Yahoo Tuning Groups Ultimate Backup tuning A notation example

I wrote a program to impliment a notation which so far as I can tell at the moment corresponds to what both Manuel and George do, but I would be interested to see any comparisons.

We may start with the more abstract creature I call a "notation", in this case <9/8, 2187/2048, 81/80, 64/63, 33/32>^(-1) =
[h7, v5-h2, -v5, -v7, v11]. What all this means is that we may express everything in the 11-limit as a product of the above intervals, to powers which are determined by applying the various "vals" such as h7 or -v5 to the rational number q in question.
Here h7 is the normal 7-et mapping, and v5 is the mapping
("5-adic valuation" being the technical name of it) which sends 5 to 1 and everything else to 0. Hence a number q in the 11-limit can be expressed as

q = (9/8)^h7(q) (2187/2048)^(v5(q)-h2(q) (81/80)^(-v5(q))
(64/63)^(-v7(q)) (33/32)^v11(q)

The first two basis intervals, the major tone and the apotome, together give us the 3-limit, and can be used therefore to notate a
Pythagorean scale. The scale 1-9/8-81/64-4/3-3/2-27/16-243/128
in this manner can be expressed as

1 [0 0] C
9/8 [1 0] D
81/64 [2 0] E
4/3 [3 -1] F
3/2 [4 -1] G
27/16 [5 -1] A
243/128 [6 -1] B

By adding floor((2 h7(q)+1)/7) to the second number, we get a system where C-B goes [0 0], [1 0], [2 0], [3 0], [4 0], [5 0], [6 0]. We now may produce a Pythagorean system for the 3-limit; if we have
[U, V] we reduce U modulo 7, and write C-B as the reduction ranges from 0 (C) to 6 (B). We then take floor(U/7)+4, and we have the IPN standard octave number. Finally, V simply gives the number of sharp or flat symbols: positive V gives us so many sharps, and negative V so many flats.

We now add symbols for 81/80, 64/63 and 33/32, which can be George's, Manual's, or whatever we prefer. I modified Manual's system, since
the backslash symbol \ is a control character, and using 7 for a septimal comma conflicts with the IPN standard notation for octaves, which uses arabic numerals also. I have my notation program at the moment set to use # for sharp, b for flat, + for 81/80 up, - for
81/80 down, > for 64/63 up, < for 64/63 down, and Manual's ^ for 33/32 up and v for 33/32 down. The Genesis scale of Partch notated in this manner would be:

1 C4
81/80 C4+
33/32 C4^
21/20 D4b+<
16/15 D4b+
12/11 D4v
11/10 D4b+^
10/9 D4-
9/8 D4
8/7 D4>
7/6 E4b<
32/27 E4b
6/5 E4b+
11/9 E4b^
5/4 E4-
14/11 F4<v
9/7 E4>
21/16 F4<
4/3 F4
27/20 F4+
11/8 F4^
7/5 G4b+<
10/7 F4#->
16/11 G4v
40/27 G4-
3/2 G4
32/21 G4>
14/9 A4b<
11/7 G4>^
8/5 A4b+
18/11 A4v
5/3 A4-
27/16 A4
12/7 A4>
7/4 B4b<
16/9 B4b
9/5 B4b+
20/11 B4-v
11/6 B4b^
15/8 B4-
40/21 B4->
64/33 C5v
160/81 C5-

In order to go from something notated in this way to a rational number as ordinarily expressed, simply reverse the process, and obtain the exponents [a, b, c, d, e] which allow you to calculate

q = (9/8)^a (2187/2048)^b (81/80)^c (64/63)^d (33/32)^e

🔗genewardsmith <genewardsmith@juno.com>

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> We now add symbols for 81/80, 64/63 and 33/32, which can be George's, Manual's, or whatever we prefer. I modified Manual's system, since
> the backslash symbol \ is a control character, and using 7 for a septimal comma conflicts with the IPN standard notation for octaves, which uses arabic numerals also. I have my notation program at the moment set to use # for sharp, b for flat, + for 81/80 up, - for
> 81/80 down, > for 64/63 up, < for 64/63 down, and Manual's ^ for 33/32 up and v for 33/32 down.

It occurred to me that it might be a good idea to have this correspond to the Sims 72-et ascii version, and I find that I seem to have accidentally lurched into the correct corresponding system already. This, incidentally, gives a canonical lifting of the 72-et to just intonation--anything notated in the Sims/Monzo system or George's sagittal should be convertable to JI in the manner I described.

Various simplifications occur in different temperaments because of the commas of those temperaments. Any meantone or other system with
81/80~1 can leave off that symbol, and any system, such as the 15, 17
22, 25, 27, 37 or 49 ets, with 64/63~1 can leave that off. A system
with (64/63)/(81/80) = 5120/5107~1, such as the 29,34,41,46,53,58,
87,94,99,111,140 or 152 ets can use the same symbol for both commas.
A system where (64/63)^2/(33/32) = 131072/130977~1, such as the 34,
41,46,53,84,87,94,99,130,140,171,224, or 270 ets can use two 64/63 symbols in place of a 33/32, and a system where (33/32)/(81/80)^2 =
2200/2187~1, such as the 34,41,46,53,80,87,94,99,121 or 140 ets can do the same with 81/80 and 33/32. We might note that far from being hard to notate, the 41-et allows considerable simplifications.

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> It occurred to me that it might be a good idea to have this
>correspond to the Sims 72-et ascii version, and I find that I seem
>to have accidentally lurched into the correct corresponding system
>already. This, incidentally, gives a canonical lifting of the 72-et
>to just intonation--anything notated in the Sims/Monzo system or
>George's sagittal should be convertable to JI in the manner I
>described.

though you'd be introducing 'wolves' -- consonant intervals off by,
for example, 225:224 -- if you did this.

> Various simplifications occur in different temperaments because of
the commas of those temperaments. Any meantone or other system with
> 81/80~1 can leave off that symbol, and any system, such as the 15,
17
> 22, 25, 27, 37 or 49 ets, with 64/63~1 can leave that off. A system
> with (64/63)/(81/80) = 5120/5107~1, such as the 29,34,41,46,53,58,
> 87,94,99,111,140 or 152 ets can use the same symbol for both commas.
> A system where (64/63)^2/(33/32) = 131072/130977~1, such as the 34,
> 41,46,53,84,87,94,99,130,140,171,224, or 270 ets can use two 64/63
symbols in place of a 33/32, and a system where (33/32)/(81/80)^2 =
> 2200/2187~1, such as the 34,41,46,53,80,87,94,99,121 or 140 ets can
do the same with 81/80 and 33/32. We might note that far from being
hard to notate, the 41-et allows considerable simplifications.

I believe George is aware of most of these facts though a few will
probably be new to him. however, i must object that for ets like 25
or even 34 you must specify which mapping you are using -- you can't
just assume that your 'canonical' one based on approximating the
primes only is going to be univerally desired -- even in the 5-limit,
it's only the third-best option for some ets, such as 64-equal, as
far as i'm concerned.