Group actions, power mean orbit size, and musical scales

This paper provides an application of group actions to the study of musical scales and uses it to motivate a new invariant, which we call the $t$-power diameter ${\operatorname{diam}}_{t}(G,S)$, defined, for any group $G$, finite $G$-set $S$, and extended real number $t\in{\mathbb{R}}\cup\{\infty,-\infty\}$, to be the size of any maximal orbit of $S$ divided by the $t$-power mean orbit size of the elements of $S$.

We may represent the chromatic scale as the set

$${\mathbb{Z}}_{12}=\{0,1,2,3,4,5,6,7,8,9,10,11\},$$

where $0$ represents the tonic, or key, of the chromatic scale, which can be any fixed pitch class. Thus, for example, if one decides to let $0$ represent the pitch class C, then $1$ represents C$\sharp$, $2$ represents D, and so on. A scale (in ${\mathbb{Z}}_{12}$) is a subset of ${\mathbb{Z}}_{12}$, while a tonic scale (in ${\mathbb{Z}}_{12}$) is a scale in ${\mathbb{Z}}_{12}$ containing $0$. A tonic scale is $k$-tonic if it consists of $k$ notes, where $k\in\{1,2,\ldots,12\}$; thus, any tonic scale $s$ is $|s|$-tonic, where $|s|$ is the cardinality of $s$. The $k$-tonic scales, respectively, for $k=1,2,\ldots,12$ are called monotonic, ditonic, tritonic, tetratonic, pentatonic, hexatonic, heptatonic, octatonic, nonatonic, decatonic, hendecatonic, and chromatic. There are a total of $2^{11}=2048$ possible tonic scales, with a total of ${11\choose k-1}$ $k$-tonic scales for each $k=1,2,3,\ldots,12$. These numbers comprise the eleventh row of Pascal’s triangle:

$$1\ \ \ 11\ \ \ 55\ \ \ 165\ \ \ 330\ \ \ 462\ \ \ 462\ \ \ 330\ \ \ 165\ \ \ 55\ \ \ 11\ \ \ 1.$$

Thus, for example, there are 462 heptatonic scales and 330 pentatonic scales. We call a scale that may not contain the tonic an atonic scale (in ${\mathbb{Z}}_{12}$). There are a total of $2^{12}=4096$ possible atonic scales (including the unique empty scale), with a total of ${12\choose k}$ $k$-atonic scales for each $k=0,1,2,3,\ldots,12$. These numbers comprise the twelfth row of Pascal’s triangle:

$$1\ \ \ 12\ \ \ 66\ \ \ 220\ \ \ 495\ \ \ 792\ \ \ 924\ \ \ 792\ \ \ 495\ \ \ 220\ \ \ 66\ \ \ 12\ \ \ 1.$$

Throughout this paper, ${\mathscr{T}}$ denotes the set of all tonic scales in ${\mathbb{Z}}_{12}$ and ${\mathscr{T}}_{k}$ the set of all $k$-tonic scales in ${\mathbb{Z}}_{12}$. The symmetric group $S_{11}$ acts naturally on the sets ${\mathscr{T}}$ and ${\mathscr{T}}_{k}$: a permutation $\sigma$ in $S_{11}$ acts on a tonic scale $s\in{\mathscr{T}}$ by $\sigma\cdot s=\sigma(s)=\{\sigma(x):x\in s\}$, where one sets $\sigma(0)=0$. Explicitly, an element $\sigma$ of $S_{11}$ maps a scale $\{0,s_{1},s_{2},\ldots,s_{k-1}\}$ to the scale $\{0,\sigma(s_{1}),\sigma(s_{2}),\ldots,\sigma(s_{k-1})\}$. This defines an action of $S_{11}$ on ${\mathscr{T}}$, and since $|\sigma(s)|=|s|$ for all $s$, the action induces an action on ${\mathscr{T}}_{k}$ for all $k\in\{1,2,\ldots,12\}$. More generally, the group $S_{12}$ acts on the set $\SS$ of all (atonic) scales in ${\mathbb{Z}}_{12}$. Moreover, if we consider $S_{11}=\{\sigma\in S_{12}:\sigma(0)=0\}$ as a subgroup of $S_{12}$, then the action of $S_{12}$ on $\SS$ descends to the action of $S_{11}$ on ${\mathscr{T}}$. Although in this paper we focus mainly on the action of $S_{11}$ on ${\mathscr{T}}$, many of our results ascend appropriately to the action of $S_{12}$ on $\SS$.

