Yahoo Tuning Groups Ultimate Backup tuning-math Re: Saggital chord symbols

Hi George,

> > Then the list of comma symbols as I have it so far is:
> > c 81/80
> > s 64/63
> > u 33/32
> > t 27/26
> > e 128/125 diesis
> > d 2048/2025 diaschisma

> We already have Sagittal up and down shorthand characters for all of
> these -- and also for many more commas to the 23-limit. You would
> only need to replace the slash character "/" that you're using as a
> note delimiter by a dash "--", since a hyphen character is not used
> in Sagittal shorthand (except in a user-defined application).

> For your reference, here is a message that contains a table showing
> the complete Sagittal ASCII shorthand:
> /tuning/topicId_54793.html#54967

Rightio, I remember now that you recommended
some single character symbols for the 72-et notation in FTS,
and I've even added them in to the curreent release candidate for
FTS, but had forgotten all about them.

Sorry, I never saw your post with the more complete list of
short ascii symbols at all, I see indirectly it was a reply to
my earlier message. .

Anyway it is an easy matter to change the sagittal symbols for the chord progression
player at this stage - that's no problem. I just have a table
of symbols with their translations and it is easy to add to it.

In fact I've added in the longer sagittal symbols already.

One thing I might do is to retain the abbreviations for a learner
mode, now that they are already programmed in. Probably
preset to show the short ascii symbols, but user can switch
to abbreviations if they find it helpful when learning the
notation.

I find it easier right now as I'm still learning the proper symbols
but will soon be able to move over to using the proper
symbols, so maybe others will be in the same situation when
they start to use the notation. As a newbie, it is easy
to remember that bc and #c mean flat or sharp by a comma
and bs and #s mean flat or sharp by a septimal diesis.
with their initial letters. It's not quite so easy to remember
what the single character symbols are, but with
repeated use one would learn them quickly of course.
For instance / and \ are easy to remember for the syntonic
comma, and then f and t for the septimal comma I find
harder to remember and get the u and n for the
tridecimal comma a bit confused because
I think of it as u for undecimal - there maybe
the abbreviations may add to the confusion so
that may be a possible reason for not using them at all
possibly.

The notation window will have a list of all the symbols
so that it is easy to look them up from within the program.
Also the user can show them on the picture of the p.c.keyboard
and swap between the various possibilities.

I do like the Sagittal system of accidental signs
and this wasn't meant as any kind of attempt to
try and change or improve on it. I just found
they weren't working with chord symbols, but your
short symbols do indeed solve that problem.

Thanks for the list of symbols. I've put most of them
in with a few left to do

Here is David Canright's thirteen limit scale using them:

http://www.robertinventor.com/sagittal.png

1 13/12 9/8 7/6 5/4 4/3 11/8 3/2 13/8 5/3 7/4 11/6 2/1

I didn't find these in your list, maybe they can be got
by combining others in the list:

128/125 diesis
25/24 chromatic semitone

For the 1053/1024 or 35/36, the chord progression
player has to play one or the other in its just intonation
realisation of the chords, so I'm not sure yet what to
do about that.

Yes I did have some slips in my e-mail. Yes I realised
that I'd mis-spelt Sagittal actually, and have corrected that.

> In the first example you gave:

> > Then, to see most of them in play in a single chord,
> > 1/1 5/4 3/2 7/4 9/4 11/4 13/4
> > could be notated as
> > C5p/bc3/bs7/9/#u11/bt13
> > just to show how the notation works.
> >
> > The p at the start there says to start from the pythagorean scale
as ones basis
> > for the tuning of the chord, as is usual in Saggital.

> By "pythagorean scale" do you mean a pythagorean major scale
> beginning on the particular tone designated as 1/1, or a pythagorean
> scale on the natural notes (i.e., major scale beginning on C)?

Surely, one on the natural notes, using Pythagorean chromatic
semitones for the accidentals. After a bit of thought that seemed
to be way it would have to be, with C as 1/1, F as 4/3,
and pure fifths up to B. But do be sure to say
if I haven't undestood how it works yet.

I've also made it so user can choose to interpret in an
n-et with the interpretation of all the commas in that
et so that the commas and dieses can vanish or change sign
etc. There's a radio button the user can use to switch
to an n-et interpretation of Sagittal for any n-et they
like. But I haven't done anything about implementing the
n-et recommendations as that would involve a lot of programming
so can do that later if it is needed.

It will avoid using negative and vanishing commas when making
accidentals up itself at least. But allow anything in user
input - so for instance you can enter a chord sequence
and then change teh n-et and see how it gets re-interpreted
with e.g. zero or negative comma etc.

> > If you left it out then
> > it would be understood as a twelve equal chord which we don't want.

> It appears that you left out a note. I believe you meant this:
> C5p/bc3/5/bs7/9/#u11/bt13

> Sagittal shorthand characters would render this as:
> C5p--\3--5--t7--9--^11--w13
> which I find much easier to read than what you have, since there are
> fewer characters to read for each note and the dashes separate the
notes more clearly.

Rightio.

Yes indeed, one idea is to add in '-' as an extra possible
note addition symbol, and maybe just say that you
can't use a / with this p notation - or else
maybe in case user wants to use a /
just say that the first occurrence of - changes
the note addition sign.

I've been using @ as note addition symbols
for this sort of situation,
so it would be
C5p@\3@5@t7@9@^11@w13

instead of
C5p--\3--5--t7--9--^11--w13

I agree, your notation is simpler,
and I'll see if I can convert the player to use it.
(retaining the @ as well for backward
compatibility).

There is one consideration there that needs some
thought - it is quite common to use
roman numerals as
I-ii-V-I
all in one row without spaces between the dashes.
and I want user to be able to paste that into the player.

I already deal with that with a pre-parse stage to check if
it looks like a roman numeral with dashes type notation
so that can probably be dealt with.

> In addition, I have a question. If that same chord were built on F,
> would your notation indicate it as this (as alterations to a
> pythagorean major scale beginning on F)?

> F5p/bc3/5/bs7/9/#u11/bt13

> Or would the unaltered 11th be a "B" (thereby requiring the 11th to
> be flatted thus)?

> F5p/bc3/5/bs7/9/#ub11/bt13

I've done it the first way so that it is an alteration
to a pythagorean major scale beginning on the F.
I think that makes more sense, otherwise the chord
symbol changes when you change the root.
Is that in accord with Sagittal?

> > The diaschisma can be used to get a 25/16 from the nearest
> pythagorean pitch, in e.g.:
> > 1/1 5/4 25/16
> > which can be notated as:Cp0/bc3/bbd6

> Should that be Cp0/bc3/bdb6? Using "b" to mean two different things
is especially confusing in this example. The existing Sagittal ASCII
> shorthand characters could express this chord in two different ways,
> with either an altered 5th or altered 6th degree of the scale:

Yes I agree, bdb6 is better. Though my player can recognise
the bbd6 here either way around

> C5p--\3--_#5
> C5p--\3--'\b6

Shouldn't that be C0p-- ...

or C5p(5)... or Cp0...

The chord progression player at present would play a
C5p/bc3/bdb6
as 1/1 5/4 3/2 25/16 2/1.

Or, am I missing something there about how the
notation works?

> where _ represents a downward alteration of 6400:6561 (twice 80:81)
> to a pythagorean G#, and
> '\ represents a downward alteration of a diaschisma to a pythagorean
> Ab. Thus you would have the ability to spell chords in more than one
> way.

I thought of another alternative too requiring
a symbol for the chromatic semitone:

1/1 5/4 25/16
=

Cpbc/#cs5

which in your notation would be
Cp/--?5
where ? is the symbol for the chromatic semitone 25/24 whatever it is.

It's a bit awkward to have the underline as an accidental here,
as I've been using underlines in my chord progression player
as a spacing character which can be ignored.

I'll need to think about whether I can change that.
Is there any alternative to it by any chance?

Also the dash too - I know it is a particularly clear notation
so that's a strong incentive to support it
- but users will be used to using for chords as in
C5/7 etc and the player supports that for all
the other chords, and I have to support that
as well certainly because one objective for the player
is that if anyone pastes any chord progression such as
they might find on a web page etc into the player,
then it should be able to play it right away with
no changes, to make it easy for complete newbies to use.

So, it is a bit awkward to have
to say that you can't use the / with the Sagittal
notation, and do a special case for it somehow.
But hopefully one can come to some suitable solution.

I've also got a more general question about Sagittal.

I've just been doing a section of the help (after
mentioning meantones and schismic temperaments)
taking 14-et as an example of qutie a bizarre tuning in the
notation because so many commas and things vanish
or change sign, even the pythagorean chromatic semitone
so ordinary sharps and flats will now have no effect
- and the chromatic semitone becomes a whole tone!

It's a tuning I've done some composing in
(well 7-et anyway and some experimenting in 14-et)
so was particularly intersted to see what happens.

Anyway I came to the conclusion that you need to use
the septimal diesis 64/63 to notate the accidentals,

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
C D E F G A B C
C#s Ebs E#s F#s G#s Abs Bbs

and I saw in your pdf that you recommend it too, looking
this et up. But then the normal way of thinking about
9?7 is as 81/64 raised by a septimal diesis. But here
our E is already a 9/7. So how does one understand
use of the septimal diesis as an accidental in
14-et.

Anyway I just wanted to check really that I'm
understanding the situation correctly, and
perhaps this is just an awkwardenss one
has to live with when working with an et
such as 14-et which doesn't fit in well with
12 tone concepts at all

But I'm interested to know what you say about it..

Also another thing I was wondering about
- the chord progression player in FTS is the
only part of it really that is restricted to
octave based tunings, because I don't know
of any standard way of notating non octave
chords. But I wondered, thinking forwards
to possiblee later developments, is there
any work done on a non octave version of
Sagittal?

For instance I suppose one might do a
3/1 based Saggital with the 2/1 in place
of 3/1 as the generating interval for the
note names or something.

How would you notate Bohlen Pierce in
Sagittal? Or Wendy Carlos's
Gamma?

Robert

🔗Robert Walker <robertwalker@ntlworld.com>

Hi George,

I've added all the sagittal one character symbols to FTS
now from that e-mail.

I wonder if you could check over this print out of the
list of symbols to see if it is correct:
Sagittal Accidentals:

Abbr. ratio cents steps sharp flat name
2187/2048 113.69 5 /||\ \!!/ Pythagorean chromatic semitone
cs 25/24 70.67 3 )||( )!!( chromatic semitone
t 27/26 65.34 3 (|\ (!/ tridecimal comma, 13L-diesis
dd 8505/8192 64.91 3 |\ |/ 35L diesis
ud 729/704 60.41 3 (|) (!) undecimal diesis, 11L-diesis
scc 4096/3969 54.53 2 (/| (\! septimal comma (2), Archytas' comma, 49M-diesis
u 33/32 53.27 2 /|\ \!/ undecimal comma, 11M-diesis
sc 36/35 48.77 2 /|) \!) septimal comma, Archytas' comma, 1/4 tone, 35M-diesis
td 1053/1024 48.35 2 /|) \!) tridecimal major diesis, 13M-diesis
td 40/39 43.83 2 /|) \!) tridecimal diesis
m 6561/6400 43.01 2 //| \\! Mathieu superdiesis, 25S-diesis
d 128/125 41.06 2 //| \\! diesis
f 45/44 38.91 2 (|( (!( 1/5 tone, 5:11S-diesis
cj 45927/45056 33.15 2 (| (! 7:11-comma
cy 55/54 31.77 1 |\ !/ 55-comma
s 64/63 27.26 1 |) !) septimal diesis, 7-comma
c% 41553/40960 24.88 1 )/| )/! 5:19-comma
c 81/80 21.51 1 /| \! syntonic comma, 5-comma
ds 2048/2025 19.55 1 '/ '\ diaschisma
vc 736/729 16.54 1 |~ !~ 23-comma, vicesimotertial comma
sd 4131/4096 14.73 1 ~|( ~!( 17-comma,
sdc 2187/2176 8.73 0 ~| ~! septendecimal comma, 17-kleisma
k 5120/5103 5.76 0 |( !( 5:7-kliesma
uc 513/512 3.38 0 )| )! undevisical comma, Boethius' Comma, 19-schisma
sa 32805/32768 1.95 0 '| '! schisma, 5-schisma
BTW, the number of steps there is the number in 53-et.

