Distributional evenness: Difference between revisions

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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 [[maximal evenness|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.

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 [[maximal evenness|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 <nowiki>[[arity]]</nowiki>: we simply extend the consideration of evenly distributing each step size to every step size.

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.

In practice, binary DE 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.

== 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 ''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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 {{Distinguish|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 [[maximal evenness|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.
+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 [[maximal evenness|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 <nowiki>[[arity]]</nowiki>: we simply extend the consideration of evenly distributing each step size to every step size.
-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.
+In practice, binary DE 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.
 == 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 ''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.)