Counting symmetric scales in equal-temperament

Jeffery Mensah • December 7, 2025∼1250 words

In this post, we give two puzzles related to a concept in music theory referred to as scale symmetry. Roughly speaking, a symmetric scale is a musical scale which does not give a strong impression of tonality. Before stating the two problems, we give a brief overview of the relevant theory and establish some terminology.

Equal-temperament

A tuning system is a method of defining which pitches to use in music and how to label them. The most common of these in use today is equal temperament, which divides the octave into $m$ (typically twelve) equally spaced notes. Theoretically, we may label the pitches in this system by integers in order of increasing pitch height.

An interval is a measure of the difference in pitch height between two pitches. In equal-temperament, these form a group isomorphic to $\mathbb{Z}$ which acts on the set of pitches by addition. The pitch class of a pitch $k \in \mathbb{Z}$ is the set of pitches which are some number of octaves apart; in other words, it is the residue class $[k]$ modulo $m$ .

Scales

A musical scale is an ordered collection of pitches spanning an octave, which defines the pitch classes in use for a piece or section of music.

The pitches in a scale may be obtained by starting at a initial note called the tonic and applying a sequence of ascending intervals that collectively span a single octave, until the tonic is reached again. For our purposes, the only intervals we consider are half steps ( $\mathrm{H}$ ) and whole steps ( $\mathrm{W}$ ), which increase the label of a pitch by $1$ and $2$ , respectively. We call the sequence used to construct the scale its type or pattern. For example, the following illustration depicts the pitch collection of a major scale with tonic $[0]$ , obtained from the interval sequence $\mathrm{WWHWWWH}$ .

When listening music using a certain scale, the brain subconsciously tries to determine the tonic, a process which forms the basis of solmization systems such as movable-do solfège. This leads to the following natural question:

When can one determine the tonic of a given scale type only by hearing the notes in its pitch collection?

For example, one can always aurally determine the tonic of a major scale from just its pitch collection, since no two major scales have identical pitch collections. However, for the whole-tone scale, there is ambiguity in the choice of tonic: music written in the $\mathrm{C}$ whole-tone scale could plausibly be heard as being in the $\mathrm{D}$ whole-tone scale, since both scales share the same notes.

Scales for which this perceptual problem does not have a unique solution are said to be tonally ambiguous. We now state the two problems and give their solutions.

Problem 1

Are there any tonally ambiguous seven-note scales in $12$ -tone equal temperament?

Solution to Problem 1

The answer is no.

Let $\mathbf{P} = \mathbb{Z}/12\mathbb{Z}$ be the set of all pitch classes, which is a homogeneous space for the cyclic group $\mathbb{Z}/12\mathbb{Z}$ . Note that a sequence of ascending intervals $j_1, \ldots, j_n \in \{1, 2\}$ defining a scale pattern may be identified with a subset $\Sigma \subset \mathbb{Z}/12\mathbb{Z}$ of size $n$ by considering the set of partial sums

\Sigma(j) = \left\{ \sum_{i=1}^{k} j_i \mid 1 \leq k \leq n \right\}.

Then given a pitch class $\pi \in \mathbf{P}$ , the set of pitch classes $\Sigma + \pi$ is a scale with type $j$ and tonic $\pi$ . If there exists a different tonic $\pi'$ producing the same pitch collection, then

\Sigma + \pi = \Sigma + \pi' \implies \Sigma + (\pi - \pi') = \Sigma \implies \pi - \pi' \in \mathrm{stab}(\Sigma).

Thus, the action of $\langle \pi - \pi' \rangle$ on $\mathbb{Z}/12\mathbb{Z}$ restricts to an action of $\langle \pi - \pi' \rangle$ on $\Sigma$ . The orbits of this action are simply the orbits of the former action, which are the cosets of $\langle \pi - \pi' \rangle$ . Since the orbits also partition $\Sigma$ , it follows that the order of $\langle \pi - \pi' \rangle$ divides $|\Sigma|$ . In particular, $\gcd(12, |\Sigma|) > 1$ , which implies that a tonally ambiguous scale cannot consist of a number of notes coprime to $12$ .