The group $S_{11}$ has $11!=39,916,800$ elements (and the group $S_{12}$ has $12!=479,001,600$ elements). A “musical” scale acted on by a randomly chosen element of $S_{11}$ is very unlikely to be very musical. For example, under the permutation $(1\ 4)(3\ 5)(8\ 7\ 9\ 10)$, the heptatonic major scale $\{0,2,4,5,7,9,11\}$ maps to the somewhat “unmusical” scale $\{0,1,2,3,9,10,11\}$. However, by contrast, some permutations preserve musicality fairly well, e.g., the permutation $(3\ 4)(8\ 9)$, which swaps the major scale and the harmonic minor scale $\{0,2,3,5,7,8,11\}$. One of the main questions we investigate in this paper is the following: are there medium-sized subgroups of $S_{11}$ whose actions on ${{\mathscr{T}}}_{7}$ preserve “musicality”? An ideal subgroup of $S_{11}$ would be one that “respects musicality” in the sense that scales of approximately the same “musicality” appear in the same orbit. As we will see, some subgroups of $S_{11}$ induce actions on the heptatonic scales that preserve musicality better than others do. One of our main claims is that the group $\Gamma$ of $S_{11}$ generated by $\{(1\ 2),(3\ 4),(5\ 6),(8\ 9),(10\ 11)\}$ is the “best” such subgroup, and the 32 scales in its unique maximal orbit, which coincide with the 32 thāts of Hindustani (North Indian) classical music, represent under a particular measure the “most musical” scales among the 462 possible heptatonic scales.

To measure the musical efficacy of a subgroup $G$ of $S_{11}$, we define the $G$-musicality of a scale $s\in{\mathscr{T}}_{k}$ to be the size $|Gs|$ of the $G$-orbit $Gs$ of $s$ in ${\mathscr{T}}_{k}$ divided by the average size of a $G$-orbit of ${\mathscr{T}}_{k}$. A consequence of this definition is that the $G$-musicality is a small as possible, namely $1$, for all scales if and only if every orbit has the same number of elements, which holds if $G=S_{11}$ or if $G$ is the trivial group. In general, the $G$-musicality of a given scale will attain a maximum for some subgroups of $S_{11}$ in between those two extremes.

The intution behind the concept of $G$-musicality is that, if a scale has a small $G$-orbit relative to the average $G$-orbit size, then it has too much symmetry relative to $G$ and thus the notes comprising the scale are more “$G$-equivalent” to each other and therefore have fewer notes that have their own individual character, whereas scales with larger orbits relative to the average orbit size are comprised of notes that can be better differentiated, or distiguished from one another, by the group $G$. The thesis of this paper is that the subgroups $G$ of $S_{11}$ that yield the largest possible $G$-musicality (relative to the other subgroups of $S_{11}$) of any $k$-tonic scale naturally lead to mathematically and musically interesting theories of $k$-tonic scales, namely, those $k$-tonic scales with the largest $G$-musicality for any subgroup $G$ of $S_{11}$.

There is a slight subtlety here, however, since, given a group $G$ and a finite $G$-set $S$, the average size of a $G$-orbit of $S$ can be measured in at least two distinct ways. One might naively define the average size of a $G$-orbit of $S$ to be

$$\frac{|S|}{|S/G|}=\frac{\sum_{i=1}^{r}|O_{i}|}{r},$$

where $S/G=\{O_{1},O_{2},\ldots,O_{r}\}$ is the set of all $G$-orbits of $S$ and where $r=|S/G|$ is the number of orbits. This represents the average number of elements in each orbit, in a naive sense. However, one may also define the average orbit size of the elements of $S$ to be

$${\operatorname{orb}}_{1}(G,S)=\frac{\sum_{s\in S}|Gs|}{|S|}=\frac{\sum_{i=1}^{r}|O_{i}|^{2}}{\sum_{i=1}^{r}|O_{i}|}.$$

This represents the expected value of $|Gs|$ for $s\in S$, where each element of $S$ is equally likely to be chosen. By contrast, the number $\frac{|S|}{|S/G|}$ previously considered represents the expected value of $|O_{i}|$, where each orbit $O_{i}$ is equally likely to be chosen. Since our focus is on the elements of $S$ rather than on the orbits, ${\operatorname{orb}}_{1}(G,S)$ is a better notion of average orbit size than is the naive definition $\frac{|S|}{|S/G|}$.

