Group actions, power mean orbit size, and musical scales

We provide an application of the theory of group actions to the study of musical scales. For any group G, finite G-set S, and real number t, we define the t-power diameter diam(t)(G, S) to be the size of any maximal orbit of S divided by the t-power mean orbit size of the elements of S. The symmetric group S-11 acts on the set of all tonic scales, where a tonic scale is a subset of Z(12) containing 0. We show that for all t is an element of [-1, 1], among all the subgroups G of S-11, the t-power diameter of the G-set of all heptatonic scales is the largest for the subgroup Gamma, and its conjugate subgroups, generated by {(1 2), (3 4), (5 6), (8 9), (10 11)}. The unique maximal Gamma-orbit consists of the 32 thats of Hindustani classical music popularized by Bhatkhande. This analysis provides a reason why these 32 scales, among all 462 heptatonic scales, are of mathematical interest. We also apply our analysis, to a lesser degree, to hexatonic and pentatonic scales.