Then with the short ascii symbols:
Sagittal Accidentals:

Abbr. ratio cents steps sharp flat name
2187/2048 113.69 5 # b Pythagorean chromatic semitone
cs 25/24 70.67 3 #cs bcs chromatic semitone
t 27/26 65.34 3 m w tridecimal comma, 13L-diesis
dd 8505/8192 64.91 3 m w 35L diesis
ud 729/704 60.41 3 @ o undecimal diesis, 11L-diesis
scc 4096/3969 54.53 2 g a septimal comma (2), Archytas' comma, 49M-diesis
u 33/32 53.27 2 ^ v undecimal comma, 11M-diesis
sc 36/35 48.77 2 n u septimal comma, Archytas' comma, 1/4 tone, 35M-diesis
td 1053/1024 48.35 2 n u tridecimal major diesis, 13M-diesis
td 40/39 43.83 2 n u tridecimal diesis
m 6561/6400 43.01 2 _ = Mathieu superdiesis, 25S-diesis
d 128/125 41.06 2 _ = diesis
f 45/44 38.91 2 q d 1/5 tone, 5:11S-diesis
cj 45927/45056 33.15 2 j ? 7:11-comma
cy 55/54 31.77 1 y k 55-comma
s 64/63 27.26 1 f t septimal diesis, 7-comma
c% 41553/40960 24.88 1 % & 5:19-comma
c 81/80 21.51 1 / \ syntonic comma, 5-comma
ds 2048/2025 19.55 1 '/ '\ diaschisma
vc 736/729 16.54 1 ~ z 23-comma, vicesimotertial comma
sd 4131/4096 14.73 1 p h 17-comma,
sdc 2187/2176 8.73 0 $ s septendecimal comma, 17-kleisma
k 5120/5103 5.76 0 r c 5:7-kliesma
uc 513/512 3.38 0 * ; undevisical comma, Boethius' Comma, 19-schisma
sa 32805/32768 1.95 0 ' . schisma, 5-schisma

There, I'm missing a Sagittal short ascii symbol for the chromatic semitone, which is whyit is shown as #cs and bcs for now.
I think it is possible I may have found a solution for the dashes and underlinesproblem.
The idea is to treat the dash note addition symbol as onethat also sets it to accept the saggital accidentals.
So for instance, can do the major chord as C-\
Minor asCm-/
Then that harmonic series chord can be:
C-\-t7-9-^11-w13
where I'll also permit double dashes
Also 1/1 5/4 25/16 as:
C-\(5)--_#5
or
C5p--\3--'\b6
- still working on it and the implications for checking forthe roman numerals and other notations recogised,but it looks promising.
Robert

----- Original Message -----
From: <tuning-math@yahoogroups.com>
To: <tuning-math@yahoogroups.com>
Sent: Tuesday, September 07, 2004 9:03 PM
Subject: [tuning-math] Digest Number 1159

