Distributional evenness: Difference between revisions

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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'''.

[[Category:Terms]]

[[Category:Scale]]

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 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>-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'''.
 [[Category:Terms]]
 [[Category:Scale]]