12 notes with 36/35 a chromatic comma

As I pointed out, for septimal systems we get meantone, pajara, injera
and tripletone in this way. Below I give the name, the wedgie, a TM
reduced scale, major and minor tetrads going around this scale. In the
case of meantone I add the cicle of fifths version.

Meantone

[1, 4, 10, 4, 13, 12]
[1, 3/2, 9/8, 5/3, 5/4, 15/8, 7/5, 21/20, 14/9, 7/6, 7/4, 21/16]
[1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8]

Tetrads around circle of fifths

0 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
2 [1, 5/4, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
3 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
4 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
6 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
7 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
8 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
9 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
10 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
11 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]

Tetrads around scale degrees

0 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
2 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
3 [1, 5/4, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
4 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
6 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
7 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
8 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
9 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
10 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
11 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]

Pajara

[2, -4, -4, -11, -12, 2]
[1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 12/7, 7/4, 15/8]

Tetrads

0 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
2 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
3 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
4 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
5 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
6 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
7 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
8 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
9 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
10 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
11 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]

Injera

[2, 8, 8, 8, 7, -4]
[1, 15/14, 9/8, 7/6, 9/7, 4/3, 7/5, 3/2, 14/9, 5/3, 9/5, 27/14]

Tetrads

0 [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
1 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
2 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
3 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
4 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
5 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
6 [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
7 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
8 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
9 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
10 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
11 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]

Tripletone

[3, 0, -6, -7, -18, -14]
[1, 16/15, 10/9, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]

Tetrads

0 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
1 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
2 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
3 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
4 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
6 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
7 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
8 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
9 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
10 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
11 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]

keep it up, gene! i think one of the more important chromatic unison
vectors kalle and carl were interested in was 49/48. the reason i
don't refer to these chromatic unison vectors as "commas" is because
my terminology is based on generalizing common-practice western
musical theory (generalizing an old subject is usually the best way
to terminologize a new one). common-practice western musical theory
is based on meantone-7, and there the chromatic unison vector is
25:24 -- known classically as the "chroma"; while the commatic unison
vector is 81:80 -- known classically as the "comma".

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> As I pointed out, for septimal systems we get meantone, pajara,
injera
> and tripletone in this way. Below I give the name, the wedgie, a TM
> reduced scale, major and minor tetrads going around this scale. In
the
> case of meantone I add the cicle of fifths version.
>
> Meantone
>
> [1, 4, 10, 4, 13, 12]
> [1, 3/2, 9/8, 5/3, 5/4, 15/8, 7/5, 21/20, 14/9, 7/6, 7/4, 21/16]
> [1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8]
>
> Tetrads around circle of fifths
>
> 0 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 2 [1, 5/4, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
> 3 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 4 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 6 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 7 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 8 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 9 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 10 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 11 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
>
> Tetrads around scale degrees
>
> 0 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 2 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 3 [1, 5/4, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
> 4 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 6 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> 7 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 8 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 9 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 10 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 11 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
>
>
> Pajara
>
> [2, -4, -4, -11, -12, 2]
> [1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 12/7, 7/4, 15/8]
>
> Tetrads
>
> 0 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> 1 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 2 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 3 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
> 4 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 5 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 6 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> 7 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 8 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 9 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
> 10 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 11 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
>
>
> Injera
>
> [2, 8, 8, 8, 7, -4]
> [1, 15/14, 9/8, 7/6, 9/7, 4/3, 7/5, 3/2, 14/9, 5/3, 9/5, 27/14]
>
> Tetrads
>
> 0 [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
> 1 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 2 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 3 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> 4 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 5 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 6 [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 5/3]
> 7 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 8 [1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 9 [1, 9/7, 32/21, 9/5] [1, 6/5, 32/21, 12/7]
> 10 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
> 11 [1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
>
>
> Tripletone
>
> [3, 0, -6, -7, -18, -14]
> [1, 16/15, 10/9, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]
>
> Tetrads
>
> 0 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
> 1 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 2 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 3 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
> 4 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
> 5 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 6 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 7 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
> 8 [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
> 9 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
> 10 [1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
> 11 [1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> keep it up, gene! i think one of the more important chromatic unison
> vectors kalle and carl were interested in was 49/48.

I think 15/14, 21/20 and 49/48 all need to be investigated, but I'm
still sorting things out, as I've just found our old friend torsion
makes the situation for schismic[24] more complicated.

Here is Blackjack, where we see the wolf fifths continue to be the
flys in the ointment in these chroma-36/35 systems:

[6, -7, -2, -25, -20, 15]

[1, 49/48, 15/14, 35/32, 8/7, 7/6, 49/40, 5/4, 21/16, 4/3, 7/5, 10/7,
3/2, 32/21, 8/5, 49/30, 12/7, 7/4, 64/35, 15/8, 49/25]

[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > keep it up, gene! i think one of the more important chromatic
unison
> > vectors kalle and carl were interested in was 49/48.
>
> I think 15/14, 21/20 and 49/48 all need to be investigated, but I'm
> still sorting things out, as I've just found our old friend torsion
> makes the situation for schismic[24] more complicated.

*now* do you understand why i said that NMOS scales have something to
do with torsion?? :)

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

> > I think 15/14, 21/20 and 49/48 all need to be investigated, but I'm
> > still sorting things out, as I've just found our old friend torsion
> > makes the situation for schismic[24] more complicated.
>
> *now* do you understand why i said that NMOS scales have something to
> do with torsion?? :)

It's just revelations left and right. It hadn't bit me before, but it
does have a bite, so you are right.

Not only is schismic[24] a bit of a pain, I found that for
diaschismic[24] I needed to go around the circle of generators, since
the scale didn't come out with the right order. Here it is; the list
of tetrads goes around a circle of 12 16/15 generators; of course
there are 12 more tetrads just like these.

Diaschismic

[2, -4, -16, -11, -31, -26]

[1, 64/63, 21/20, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 32/25, 4/3,
27/20, 45/32, 10/7, 40/27, 3/2, 25/16, 8/5, 5/3, 27/16, 16/9, 9/5,
15/8, 40/21]

[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 9/5] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 35/24, 7/4] [1, 7/6, 35/24, 5/3]

>Below I give the name, the wedgie, a TM reduced scale, major and
>minor tetrads going around this scale.

Bingo!

-C.