Interleaving: Difference between revisions
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# 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). | # 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 of the floughtenability condition === | === Proof of the floughtenability condition === | ||
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 | 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. | ||
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> | 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> | ||