Step variety
An n-ary scale is a scale with exactly n distinct step sizes; a scale's arity is the number of distinct step sizes it has. Unary, binary and ternary scales are scales with exactly 1, 2 and 3 step sizes, respectively.
Unary scales are equal tunings. The class of binary scales consists of all MOS scales and every alteration-by-permutation of a MOS scale, but do not include altered mosses such as the harmonic minor scale, 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).
History of the term
The terms binary and ternary are already used in some academic literature in reference to scales; see e.g. Bulgakova, Buzhinsky and Goncharov (2023), "On balanced and abelian properties of circular words over a ternary alphabet".
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 generically rank-n (the group generated by the n step sizes Xi > 0, i = 1, ..., n, has rank n, not lower, for almost all choices of Xi, in the same sense that almost all real numbers between 0 and 1 are irrational).