  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.
