Interleaving: Difference between revisions

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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 offsets. 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) = 1/1, ''S''2(0) = δ, the scale ''F'' is the union of ''S''1 and ''S''2, and ''F''(0) = 1/1. Let <math>m_k = \min \mathcal{D}_k(S)</math> and <math>M_k = \max \mathcal{D}_k(S).</math>

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

Suppose δ is not in any intervals [''m''''k'', ''M''''k''], 1 ≤ ''k'' ≤ ''n'' − 1, ''n'' = len(''S''). Then for any ''k'', ''S''1(''k'') falls between adjacent notes of ''S''2. The same holds when we reverse the roles of ''S''1 and ''S''2 and use the offset ''E'' − δ; since the union of the [''m''''k'', ''M''''k''] is invariant under taking equave complements, neither is ''E'' − δ within any [''m''''k'', ''M''''k'']. The reverse implication follows.

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 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 offsets. 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) = 1/1, ''S''<sub>2</sub>(0) = δ, the scale ''F'' is the union of ''S''<sub>1</sub> and ''S''<sub>2</sub>, and ''F''(0) = 1/1. 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) = 1/1, ''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) = 1/1. Let <math>m_k = \min \mathcal{D}_k(S)</math> and <math>M_k = \max \mathcal{D}_k(S).</math>
 Suppose δ is not in any intervals [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>], 1 &le; ''k'' &le; ''n'' &minus; 1, ''n'' = len(''S''). Then for any ''k'', ''S''<sub>1</sub>(''k'') falls between adjacent notes of ''S''<sub>2</sub>. The same holds when we reverse the roles of ''S''<sub>1</sub> and ''S''<sub>2</sub> and use the offset ''E'' &minus; δ; since the union of the [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>] is invariant under taking equave complements, neither is ''E'' &minus; δ within any [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>]. The reverse implication follows.