Group actions, power mean orbit size, and musical scales

TL;DR

This paper applies group action theory to musical scales, introducing the $t$-power diameter to identify the most significant scale orbits, notably explaining the prominence of Hindustani classical music scales.

Contribution

It introduces the $t$-power diameter concept and demonstrates its use in analyzing the structure of musical scales under group actions, highlighting the special role of certain subgroups.

Findings

The $t$-power diameter is maximized for specific subgroups related to Hindustani scales.

32 Hindustani scales form the unique maximal orbit under the subgroup $ ext{Gamma}$.

The analysis extends to hexatonic and pentatonic scales, providing broader insights.

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.