Sagittal notation for Carlos Alpha

I think this might work as a good notation for Wendy Carlos' Alpha scale. First, the key is that Alpha has a very good 3/2 approximation. So, the scale can be considered as an equal division of 3/2, and you only need to notate 9 pitches. Everything else is a transposition by fifths, which is easy in a notation system based on a chain of fifths.

The next intervals to notate are the major and minor thirds, which can be notated in the usual way. I'll use D - A as a reference interval for the fifth. A minor third above D (or a major third below A is notated as F/| in Sagittal notation, and a major third above D (or a minor third below A) is notated as F||\ .

So there are 3 gaps between D and F/| to fill, and 3 gaps between F||\ and A. It will be convenient to fill these in a symmetrical way. 3 steps of Alpha approximates 8/7, and an 8/7 above D is E|) . The same interval below A (21/16 above D) is then notated as G!) . A 7-step interval is close to 11/8, which is notated as G/|\ above D, so 2 steps above D can be notated as 12/11 or E\!/ for symmetry.

Now the question is what to do about the 1 step above D or below A. Two possibilities are: notate 1 step as 21/20 (which results in 8 steps being notated as 10/7), or notate 1 step as 22/21 (or 63/44 for 8 steps). Although 21/20 uses an accented symbol .||) , there's an alternative )/|| (which actually stands for 5103/4864, if you go all the way to the 19-limit, but there's no danger of confusion in the 11-limit).

The pattern can be extended in both directions.

F F)/|| G\!/ G|) A!!/ A\! B!!!) B(!) C)\!! C
C C)/|| D\!/ D|) E!!/ E\! F!) F/|\ G)\!! G
G G)/|| A\!/ A|) B!!/ B\! C!) C/|\ D)\!! D
D D)/|| E\!/ E|) F/| F||\ G!) G/|\ A)\!! A
A A)/|| B\!/ B|) C/| C||\ D!) D/|\ E)\!! E
E E)/|| F(|) F|||) G/| G||\ A!) A/|\ B)\!! B

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> I think this might work as a good notation for Wendy Carlos' Alpha
> scale. First, the key is that Alpha has a very good 3/2
approximation.
> So, the scale can be considered as an equal division of 3/2, and
you
> only need to notate 9 pitches. Everything else is a transposition
by
> fifths, which is easy in a notation system based on a chain of
fifths.
>
> The next intervals to notate are the major and minor thirds, which
can
> be notated in the usual way. I'll use D - A as a reference interval
for
> the fifth. A minor third above D (or a major third below A is
notated as
> F/| in Sagittal notation, and a major third above D (or a minor
third
> below A) is notated as F||\ .
>
> So there are 3 gaps between D and F/| to fill, and 3 gaps between
F||\
> and A. It will be convenient to fill these in a symmetrical way. 3
steps
> of Alpha approximates 8/7, and an 8/7 above D is E|) . The same
interval
> below A (21/16 above D) is then notated as G!) . A 7-step interval
is
> close to 11/8, which is notated as G/|\ above D, so 2 steps above D
can
> be notated as 12/11 or E\!/ for symmetry.
>
> Now the question is what to do about the 1 step above D or below A.
Two
> possibilities are: notate 1 step as 21/20 (which results in 8 steps
> being notated as 10/7), or notate 1 step as 22/21 (or 63/44 for 8
> steps). Although 21/20 uses an accented symbol .||) , there's an
> alternative )/|| (which actually stands for 5103/4864, if you go
all the
> way to the 19-limit, but there's no danger of confusion in the 11-
limit).
>
> The pattern can be extended in both directions.
>
> F F)/|| G\!/ G|) A!!/ A\! B!!!) B(!) C)\!! C
> C C)/|| D\!/ D|) E!!/ E\! F!) F/|\ G)\!! G
> G G)/|| A\!/ A|) B!!/ B\! C!) C/|\ D)\!! D
> D D)/|| E\!/ E|) F/| F||\ G!) G/|\ A)\!! A
> A A)/|| B\!/ B|) C/| C||\ D!) D/|\ E)\!! E
> E E)/|| F(|) F|||) G/| G||\ A!) A/|\ B)\!! B

Hi Herman,

In notating a temperament, the first thing I'd look for is an EDO of
reasonable number that closely approximates it. In the case of the
Carlos temperaments:

Alpha: generator of 78c ~5deg77 (77.922c)
Beta: generator of 63.8c ~5deg94 (63.830c)
Gamma: generator of 35.1c ~5deg171 (35.088c)

Then look up the symbol sequences for those EDO's:

77: /| |) /|\ (|) ||) ||\ /||\
94: ~|( /| (|( /|\ (|) ~||( ||\ (||( /||\
171: |( ~|( /| |) |\ //| /|) /|\ (|\ )||( /|| ||) ||\
(||( /||) /||\

The 77-EDO symbol sequence agrees with your notation, except that ||)
is used instead of )/||; you observed that .||) (20:21) would be
technically correct, so if the 5-schisma vanishes in Carlos-alpha,
then it's okay to drop the left accent to arrive at a simpler
notation.

Note: The notation for 171 above is what appears in the XH18
Sagittal paper:
http://dkeenan.com/sagittal/Sagittal.pdf

However, in my personal notes I've marked 171 as subject to review,
since I'm now thinking that it should be notated entirely with
symbols that represent 7-limit intervals:
171b: |( ~|( /| |) ~|) //| /|) )/|\ (|\ )||( ~~|| ||)
||\ (||( /||) /||\
where
~|) is defined as 48:49, replacing |\ (defined as 54:55);
~|( is defined as 4096:4131, but would be interpreted as 125:126
and/or 243:245;
)/|\ is defined as 392:405, replacing /|\ (defined as 32:33).

--George

George D. Secor wrote:

> The 77-EDO symbol sequence agrees with your notation, except that ||) > is used instead of )/||; you observed that .||) (20:21) would be > technically correct, so if the 5-schisma vanishes in Carlos-alpha, > then it's okay to drop the left accent to arrive at a simpler > notation.

Technically it doesn't vanish, but it's 77 steps up (minus 5 octaves, but Alpha doesn't have octaves). With the mapping of 5/4 as 5 steps and 3/2 as 9 steps, the comma tempered out by Alpha is [13, 5, -9> (1990656/1953125). In any case, I guess it's safe to ignore schisma accents in any scale with steps this large (78 cents).