Distributional evenness

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Not to be confused with Maximal evenness.

A scale with two step sizes is distributionally even (DE) if it has its two step sizes distributed as evenly as possible (i.e. each step size is distributed in a maximally even pattern among the steps of the scale). This turns out to be equivalent to the property of having maximum variety 2; that is, each interval class ("seconds", "thirds", and so on) contains no more than two sizes. Though the term as originally defined is limited to scales with two step sizes, distributional evenness has an obvious generalization to scales of arbitrary arity: we simply require that each of the three or more step sizes be evenly distributed.

In practice, binary DE scales are often referred to as "MOS scales", but some consider this usage to be technically incorrect because a MOS as defined by Erv Wilson was to have exactly two specific intervals for each class other than multiples of the octave. When Wilson discovered MOS scales and found numerous examples, DE scales with period a fraction of an octave such as pajara, augmented, diminished, etc. were not among them.

Extended definition

Let r ≥ 2 and let [math]S: \mathbb{Z}\to\mathbb{R}[/math] be an r-ary periodic scale with length n (i.e. S(kn) = kP where P is the period), with step sizes x1, ..., xr, i.e. such that [math]\Delta S(i) := S(i+1)-S(i)\in \{x_1, ..., x_r\} \forall i \in \mathbb{Z}.[/math] The scale S is distributionally even if for every i ∈ {1, ..., r}, (ΔS)−1(xi) mod n is a maximally even subset of [math]\mathbb{Z}/n.[/math] (For the original definition of DE, simply set r = 2.)

Using this definition, a scale word on r letters x1, ..., xr is DE if and only if for every i ∈ {1, ..., r}, the binary scale obtained by equating all step sizes except xi is DE. This generalization of DE is thus an extraordinarily strong property: distributionally even scales over r letters are a subset of product words of r − 1 MOS scales, which can be thought of as temperament-agnostic Fokker blocks.

All DE scales in this extended sense are also billiard scales.[1]

Related topics


  1. Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.