Leveraging a Synergy with Diaschismic 5-limit Tuning in 12-tET (and 22-tET)

Thought I would start fresh. Sorry for the hackneyed title.

This is my attempt to combine musical set theory with tuning theory via Steiner systems.

The one I am using is the M5 version of the Elkies Double Steiner System. (This forms the group M12)

http://www.math.harvard.edu/~elkies/m12.pdf

(I am also into the M13 psuedo group). Anyway, taking the M5 version just converts semitones to fifths and vice versa, and you still canvass all the pentachords twice.

Now a nice feature is that this has transpositional symmetry, so you get all twelve transpositions of each pentad class from transpositions of Steiner hexads. (It has to be double though, so each pentad has two hexad parents). The nice feature is that, taking a pentad, and the elements from each parent, you form a septad such that the pentad complement (which may be just the inverse, or a different pentad class) will fit into the the complements of the two Steiners from the first line. (This is pretty easy to verify). So it ties things together nicely).

Now I can join Steiners in pairs (in many different ways) and end up with 5 septads, using B5V1, NH, OX, R1F, -ZR1-ZF, and E-E which tidies upthe odd one, which is it's own complement, then you can take either inverses or complements of the first five pairs; either way works. So we only need to consider the first 6 pairs. Now, modelling after diaschismic the requirement is to have 3,3,3,3,5,5 in just fractions, withe the stipulation that moving rows and columns around gives this generating set. (3^4 5^2 shape makes the octave). The rule being that
you move rows and columns, not elements themselves HOWEVER any element moved twice is eliminated, reducing our septad to the main hexad.

I have all six pairs worked out, here is one of them, which is B5V1

0,2,3,4,5,7,9:

0-7-2; 0-4; 7-11; 11-6-1. Move 11-6-1 over 2 and also move 7-11 over 0, and of course 11 must be eliminated, (put this over both 2 and 0) creating

0-7-2-9-4; 8-0-4 as required, except that you really want to end up with 0-7-2-9-4-8-0, so move this once more from 8-0-4 to 4-8-0 and you are done. This works with all 6 types (and their inverses, with E-E going to itself). This cuts up the rug of the space.

I guess I'm a 5-limit C4 X C3 12-tET guy which makes me a poor contributor to the UnTwelve group on Facebook, but at least this diaschismic idea does extend to a double Steiner system in M22 (22-tET) using hexads and their trio subsets as I described in aforementioned posts.

PGH