Nevertheless, both of these measures of “average orbit size” have mathematical merit, and this is supported by the observation that, using power means, one may continuously deform one of these two means to the other, as follows. For any $t\in{\mathbb{R}}-\{0\}$, we define the $t$-power mean orbit size of the elements of $S$ to be

$${\operatorname{orb}}_{t}(G,S)=\left(\frac{\sum_{s\in S}|Gs|^{t}}{|S|}\right)^{1/t}.$$

This represents the $t$-power mean of $|Gs|$ over all $s\in S$. Clearly, ${\operatorname{orb}}_{t}(G,S)$ for $t=1$ is the average orbit size of the elements of $S$. Moreover, for $t=-1$, we have

$${\operatorname{orb}}_{-1}(G,S)=\left(\frac{\sum_{s\in S}|Gs|^{-1}}{|S|}\right)^{-1}=\frac{|S|}{\sum_{O\in S/G}|O||O|^{-1}}=\frac{|S|}{|S/G|},$$

and therefore ${\operatorname{orb}}_{-1}(G,S)$ is the average number of elements of $S$ in each orbit. Taking appropriate limits at $t=0,\pm\infty$, one can define ${\operatorname{orb}}_{t}(G,S)$ for all $t\in[-\infty,\infty]$, and then

$${\operatorname{orb}}_{\infty}(G,S)=\max\{|Gs|:s\in S\}$$

is the maximal orbit size of $S$, and

$${\operatorname{orb}}_{-\infty}(G,S)=\min\{|Gs|:s\in S\}$$

is the minimal orbit size. Clearly, then, every $G$-orbit of $S$ has the same size if and only if the function ${\operatorname{orb}}_{t}(G,S)$ is constant with respect to $t$. One can use calculus to show that, if the function ${\operatorname{orb}}_{t}(G,S)$ is not constant, then it is bounded and has positive derivative everywhere. Moreover, from $|S|$ and the function ${\operatorname{orb}}_{t}(G,S)$ for $t\in[0,1]$, one can recover all of the orbit sizes. Our general philosophy is that the critical region of interest of the function ${\operatorname{orb}}_{t}(G,S)$ is the interval $t\in[-1,1]$, with the value at $t=1$ being the most important.

For any finite $G$-set $S$, we define the $t$-power diameter ${\operatorname{diam}}_{t}(G,S)$ of $S$ to be

$${\operatorname{diam}}_{t}(G,S)=\frac{{\operatorname{orb}}_{\infty}(G,S)}{{\operatorname{orb}}_{t}(G,S)}=\frac{\max\{|Gs|:s\in S\}}{{\operatorname{orb}}_{t}(G,S)}.$$

In other words, ${\operatorname{diam}}_{t}(G,S)$ is the ratio of the maximal $G$-orbit size of $S$ to the $t$-power mean orbit size of the elements of $S$. The function ${\operatorname{diam}}_{t}(G,S)$, if not identically $1$, has negative derivative with respect to $t$ and has limiting values of $1$ and $\frac{\max\{|Gs|:s\in S\}}{\min\{|Gs|:s\in S\}}$ at $t=\infty$ and $t=-\infty$, respectively. The $t$-power diameter ${\operatorname{diam}}_{t}(G,S)$ represents in an intuitive sense the amount of spread in the “kinetic energy” or “entropy” of the elements of $S$ under the action of $G$, where elements with larger orbits, or equivalently with smaller stabilizers, are considered to have more kinetic energy. The mathematical problem we pose here is the following.

Such subgroups $H$ of $G$ maximize the spread of the “kinetic energy” of the elements of $S$ under the induced goup action.

The following is our main result regarding heptatonic scales in ${\mathbb{Z}}_{12}$.

The theorem can be proved numerically using GAP and SAGE, as follows. First, note that, for any group $G$ and any finite $G$-set $S$, the number ${\operatorname{diam}}_{t}(H,S)$ for any subgroup $H$ of $G$ depends only on the conjugacy class of $H$. The group $S_{11}$ has 3094 subgroups up to conjugacy. (Those that are cyclic have order $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $11$, $12$, $14$, $15$, $18$, $20$, $21$, $24$, $28$, or $30$.) Using GAP and SAGE, one can compute generators for representatives of all 3094 conjugacy classes, and for each of these representatives $G$ one can compute the $G$-orbits of ${\mathscr{T}}_{7}$. Then the functions ${\operatorname{diam}}_{t}(G,{\mathscr{T}}_{7})$ can be plotted and verified to achieve a maximum for $G=\Gamma$ for all $t$ in the interval $[-1,1]$.

