Superparticular 5-limit scales

The three smallest 5-limit superparticulars are 81/80, 25/24, and 16/15. Putting these into the form of a matrix and inverting gives us
[h3 h5 h7], and hence (81/80)^3 (25/24)^5 (16/15)^7 = 2. We can arrange these 15 scale steps in a number of ways given by the multinomial coefficient 15!/(3! 5! 7!) = 360360, which rotations and inversions would reduce further.

All of these scales are epimorphic, with defining val h15 = h3+h5+h7, so singling out the interesting ones means putting on additional contraints; convexity and connectedness suggest themselves, of course.

Anyone care to take a shot at it?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> The three smallest 5-limit superparticulars are 81/80, 25/24, and
16/15. Putting these into the form of a matrix and inverting gives us
> [h3 h5 h7], and hence (81/80)^3 (25/24)^5 (16/15)^7 = 2. We can
arrange these 15 scale steps in a number of ways given by the
multinomial coefficient 15!/(3! 5! 7!) = 360360, which rotations and
inversions would reduce further.
>
> All of these scales are epimorphic, with defining val h15 =
h3+h5+h7, so singling out the interesting ones means putting on
additional contraints; convexity and connectedness suggest
themselves, of course.
>
> Anyone care to take a shot at it?

Sure -- let's adopt convexity and connectedness (via consonances).