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. | 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. Distributionally even scales of arbitrary arity are, in fact, temperament-agnostic [[Fokker block]]s. | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Scale]] | [[Category:Scale]] | ||