Distributional evenness: Difference between revisions

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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.)

Using this definition, an ''r''-ary scale word in ''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 shows that distributionally even scales of arbitrary arity ''r'' are a subset ~~of temperament-agnostic [[Fokker block]]s, i.e.~~ [[product word]]s of ''r'' − 1 MOS scales.

Using this definition, an ''r''-ary scale word in ''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 shows that distributionally even scales of arbitrary arity ''r'' are a subset [[product word]]s of ''r'' − 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.

[[Category:Terms]]

[[Category:Scale]]

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 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.)
-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 ''r'' are a subset of temperament-agnostic [[Fokker block]]s, i.e. [[product word]]s of ''r'' &minus; 1 MOS scales.
+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 ''r'' are a subset [[product word]]s of ''r'' &minus; 1 MOS scales, which can be thought of as temperament-agnostic [[Fokker block]]s.
 [[Category:Terms]]
 [[Category:Scale]]