>
> There are 7 messages in this issue.
>
> Topics in this digest:
>
> 1. Marvellous ellipses
> From: "Gene Ward Smith" <gwsmith@svpal.org>
> 2. Re: Marvellous ellipses
> From: "Gene Ward Smith" <gwsmith@svpal.org>
> 3. Re: Re: 7-limit 12-note epimorphic scales containing a hexany
> From: "Manuel Op de Coul" <manuel.op.de.coul@eon-benelux.com>
> 4. Re: Re: 7-limit 12-note epimorphic scales containing a hexany
> From: Carl Lumma <ekin@lumma.org>
> 5. TOP
> From: "Manuel Op de Coul" <manuel.op.de.coul@eon-benelux.com>
> 6. scala suggestion
> From: Carl Lumma <ekin@lumma.org>
> 7. Re: 7-limit 12-note epimorphic scales containing a hexany
> From: "Carl Lumma" <ekin@lumma.org>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 1
> Date: Tue, 07 Sep 2004 03:31:46 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
> Subject: Marvellous ellipses
>
> If |a b c> is a 5-limit monzo, we may define a quadratic form on
> note-classes by ell(|a b c>) = 4b^2 - 7bc + 4c^2. The discriminant of
> this is (-7)^2 - 4*4*4 = -15, which is negative, so setting ell to a
> constant gives us ellipses. ell(3) = ell(5) = ell(225/32) = 4; the
> last is the 5-limit marvel projection of 7, and so one of the ellipses
> has the class for the marvel versions of 3, 5, and 7 on its boundry.
>
> A marvellous ellipse is simply a set of notes 1 <= q < 2 such that
> ell(q) <= N for some bounding value N. These turn out to define
> interesting scales. It is clear from the manner of their construction
> that they are not only inversely symmetric but also have a 3<==>5
> symmetry. They also have an odd number of notes to an octave. What
> isn't clear, but which seems to be true, is that they have a tendency
> to be permutation epimorphic. It might also be noted that attempting
> to do the same for the 9-limit, and have an ellipse with 5/4, 225/128
> and 9/8 on the boundry, will not work, since 5/4, 9/8, and 256/225 lie
> along a line.
>
> I've uploaded some marvelous ellipses--of sizes 9, 13, 15, 21, and
> 31--to the files section:
>
> /tuning-math/files/marvell/
>
> The 9-note scale is epimorphic, and 13, 15 and 19 are all permutation
> epimorphic.
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 2
> Date: Tue, 07 Sep 2004 08:05:10 -0000
> From: "Gene Ward Smith" <gwsmith@svpal.org>
> Subject: Re: Marvellous ellipses
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > The 9-note scale is epimorphic, and 13, 15 and 19 are all permutation
> > epimorphic.
>
> I don't know the reason for it, but it is clear there is one. I looked
> at using 2401/2400 with {5,7} note classes, and 3 represented by
> 2401/800. Trying to get 5, 7, and 2401/800 on the boundry of a conic
> section centered at 1 leads to a hyperbola, so I tried
> 8x^2 + 15xy + 8y^2 instead. This didn't net me as many complete
> tetrads as I might like (though of course very many 4:5:7 chords and
> the like) but it did, once again, turn up with a lot of epimorphic and
> permuation epimorphic offerings. Perhaps some theory for ellipsoids
> could be developed to go with Fokker blocks.
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 3
> Date: Tue, 7 Sep 2004 16:14:26 +0200
> From: "Manuel Op de Coul" <manuel.op.de.coul@eon-benelux.com>
> Subject: Re: Re: 7-limit 12-note epimorphic scales containing a hexany
>
>
> Carl wrote:
>
> >By the way, Scala's 'compare scale' feature is broken in
> >several ways, so I wouldn't trust it.
>
> Still haven't seen anything substantiating this allegation.
>
> Manuel
>
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 4
> Date: Tue, 07 Sep 2004 08:43:50 -0700
> From: Carl Lumma <ekin@lumma.org>
> Subject: Re: Re: 7-limit 12-note epimorphic scales containing a hexany
>
> >Carl wrote:
> >
> >>By the way, Scala's 'compare scale' feature is broken in
> >>several ways, so I wouldn't trust it.
> >
> >Still haven't seen anything substantiating this allegation.
> >
> >Manuel
>
> Thanks for reminding me. I'll get this to you this week.
>
> -Carl
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 5
> Date: Tue, 7 Sep 2004 17:57:46 +0200
> From: "Manuel Op de Coul" <manuel.op.de.coul@eon-benelux.com>
> Subject: TOP
>
>
> I've added TOP tempering to Scala, Gene's webpage was very helpful
> in getting it right in one try.
> The command is PROJECT/TEMPER/TOP and the dialog Modify:Temper.
>
> http://www.xs4all.nl/~huygensf/software/Scala_Setup.exe
>
> Manuel
>
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 6
> Date: Tue, 07 Sep 2004 11:11:31 -0700
> From: Carl Lumma <ekin@lumma.org>
> Subject: scala suggestion
>
> versioning for the scale archive
>
> -C.
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 7
> Date: Tue, 07 Sep 2004 18:12:37 -0000
> From: "Carl Lumma" <ekin@lumma.org>
> Subject: Re: 7-limit 12-note epimorphic scales containing a hexany
>
> > >Carl wrote:
> > >
> > >>By the way, Scala's 'compare scale' feature is broken in
> > >>several ways, so I wouldn't trust it.
> > >
> > >Still haven't seen anything substantiating this allegation.
> > >
> > >Manuel
> >
> > Thanks for reminding me. I'll get this to you this week.
>
> The problem is, I forget what I was doing. Kurt saw it;
> maybe he remembers.
>
> -Carl
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
>
>
> ------------------------------------------------------------------------
> Yahoo! Groups Links
>
>
>
>
> ------------------------------------------------------------------------
>
>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:
> [gs:]
> > For your reference, here is a message that contains a table
showing
> > the complete Sagittal ASCII shorthand:
> > /tuning/topicId_54793.html#54967
>
> Rightio, I remember now that you recommended
> some single character symbols for the 72-et notation in FTS,
> and I've even added them in to the curreent release candidate for
> FTS, but had forgotten all about them.
>
> Sorry, I never saw your post with the more complete list of
> short ascii symbols at all, I see indirectly it was a reply to
> my earlier message. .
>
> Anyway it is an easy matter to change the sagittal symbols for the
chord progression
> player at this stage - that's no problem. I just have a table
> of symbols with their translations and it is easy to add to it.
>
> In fact I've added in the longer sagittal symbols already.

That's good, because there are some that don't have a shorthand.
Have you added them in both the pure and mixed versions?

> One thing I might do is to retain the abbreviations for a learner
> mode, now that they are already programmed in. Probably
> preset to show the short ascii symbols, but user can switch
> to abbreviations if they find it helpful when learning the
> notation.

Just so you make it very clear which is which, so as not to confuse
the user.

> ... It's not quite so easy to remember
> what the single character symbols are, but with
> repeated use one would learn them quickly of course.
> For instance / and \ are easy to remember for the syntonic
> comma, and then f and t for the septimal comma I find
> harder to remember and get the u and n for the
> tridecimal comma a bit confused because
> I think of it as u for undecimal - there maybe
> the abbreviations may add to the confusion so
> that may be a possible reason for not using them at all
> possibly.

We devised the character shorthand because we realized that there
would be instances (such as lattice diagrams) in which a very concise
form of the notation would be required. We chose pairs of characters
that would most closely resemble the actual symbols, giving priority
to the intervals that would be used the most. There are no word-
associations intended for any of the letters.

> ...
> Thanks for the list of symbols. I've put most of them
> in with a few left to do
>
> Here is David Canright's thirteen limit scale using them:
>
> http://www.robertinventor.com/sagittal.png
>
> 1 13/12 9/8 7/6 5/4 4/3 11/8 3/2 13/8 5/3 7/4 11/6 2/1

Very nice. But 5/4 and 5/3 should be E\ and A\ (backslashes, with
downward slope), respectively. Also, 11/6 should be Bo rather than
B@, since the alteration is *downward* from a pythagorean B. It does
take a little practice to read these quickly, but only a little.
Even though I haven't used the shorthand very much, I was able to
spot these right away.