Problem 2

Provide a general formula for the number of tonally ambiguous scale types in $m$ -tone equal temperament.

Solution to Problem 2

Let $\mathcal{J}_{n, m}$ be the set of $n$ -note scale patterns in $m$ -tone equal temperament, that is,

\mathcal{J}_{n, m} = \left\{ (j_1, \ldots, j_n) \bigm| j_i \in \{1, 2\} \text{ and } \sum_{i=1}^{n} j_i = m \right\},

and let $\mathcal{J}_m = \bigcup_{n} \mathcal{J}_{n, m}$ . We may identify any sequence $j \in \mathcal{J}_{n, m}$ with a subset of $\mathbb{Z}/m\mathbb{Z}$ by considering its partial sums

\Sigma(j) = \left\{ \sum_{i=1}^{k} j_i \mid 1 \leq k \leq n \right\}.

As shown previously, a scale type $j \in \mathcal{J}_{m}$ is ambiguous if and only if $\Sigma(j)$ has a nontrivial stabilizer or, equivalently, the orbit of $\Sigma(j)$ has size less than $m$ . So, let $\mathcal{S}_{k, m}$ be the number of scale patterns $\Sigma \subseteq \mathbb{Z}/m\mathbb{Z}$ such that $|\mathrm{orb}(\Sigma)| = k$ . For an integer $k$ dividing $m$ , we show that there exists a bijection between scale patterns in $k$ -tone equal temperament and scale patterns in $m$ -tone equal temperament whose orbit has size dividing $k$ . To this end, consider the map

\Phi \colon \mathcal{J}_{k} \to \bigcup_{d \mid k} \mathcal{S}_{d, m}; \quad \quad j\, \overset{\Phi}{\longmapsto} \Sigma\big(\underbrace{\,j \cdots j\,}_{m/k\text{ times}}\big).

This map is well-defined: if $j \in \mathcal{J}_{k}$ , then repeating it $m/k$ times yields a sequence with a sum of $m$ , and by construction, $k + \Phi(j) = \Phi(j)$ , implying the orbit of $\Phi(j)$ has size dividing $k$ .

Since $\Phi$ is defined by repetition, it is injective. To show surjectivity, note that any pattern $\Sigma(j)$ whose orbit has size dividing $k$ must be invariant under translation by $k$ . By definition, $0$ is a partial sum of $j$ , which implies that $k$ must also be a partial sum. Thus, $j$ contains a prefix $j' \in \mathcal{J}_k$ such that $j = \Phi(j')$ , so $\Phi$ is also surjective. It follows from the Möbius inversion formula that

|\mathcal{S}_{k, m}| = \sum_{d \mid k} \mu(d) |\mathcal{J}_{k/d}|,

where $\mu \colon \mathbb{N} \to \mathbb{Z}$ is the Möbius function. The size of $\mathcal{J}_m$ is well-known: removing the last term of a sequence in $\mathcal{J}_m$ yields a sequence in either $\mathcal{J}_{m-1}$ or $\mathcal{J}_{m-2}$ . Since the initial values are $|\mathcal{J}_1| = |\mathcal{J}_2| = 1$ , this implies that $|\mathcal{J}_m|$ is the $m$ th Fibonacci number, $F_m$ . Therefore, the number of ambiguous scales in $m$ -tone equal temperament is

\#\text{ambiguous} = \left| \mathcal{J}_m - \mathcal{S}_{m, m}\right| = - \vphantom{\sum^{.}_{,}}\sum_{\substack{d\hspace{0.5pt}\mid m \\ d \neq 1}} \mu(d) F_{m/d}\,.

For example, in ordinary $12$ -tone equal temperament, there are $1 \cdot 8 + 1 \cdot 3 + 0 \cdot 2 - 1 \cdot 1 + 0 \cdot 1 = 10$ types of tonally ambiguous scales.