number of generators (which leaves something to be
> addressed), but this is more than half the distance to the next
> higher nominal, so it is not necessarily a fatal error.

Do you think nominals should be separated by fifths, by periods, or
could they be fifths lowered by 225/224, the 112/75 fifth?

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> I don't see why you need to go that far. My present theory as to how
> it should work is that -9 to 9 generators are notated first, giving a
> notation for 171. The 99 and 72 would be strict subsets of that. 270,
> 441 and 612 would be notated purely by using the schisma.

I would suggest the following as operating principles:

(1) Nine nominals, A-I. These to be arranged so that the "fifth" is
5/9 of an octave, or 666 2/3 cents. The nominals now have the Bach
property, that BACH spells out something somewhat like what it is
supposed to.

(2) Single symbol pairs using ISO-Latin-1 characters for each
generator step from 1 to 9. These would represent items on the
following list:

1: {36/35, 250/243, 21/20}
2: {49/48, 50/49}
3: {245/243, 126/125, 1728/1715}
4: {28/27, 25/24}
5: {875/864, 3125/3087, 81/80}
6: {686/675, 64/63}
7: {392/375, 256/245, 405/392}
8: {1029/1024, 225/224}
9: {128/125}

The most logical choices seem to me to be 36/35, 49/48, 126/125,
25/24, 81/80, 64/63, 256/245, 225/224, 128/125. However if you are
allergic to 126/125 and 225/224, you can substitute 245/243 or
1728/1715 for 126/125, and 1029/1024 for 225/224.

(3) A symbol pair, of course, for the schisma, but possibly for two
and three schismas also.

(4) The generator should be thought of in terms of 6/5.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > I'm still trying to figure out exactly what "Scala-compatible"
means,
> > because I'm not very familiar with the E-notations in Scala.
>
> I'm afraid I don't know what it means either. My working definition
is
> anything Manuel is willing to install as a Scala notation.

Then you'll have to decide whether you want the set of character-
pairs to be based on an expansion of those in the E-notations or on
those in Sagittal shorthand. The philosophical difference between
the two approaches is that E-notation uses combinations of character
symbolizing ratios, whereas Sagittal uses single accidentals for
ratios and permits multiple shorthand characters only for schismas
and/or schisminas.

> > The semantical problem is that the symbol arithmetic must be
correct
> > not only in the number of scale degrees, but also in the number
of
> > generators indicated by the character components. If this is to
be a
> > true ennealimal notation, then it must be valid not only for 441-
ET,
> > but also 270, 171, 99, etc., and that will be the case only if
you
> > don't go beyond +-24 generators.
>
> I don't see why you need to go that far.

I have a suspicion that if you aren't careful enough to notate all of
the vectors consistently across the ennealimal family, then something
is liable to come out and bite you later on. If you devise your
character combinations (or symbols) by adding together degrees of an
octave-division, your notation may not be completely valid for the
other divisions in the ennealimmal family. But if you do it by
adding together numbers of ennealimal generators, then the notation
will be guaranteed to be valid across the family.

> My present theory as to how
> it should work is that -9 to 9 generators are notated first, giving
a
> notation for 171. The 99 and 72 would be strict subsets of that.
270,
> 441 and 612 would be notated purely by using the schisma.

That's the way I would expect it to work, too. I really should have
selected the accidentals in the table I gave with all of these
divisions in mind, so what I've already given will have to be
regarded as tenative.

With Sagittal accidentals it's possible that one or two of the
accidentals within a +-9 generator range would be accented symbols;
however, there would not be any two accidentals differing by a
schisma-accent. Like you, I would rather not see any accented
symbols for 99 or 171, so I'll have to go back to the drawing board
to see what I can come up with.

> > There's one instance (25deg) where the Sagittal syntax doesn't
give
> > the right number of generators (which leaves something to be
> > addressed), but this is more than half the distance to the next
> > higher nominal, so it is not necessarily a fatal error.
>
> Do you think nominals should be separated by fifths, by periods, or
> could they be fifths lowered by 225/224, the 112/75 fifth?

I would expect it to be by periods. That's about all I have to say
regarding that, because I've left the matter of non-diatonic nominals
in Sagittal to Dave and Herman, preferring to concentrate on the
accidentals.

> I would suggest the following as operating principles:
> ...
> (3) A symbol pair, of course, for the schisma, but possibly for two
> and three schismas also.

I doubt that you'll need to alter by two schismas. In 612 the period
is 68deg, so you'll cover all of the tones in +-34 generators. The
schisma is -19 generators, so two schismas will take you -38
generators from a nominal.

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> >
> > > I'm still trying to figure out exactly what "Scala-compatible"
means,
> > > because I'm not very familiar with the E-notations in Scala.
> >
> > I'm afraid I don't know what it means either. My working
definition is
> > anything Manuel is willing to install as a Scala notation.
>
> Then you'll have to decide whether you want the set of character-
> pairs to be based on an expansion of those in the E-notations or on
> those in Sagittal shorthand. The philosophical difference between
> the two approaches is that E-notation uses combinations of
character
> symbolizing ratios, whereas Sagittal uses single accidentals for
> ratios and permits multiple shorthand characters only for schismas
> and/or schisminas.

