Distributional evenness

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.

Formal definition

Let r ≥ 2 and let S be an r-ary periodic scale with length n (i.e. S(kn) = kP where P is the period), with step sizes x₁, ..., x_r, i.e. such that ΔS(i) := S(i+1) − S(i) ∈ {x₁, ..., x_r} ∀i ∈ Z. The scale S is distributionally even if for every i ∈ {1, ..., r}, (ΔS)⁻¹(x_i) is a maximally even MOS in Z/nZ. (For the original definition of DE, simply set r = 2.)

Using this definition, an r-ary scale word in x₁, ..., x_r is DE if and only if for every i ∈ {1, ..., r}, the binary scale obtained by equating all step sizes except x_i is DE. This shows that distributionally even scales of arbitrary arity r are a subset of product words of r − 1 MOS scales, which can be thought of as temperament-agnostic Fokker blocks.