Interleaving: Difference between revisions

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A scale is (''k''-)'''flought''' (/flɔːt/, rhymes with ''bought'') if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', where ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls between two notes of the other copy. The set of offsets that separate the strands from a fixed strand is a chord called the ''polyoffset'', which is determined up to inversion and [[equave]]-equivalence. A flought scale is thus a [[cross-set]] with a little additional structure. One can '''floughten''' a scale ''s'' with a certain polyoffset Δ (or: "~~a polyoffset~~ ''floughtens'' ''s''" or "''s'' is ''floughtenable'' with Δ") if ''s'' is the strand scale of a flought scale with polyoffset Δ. Such a scale is denoted Fl(''s''; Δ). The concept of flought scales is a generalization of [[dipentatonic scale]]s and (even-length) [[generator-offset]] scales.

A scale is (''k''-)'''flought''' (/flɔːt/, rhymes with ''bought'') if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', where ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls between two notes of the other copy. The set of offsets that separate the strands from a fixed strand is a chord called the ''polyoffset'', which is determined up to inversion and [[equave]]-equivalence. A flought scale is thus a [[cross-set]] with a little additional structure. One can '''floughten''' a scale ''s'' with a certain polyoffset Δ (or: "Δ ''floughtens'' ''s''" or "''s'' is ''floughtenable'' with Δ") if ''s'' is the strand scale of a flought scale with polyoffset Δ. Such a scale is denoted Fl(''s''; Δ). The concept of flought scales is a generalization of [[dipentatonic scale]]s and (even-length) [[generator-offset]] scales.

[[Blackdye]], [[Zil]][14], and [[bicycle]] are examples of flought scales, because they each have two interleaved strands, respectively Pyth[5], Zarlino, and 8:9:10:11:13:14. The terminology, however, is intended to cover any number of strands and any choice of strand scale.

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* Δ be a chord such that every interval of Δ falls within (0, ''P'').

Then ''s'' ~~can be floughtened with the polyoffset chord Δ~~ if and only if no nonunison (positive) interval in Δ falls within

Then the polyoffset chord Δ floughtens ''s'' if and only if no nonunison (positive) interval in Δ falls within

<math>\bigcup_{i=0}^{\mathrm{len}(s) - 1} [\min \mathcal{D}_k(s), \max \mathcal{D}_k(s)].</math>

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-A scale is (''k''-)'''flought''' (/flɔːt/, rhymes with ''bought'') if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', where ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls between two notes of the other copy. The set of offsets that separate the strands from a fixed strand is a chord called the ''polyoffset'', which is determined up to inversion and [[equave]]-equivalence. A flought scale is thus a [[cross-set]] with a little additional structure. One can '''floughten''' a scale ''s'' with a certain polyoffset Δ (or: "a polyoffset ''floughtens'' ''s''" or "''s'' is ''floughtenable'' with Δ") if ''s'' is the strand scale of a flought scale with polyoffset Δ. Such a scale is denoted Fl(''s''; Δ). The concept of flought scales is a generalization of [[dipentatonic scale]]s and (even-length) [[generator-offset]] scales.
+A scale is (''k''-)'''flought''' (/flɔːt/, rhymes with ''bought'') if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', where ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls between two notes of the other copy. The set of offsets that separate the strands from a fixed strand is a chord called the ''polyoffset'', which is determined up to inversion and [[equave]]-equivalence. A flought scale is thus a [[cross-set]] with a little additional structure. One can '''floughten''' a scale ''s'' with a certain polyoffset Δ (or: "Δ ''floughtens'' ''s''" or "''s'' is ''floughtenable'' with Δ") if ''s'' is the strand scale of a flought scale with polyoffset Δ. Such a scale is denoted Fl(''s''; Δ). The concept of flought scales is a generalization of [[dipentatonic scale]]s and (even-length) [[generator-offset]] scales.
 [[Blackdye]], [[Zil]][14], and [[bicycle]] are examples of flought scales, because they each have two interleaved strands, respectively Pyth[5], Zarlino, and 8:9:10:11:13:14. The terminology, however, is intended to cover any number of strands and any choice of strand scale.
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 * Δ be a chord such that every interval of Δ falls within (0, ''P'').
-Then ''s'' can be floughtened with the polyoffset chord Δ if and only if no nonunison (positive) interval in Δ falls within
+Then the polyoffset chord Δ floughtens ''s'' if and only if no nonunison (positive) interval in Δ falls within
 <math>\bigcup_{i=0}^{\mathrm{len}(s) - 1} [\min \mathcal{D}_k(s), \max \mathcal{D}_k(s)].</math>