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).