The fact that ${\operatorname{diam}}_{t}(G,{\mathscr{T}}_{7})$ attains a maximum on the entire interval $[-1,1]$ for a single conjugacy class of subgroups of $S_{11}$ is in itself a surprising result. One has

$${\operatorname{diam}}_{1}(\Gamma,{\mathscr{T}}_{7})\approx 3.5250,$$

$${\operatorname{diam}}_{0}(\Gamma,{\mathscr{T}}_{7})\approx 4.8324,$$

$${\operatorname{diam}}_{-1}(\Gamma,{\mathscr{T}}_{7})\approx 6.6494.$$

Thus, for example, a consequence of Theorem 1.2 is that, for any subgroup $G$ of $S_{11}$, the maximal $G$-orbit size of ${\mathscr{T}}_{7}$ is at most $3.5251$ times the average $G$-orbit size of the elements of ${\mathscr{T}}_{7}$, and the maximum ratio possible is obtained precisely by the group $G=\Gamma$ and its conjugates.

Given a subgroup $G$ of $S_{11}$ and a scale $s\in{\mathscr{T}}$, we define the $(G,t)$-musicality of $s$ to be the quantity

$$m(G,t,s)=\frac{|Gs|}{{\operatorname{orb}}_{t}(G,{\mathscr{T}}_{|s|})},$$

which is the size of the $G$-orbit of $s$ relative to the $t$-power mean orbit size of the elements of ${\mathscr{T}}_{|s|}$. Thus, the quantity

$${\operatorname{diam}}_{t}(G,{\mathscr{T}}_{k})=\max\{m(G,t,s):s\in{\mathscr{T}}_{k}\}$$

represents the largest possible $(G,t)$-musicality of any $k$-tonic scale. Theorem 1.2 says that the heptatonic scales with the largest possible $(G,t)$-musicality for any subgroup $G$ of $S_{11}$ and any $t\in[-1,1]$ occur precisely for the group $G=\Gamma$ and its conjugates. These scales comprise the unique maximal $\Gamma$-orbit of ${\mathscr{T}}_{7}$ and consist of the 32 heptatonic scales

$$\left\{0,{1\atop 2},{3\atop 4},{5\atop 6},7,{8\atop 9},{10\atop 11}\right\},$$

or equivalently, starting, say, at C, the 32 scales

$$\left\{\mbox{C},{\mbox{D}\flat\atop\mbox{D}},{\mbox{E}\flat\atop\mbox{E}},{\mbox{F}\atop\mbox{F}\sharp},\mbox{G},{\mbox{A}\flat\atop\mbox{A}},{\mbox{B}\flat\atop\mbox{B}}\right\}$$

listed in Table 1. This orbit contains the major scale, the harmonic and melodic minor scales, and many other heptatonic scales that figure prominently in Western and Indian classical music. In fact, all 32 of these scales are among the 72 mēḷakarta ragas of Carnatic (South Indian) classical music standardized by Govindacharya in the 18th century and coincide with the 32 thāts of Hindustani classical music popularlized by the system created by Vishnu Narayan Bhatkhande (1860–1936), one of the most influential musicologists in the field of Hindustani classical music in the twentieth century.

The remainder of this paper is organized as follows. In Section 2 we discuss power means and expected values, and in Section 3 we apply Section 2 to the study of the power mean orbit size and diameter of a finite $G$-set. In Section 4 we apply Section 3 to the study of scales in ${\mathbb{Z}}_{12}$. In Section 5 we focus on heptatonic scales in particular, while in Section 6 we briefly study hextonic scales, and in Section 7 we study pentatonic scales.

I would like to thank the two reviewers for their thoughtful and invaluable input on the first draft of this paper. As one of the reviewers pointed out, it is likely that the methods of this paper can be combined synergistically with other ways of understanding musical scales, such as those developed in [1], [2], [3], [4], [5], [6], [7], [8], and [9]. It is not my intention that the results in this paper be definitive. My aim is merely to provide yet another perspective on the already well-developed mathematical theories of musical scales, one that, in my view, has also inspired some new and interesting problems regarding the theory of group actions. Based on the reviewers comments, I also discuss some ways in which the theory might be amended or developed further.

The research for this paper was conducted with undergraduate students James Allen and Paul Estrada and MS student Michael McCann at California State University, Channel Islands, in the academic year 2016–17, under the supervision of the author. The idea for the project began with conversations between the author and undergraduate student Vickie Chen during a semester-long project on group theory in music for a first course in abstract algebra. It is in those conversations that Vickie and I first came up with the idea of examining the maximal $\Gamma$-orbit of ${\mathscr{T}}_{7}$, an idea that, to our pleasant surprise, eventually led to Theorem 1.2.

