Filling the idea pool with Tenney reduced scales

This business of finding the Tenney-reduced epimorphic scale for various prime limits and numbers of steps looks like another useful project. Here is 11-limit, 41-et:

[1, 56/55, 28/27, 21/20, 15/14, 12/11, 10/9, 9/8, 8/7, 7/6, 25/21,
6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 15/11, 11/8, 7/5, 10/7, 16/11, 22/15, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 11/6, 15/8, 21/11, 27/14, 49/25]

By way of comparison, here is Genesis Minus:

[1, 81/80, 33/32, 21/20, 16/15, 12/11, 10/9, 9/8, 8/7, 7/6, 32/27,
6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 27/20, 11/8, 7/5, 10/7, 16/11, 40/27, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 11/6, 15/8, 40/21, 64/33, 160/81]

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> This business of finding the Tenney-reduced epimorphic scale for
various prime limits and numbers of steps looks like another useful
project. Here is 11-limit, 41-et:
>
> [1, 56/55, 28/27, 21/20, 15/14, 12/11, 10/9, 9/8, 8/7, 7/6, 25/21,
> 6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 15/11, 11/8, 7/5, 10/7,
16/11, 22/15, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7,
7/4, 16/9, 9/5, 11/6, 15/8, 21/11, 27/14, 49/25]
>
> By way of comparison, here is Genesis Minus:
>
> [1, 81/80, 33/32, 21/20, 16/15, 12/11, 10/9, 9/8, 8/7, 7/6, 32/27,
> 6/5, 11/9, 5/4, 14/11, 9/7, 21/16, 4/3, 27/20, 11/8, 7/5, 10/7,
16/11, 40/27, 3/2, 32/21, 14/9, 11/7, 8/5, 18/11, 5/3, 27/16, 12/7,
7/4, 16/9, 9/5, 11/6, 15/8, 40/21, 64/33, 160/81]

how about an evangelina example: 19-limit, 22-tone where 1216:1215
and 57:56 are unison vectors?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> how about an evangelina example: 19-limit, 22-tone where 1216:1215
> and 57:56 are unison vectors?

This is a bizarre request--1216/1215 comes out a comma if you map
19 to 94, and 57/56 if you map 19 to 93. To get them both to be commas, you need to screw the mapping of 7.

5-limit:

1, 25/24, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 32/25, 4/3, 25/18,
45/32, 36/25, 3/2, 25/16, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 48/25

7-limit:

1, 25/24, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 25/18, 7/5,
35/24, 3/2, 14/9, 8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 27/14

11-limit:

1, 22/21, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5,
16/11, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 21/11

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> 5-limit:
>
> 1, 25/24, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 32/25, 4/3, 25/18,
> 45/32, 36/25, 3/2, 25/16, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 48/25
>
> 7-limit:
>
> 1, 25/24, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 25/18, 7/5,
> 35/24, 3/2, 14/9, 8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 27/14
>
> 11-limit:
>
> 1, 22/21, 15/14, 10/9, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5,
> 16/11, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 15/8, 21/11

it might make more sense to reduce by odd-limit, not tenney height,
because these represent octave-repeating scales. 27/14 is not truly
simpler than 48/25 in this context.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> it might make more sense to reduce by odd-limit, not tenney height,
> because these represent octave-repeating scales. 27/14 is not truly
> simpler than 48/25 in this context.

I wanted something which would be better at breaking ties, but it occurs to me that tenney height itself could be the tie-breaker, so maybe I should do that instead. Any other comments about this project?

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

I wasn't too happy with the odd-limit results--here they are for
8 tones:

5-limit

1, 32/27, 9/8, 4/3, 81/64, 3/2, 16/9, 27/16

7-limit, h8

1, 7/6, 8/7, 4/3, 49/32, 3/2, 7/4, 12/7

7-limit, h8+v7

1, 8/7, 9/8, 4/3, 21/16, 3/2, 16/9, 7/4

None of these are epimorphic, and none use any 5s. I'll try the symmetric octave-class lattice distance next.