Interleaving: Difference between revisions
| Line 10: | Line 10: | ||
# 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 === | ||
{{Collapsible box|header=Proof| | |||
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. | 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. | ||
| Line 19: | Line 20: | ||
Now assume 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'' − 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. | Now assume 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'' − 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 == | == Some flought scales == | ||