Interleaving: Difference between revisions

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# 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 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 interleaves ''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 Interleaved(''S''; δ) exists, then Interleaved(''S''; δ) = Interleaved(''S''; E - δ) = Interleaved(''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.

# 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 ~~interleaveing~~ ''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 Interleaved(''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 Interleaved(''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 Interleaved(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 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 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 interleaves ''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 Interleaved(''S''; δ) exists, then Interleaved(''S''; δ) = Interleaved(''S''; E - δ) = Interleaved(''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.
-# 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 interleaveing ''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 Interleaved(''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 Interleaved(''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 Interleaved(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=