Interleaving - Revision history

Inthar: /* Proof of the offset constraints */

2026-04-20T14:38:34Z

Proof of the offset constraints

← Older revision		Revision as of 14:38, 20 April 2026
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	For the forward implication, we wish to show that the interleaving condition is violated if ''m''<sub>''k''</sub> < ''M''<sub>''k''</sub> and δ ∈ [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>] for some ''k'', 1 ≤ ''k'' ≤ ''n'' − 1. We first observe that if ''m''<sub>''k''</sub> < ''M''<sub>''k''</sub>, then ''S'' has some pair of stacked ''k''-steps, say (''S''(''n''<sub>0</sub>), ''S''(''n''<sub>0</sub> + ''k'')) (''S''(''n''<sub>0</sub> + ''k''), ''S''(''n''<sub>0</sub> + 2''k'')), whose sizes ''t''<sub>0</sub>, ''t''<sub>1</sub> are unequal and both contained in [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>]. Moreover, such closed intervals [''t''<sub>0</sub>, ''t''<sub>1</sub>] or [''t''<sub>1</sub>, ''t''<sub>0</sub>], taken over all non-ed''E'' subsets comprised of stacked ''k''-steps in ''S'', must cover [''m''<sub>''k''</sub>, ''M''<sub>k</sub>]. Indeed, if such a subset in ''S'' has the ''k''-step ''M''<sub>''k''</sub>, that subset must also have a ''k''-step smaller than ''k''/gcd(''n'', ''k'') steps of ''n''/gcd(''n'', ''k'')-ed''E'', and by symmetry, the previous clause also holds when "''M''<sub>''k''</sub>" and "smaller" are replaced with "''m''<sub>''k''</sub>" and "larger".		For the forward implication, we wish to show that the interleaving condition is violated if ''m''<sub>''k''</sub> < ''M''<sub>''k''</sub> and δ ∈ [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>] for some ''k'', 1 ≤ ''k'' ≤ ''n'' − 1. We first observe that if ''m''<sub>''k''</sub> < ''M''<sub>''k''</sub>, then ''S'' has some pair of stacked ''k''-steps, say (''S''(''n''<sub>0</sub>), ''S''(''n''<sub>0</sub> + ''k'')) (''S''(''n''<sub>0</sub> + ''k''), ''S''(''n''<sub>0</sub> + 2''k'')), whose sizes ''t''<sub>0</sub>, ''t''<sub>1</sub> are unequal and both contained in [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>]. Moreover, such closed intervals [''t''<sub>0</sub>, ''t''<sub>1</sub>] or [''t''<sub>1</sub>, ''t''<sub>0</sub>], taken over all non-ed''E'' subsets comprised of stacked ''k''-steps in ''S'', must cover [''m''<sub>''k''</sub>, ''M''<sub>k</sub>]. Indeed, if such a subset in ''S'' has the ''k''-step ''M''<sub>''k''</sub>, that subset must also have a ''k''-step smaller than ''k''/gcd(''n'', ''k'') steps of ''n''/gcd(''n'', ''k'')-ed''E'', and by symmetry, the previous clause also holds when "''M''<sub>''k''</sub>" and "smaller" are replaced with "''m''<sub>''k''</sub>" and "larger".

	The covering of [''m''<sub>''k''</sub>, ''M''<sub>k</sub>] constructed above grants us a stacked pair ''t''<sub>0</sub>, ''t''<sub>1</sub> of unequal ''k''-steps in ''S'' such that δ ∈ [''t''<sub>0</sub>, ''t''<sub>1</sub>] ⊆ [''m''<sub>''k''</sub>, ''M''<sub>k</sub>]. Assume ''t''<sub>0</sub> < ''t''<sub>1</sub>. (If ''t''<sub>0</sub> > ''t''<sub>1</sub>, take equave complements and use the offset ''E'' − δ.) Then the corresponding occurrence of the ''k''-step ''t''<sub>0</sub> in ''S''<sub>2</sub> is shifted into the closed interval ''I'' corresponding to the ''k''-step ''t''<sub>1</sub> in ''S''<sub>1</sub>. But we then have ''k'' + 1 notes of ''S''<sub>2</sub> within ''I''. Assuming none of these notes coincide with a note of ''S''<sub>1</sub> (otherwise, interleaving would be violated), each of the ''k'' + 1 notes must fall within one of the ''k'' scale steps subtended by ''t''<sub>0</sub> in ''S''<sub>1</sub>. By the pigeonhole principle, at least one of these steps in ''S''<sub>1</sub> must contain two consecutive notes of ''S''<sub>2</sub> in its interior, breaking the interleaving condition as desired.}}		The covering of [''m''<sub>''k''</sub>, ''M''<sub>k</sub>] constructed above grants us a stacked pair ''t''<sub>0</sub>, ''t''<sub>1</sub> of unequal ''k''-steps in ''S'' such that δ ∈ [''t''<sub>0</sub>, ''t''<sub>1</sub>] ⊆ [''m''<sub>''k''</sub>, ''M''<sub>k</sub>]. Assume ''t''<sub>0</sub> < ''t''<sub>1</sub>. (If ''t''<sub>0</sub> > ''t''<sub>1</sub>, take equave complements and use the offset ''E'' − δ.) Then the corresponding occurrence of the ''k''-step ''t''<sub>0</sub> in ''S''<sub>2</sub> is shifted into the closed interval ''I'' corresponding to the ''k''-step ''t''<sub>1</sub> in ''S''<sub>1</sub>. But we then have ''k'' + 1 notes of ''S''<sub>2</sub> within ''I''. Assuming none of these notes coincide with a note of ''S''<sub>1</sub> (otherwise, interleaving would be violated), each of the ''k'' + 1 notes must fall within one of the ''k'' scale steps subtended by ''t''<sub>0</sub> in ''S''<sub>1</sub>. By the pigeonhole principle, at least one of these steps in ''S''<sub>1</sub> must contain two consecutive notes of ''S''<sub>2</sub> in its interior, breaking the interleaving condition as desired.