Arithmetic means are generalized by what are known as power means. If $S=\{s_{1},\ldots,s_{r}\}$ is a finite set of cardinality $r=|S|$ and $X:S\longrightarrow{\mathbb{R}}_{>0}$ a positive real-valued random variable on $S$ (with the uniform distribution on $S$), then, for any nonzero $t\in{\mathbb{R}}$, the $t$-power mean of $X$ is defined to be the positive real number

$${\mathbb{M}}_{t}(X)={\mathbb{M}}_{t}(X(s):s\in S)={\mathbb{M}}_{t}(x_{1},\ldots,x_{r})=\left(\frac{\sum_{i=1}^{r}x_{i}^{t}}{r}\right)^{1/t},$$

where $x_{i}=X(s_{i})$ for all $i$. Equivalently, the arithmetic mean of $X$ is just ${\mathbb{M}}_{1}(X)$, and one sets

$${\mathbb{M}}_{t}(X)={\mathbb{M}}_{1}(X^{t})^{1/t}.$$

For $a=0,\pm\infty$, one defines

$${\mathbb{M}}_{a}(X)={\mathbb{M}}_{a}(X(s):s\in S)={\mathbb{M}}_{a}(x_{1},\ldots,x_{r})=\lim_{t\rightarrow a}{\mathbb{M}}_{t}(x_{1},\ldots,x_{r}).$$

It is well known that

$${\mathbb{M}}_{0}(x_{1},\ldots,x_{r})=\left(\prod_{i=1}^{r}x_{i}\right)^{1/r}$$

is the geometric mean of the $x_{i}$, while

$${\mathbb{M}}_{\infty}(x_{1},\ldots,x_{r})=\max(x_{1},\ldots,x_{r})$$

and

$${\mathbb{M}}_{-\infty}(x_{1},\ldots,x_{r})=\min(x_{1},\ldots,x_{r})$$

are the maximum and minimum, respectively, of the $x_{i}$.

Let $[-\infty,\infty]={\mathbb{R}}\cup\{\infty,-\infty\}$ denote the set of all extended real numbers. For all $t\in[-\infty,\infty]$ one has

$${\mathbb{M}}_{t}(cX)=c{\mathbb{M}}_{t}(X)$$

for all $c>0$ and

$${\mathbb{M}}_{t}(X^{-1})={\mathbb{M}}_{-t}(X)^{-1}.$$

It follows that, if $XY=c$, that is, if $Y=c/X$, then

$${\mathbb{M}}_{t}(X){\mathbb{M}}_{-t}(Y)=c$$

for all $t$.

We may generalize the definition of ${\mathbb{M}}_{t}(X)$ by assuming that $S$ is a finite probability space with probability distribution ${\mathbb{P}}:S\longrightarrow[0,1]$, which we assume is nonzero at all elements of $S$, and $X:S\longrightarrow{\mathbb{R}}_{>0}$ is a positive real-valued random variable on $S$. (Previously we implicitly assumed that ${\mathbb{P}}(s)=\frac{1}{|S|}$ for all $s\in S$.) We may define the $t$-power expected value of $X$ to be

$${\mathbb{E}}_{t}(X)={\mathbb{E}}_{t}(X(s):s\in S)=\left(\sum_{s\in S}{\mathbb{P}}(s)X(s)^{t}\right)^{1/t}.$$

Equivalently, the expected value of $X$ is just ${\mathbb{E}}_{1}(X)$, and one sets

$${\mathbb{E}}_{t}(X)={\mathbb{E}}_{1}(X^{t})^{1/t}.$$

For $a=0,\pm\infty$, one defines

$${\mathbb{E}}_{a}(X)=\lim_{t\rightarrow a}{\mathbb{E}}_{t}(X).$$

One has

$${\mathbb{E}}_{0}(X)=\prod_{s\in S}X(s)^{{\mathbb{P}}(s)},$$

while

$${\mathbb{E}}_{\infty}(X)=\max X(S)$$

and

$${\mathbb{E}}_{-\infty}(X)=\min X(S).$$

If $X$ is constant, then clearly ${\mathbb{E}}_{t}(X)$ is a constant function of $t$ and one has ${\mathbb{E}}_{t}(X)=X(s)$ for all $t$ and all $s\in S$. Conversely, if ${\mathbb{E}}_{t}(X)$ is a constant function of $t$, then $\max X(S)=\min X(S)$, whence $X$ must be constant.

