Interleaving: Difference between revisions

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A scale is a(n) (''k''-)'''interleaving''' or a (''k''-)'''floughtening''' (''flought'' is pronounced /flɔːt/, rhymes with ''bought''; this is a neo-Anglish term cognate with German ''geflochten'' 'braided') 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. We also say that the scale is (''k''-)''interleaved'' ''s''. 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 for a given interleaved scale. An interleaved scale is thus a [[cross-set]] with a little additional structure. One can '''interleave''' or '''floughten''' a scale ''s'' by a certain ~~polyoffset~~ Δ (or: "Δ ''interleaves'' ''s''" or "''s'' is ''interleavable'' by Δ") if ''s'' is the strand scale of an interleaved scale with ~~polyoffset~~ Δ. Such a scale is denoted {{nowrap|Interleave(''s''; Δ)}}. The concept of interleaved scales is a generalization of [[bipentatonic scale]]s.

A scale is a(n) (''k''-)'''interleaving''' or a (''k''-)'''floughtening''' (''flought'' is pronounced /flɔːt/, rhymes with ''bought''; this is a neo-Anglish term cognate with German ''geflochten'' 'braided') 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. We also say that the scale is (''k''-)''interleaved'' ''s''. The set of offsets that separate the strands from a fixed strand is a chord called the ''offset chord'', which is determined up to inversion and [[equave]]-equivalence for a given interleaved scale. An interleaved scale is thus a [[cross-set]] with a little additional structure. One can '''interleave''' or '''floughten''' a scale ''s'' by a certain offset chord Δ (or: "Δ ''interleaves'' ''s''" or "''s'' is ''interleavable'' by Δ") if ''s'' is the strand scale of an interleaved scale with offset chord Δ. Such a scale is denoted {{nowrap|Interleave(''s''; Δ)}}. The concept of interleaved scales is a generalization of [[bipentatonic scale]]s.

[[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.

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* Interleave(9/8-14/11-4/3-3/2-56/33-21/11-2/1; 9/7)

== Properties ==

# The following is a necessary and sufficient condition for interleavability. Let ''S'' be a scale with equave E, <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step intervals of ''S'', and Δ be a chord such that every interval of Δ falls within the open interval (0, E). Then the ~~polyoffset~~ chord Δ interleaves ''S'' if and only if no nonunison (positive) interval in Δ falls within <math> [\min \mathcal{D}_k(S), \max \mathcal{D}_k(S)]</math> for any ''k'' ∈ {0, ... len(''S'') - 1}.

# The following is a necessary and sufficient condition for interleavability. Let ''S'' be a scale with equave E, <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step intervals of ''S'', and Δ be a chord such that every interval of Δ falls within the open interval (0, E). Then the offset chord Δ interleaves ''S'' if and only if no nonunison (positive) interval in Δ falls within <math> [\min \mathcal{D}_k(S), \max \mathcal{D}_k(S)]</math> for any ''k'' ∈ {0, ... len(''S'') - 1}.

# For any periodic scale ''S'' with equave E, if δ is an offset and Interleave(''S''; δ) exists, then Interleave(''S''; δ) = Interleave(''S''; E - δ) = Interleave(''S''; δ + E). Thus, taking the equave complement of an offset in ~~a polyoffset~~ does not change the interleaved scale, nor does shifting any individual offset by equaves.

# For any periodic scale ''S'' with equave E, if δ is an offset and Interleave(''S''; δ) exists, then Interleave(''S''; δ) = Interleave(''S''; E - δ) = Interleave(''S''; δ + E). Thus, taking the equave complement of an offset in an offset chord does not change the interleaved scale, nor does shifting any individual offset by equaves.

# Given an E-equivalent scale ''S'', offsets δ within the open interval (0, min({step sizes in ''S''})) are called ''small'' in the context of interleaving ''S''. Small offsets are significant because the resulting interleaved scale has a structure that closely mimics the underlying scale structure: if ''S'' is a circular word <math>w(a_1, a_2, ..., a_n)</math> then Interleave(''s''; δ) uses the same circular word but with δ followed by the difference between δ and every step size in w, namely <math>w(\delta b_1, \delta b_2, ..., \delta b_n)</math> where <math>b_i = a_i - \delta</math>.

# An interleaved scale is not always CS, even when the strand is CS and the scale has a [[generator sequence]] where every generator subtends the same number of steps. One such scale is Interleave(Zarlino; 32/25) = 25/24 9/8 75/64 5/4 125/96 4/3 375/256 3/2 25/16 5/3 225/128 15/8 125/64 2/1 which has [[GS]](32/25 125/96 32/25 5/4).

{{proof|title=Proof of the offset constraints|contents=

If the ~~polyoffset~~ has more than two notes, the interleaving condition only needs to hold for ''pairs'' of distinct strands, and hence the above property only needs to hold for pairs of notes in the ~~polyoffset~~. This reduces the proof to the case of one offset δ.

If the offset chord has more than two notes, the interleaving condition only needs to hold for ''pairs'' of distinct strands, and hence the above property only needs to hold for pairs of notes in the offset chord. This reduces the proof to the case of one offset δ.