> I didn't find these in your list, maybe they can be got
> by combining others in the list:
>
> 128/125 diesis

Yes, but a shorthand character may be combined only with a 5-schisma
accent mark. The 125S-diesis is equal to two 5-commas 80:81 less a 5-
schisma (32768:32805). The Sagittal symbols for this are .//| up
and '\\! down, which is .= and '_ , respectively, in the shorthand
notation. We used the equal sign for the up-symbol because it has
two strokes, and the best thing we could find for the down-symbol was
the underscore.

> 25/24 chromatic semitone

Except for sharp, flat, and double-sharp (# b x), we have no single-
character notation for any interval larger than the diesis category
(upper boundary ~68.57c), so 25/24 would be a combination of
characters: either an apotome reduced by \\!, or a minor 2nd reduced
by a diaschisma, '\! . These pitches would then be notated as C)||(
and D'||\ (pure), C#\\! and Db'\! (mixed), or C#_ and Db'\
(shorthand), respectively, taking C as 1/1.

In the shorthand version of your "Sagittal Accidentals" table you
should replace the "#cs" and "bcs" entries for 25/24 with #_ and b=,
but the long entries you have for 25/24 in the upper table are
correct. In the two tables I suggest that you replace the column
headings "sharp" and "flat" with "up" and "down". The Sagittal
entries for 40/39 should be //| and \\! (long), = and ^ (short); we
call this one the 5:13S-diesis. Should you be curious about the
exact boundaries between small, medium, and large dieses, etc., see
the following:

/tuning/topicId_53869.html#54584

> For the 1053/1024 or 35/36, the chord progression
> player has to play one or the other in its just intonation
> realisation of the chords, so I'm not sure yet what to
> do about that.

There is also 250/243 (an apotome less three 5-commas) to be taken
into account. We have given the symbol /|) a primary (or exact)
definition of 35:36 and have allowed it to approximate the other two
ratios (with errors ~0.4c). Similarly, we define (|\ as exactly
8192:8505, and that symbol will also approximate 26:27 and
51200:531441. I rather doubt that there are any JI fanatics out
there that will be able to hear a 0.4-cent error. (But in case there
are, please be advised that we, with a lot of help from Gene,
recently laid the groundwork for the extreme-precision version of the
Sagittal notation that has resolution fine enough to notate 2460-ET,
or steps of less than 1/2 cent! Scary, eh?)

> ...
> I've also made it so user can choose to interpret in an
> n-et with the interpretation of all the commas in that
> et so that the commas and dieses can vanish or change sign
> etc. There's a radio button the user can use to switch
> to an n-et interpretation of Sagittal for any n-et they
> like. But I haven't done anything about implementing the
> n-et recommendations as that would involve a lot of programming
> so can do that later if it is needed.

Have you looked at the file sag_et.par that comes with Scala? This
is one of the files that Dave and I maintain and update
periodically. It contains the Sagittal symbol sequences for various
ET's. You're welcome to use it for FTS, if you wish. (This reminds
me that the 14-ET symbol sequence in it needs to be revised, as I
note below.)

> It will avoid using negative and vanishing commas when making
> accidentals up itself at least. But allow anything in user
> input - so for instance you can enter a chord sequence
> and then change teh n-et and see how it gets re-interpreted
> with e.g. zero or negative comma etc.

Sounds pretty nifty!

> > > If you left it out then
> > > it would be understood as a twelve equal chord which we don't
want.
>
> > It appears that you left out a note. I believe you meant this:
> > C5p/bc3/5/bs7/9/#u11/bt13
>
> > Sagittal shorthand characters would render this as:
> > C5p--\3--5--t7--9--^11--w13
> > which I find much easier to read than what you have, since there
are
> > fewer characters to read for each note and the dashes separate the
> notes more clearly.
>
> Rightio.
>
> Yes indeed, one idea is to add in '-' as an extra possible
> note addition symbol, and maybe just say that you
> can't use a / with this p notation - or else
> maybe in case user wants to use a /
> just say that the first occurrence of - changes
> the note addition sign.
>
> I've been using @ as note addition symbols
> for this sort of situation,
> so it would be
> C5p@\3@5@t7@9@^11@w13
>
> instead of
> C5p--\3--5--t7--9--^11--w13
>
> I agree, your notation is simpler,
> and I'll see if I can convert the player to use it.
> (retaining the @ as well for backward
> compatibility).

We do use the @ character for the 11L-diesis (704:729) up, but I
don't think that one would be used very often. But when you parse
this, you could write the code to interpret the second of two @
characters without an intervening numeric character as a Sagittal
character.

> There is one consideration there that needs some
> thought - it is quite common to use
> roman numerals as
> I-ii-V-I
> all in one row without spaces between the dashes.
> and I want user to be able to paste that into the player.
>
> I already deal with that with a pre-parse stage to check if
> it looks like a roman numeral with dashes type notation
> so that can probably be dealt with.
>
> > In addition, I have a question. If that same chord were built on
F,
> > would your notation indicate it as this (as alterations to a
> > pythagorean major scale beginning on F)?
>
> > F5p/bc3/5/bs7/9/#u11/bt13
>
> > Or would the unaltered 11th be a "B" (thereby requiring the 11th
to
> > be flatted thus)?
>
> > F5p/bc3/5/bs7/9/#ub11/bt13
>
> I've done it the first way so that it is an alteration
> to a pythagorean major scale beginning on the F.
> I think that makes more sense, otherwise the chord
> symbol changes when you change the root.

Yes, I agree.

> Is that in accord with Sagittal?

It doesn't matter.

> > > The diaschisma can be used to get a 25/16 from the nearest
pythagorean pitch, in e.g.:
> > > 1/1 5/4 25/16
> > > which can be notated as:Cp0/bc3/bbd6
>
> > Should that be Cp0/bc3/bdb6? Using "b" to mean two different
things
> is especially confusing in this example. The existing Sagittal ASCII
> > shorthand characters could express this chord in two different
ways,
> > with either an altered 5th or altered 6th degree of the scale:
>
> Yes I agree, bdb6 is better. Though my player can recognise
> the bbd6 here either way around

My immediate impression of "bb" is as a double-flat, another reason
why I would discourage using it in combination with a comma-character
to indicate direction.

