Group actions, power mean orbit size, and musical scales

Abstract

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 {\it $t$-power diameter} $\operatorname{diam}_t(G,S)$ 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 {\it tonic scale} is a subset of $\mathbb{Z}_{12}$ containing $0$. We show that, for all $t \in [-1,1]$, among all the subgroups $G$ of $S_{11}$, the $t$-power diameter of the $G$-set of all heptatonic scales is 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 th\=ats 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.