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.

In practice, such 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.

Definition and generalization

Distributional evenness has an obvious generalization to scales of arbitrary arity: we simply extend the consideration of evenly distributing each step size to every step size of an arbitrary scale.

Formally, 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. For each i ∈ {1, ..., r}, define T_i = (ΔS)⁻¹(x_i), naturally viewed as a subset of Z/nZ. The scale S is distributionally even if for every i ∈ {1, ..., r}, T_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.