	== Ternary interleaved scales ==		== Ternary interleaved scales ==

Inthar: /* Properties */

2026-04-15T13:18:11Z

Properties

← Older revision		Revision as of 13:18, 15 April 2026
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	# 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>.		# 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).		# 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~~=
			=== Proof of the offset constraints ===
	The interleaving condition only quantifies over ''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.		The interleaving condition only quantifies over ''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.

Inthar: /* Properties */

2026-01-03T04:48:23Z

Properties

← Older revision		Revision as of 04:48, 3 January 2026
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	# 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).		# 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=		{{proof\|title=Proof of the offset constraints\|contents=
	~~If the offset chord has more than two notes, the~~ interleaving condition only quantifies over ''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.		The interleaving condition only quantifies over ''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>		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>

Inthar: /* Properties */

2026-01-03T04:47:42Z

Properties

← Older revision		Revision as of 04:47, 3 January 2026
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	# 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).		# 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=		{{proof\|title=Proof of the offset constraints\|contents=
	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.		If the offset chord has more than two notes, the interleaving condition only quantifies over ''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>		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>

Inthar at 00:51, 1 January 2026

2026-01-01T00:51:04Z

← Older revision		Revision as of 00:51, 1 January 2026
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	A scale is a(n) (''k''-)'''interleaving''' 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.		A scale is a(n) (''k''-)'''interleaving''' 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''' 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.		[[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.

Inthar at 00:50, 1 January 2026

2026-01-01T00:50:32Z

← Older revision		Revision as of 00:50, 1 January 2026
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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 ''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.		A scale is a(n) (''k''-)'''interleaving''' 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.		[[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.

Inthar: /* Mutual interleavability */

2026-01-01T00:49:18Z

Mutual interleavability

← Older revision		Revision as of 00:49, 1 January 2026
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	Two periodic scales <math>S, T : \mathbb{Z} \to \mathbb{R}</math> of the same length and equave are ''mutually interleavable'' if there exists <math>\delta\in\mathbb{R}</math> such that ''S'' and ''T'' + δ are interleaved. Note that though a given 2''n''-note scale being a mutually interleaved result of ''some'' pair of scales may be trivial, a ''given'' pair of scales being mutually interleavable is less so: for example, '''MMMM''' and '''Lsss''' are not mutually interleavable when '''s''' is too small.		Two periodic scales <math>S, T : \mathbb{Z} \to \mathbb{R}</math> of the same length and equave are ''mutually interleavable'' if there exists <math>\delta\in\mathbb{R}</math> such that ''S'' and ''T'' + δ are interleaved. Note that though a given 2''n''-note scale being a mutually interleaved result of ''some'' pair of scales may be trivial, a ''given'' pair of scales being mutually interleavable is less so: for example, '''MMMM''' and '''Lsss''' are not mutually interleavable when '''s''' is too small.

	A ''~~contrainterleaved~~'' ~~scale~~ is a mutually interleaved pair of the two chiralities of a [[chiral scale]].		A ''contrainterleaving'' is a mutually interleaved pair of the two chiralities of a [[chiral scale]].
	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Pages with open problems]]		[[Category:Pages with open problems]]

Inthar: /* Mutual interleavability */

2025-12-31T23:49:28Z

Mutual interleavability

← Older revision		Revision as of 23:49, 31 December 2025
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	Two periodic scales <math>S, T : \mathbb{Z} \to \mathbb{R}</math> of the same length and equave are ''mutually interleavable'' if there exists <math>\delta\in\mathbb{R}</math> such that ''S'' and ''T'' + δ are interleaved. Note that though a given 2''n''-note scale being a mutually interleaved result of ''some'' pair of scales may be trivial, a ''given'' pair of scales being mutually interleavable is less so: for example, '''MMMM''' and '''Lsss''' are not mutually interleavable when '''s''' is too small.		Two periodic scales <math>S, T : \mathbb{Z} \to \mathbb{R}</math> of the same length and equave are ''mutually interleavable'' if there exists <math>\delta\in\mathbb{R}</math> such that ''S'' and ''T'' + δ are interleaved. Note that though a given 2''n''-note scale being a mutually interleaved result of ''some'' pair of scales may be trivial, a ''given'' pair of scales being mutually interleavable is less so: for example, '''MMMM''' and '''Lsss''' are not mutually interleavable when '''s''' is too small.

	A ''contrainterleaved'' scale is a mutually interleaved pair of the two chiralities of a [[chiral scale]]~~. Conjecture: All [[even-regular]] MV3 scales are contrainterleaved~~.		A ''contrainterleaved'' scale is a mutually interleaved pair of the two chiralities of a [[chiral scale]].
	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Pages with open problems]]		[[Category:Pages with open problems]]

Inthar at 20:11, 17 December 2025

2025-12-17T20:11:33Z

← Older revision		Revision as of 20:11, 17 December 2025
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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.		[[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)		* Interleave(9/8-14/11-4/3-3/2-56/33-21/11-2/1; 9/7)
	== Properties ==		== 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>.		# 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).		# 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=		{{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>		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>

Inthar at 17:28, 10 January 2025

2025-01-10T17:28:34Z

← Older revision		Revision as of 17:28, 10 January 2025
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	A scale is a (''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 ''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.

	[[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.		[[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.