Interleaving: Difference between revisions

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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).

{{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''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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 # 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 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>