# maximal evenness

[from Paul Erlich, private communication]

"Maximal evenness" assumes that the universe of possible intervals and pitches forms a finite set, and that all musical materials will be drawn from, and evaluated relative to, this set. (As you might expect, I'm philosophically opposed to this view to a strong extent.)

Examples of maximally even scales in a universe of 12 pcs include 2-1-2-1-2-1-2-1 and 2-2-1-2-2-2-1, because they are, in a particular sense, the best approximations of 8-equal and 7-equal available in 12-equal.

The diatonic scale in 31-equal is not maximally even, which in my opinion, shows how un-useful the maximal evenness concept is :) The maximally even 7-note scale in 31-equal is 4-4-5-4-5-4-5, not the diatonic 5-5-3-5-5-5-3.

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[Manuel Op de Coul, Scala tips.par file]

A scale is maximally even if there are no more than two different interval sizes for each interval class (this can be seen in Scala with SHOW INTERVALS), and if the difference between each size of interval per interval class is less than one scale step. Originally, this one scale step was intended to be the step of some equal tempered scale from which a subset was taken. (Both versions of this property are shown by SHOW DATA.)

If the scale is a subset of an equal temperament with less than 1200 tones and maximally even, it's said that it's ME for that particular equal temperament. For the more general version, it's said that it's ME for L / S <= 2, the ratio of the size of larger step (step being interval of class 1) to the size of the smallest step must be less than or equal to two. (See also [Scala command] FIT/MODE.)

A weaker form is distributional evenness, which see.

For maximal evenness, the definition here is in one respect different than with SHOW DATA: the difference in step sizes may not be greater than 1. This is the original definition of John Clough. For distributional evenness, Clough's definition is not restricted to integer step size proportions, so there is no difference of definition there.

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