Step variety

The step variety (or arity) of a scale is the number of distinct step sizes it has. Unary, binary, ternary, and quaternary scales are scales with exactly 1, 2, 3, and 4 step sizes, respectively. An n-ary scale is a scale with exactly n distinct step sizes.

Unary scales are equal tunings. The class of binary scales consists of all MOS scales and every alteration-by-permutation of an MOS scale, but do not include altered MOS scales such as the harmonic minor scale (abstract step pattern: MsMMsLs), which gain additional step sizes from the alteration. Ternary scales are much less well-understood than binary ones, but one well-studied type of ternary scales is the class of generator-offset scales. Most known facts about ternary scales on the wiki can be found on the page rank-3 scale (which is mostly about specifically ternary scales).

Etymology

The terms binary and ternary are already used in some academic literature in reference to words over an alphabet, in particular to circular words that represent abstract scales; see e.g. Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet". The use of the term arity borrows an existing technical term and generalizes from this use of binary, ternary, and n-ary to refer to the number of letters in an alphabet in combinatorics on words; combinatorics-on-words literature often instead uses "word on n letters" or "alphabet with n letters" in the arbitrary-n case.

The term step variety, coined by Frédéric Gagné, is in analogy with interval variety for the number of distinct interval sizes in each interval class.

Difference from scale rank

Certain abstract scale theorists in the xen community have taken to using the n-ary terminology in view of the subtlety of the notion of a scale's rank. Examples of this subtlety are:

Equal tunings contain MOS scales and ternary scales, but the group generated by the step sizes in these tunings of the scales must be rank 1.
Certain chroma-altered MOS scales, which are contained in the group generated by the period and the generator of the unaltered MOS are ternary. An example is harmonic minor in any non-edo diatonic tuning, a chroma-alteration of the diatonic MOS with step pattern msmmsLs.

The term n-ary disregards the rank of the group generated by the step sizes, although an n-ary scale is still, in a probabilistic sense, generically rank-n (the group generated by the n step sizes X_i > 0, i = 1, ..., n, has rank n, not lower, for almost all choices of X_i, in the same sense that almost all real numbers between 0 and 1 are irrational).

Mathematical facts

Counting scales of a given size on a given number of letters

For r ≥ 1, the number of possible patterns (up to rotation) for periodic scales of size n ≥ r on r ordered step sizes x₁ > x₂ > ... > x_r is

[math]\displaystyle{\dfrac{1}{n} \sum_{d\mid n} \phi(d) \sum_{j=1}^r (-1)^{r-j} {r \choose j} j^{n/d}} \\ =\displaystyle{\dfrac{r!}{n} \sum_{d\mid n} \phi(d) S(n/d, r)}[/math]

where [math]\phi[/math] is the Euler totient function and [math]S(n, r)[/math] is the Stirling number of the second kind which counts ways to partition an n-element set into r distinguished parts.

List of named ternary scales

The following is a list of (temperament-agnostic) names that have been given to ternary scales. We ignore the exact arrangement of scale words here.

7 notes

nicetone (3L 2M 2S)
omnidiatonic (2L 3M 2S)
smicot (3L 1M 3S)
dualdye (5L 1M 1S)

8 notes

porcusmine (4L 3M 1S)
pinedye (5L 2M 1S)

9 notes

diasem (5L 2M 2S)
armozan (2L 5M 2S)
semizan (2L 3M 4S)
balgram (2L 2M 5S)

10 notes

blackbuzz (5L 3M 2S)
blackdye (5L 2M 3S)
lemmagic (3L 3M 4S)
blackville (2L 5M 3S)
jarlem (2L 4M 4S)

11 notes

diamech (5L 2M 4S)
semion (5L 3M 3S)

12 notes

diachromedye (5L 2M 5S)

14 notes

whitedye (5L 2M 7S)
chromatic crossdye (5L 7M 2S)