Maximal subgroups for N-edo belonging to kN-edo

If you look at 18 in the 17-limit, it gives good tunings on the 17-limit subgroup 2.9.75.21.55.39.51. It turns out this is not surprising, as it is the largest subgroup of the 17-limit on which 18 acts like 72. The proof of this is that the kernel, ie the commas of 72 in the 17-limit, are expressible in this subgroup; they define both 72 in the 17-limit and 18 on the subgroup.

A problem here is how to find such a maximal subgroup. Simply taking the group generated by the elements in the corresponding tonality diamond which belong to 18 turns out not to be strong enough: in this instance, we get 2.9.21.33/5.39.51, which is a proper subgroup of the maximal one. It is a large enough subgroup to represent everything in the diamond, but still not maximal; in fact, from it we get a rank two temperament instead of a rank one temperament like 72.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> A problem here is how to find such a maximal subgroup.

What seems to work in practice is this: you find the elements of the tonality diamond for kN-edo which also belong to N-edo, and to these you add a basis for the commas of kN-edo. The subgroup these generate is apparently probably what you want.