> > C5p--\3--_#5
> > C5p--\3--'\b6
>
> Shouldn't that be C0p-- ...
>
> or C5p(5)... or Cp0...

Since I'm not familiar with FTS syntax, I don't know what it should
be -- I was thinking that the "5" in "C5p" is an octave number, but
perhaps I am mistaken. My main concern was with the other notes
(those above the root of the chord).

> The chord progression player at present would play a
> C5p/bc3/bdb6
> as 1/1 5/4 3/2 25/16 2/1.

I don't follow this. I see two /-delimiters, hence three notes
specified, yet there are ratios for five notes?

> Or, am I missing something there about how the
> notation works?

The Sagittal accidentals were intended to be used for pitch rather
than interval or chord notation, so it's pretty much up to you as to
how to implement the ascii-shorthand version of these for that
purpose. We use "c" for 5103:5120-down and "d" for 44:45-down, so
what you have in that example is not Sagittal.

> > where _ represents a downward alteration of 6400:6561 (twice
80:81)
> > to a pythagorean G#, and
> > '\ represents a downward alteration of a diaschisma to a
pythagorean
> > Ab. Thus you would have the ability to spell chords in more than
one
> > way.
>
> I thought of another alternative too requiring
> a symbol for the chromatic semitone:
>
> 1/1 5/4 25/16
> =
>
> Cpbc/#cs5
>
> which in your notation would be
> Cp/--?5
> where ? is the symbol for the chromatic semitone 25/24 whatever it
is.

You've lost me! However, I did cover notation of 25/24 above.

> It's a bit awkward to have the underline as an accidental here,
> as I've been using underlines in my chord progression player
> as a spacing character which can be ignored.
>
> I'll need to think about whether I can change that.
> Is there any alternative to it by any chance?

No. I explained above how we arrived at it. Could the presence of a
dash be used as an indication *not* to ignore an underscore character?

> Also the dash too - I know it is a particularly clear notation
> so that's a strong incentive to support it
> - but users will be used to using for chords as in
> C5/7 etc and the player supports that for all
> the other chords, and I have to support that
> as well certainly because one objective for the player
> is that if anyone pastes any chord progression such as
> they might find on a web page etc into the player,
> then it should be able to play it right away with
> no changes, to make it easy for complete newbies to use.

Aha!

> So, it is a bit awkward to have
> to say that you can't use the / with the Sagittal
> notation, and do a special case for it somehow.
> But hopefully one can come to some suitable solution.

Perhaps you could write a scan-and-replace routine that would convert
those slashes to dashes (or whatever you settle on for note
delimiters). Otherwise, I have some other ideas:

1) We kept "<" and ">" free for user-defined applications, so
perhaps ">" could be used for /| and "<" for \! . However, the
directionality of the symbols is not very intuitive, and this
character pair is presently used for the 7-comma in HEWM notation, so
this would not seem to be a very good solution.

2) Use "+" and "-" for the 5-comma up and down and keep the slashes
for your note delimiters. The directionality is certainly easy to
remember, and the characters would tie in with both HEWM notation and
the Sagittal-Wilson option given near the end of our paper.

3) Use "--" as a sort of delimiter to indicate that whatever
characters are included in a chord containing "--" should, whenever
possible, be interpreted as Sagittal shorthand characters. (You
mentioned something like this to me off-list.)

> I've also got a more general question about Sagittal.
>
> I've just been doing a section of the help (after
> mentioning meantones and schismic temperaments)
> taking 14-et as an example of qutie a bizarre tuning in the
> notation because so many commas and things vanish
> or change sign, even the pythagorean chromatic semitone
> so ordinary sharps and flats will now have no effect
> - and the chromatic semitone becomes a whole tone!
>
> It's a tuning I've done some composing in
> (well 7-et anyway and some experimenting in 14-et)
> so was particularly intersted to see what happens.
>
> Anyway I came to the conclusion that you need to use
> the septimal diesis 64/63 to notate the accidentals,
>
> 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
> C D E F G A B C
> C#s Ebs E#s F#s G#s Abs Bbs
>
> and I saw in your pdf that you recommend it too, looking
> this et up. But then the normal way of thinking about
> 9?7 is as 81/64 raised by a septimal diesis. But here
> our E is already a 9/7. So how does one understand
> use of the septimal diesis as an accidental in
> 14-et.

We have two different notations for 14-ET. One approach notates 14-
ET as a subset of 56-ET (for which the symbol sequence is given in
Figure 8 of the Sagittal paper). This is evidently the approach you
are referring to, since a chain of four 5ths of 56-ET brings us to
20deg56, which is 5deg14. In 56-ET |) is calculated to be 1deg (by
going two best 4ths of 23deg up, then down 1deg to 45deg56, which is
the closest approximation to 7/4), but the wide fifths of 56-ET do
not result in E|) being the best approximation to 9/7. In essence,
the problem is that 56-ET is not consistent at the 9-limit.

The other approach uses a chain of native fifths for the 7 naturals
(i.e., a 7-ET circle), and a single pair of Sagittal accidentals is
then used to alter these up or down to indicate the other 7-ET
circle. With a chain of 4 fifths and octave reduction, the E is 4
degrees (~343c) above C, and raising it by one degree would give us
E|), ~429c. So the 14-ET native-fifth notation would give you what
you want. However, since releasing our theoretical paper, we have
concluded that to make the notation consistent with a 28-ET native-
fifth notation we should instead use the symbol pair |\ and !/ for a
single degree of 14-ET. The single degree of 28-ET then gets the
symbol pair |) and !) , which was determined by going upward by two
best 4ths (of 12deg28) to Bb (~1029c) and then lowering that by
1deg28 to arrive at the best approximation of 7/4 in 28-ET (23deg),
which is then notated as Bb!) . Due to the fact that that 28-ET is
not 9-limit consistent, this does not result in E|) being the best
approximation to 9/7 in 28-ET.

