Interleaving: Difference between revisions
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[[Blackdye]], [[Zil]][14], and [[bicycle]] are examples of interleaved scales, because they each have two interleaved strands, respectively Pyth[5], Zarlino, and 8:9:10:11:13:14. The terminology, however, is intended to cover any number of strands and any choice of strand scale. | [[Blackdye]], [[Zil]][14], and [[bicycle]] are examples of interleaved scales, because they each have two interleaved strands, respectively Pyth[5], Zarlino, and 8:9:10:11:13:14. The terminology, however, is intended to cover any number of strands and any choice of strand scale. | ||
== Some interleaved scales == | == Some interleaved scales == | ||
Interleaved scales can easily be built from a harmonic series mode as the strand: for example, if ''n''::2''n'' is the strand, then (2''n'' + 1)/''2n'' always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples: | Interleaved scales can easily be built from a harmonic series mode as the strand: for example, if ''n''::2''n'' is the strand, then (2''n'' + 1)/''2n'' always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples: | ||
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The covering of [''m''<sub>''k''</sub>, ''M''<sub>k</sub>] constructed above grants us 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>] ⊆ [''m''<sub>''k''</sub>, ''M''<sub>k</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.}} | The covering of [''m''<sub>''k''</sub>, ''M''<sub>k</sub>] constructed above grants us 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>] ⊆ [''m''<sub>''k''</sub>, ''M''<sub>k</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.}} | ||
== Ternary interleaved scales == | |||
Given a [[ternary]] [[step signature]] of the form {{nowrap|''a'''''X'''''b'''''Y'''(''a'' + ''b'')'''Z'''}}, there exists a unique single-period (abstractly) 2-interleaved ternary [[word|scale word]] with that step signature. This ternary scale word consists of ''a''-many '''XZ''' subwords and ''b''-many '''YZ''' subwords arranged in a MOS pattern (like the steps of ''a'''''L'''''b'''''s''') and consists of an interleaved pair of two {{nowrap|''a''('''X''' + '''Z''')''b''('''Y''' + '''Z''')}} subsets, offset by '''Z'''. [[Blackdye]] ('''sLmLsLmLsL''', 5'''L'''2'''m'''3'''s''') and [[whitedye]] ('''LsLsLsmsLsLsms''', 5'''L'''2'''m'''7'''s''') are examples of this. | |||
Proof: Suppose the scale is made of two interleaved subsets offset by the abstract interval '''δ''', which must subtend an odd number of steps, say 1 ≤ ''k'' ≤ (''a'' + ''b'')/2. | |||
Case 1: ''k'' = 1. Any '''Z'''<sup>''q''</sup> a maximal subword of consecutive '''Z'''s has ''q'' odd, and they all must be the same length and separated by one non-'''Z''' letter '''W''': | |||
'''ZZ ZZ ... ZW''' (in strand ''S''<sub>1</sub>) | |||
'''Z ZZ ... ZZ WZ''' (in strand ''S''<sub>2</sub>) | |||
Moreover, the offset '''δ''' = '''Z'''. | |||
If any maximal subword of consecutive '''Z'''s has ''q'' > 1, then the scale can be split into two subwords, ''w''<sub>1</sub> with consecutive '''Z''''s and ''w''<sub>2</sub> with consecutive non-'''Z''' letters. Consecutive non-'''Z''' letters contradict '''δ''' = '''Z''': Let ''r'' be the length of a maximal subword of non-'''Z''' letters in ''w''<sub>2</sub>. | |||
'''W'''<sub>1</sub>...'''W'''<sub>''r''</sub> (all non-'''Z'''s) | |||
(Assume WOLOG) '''W'''<sub>1</sub> + '''W'''<sub>2</sub> (in strand ''S''<sub>'1</sub>) = '''Z''' + '''W'''<sub>1</sub> (in strand ''S''<sub>2</sub>) | |||
'''W'''<sub>2</sub> = '''Z''' (contradiction) | |||
== Generalizations == | == Generalizations == | ||