Might I add that the great advantage with the Sagittal approach is
that there's no need to do mental arithmetic with a string of
accidentals in order to determine the size of the alteration -- very
helpful when reading in real time.

> > > The semantical problem is that the symbol arithmetic must be
correct
> > > not only in the number of scale degrees, but also in the number
of
> > > generators indicated by the character components. If this is
to be a
> > > true ennealimal notation, then it must be valid not only for
441-ET,
> > > but also 270, 171, 99, etc., and that will be the case only if
you
> > > don't go beyond +-24 generators.
> >
> > I don't see why you need to go that far.
>
> I have a suspicion that if you aren't careful enough to notate all
of
> the vectors consistently across the ennealimal family, then
something
> is liable to come out and bite you later on. If you devise your
> character combinations (or symbols) by adding together degrees of
an
> octave-division, your notation may not be completely valid for the
> other divisions in the ennealimmal family. But if you do it by
> adding together numbers of ennealimal generators, then the notation
> will be guaranteed to be valid across the family.
>
> > My present theory as to how
> > it should work is that -9 to 9 generators are notated first,
giving a
> > notation for 171. The 99 and 72 would be strict subsets of that.
270,
> > 441 and 612 would be notated purely by using the schisma.
>
> That's the way I would expect it to work, too. I really should
have
> selected the accidentals in the table I gave with all of these
> divisions in mind, so what I've already given will have to be
> regarded as tenative.
>
> With Sagittal accidentals it's possible that one or two of the
> accidentals within a +-9 generator range would be accented symbols;
> however, there would not be any two accidentals differing by a
> schisma-accent. Like you, I would rather not see any accented
> symbols for 99 or 171, so I'll have to go back to the drawing board
> to see what I can come up with.

It turns out that about half are accented. (See table below.)

> ...
> > I would suggest the following as operating principles:
> > ...
> > (3) A symbol pair, of course, for the schisma, but possibly for
two
> > and three schismas also.
>
> I doubt that you'll need to alter by two schismas.

As it turned out, this was not the case. But rather than use double-
schismas (not permitted in Sagittal), I chose to notate 612 with
additional symbols that would be appropriate for (11-limit)
hemiennealimmal.

> In 612 the period
> is 68deg, so you'll cover all of the tones in +-34 generators. The
> schisma is -19 generators, so two schismas will take you -38
> generators from a nominal.

Here's what I came up with for Sagittal accidentals to these
ennealimmal divisions, using symbols that require the least number of
generators:

deg chars
72 99 171 270 441 612 dn up ratio(s) gens.
-- -- --- --- --- --- -- -- ---------------- -----
1 1 . ' 32768:32805 19G
2 .c 'r +30G
1 2 3 c r 5103:5120 +11G
1 2 3 4 'c .r 224:225 -8G
5 i * (891:896) -27G
4 6 ¦ ¡ 1594323:1605632 +22G
1 1 2 3 5 7 'z .~ 243:245, 125:126 +3G
6 8 z ~ 32768:33075 16G
9 § (99:100) +33G
4 7 10 '\ ./ 2025:2048 +14G
2 3 5 8 11 \ / 80:81 -5G
9 12 .\ '/ pyth. comma -24G
13 't .f +25G
4 6 10 14 t f 63:64 +6G
7 11 15 .t 'f 3584:3645 -13G
16 k y (54:55) -32G
12 17 '¿ .ç 98415:100352 +17G
2 3 5 8 13 18 ¿ ç 48:49, 49:50 -2G
14 19 .¿ 'ç 524288:535815 -21G
20 d q (44:45) +28G
6 9 15 21 '_ .= 125:128 +9G
10 16 22 _ = 6400:6561 -10G
23 ._ '= -29G
17 24 'u .n 127575:131072 +20G
3 4 7 11 18 25 u n 35:36 +1G
19 26 .u 'n 57344:59049 -18G
27 'a .g +31G
12 20 28 a g 3969:4096 +12G
8 13 21 29 .a 'g 392:405 -7G
30 ¤ ° 49L-diesis -26G
22 31 o @ (704:729) +23G
4 5 9 14 23 32 'w .m 27:28 +4G
15 24 33 w m 8192:8505 -15G
34 .w 'm -34G

I've put the 11-ratios in parentheses. I used 704:729 for both 441
and 612 (the only 11-ratio needed for 441, and valid in the +23G
position for both divisions). I found that 32:33 wasn't valid in the
+31G position for both 441 and 612 (though it technically isn't
required for the former), so I chose an accented (7-limit) symbol
pair instead. (But I might reconsider, since not using 32:33 would
contribute to a more symmetrical symbol sequence.)

Gene, should you choose to go with an expanded E-notation, then I
think that most of the above would apply, only that the characters
would be different.

--George