One can show that the function ${\mathbb{E}}_{t}(X)$ of $t$ is differentiable with nonnegative derivative, and is therefore nondecreasing, with respect to $t$. Thus, one has

$$\min X(S)\leq{\mathbb{E}}_{t}(X)\leq\max X(S)$$

for all $t$. Moreover, if $X$ is nonconstant, then ${\mathbb{E}}_{t}(X)$ has positive derivative, and therefore is strictly increasing, with respect to $t$. In other words, if nonconstant, the function ${\mathbb{E}}_{t}(X)$ of $t$ is a sigmoid function, that is, it is a bounded differentiable function from ${\mathbb{R}}$ to ${\mathbb{R}}$ whose derivative is everywhere positive. Thus it has horizontal asymptotes, specifically at $y=\max X(S)$ and $y=\min X(S)$ at $\infty$ and $-\infty$, respectively. Thus, its graph is an “S-shaped” curve. For an explicit example using the uniform probability distribution, see Figure 1.

Throughout this section, $G$ denotes a group and $S$ a finite $G$-set. One may readily generalize all of what follows to the situation where $S$ is also assumed to be a probability space; in that case, one simply replaces all $t$-power means with $t$-power expected values.

For any $t\in[-\infty,\infty]$, we define the $t$-power mean orbit size of the elements of $S$ to be

$${\operatorname{orb}}_{t}(G,S)={\mathbb{M}}_{t}(|Gs|:s\in S),$$

which, for $t\neq 0,\pm\infty$ is given by

$${\operatorname{orb}}_{t}(G,S)=\left(\frac{\sum_{s\in S}|Gs|^{t}}{|S|}\right)^{1/t}=\left(\frac{\sum_{O\in S/G}|O|^{t+1}}{|S|}\right)^{1/t}.$$

As observed in the introduction, ${\operatorname{orb}}_{1}(G,S)$ is the average orbit size of the elements of $S$, and ${\operatorname{orb}}_{-1}(G,S)=\frac{|S|}{|S/G|}$ is the average number of elements of $S$ in each orbit, while ${\operatorname{orb}}_{\infty}(G,s)=\max\{|Gs|:s\in S\}$ is the maximal orbit size of $S$ and ${\operatorname{orb}}_{-\infty}(G,s)=\min\{|Gs|:s\in S\}$ is the minimal orbit size.

For any $s\in S$, we define the $t$-power relative size of $s$ to be

$$|s|_{G,S,t}=\frac{|Gs|}{{\operatorname{orb}}_{t}(G,S)}.$$

The $t$-power relative size of $s$ is the size of the orbit of $s$ relative to (or normalized with respect to) the $t$-power mean orbit size of the elements of $S$. The $t$-power mean of the orbit sizes of the elements of $S$ is equal to

$${\mathbb{M}}_{t}(|Gs|:s\in S)={\operatorname{orb}}_{t}(G,S)$$

of $S$, while the $t$-power mean of the $t$-power relative sizes of the elements of $S$ is equal to $1$:

$${\mathbb{M}}_{t}(|s|_{G,S,t}:s\in S)=1.$$

Because of this normalization property, if $H$ and $K$ are subgroups of $G$, then it makes sense to compare the values of $|s|_{H,S,t}$ and $|s|_{K,S,t}$ with each other.

The $t$-power diameter ${\operatorname{diam}}_{t}(G,S)$ of $S$, as defined in the introduction, is equivalently the maximal $t$-power relative size of an element of $S$, that is, one has

$${\operatorname{diam}}_{t}(G,S)=\max\{|s|_{G,S,t}:s\in S\}=\frac{\max\{|Gs|:s\in S\}}{{\operatorname{orb}}_{t}(G,S)}=\frac{{\operatorname{orb}}_{\infty}(G,S)}{{\operatorname{orb}}_{t}(G,S)}.$$

The function ${\operatorname{diam}}_{t}(G,S)$, if not identically $1$, has negative derivative with respect to $t$ and has limiting values of $1$ and $\frac{\max\{|Gs|:s\in S\}}{\min\{|Gs|:s\in S\}}$ at $t=\infty$ and $t=-\infty$, respectively.

For example, Figure 2 provides the graph of ${\operatorname{orb}}_{t}(G,S)$ and ${\operatorname{diam}}_{t}(G,S)$ for any $G$-set $S$ with orbit sizes $2$, $2$, $7$, $8$, and $10$.

