Interleaving: Difference between revisions

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A scale is (''k''-)'''interleaved''' ~~(/flɔːt/, rhymes with ''bought'')~~ if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', and ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls strictly 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. An interleaved scale is thus a [[cross-set]] with a little additional structure. One can '''interleave''' or '''interleaved''' a scale ''s'' with a certain polyoffset Δ (or: "Δ ''interleaves'' ''s''" or "''s'' is ''interleavable'' with Δ") if ''s'' is the strand scale of an interleaved scale with polyoffset Δ. Such a scale is denoted Interleaved(''s''; Δ). The concept of interleaved scales is a generalization of [[bipentatonic scale]]s and (even-length) [[generator-offset]] scales.

A scale is (''k''-)'''interleaved''' if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', and ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls strictly 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. An interleaved scale is thus a [[cross-set]] with a little additional structure. One can '''interleave''' or '''interleaved''' a scale ''s'' with a certain polyoffset Δ (or: "Δ ''interleaves'' ''s''" or "''s'' is ''interleavable'' with Δ") if ''s'' is the strand scale of an interleaved scale with polyoffset Δ. Such a scale is denoted Interleaved(''s''; Δ). The concept of interleaved scales is a generalization of [[bipentatonic scale]]s and (even-length) [[generator-offset]] scales.

[[Blackdye]], [[Zil]][14], and [[bicycle]] are examples of interleaved 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.

This concept was formerly known as '''flought scales''' (/flɔːt/, rhymes with ''bought'').

== Some interleaved scales ==

Interleaved scales can easily be built from a harmonic series mode as the strand: for example, if ''n''::2''n'' is the strand, then (2''n'' + 1)/''2n'' always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples:

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-A scale is (''k''-)'''interleaved''' (/flɔːt/, rhymes with ''bought'') if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', and ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls strictly 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. An interleaved scale is thus a [[cross-set]] with a little additional structure. One can '''interleave''' or '''interleaved''' a scale ''s'' with a certain polyoffset Δ (or: "Δ ''interleaves'' ''s''" or "''s'' is ''interleavable'' with Δ") if ''s'' is the strand scale of an interleaved scale with polyoffset Δ. Such a scale is denoted Interleaved(''s''; Δ). The concept of interleaved scales is a generalization of [[bipentatonic scale]]s and (even-length) [[generator-offset]] scales.
+A scale is (''k''-)'''interleaved''' if it is made of ''k'' > 1 copies (called ''strands'') of an ''n''-note [[periodic scale]] ''s'', and ''any two copies'' of ''s'' are interleaved so that any note of the first copy falls strictly 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. An interleaved scale is thus a [[cross-set]] with a little additional structure. One can '''interleave''' or '''interleaved''' a scale ''s'' with a certain polyoffset Δ (or: "Δ ''interleaves'' ''s''" or "''s'' is ''interleavable'' with Δ") if ''s'' is the strand scale of an interleaved scale with polyoffset Δ. Such a scale is denoted Interleaved(''s''; Δ). The concept of interleaved scales is a generalization of [[bipentatonic scale]]s and (even-length) [[generator-offset]] scales.
 [[Blackdye]], [[Zil]][14], and [[bicycle]] are examples of interleaved 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.
+This concept was formerly known as '''flought scales'''  (/flɔːt/, rhymes with ''bought'').
 == Some interleaved scales ==
 Interleaved scales can easily be built from a harmonic series mode as the strand: for example, if ''n''::2''n'' is the strand, then (2''n'' + 1)/''2n'' always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples: