Distributional evenness: Difference between revisions
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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) − ''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>−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.) | 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) − ''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>−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.) | ||
Using this definition, an ''r''-ary scale word in ''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 shows that distributionally even scales of arbitrary arity are in fact the same concept as | Using this definition, an ''r''-ary scale word in ''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 shows that distributionally even scales of arbitrary arity are in fact the same concept as JI-agnostic [[Fokker block]]s. | ||
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[[Category:Scale]] | [[Category:Scale]] | ||
Revision as of 00:32, 27 August 2023
- 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, ..., xr, i.e. such that ΔS(i) := S(i+1) − S(i) ∈ {x1, ..., xr} ∀i ∈ Z. The scale S is distributionally even if for every i ∈ {1, ..., r}, (ΔS)−1(xi) 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, ..., xr is DE if and only if for every i ∈ {1, ..., r}, the binary scale obtained by equating all step sizes except xi is DE. This shows that distributionally even scales of arbitrary arity are in fact the same concept as JI-agnostic Fokker blocks.