Interleaving: Difference between revisions

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The term ''flought'' was coined by [[Inthar]] by evolving the Old English past participle ''(ġe)flohten'' of the verb ''fleohtan'' 'to weave; to plait' into a hypothetical native Modern English cognate to the words ''plait'' and ''plexus''.

== Properties ==

# The following is a necessary and sufficient condition for floughtenability. Let ''S'' be a scale with equave ''E'', <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within (0, ''E''). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) dyad 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 floughtenability. Let ''S'' be a scale with equave ''E'', <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within the open interval (0, ''E''). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) dyad 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 Fl(''S''; δ) exists, then Fl(''S''; δ) = Fl(''S''; E - δ) = Fl(''S''; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the flought scale, nor does shifting any individual offset by equaves.

# Given an E-equivalent scale ''S'', offsets δ within (0, min({step sizes in ''S''})) are called ''small'' in the context of floughtening ''S''. Small offsets are significant because the resulting flought 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 Fl(''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 floughtening ''S''. Small offsets are significant because the resulting flought 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 Fl(''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>.

# A flought 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 Fl(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=

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

== Some flought scales ==

Flought 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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 The term ''flought'' was coined by [[Inthar]] by evolving the Old English past participle ''(ġe)flohten'' of the verb ''fleohtan'' 'to weave; to plait' into a hypothetical native Modern English cognate to the words ''plait'' and ''plexus''.
 == Properties ==
-# The following is a necessary and sufficient condition for floughtenability. Let ''S'' be a scale with equave ''E'', <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within (0, ''E''). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) dyad 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 floughtenability. Let ''S'' be a scale with equave ''E'', <math>\mathcal{D}_k(S)</math> be the set of all ''k''-step dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within the open interval (0, ''E''). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) dyad 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 Fl(''S''; δ) exists, then Fl(''S''; δ) = Fl(''S''; E - δ) = Fl(''S''; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the flought scale, nor does shifting any individual offset by equaves.
-# Given an E-equivalent scale ''S'', offsets δ within (0, min({step sizes in ''S''})) are called ''small'' in the context of floughtening ''S''. Small offsets are significant because the resulting flought 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 Fl(''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 floughtening ''S''. Small offsets are significant because the resulting flought 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 Fl(''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>.
 # A flought 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 Fl(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=
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 The cover 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>]. 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'' &minus; 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.}}
 == Some flought scales ==
 Flought 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: