论文标题
小组动作，力量平均轨道规模和音乐秤
Group actions, power mean orbit size, and musical scales
论文作者
论文摘要
我们将小组行动理论的应用在音乐量表的研究中。对于任何组$ g $，有限的$ g $ -set $ s $和实际数字$ t $，我们定义{\ it $ t $ - poppathimeter} $ \ perperatorname {diym} _t（g，s）$的大小为$ t $ s $的最大轨道的大小，由$ t $ t $ t $ - popper neppoper-popper nepoppoper Orbit Orbit Orbit orbit oerbit obit s $ s $ s $ s。对称组$ s_ {11} $在所有补品秤的集合上作用，其中{\ it Tonic scale}是$ \ mathbb {z} _ {12} $的子集，其中包含$ 0 $。我们表明，对于所有$ t \在[-1,1] $中的所有$ t \，在所有子组$ g $ $ s_ {11} $中，$ t $ - $ g $的$ g $ - $ g $的直径 - 所有伸型量表的$ g $ set均为亚组$γ$的最大，以及其偶联的子组，以及其偶性子级，由$ \ \ 2）（3）（3）（3） 6），（8 \ 9），（10 \ 11）\} $。独特的最大$γ$ -Orbit由Bhatkhande普及的印度斯坦古典音乐的32吨。该分析提供了一个原因，使这32个量表（在所有462个七含量量表中）具有数学利益。我们还将分析在较小程度上应用于己二式和五音阶量表。
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 $Γ$, and its conjugate subgroups, generated by $\{(1 \ 2),(3 \ 4),(5 \ 6),(8 \ 9),(10 \ 11)\}$. The unique maximal $Γ$-orbit consists of the 32 thāts 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.
