Some 12-tone meantone scales/temperaments

Here are up to isomorphism by mode and inversion all of the meantone scales of twelve tones which have a 7-limit edge-connectivity greater than two. While the usual meantone scale (with a connectivity of six) wins, it does not dominate, and the other scales/temperaments are worth considering. While the results are given in terms of the 31-et, they do not depend on the precise tuning, and are generic meantone results.

I am not aware if this sort of thing has ever been investigated, but it certainly seems worth pursuing.

[0, 2, 5, 8, 10, 13, 15, 18, 20, 23, 26, 28]
[2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3] 6

[0, 3, 6, 8, 11, 13, 16, 19, 21, 23, 26, 29]
[3, 3, 2, 3, 2, 3, 3, 2, 2, 3, 3, 2] 5

[0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 29]
[3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2] 5

[0, 3, 6, 8, 10, 13, 16, 19, 21, 23, 26, 29]
[3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2] 5

[0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 28]
[3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3] 4

[0, 3, 6, 8, 11, 13, 16, 18, 21, 24, 26, 28]
[3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3] 4

[0, 2, 5, 8, 10, 13, 15, 17, 20, 23, 25, 28]
[2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3] 4

[0, 3, 6, 8, 11, 13, 15, 18, 21, 23, 26, 28]
[3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3] 3

[0, 3, 6, 8, 11, 13, 16, 18, 20, 23, 26, 28]
[3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3] 3

[0, 3, 6, 9, 11, 13, 15, 18, 21, 24, 26, 28]
[3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 2, 3] 3

[0, 3, 6, 9, 11, 13, 15, 17, 19, 22, 25, 28]
[3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3] 3

[0, 2, 5, 8, 10, 12, 15, 18, 20, 22, 25, 28]
[2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3] 3

[0, 2, 5, 8, 11, 13, 15, 17, 20, 23, 26, 28]
[2, 3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3] 3

[0, 3, 5, 7, 10, 13, 15, 18, 20, 22, 25, 28]
[3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 3, 3] 3

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here are up to isomorphism by mode and inversion all of the
>meantone scales of twelve tones which have a 7-limit edge-
>connectivity greater than two. While the usual meantone scale (with
>a connectivity of six) wins, it does not dominate, and the other
>scales/temperaments are worth considering.

One of them ought to be the Keenan scale, which is the meantonized
Lumma/Fokker scale. Which one? I'm suprised no one said anything.

>While the results are given in terms of the 31-et, they do not
>depend on the precise tuning, and are generic meantone results.
>
> I am not aware if this sort of thing has ever been investigated,
>but it certainly seems worth pursuing.

Oh yes.
>
> [0, 2, 5, 8, 10, 13, 15, 18, 20, 23, 26, 28]
> [2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3] 6
>
> [0, 3, 6, 8, 11, 13, 16, 19, 21, 23, 26, 29]
> [3, 3, 2, 3, 2, 3, 3, 2, 2, 3, 3, 2] 5
>
> [0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 29]
> [3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2] 5
>
> [0, 3, 6, 8, 10, 13, 16, 19, 21, 23, 26, 29]
> [3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2] 5
>
> [0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 28]
> [3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3] 4
>
> [0, 3, 6, 8, 11, 13, 16, 18, 21, 24, 26, 28]
> [3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3] 4
>
> [0, 2, 5, 8, 10, 13, 15, 17, 20, 23, 25, 28]
> [2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3] 4
>
> [0, 3, 6, 8, 11, 13, 15, 18, 21, 23, 26, 28]
> [3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3] 3
>
> [0, 3, 6, 8, 11, 13, 16, 18, 20, 23, 26, 28]
> [3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3] 3
>
> [0, 3, 6, 9, 11, 13, 15, 18, 21, 24, 26, 28]
> [3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 2, 3] 3
>
> [0, 3, 6, 9, 11, 13, 15, 17, 19, 22, 25, 28]
> [3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3] 3
>
> [0, 2, 5, 8, 10, 12, 15, 18, 20, 22, 25, 28]
> [2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3] 3
>
> [0, 2, 5, 8, 11, 13, 15, 17, 20, 23, 26, 28]
> [2, 3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3] 3
>
> [0, 3, 5, 7, 10, 13, 15, 18, 20, 22, 25, 28]
> [3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 3, 3] 3