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~~. ~~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~~.

A scale is '''distributionally even''' if equating all step sizes except one will always result in a MOS. MOSses are the only distributionally even binary scales. The term was originally defined as a generalization of [[maximal evenness]] specifically for binary scales; this is the most convenient generalization.

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.

== Technical definition ==

== ~~Extended~~ definition ==

Let ''r'' ≥ 2 and let <math>S: \mathbb{Z}\to\mathbb{R}</math> 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 <math>\Delta S(i) := S(i+1)-S(i)\in \{x_1, ..., x_r\} \forall i \in \mathbb{Z}.</math> The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''}, (Δ''S'')−1(''x''''i'') mod ''n'' is a [[maximally even]] subset of <math>\mathbb{Z}/n.</math> (For the original definition of DE, simply set ''r'' = 2.)

Using this definition, a scale word on ''r'' letters ''x''1, ..., ''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 generalization of DE is thus an extraordinarily strong property: distributionally even scales over ''r'' letters are a subset of [[product word]]s of ''r'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.

Distributionally even scales over ''r'' letters are a subset of [[product word]]s of ''r'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s. All DE scales in this extended sense are also [[billiard scales]].<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref>

All DE scales in this extended sense are also [[billiard scales]].<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref>

== List of distributionally even circular words ==

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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. 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.
+A scale is '''distributionally even''' if equating all step sizes except one will always result in a MOS. MOSses are the only distributionally even binary scales. The term was originally defined as a generalization of [[maximal evenness]] specifically for binary scales; this is the most convenient generalization.
-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.
+== Technical definition ==
-== Extended definition ==
 Let ''r'' ≥ 2 and let <math>S: \mathbb{Z}\to\mathbb{R}</math> 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 <math>\Delta S(i) := S(i+1)-S(i)\in \{x_1, ..., x_r\} \forall i \in \mathbb{Z}.</math> The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''},  (Δ''S'')<sup>&minus;1</sup>(''x''<sub>''i''</sub>) mod ''n'' is a [[maximally even]] subset of <math>\mathbb{Z}/n.</math> (For the original definition of DE, simply set ''r'' = 2.)
-Using this definition, a scale word on ''r'' letters ''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub> is DE if and only if for every ''i'' ∈ {1, ..., ''r''}, the binary scale obtained by equating all step sizes except ''x''<sub>''i''</sub> is DE. This generalization of DE is thus an extraordinarily strong property: distributionally even scales over ''r'' letters are a subset of [[product word]]s of ''r'' &minus; 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.
+Distributionally even scales over ''r'' letters are a subset of [[product word]]s of ''r'' &minus; 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s. All DE scales in this extended sense are also [[billiard scales]].<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref>
-All DE scales in this extended sense are also [[billiard scales]].<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref>
 == List of distributionally even circular words ==