> Also another thing I was wondering about
> - the chord progression player in FTS is the
> only part of it really that is restricted to
> octave based tunings, because I don't know
> of any standard way of notating non octave
> chords. But I wondered, thinking forwards
> to possiblee later developments, is there
> any work done on a non octave version of
> Sagittal?

We've looked at a few non-octave temperaments to see how these might
be notated.

> For instance I suppose one might do a
> 3/1 based Saggital with the 2/1 in place
> of 3/1 as the generating interval for the
> note names or something.
>
> How would you notate Bohlen Pierce in
> Sagittal? Or Wendy Carlos's
> Gamma?

A non-octave ET can be approximated by an edo if one goes high
enough. When Dave Keenan looked at Bohlen-Pierce, he concluded that
41-ET notation would suffice.

I'm not very familiar with Gamma, but I know that 718-ET (which we
can easily notate in Sagittal) comes very close. However, I would
imagine that there's a simpler (lower ET) notation that would do it
justice.

When you have so many Sagittal symbols to choose from, the problem is
not so much figuring out how to notate these tunings, but rather
finding the most logical way to notate them. The choice of symbols
often depends on how other (similar) tunings will be notated, so as
to achieve some sort of notational consistency among families of
temperaments. There will not be a final answer to these last two
questions of yours until we take the time to look at some of these
tunings more closely.

--George

🔗George D. Secor <gdsecor@yahoo.com>

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

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:
> For instance / and \ are easy to remember for the syntonic
> comma, and then f and t for the septimal comma I find
> harder to remember and get the u and n for the
> tridecimal comma a bit confused because
> I think of it as u for undecimal - there maybe
> the abbreviations may add to the confusion so
> that may be a possible reason for not using them at all
> possibly.

The primary consideration in choosing single-ASCII characters was
that they should _look_ as much as possible like the actual
graphical or true-type symbol. The graphical symbols are primary.
The assumption was that the reader was already familiar with the
graphical symbols or soon would be, and that they would eventually
just read the ASCII _as_ the corresponding graphical. i.e. they
would look at the ASCII, but see an image of the graphical symbol in
their "mind's eye". I guess we should put together a chart to
facilitate that.

But for now, here are a few rules to help remember at least the
direction up or down.

First try to see the ascii character as consiting of a single
vertical shaft with a flag or flags on it, chosen from straight,
convex, concave and wavy (although sometimes there is no part of the
character corresponding to the shaft). That often gives you the
direction up or down. For example the following are all up

?frnmpyqg

and these are all down

juvwthkdb

Another way of looking at these is if you were to cut the lowercase
letters out of cardboard (in a san serif font such as Arial or
Helvetica) and try to balance them about their midline, would they
fall toward their top side? i.e. are they top-heavy? If so, then
they are upward pointing, otherwise (whether they would fall to the
bottom or balance perfectly) they are downward. This is already the
case with "b" as the flat symbol.

If a pair consists of a lowercase letter and a special character.
The lowercase letter is always down and the special character is
always up. This is already the case with b for flat and # for sharp.
These pairs are

b#
s$
i*
z~
j?
v^
o@

In a pair consisting of two special characters, the vertical
placement of the character gives the direction. The high placed ones
are up while the low placed are down. As in these pairs

,`
.'
;"
_=

> The notation window will have a list of all the symbols
> so that it is easy to look them up from within the program.
> Also the user can show them on the picture of the p.c.keyboard
> and swap between the various possibilities.

Great. It would be good if you can display the actual true-type
symbols as well as the ASCII.

> I do like the Sagittal system of accidental signs
> and this wasn't meant as any kind of attempt to
> try and change or improve on it. I just found
> they weren't working with chord symbols, but your
> short symbols do indeed solve that problem.

Great.

> Here is David Canright's thirteen limit scale using them:
>
> http://www.robertinventor.com/sagittal.png
>
> 1 13/12 9/8 7/6 5/4 4/3 11/8 3/2 13/8 5/3 7/4 11/6 2/1

Very nice. But since this is a graphic, there's really no excuse for
not using the actual sagittal symbols rather than the ASCII
characters, is there?

Also, I expect George has pointed this out, but in the mixed
notation the flats and sharps are always closer to the letter-name
or notehead than are the sagittals, so we always go from large pitch
difference (the letter nominal or staff position) to medium pitch
difference (the sharp or flat or doubles thereof) to small pitch
difference (sagittal accidental). On the staff this sequence runs
right to left but in text with letter names it runs left to right.
So Ebt rather than Etb. And so it is naturally read as "E flat 7-
comma down" rather than "E 7-comma down flat".

Combining sagittals with diatonic interval numbers isn't something
George and I have considered before. But since it is traditional for
the flat or sharp to be on the right in that case, it makes sense to
me that the sagittal should be leftmost in this case. So for example
tb3 pronounced naturally as "7-comma down minor third" or
indeed "sub minor third".

> I didn't find these in your list, maybe they can be got
> by combining others in the list:
>
> 128/125 diesis
> 25/24 chromatic semitone

24:25 is )||( or \\!# or =#

125:128 is .//|
It has no shorthand but you could either use one of the user-
definable pairs or you could use the approximation of the double 5-
comma symbol //| whose shorthand is _ and =.

You could decide between 125:128 versus (80:81)^2 on the basis of
which gives the simplest ratio relative to the root. So when applied
to certain intervals it would be interpreted one way, and when
applied to others it would be interpreted the other way. Or you
could get fancy and look at relationships to notes other than the
root as well, but that sounds too hard.

> For the 1053/1024 or 35/36, the chord progression
> player has to play one or the other in its just intonation
> realisation of the chords, so I'm not sure yet what to
> do about that.

As above, I suggest looking at the resulting ratio (to the root)
when the nominals (diatonic interval numbers) and sharps and flats
are taken into account and play the simplest one (the one with the
lowest product of numerator and denominator). In this case it will
be chalk and cheese.

