Distributional evenness: Difference between revisions

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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.

== ~~Generalization~~ ==

== 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, consider an ''r''-ary periodic scale ''S'' 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'''. For each ''i'' ∈ {1, ..., ''r''}, define ''T''''i'' = Δ''S''−1(''x''''i''), naturally viewed as a subset of '''Z'''/''n'''''Z'''. The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''}, ''T''''i'' is a rotation of the [[maximally even]] MOS of |''T''i| notes in '''Z'''/''n'''''Z'''.

Formally, consider an ''r''-ary periodic scale ''S'' 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'''. For each ''i'' ∈ {1, ..., ''r''}, define ''T''''i'' = Δ''S''−1(''x''''i''), naturally viewed as a subset of '''Z'''/''n'''''Z'''. The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''}, ''T''''i'' is a rotation of the [[maximally even]] MOS of |''T''i| notes 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.

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 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.
-== Generalization ==
+== 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, consider an ''r''-ary periodic scale ''S'' 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'''. For each ''i'' ∈ {1, ..., ''r''}, define ''T''<sub>''i''</sub> = Δ''S''<sup>&minus;1</sup>(''x''<sub>''i''</sub>), naturally viewed as a subset of '''Z'''/''n'''''Z'''. The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''}, ''T''<sub>''i''</sub> is a rotation of the [[maximally even]] MOS of |''T''<sub>i</sub>| notes in '''Z'''/''n'''''Z'''.
+Formally, consider an ''r''-ary periodic scale ''S'' 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'''. For each ''i'' ∈ {1, ..., ''r''}, define ''T''<sub>''i''</sub> = Δ''S''<sup>&minus;1</sup>(''x''<sub>''i''</sub>), naturally viewed as a subset of '''Z'''/''n'''''Z'''. The scale ''S'' is ''distributionally even'' if for every ''i'' ∈ {1, ..., ''r''}, ''T''<sub>''i''</sub> is a rotation of the [[maximally even]] MOS of |''T''<sub>i</sub>| notes 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.