Let ''S''1, ''S''2 denote the two copies of ''S'' separated by δ, where ''S''1(0) = '''0''' (the unison), ''S''2(0) = δ. Assume that the scale ''F'' is the union of ''S''1 and ''S''2, and ''F''(0) = '''0'''. Let <math>m_k = \min \mathcal{D}_k(S)</math> and <math>M_k = \max \mathcal{D}_k(S).</math>

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-A scale is a(n) (''k''-)'''interleaving''' or a (''k''-)'''floughtening'''  (''flought'' is pronounced /flɔːt/, rhymes with ''bought''; this is a neo-Anglish term cognate with German ''geflochten'' 'braided') 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. We also say that the scale is (''k''-)''interleaved'' ''s''. 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 for a given interleaved scale. An interleaved scale is thus a [[cross-set]] with a little additional structure. One can '''interleave''' or '''floughten''' a scale ''s'' by a certain polyoffset Δ (or: "Δ ''interleaves'' ''s''" or "''s'' is ''interleavable'' by Δ") if ''s'' is the strand scale of an interleaved scale with polyoffset Δ. Such a scale is denoted {{nowrap|Interleave(''s''; Δ)}}. The concept of interleaved scales is a generalization of [[bipentatonic scale]]s.
+A scale is a(n) (''k''-)'''interleaving''' or a (''k''-)'''floughtening'''  (''flought'' is pronounced /flɔːt/, rhymes with ''bought''; this is a neo-Anglish term cognate with German ''geflochten'' 'braided') 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. We also say that the scale is (''k''-)''interleaved'' ''s''. The set of offsets that separate the strands from a fixed strand is a chord called the ''offset chord'', which is determined up to inversion and [[equave]]-equivalence for a given interleaved scale. An interleaved scale is thus a [[cross-set]] with a little additional structure. One can '''interleave''' or '''floughten''' a scale ''s'' by a certain offset chord Δ (or: "Δ ''interleaves'' ''s''" or "''s'' is ''interleavable'' by Δ") if ''s'' is the strand scale of an interleaved scale with offset chord Δ. Such a scale is denoted {{nowrap|Interleave(''s''; Δ)}}. The concept of interleaved scales is a generalization of [[bipentatonic scale]]s.
 [[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.
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 * Interleave(9/8-14/11-4/3-3/2-56/33-21/11-2/1; 9/7)
 == Properties ==
-# The following is a necessary and sufficient condition for interleavability. Let ''S'' be a scale with equave E, <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step intervals of ''S'', and Δ be a chord such that every interval of Δ falls within the open interval (0, E). Then the polyoffset chord Δ interleaves ''S'' if and only if no nonunison (positive) interval in Δ falls within <math> [\min \mathcal{D}_k(S), \max \mathcal{D}_k(S)]</math> for any ''k'' ∈ {0, ... len(''S'') - 1}.
+# The following is a necessary and sufficient condition for interleavability. Let ''S'' be a scale with equave E, <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step intervals of ''S'', and Δ be a chord such that every interval of Δ falls within the open interval (0, E). Then the offset chord Δ interleaves ''S'' if and only if no nonunison (positive) interval in Δ falls within <math> [\min \mathcal{D}_k(S), \max \mathcal{D}_k(S)]</math> for any ''k'' ∈ {0, ... len(''S'') - 1}.
-# For any periodic scale ''S'' with equave E, if δ is an offset and Interleave(''S''; δ) exists, then Interleave(''S''; δ) = Interleave(''S''; E - δ) = Interleave(''S''; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the interleaved scale, nor does shifting any individual offset by equaves.
+# For any periodic scale ''S'' with equave E, if δ is an offset and Interleave(''S''; δ) exists, then Interleave(''S''; δ) = Interleave(''S''; E - δ) = Interleave(''S''; δ + E). Thus, taking the equave complement of an offset in an offset chord does not change the interleaved scale, nor does shifting any individual offset by equaves.
 # Given an E-equivalent scale ''S'', offsets δ within the open interval (0, min({step sizes in ''S''})) are called ''small'' in the context of interleaving ''S''. Small offsets are significant because the resulting interleaved scale has a structure that closely mimics the underlying scale structure: if ''S'' is a circular word <math>w(a_1, a_2, ..., a_n)</math> then Interleave(''s''; δ) uses the same circular word but with δ followed by the difference between δ and every step size in w, namely <math>w(\delta b_1, \delta b_2, ..., \delta b_n)</math> where <math>b_i = a_i - \delta</math>.
 # An interleaved scale is not always CS, even when the strand is CS and the scale has a [[generator sequence]] where every generator subtends the same number of steps. One such scale is Interleave(Zarlino; 32/25) = 25/24 9/8 75/64 5/4 125/96 4/3 375/256 3/2 25/16 5/3 225/128 15/8 125/64 2/1 which has [[GS]](32/25 125/96 32/25 5/4).
 {{proof|title=Proof of the offset constraints|contents=
-If the polyoffset has more than two notes, the interleaving condition only needs to hold for ''pairs'' of distinct strands, and hence the above property only needs to hold for pairs of notes in the polyoffset. This reduces the proof to the case of one offset δ.
+If the offset chord has more than two notes, the interleaving condition only needs to hold for ''pairs'' of distinct strands, and hence the above property only needs to hold for pairs of notes in the offset chord. This reduces the proof to the case of one offset δ.
 Let ''S''<sub>1</sub>, ''S''<sub>2</sub> denote the two copies of ''S'' separated by δ, where ''S''<sub>1</sub>(0) = '''0''' (the unison), ''S''<sub>2</sub>(0) = δ. Assume that the scale ''F'' is the union of ''S''<sub>1</sub> and ''S''<sub>2</sub>, and ''F''(0) = '''0'''. Let <math>m_k = \min \mathcal{D}_k(S)</math> and <math>M_k = \max \mathcal{D}_k(S).</math>