Throughout this section, $G$ denotes a subgroup of $S_{11}$, which acts on the set ${\mathscr{T}}$ of all $2^{11}$ possible tonic scales, as well as on the subsets ${\mathscr{T}}_{k}$ of all ${11\choose k-1}$ possible $k$-tonic scales, for all $k=1,2,\ldots,12$, as described in the introduction, and $t$ denotes a number or variable with values in the extended reals $[-\infty,\infty]$.

For any $s\in{\mathscr{T}}$, that is, for any tonic scale $s$, we define the $(G,t)$-musicality of $s$, or of the orbit $Gs$, to be

$$m(G,t,s)=|s|_{G,{\mathscr{T}}_{|s|},t}=\frac{|Gs|}{{\operatorname{orb}}_{t}(G,{\mathscr{T}}_{|s|})},$$

which is the $t$-power relative size of $s$ in ${\mathscr{T}}_{|s|}$ (not in ${\mathscr{T}}$). Musicality defines a natural function

$$m:\operatorname{Subgp}(S_{11})\times[-\infty,\infty]\times{\mathscr{T}}\longrightarrow[1,\infty),$$

where $\operatorname{Subgp}(S_{11})$ denotes the lattice of all subgroups of $S_{11}$. For a fixed $G$, $t$, and $k$, the $t$-power mean ${\mathbb{M}}_{t}(m(G,t,s):s\in{\mathscr{T}}_{k})$ of $m(G,t,s)$ over all $k$-tonic scales $s$ is equal to $1$. The $(G,t)$-musicality $m(G,t,s)$ of a $k$-tonic scale $s$ is directly proportional to the size of its $G$-orbit. The constant of proportionality depends on $t$ and is defined natually in such a way that one can meaningfully compare the values for various subgroups $G$ of $S_{11}$ for a fixed $s$, or compare the values for various scales $s$ for a fixed group $G$.

Observe that

$${\operatorname{diam}}_{t}(G,{\mathscr{T}}_{k})=\frac{\max\{|Gs|:s\in S\}}{{\operatorname{orb}}_{t}(G,{\mathscr{T}}_{k})}=\max\{m(G,t,s):s\in{\mathscr{T}}_{k}\}.$$

Thus ${\operatorname{diam}}_{t}(G,{\mathscr{T}}_{k})$ is the largest possible $(G,t)$-musicality of a scale in ${\mathscr{T}}_{k}$, or equivalently it is the $(G,t)$-musicality of any maximal $G$-orbit of ${\mathscr{T}}_{k}$. If we let ${\mathscr{T}}_{k}//G$ denote the set of all maximal $G$-orbits of ${\mathscr{T}}_{k}$, then the union ${\mathscr{T}}_{k,G}=\bigcup({\mathscr{T}}_{k}//G)$ is the set of all scales in ${\mathscr{T}}_{k}$ that have the largest possible $(G,t)$-musicality (namely, ${\operatorname{diam}}_{t}(G,{\mathscr{T}}_{k})$) for any $t$. We call the scales in the set ${\mathscr{T}}_{k,G}$ the $k$-tonic scales of $G$. Our philosophy is that the scales in the set ${\mathscr{T}}_{k,G}$ should be regarded as the optimally musical $k$-tonic scales relative to $G$.

Let $G$ be a subgroup of $S_{n}$. We define a signature of $G$ (in $S_{n}$) to be a list $(n_{1},n_{2},\ldots,n_{k})$ of the $G$-orbit sizes of $\{\{1\},\{2\},\ldots,\{n\}\}$, listed in any particular order. In particular, one has $n=n_{1}+n_{2}+\cdots+n_{k}$ for any signature $(n_{1},n_{2},\ldots,n_{k})$ of $G$ in $S_{n}$. We say that $G$ acts without crossings (in $S_{n}$), if the $G$-orbits of the $G$-set $\{\{1\},\{2\},\ldots,\{n\}\}$ are all of the form $\{\{a\},\{a+1\},\ldots,\{a+k\}\}$, where the addition here is ordinary addition of positive integers, not addition modulo $n$. (In the case at hand, $n=11$, and $0$ is omitted from the discussion because we have chosen to leave $0$ fixed by $S_{11}$.) It is clear that every subgroup of $S_{n}$ is conjugate to a subgroup that acts without crossings. In fact, the conjugates of $G$ that act without crossings in $S_{n}$ are in one-to-one correspondence with the signatures of $G$ in $S_{n}$. If $G$ acts without crossings in $S_{n}$, we define the signature of $G$ (in $S_{n}$) to be the list $(n_{1},n_{2},\ldots,n_{k})$ of the orbit sizes of $\{\{1\},\{2\},\ldots,\{n\}\}$, listed in order so that the orbits are $\{\{1\},\{2\},\ldots,\{n_{1}\}\}$, $\{\{n_{1}+1\},\{n_{1}+2\},\ldots,\{n_{1}+n_{2}\}\}$, etc.

Let $n_{1},n_{2},\ldots,n_{d}$ be a sequence of positive integers whose sum is $11$. Then the group $S_{n_{1}}\times S_{n_{2}}\times\cdots\times S_{n_{d}}$ naturally embeds into $S_{11}$ in the following way. The first factor $S_{n_{1}}$ acts on the first $n_{1}$ numbers, $1,2,\ldots,n_{1}$. The next factor $S_{n_{2}}$ acts on the next $n_{2}$ numbers, $n_{1}+1,n_{1}+2,\ldots,n_{1}+n_{2}$. And so onward to the last factor $S_{n_{d}}$, which acts on the last $n_{d}$ numbers, $n_{1}+\cdots+n_{d-1}+1$ to $11$. We denote the image of the embedding

$$\Phi:S_{n_{1}}\times S_{n_{2}}\times\cdots\times S_{n_{d}}\longrightarrow S_{11}$$

described above by $S_{n_{1},n_{2},\ldots,n_{d}}$. We say that a twelve tone group of signature at most $(n_{1},n_{2},\ldots n_{d})$ is a subgroup of $S_{n_{1},n_{2},\ldots,n_{d}}$ of the form $\Phi(G_{1}\times G_{2}\times\cdots\times G_{d})$, where $G_{i}$ is a subgroup of $S_{n_{i}}$ for all $i$. Note that the group $S_{n_{1},n_{2},\ldots,n_{d}}\cong S_{n_{2}}\times\cdots\times S_{n_{d}}$ is the largest twelve tone group of signature at most $(n_{1},n_{2},\ldots n_{d})$ in the sense that it contains every twelve tone group of signature at most $(n_{1},n_{2},\ldots n_{d})$. For this reason we call it the maximal twelve tone group of signature $(n_{1},n_{2},\ldots n_{d})$. For example, the group $S_{11}$ the maximal twelve tone group of signature $(11)$, and therefore every subgroup of $S_{11}$ is a twelve tone group of signature $(11)$. Note that all twelve tone groups act without crossings, and the signature of the maximal twelve tone group of signature $(n_{1},n_{2},\ldots n_{d})$ is $(n_{1},n_{2},\ldots n_{d})$.

Of course, there are other natural embeddings of $S_{n_{1}}\times S_{n_{2}}\times\cdots\times S_{n_{d}}$ in $S_{11}$. For example, $S_{4}\times S_{7}$ can be embedded in $S_{11}$ by allowing the first factor to act, say, on $\{2,4,7,8\}$ and the second factor on $\{1,3,5,6,9,10,11\}$. Such an embedding does not yield a twelve tone group with signature $(4,7)$. In loose terminology, twelve tone groups do not allow the factors to act in a ways that are “intertwined”: no “crossing” is allowed. Philosophically, this restriction can be motivated as follows. The chromatic scale has a linear ordering, and our goal is to understand how various scales may be transformed from one to the other. The most obvious and most common way in which this is done is by changing various notes of the scale by applying operations $\natural,\sharp,\flat$, which can be modeled by permuting neighboring notes. The notion of a twelve tone group is meant to capture this notion of locality.

Even without these locality restrictions, our mathematical analysis of twelve tone groups will apply equally well to any of the embeddings of $S_{n_{1}}\times S_{n_{2}}\times\cdots\times S_{n_{d}}$ in $S_{11}$ that allow crossings, because we can simply relabel the numbers $0$ through $11$ so that there are no crossings. We say that a maximal twelve tone group of signature $(n_{1},n_{2},\ldots,n_{d})$ with or without crossings is a subgroup of $S_{11}$ that is the result of applying some (inner) automorphism of $S_{11}$ to the maximal twelve tone group $S_{n_{1},n_{2},\ldots,n_{d}}$. We note the following proposition, whose proof is elementary.

In this section we are primarily interested in the action of subgroups of $S_{11}$ on the set ${\mathscr{T}}_{7}$ of all heptatonic (tonic) scales, which has 462 elements.

The following proposition gives a formula for ${\operatorname{diam}}_{t}(G,{\mathscr{T}}_{7})$ for any maximal twelve tone group $G$. Multisets are a generalization of sets where, as with tuples, repetition is allowed, but, as with sets, order doesn’t matter.