Distributional evenness: Difference between revisions

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In practice, such scales are often referred to as "[[MOS scale]]s", 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.

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 extend the consideration of evenly distributing each step size to every step size.

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 ''x''1, ..., ''x''''r'', i.e. such that Δ''S''(''i'') := ''S''(''i''+1) − ''S''(''i'') ∈ {''x''1, ..., ''x''''r''} ∀''i'' ∈ '''Z'''. The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''}, (Δ''S'')−1(''x''''i'') is a [[maximally even]] MOS in '''Z'''/''n'''''Z'''. (For the original definition of DE, simply set ''r'' = 2.)

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 In practice, such scales are often referred to as "[[MOS scale]]s", 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.
+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 extend the consideration of evenly distributing each step size to every step size.
 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 ''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>, i.e. such that Δ''S''(''i'') := ''S''(''i''+1) &minus; ''S''(''i'') ∈ {''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>} ∀''i'' ∈ '''Z'''. The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''},  (Δ''S'')<sup>&minus;1</sup>(''x''<sub>''i''</sub>) is a [[maximally even]] MOS in '''Z'''/''n'''''Z'''. (For the original definition of DE, simply set ''r'' = 2.)