> Yes indeed, one idea is to add in '-' as an extra possible
> note addition symbol, and maybe just say that you
> can't use a / with this p notation - or else
> maybe in case user wants to use a /
> just say that the first occurrence of - changes
> the note addition sign.
>
> I've been using @ as note addition symbols
> for this sort of situation,
> so it would be
> C5p@\3@5@t7@9@^11@w13

"@" is 11-large-diesis up (704:729).

The colon ":" was not used as an ASCII sagittal accidental so that
it could be used for just this purpose - separating notes in chords.

Using colon elsewhere would allow this function of the hyphen to be
easily retained too.

Only the user-definable pairs
,`
-+
<>
[]
{}

Well I suppose you could use -+ instead of \/ for the 5-comma
symbols since there is a precedent for that.

If you also need something for the 125-S-diesis (125:128) and you
are already using [] for something else then I guess it would have
to be <>. This would leave only {} for the 25-S-diesis (double 5-
comma) if you don't want to use _=.

> Anyway I came to the conclusion that you need to use
> the septimal diesis 64/63 to notate the accidentals,
> and I saw in your pdf that you recommend it too, looking
> this et up. But then the normal way of thinking about
> 9?7 is as 81/64 raised by a septimal diesis. But here
> our E is already a 9/7. So how does one understand
> use of the septimal diesis as an accidental in
> 14-et.

One should understand 14-ET as inherently evil and steer well clear
of it. ;-)

Seriously. I think you goofed in figuring that the E is already a
7:9 from C? It is 4/14-oct which is 343 cents. A 7:9 is 435 cents
and corresponds to Ef as you would expect.

> But I wondered, thinking forwards
> to possiblee later developments, is there
> any work done on a non octave version of
> Sagittal?
>
> For instance I suppose one might do a
> 3/1 based Saggital with the 2/1 in place
> of 3/1 as the generating interval for the
> note names or something.

No need.

> How would you notate Bohlen Pierce in
> Sagittal? Or Wendy Carlos's
> Gamma?

Easy.

The BP generator is 3^(1/13) or about 146.3 cents. There are about
8.2 of these in an octave. We can get rid of that .2 by multiplying
by 5. 8.2 * 5 = 41. So we would notate BP in the same way as we
would notate every fifth step of 41-ET. The notational octave in
this case would be 3^(41/65) (~1199.7 cents) and the "notational
fifth" would be 3^(24/65) (~702.3 cents).

The gamma generator is 35.1 cents. There are about 34.188 of these
to a just octave. That .188 is close enough to 0.2 or 1/5. If we
multiply 34.2 by 5 we get 171. So Wendy Carlos' Gamma scale would be
notated the same as every fifth step of 171-ET. The notational
octave is exactly 34.2 gamma generators and the notational fifth is
21 gamma generators.

Notice that we are not approximating BP with 41-ET or Gamma with 171-
ET but we are notating these scales _exactly_ by correctly
specifying our notational octave and fifth.

-- Dave Keenan

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

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:
>
> > I do like the Sagittal system of accidental signs
> > and this wasn't meant as any kind of attempt to
> > try and change or improve on it. I just found
> > they weren't working with chord symbols, but your
> > short symbols do indeed solve that problem.
>
> Great.
> ...
> Combining sagittals with diatonic interval numbers isn't something
> George and I have considered before. But since it is traditional
for
> the flat or sharp to be on the right in that case, it makes sense
to
> me that the sagittal should be leftmost in this case.

I agree.

> So for example
> tb3 pronounced naturally as "7-comma down minor third" or
> indeed "sub minor third".
>
> > I didn't find these in your list, maybe they can be got
> > by combining others in the list:
> >
> > 128/125 diesis
> > ...
>
> 125:128 is .//|
> It has no shorthand but you could either use one of the user-
> definable pairs or you could use the approximation of the double 5-
> comma symbol //| whose shorthand is _ and =.

I beg to differ on that point. The ascii shorthand notation has .
and ' as the 5-schisma down and up characters, respectively. As I
already wrote off-list, in Sagittal character shorthand we do not
combine any Sagittal shorthand comma-characters to notate intervals,
except:

1) in combination with the sharp, flat, double-sharp, and double-flat
characters (# b x bb), or
2) in combination with the 5-schisma (32768:32805) characters that
represent the accent marks used in high-precision Sagittal JI: an
apostrophe ' for a 5-schisma up and period . for a 5-schisma down.

Therefore, using shorthand we can exactly symbolize such intervals as:

Longhand shorthand
down up down up comma
---- ---- ---- ---- ----------
'\! ./| '\ ./ diaschisma
.\! '/| .\ '/ pythagorean comma
'\\! .//| '_ .= meantone diesis 125:128
.!( '|( .c 'r 224:225
'(!/ .(|\ 'w .m 27:28
'!!) .||) b'f #.t 20:21

Using accent characters in combination with the other shorthand
characters will give you JI pitch increments of around 2 cents, and I
expect that you would have many more intervals possible than you will
have interval names for. But the use of the accent characters would
enable the FTS player to produce many more pitches with high
precision.

> ...
> > Anyway I came to the conclusion that you need to use
> > the septimal diesis 64/63 to notate the accidentals,
> > and I saw in your pdf that you recommend it too, looking
> > this et up. But then the normal way of thinking about
> > 9?7 is as 81/64 raised by a septimal diesis. But here
> > our E is already a 9/7. So how does one understand
> > use of the septimal diesis as an accidental in
> > 14-et.
>
> One should understand 14-ET as inherently evil and steer well clear
> of it. ;-)
>
> Seriously. I think you goofed in figuring that the E is already a
> 7:9 from C? It is 4/14-oct which is 343 cents. A 7:9 is 435 cents
> and corresponds to Ef as you would expect.

I believe that Robert might have been referring to the subset-of-56-
ET notation, while you're referring to the native 14-ET notation.
See my reply to this point (near the bottom):

/tuning-math/message/11500

Also, please observe that we've left the issue of multiples of 7-ET
as unfinished business and should at least agree that we should
update the 14-ET native notation by replacing |) with |\ in the
sag_et.par file.

--George