Interleaving: Difference between revisions

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Suppose δ is not in any intervals [''m''''k'', ''M''''k'']. Then for any ''k'', ''S''1(''k'') falls between adjacent notes of ''S''2. 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''], and the same holds when we reverse the roles of ''S''1 and ''S''2 with offset ''E'' − δ. The reverse direction follows.

For the forward direction, we wish to show that the interleaving condition is violated if ''m''''k'' < ''M''''k'' and δ ∈ [''m''''k'', ''M''''k''] for some ''k'', 1 ≤ ''k'' ≤ ''n'' − 1, ''n'' = len(''S''). We assert that if this holds, then ''S'' has some pair of stacked ''k''-steps, say ''S''(''n''0), ''S''(''n''0 + ''k'')) ''S''(''n''0 + ''k''), ''S''(''n''0 + 2''k''), whose sizes ''t''0, ''t''1 are unequal and both contained in [''m''''k'', ''M''''k''] This is because such intervals [''t''0, ''t''1] or [''t''1, ''t''0] must cover [''m''''k'', ''M''k]. Indeed, if a circle of ''k''-steps in ''S'' has the ''k''-step ''M''''k'', that circle must also have a ''k''-step smaller than ''k''/gcd(''n'', ''k'') steps of ''n''/gcd(''n'', ''k'')-ed''E'', and by symmetry, the previous clause holds when "''M''''k''" and "smaller" are replaced with "''m''''k''" and "larger".

For the forward direction, we wish to show that the interleaving condition is violated if ''m''''k'' < ''M''''k'' and δ ∈ [''m''''k'', ''M''''k''] for some ''k'', 1 ≤ ''k'' ≤ ''n'' − 1, ''n'' = len(''S''). We assert that if this holds, then ''S'' has some pair of stacked ''k''-steps, say (''S''(''n''0), ''S''(''n''0 + ''k'')) (''S''(''n''0 + ''k''), ''S''(''n''0 + 2''k'')), whose sizes ''t''0, ''t''1 are unequal and both contained in [''m''''k'', ''M''''k''] This is because such intervals [''t''0, ''t''1] or [''t''1, ''t''0] must cover [''m''''k'', ''M''k]. Indeed, if a circle of ''k''-steps in ''S'' has the ''k''-step ''M''''k'', that circle must also have a ''k''-step smaller than ''k''/gcd(''n'', ''k'') steps of ''n''/gcd(''n'', ''k'')-ed''E'', and by symmetry, the previous clause holds when "''M''''k''" and "smaller" are replaced with "''m''''k''" and "larger".

Now assume a stacked pair ''t''0, ''t''1 of unequal ''k''-steps in ''S''. Assume ''t''0 < ''t''1 and δ ∈ [''t''0, ''t''1] (If ''t''0 > ''t''1, take equave complements and use the offset ''E'' − δ.) Then the corresponding occurrence of the ''k''-step ''t''0 in ''S''2 is shifted into the closed interval ''I'' corresponding to the ''k''-step ''t''1 in ''S''1. But we then have ''k'' + 1 notes of ''S''2 within ''I''. Assuming none of these notes coincide with a note of ''S''1, each of them must fall within one of the ''k'' scale steps subtended by ''t''0 in ''S''1. By the pigeonhole principle, at least one of these steps in ''S''1 must contain two consecutive notes of ''S''2 in its interior, breaking the interleaving condition as desired.

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 Suppose δ is not in any intervals [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>]. Then for any ''k'', ''S''<sub>1</sub>(''k'') falls between adjacent notes of ''S''<sub>2</sub>. 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>], and the same holds when we reverse the roles of ''S''<sub>1</sub> and ''S''<sub>2</sub> with offset ''E'' &minus; δ. The reverse direction follows.
-For the forward direction, we wish to show that the interleaving condition is violated if ''m''<sub>''k''</sub> < ''M''<sub>''k''</sub> and δ ∈ [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>] for some ''k'', 1 &le; ''k'' &le; ''n'' &minus; 1, ''n'' = len(''S''). We assert that if this holds, then ''S'' has some pair of stacked ''k''-steps, say ''S''(''n''<sub>0</sub>),&nbsp;''S''(''n''<sub>0</sub>&nbsp;+&nbsp;''k''))&nbsp;''S''(''n''<sub>0</sub>&nbsp;+&nbsp;''k''), ''S''(''n''<sub>0</sub>&nbsp;+&nbsp;2''k''), whose sizes ''t''<sub>0</sub>, ''t''<sub>1</sub> are unequal and both contained in [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>] This is because such intervals [''t''<sub>0</sub>, ''t''<sub>1</sub>] or [''t''<sub>1</sub>, ''t''<sub>0</sub>] must cover [''m''<sub>''k''</sub>, ''M''<sub>k</sub>]. Indeed, if a circle of ''k''-steps in ''S'' has the ''k''-step ''M''<sub>''k''</sub>, that circle must also have a ''k''-step smaller than ''k''/gcd(''n'', ''k'') steps of ''n''/gcd(''n'', ''k'')-ed''E'', and by symmetry, the previous clause holds when "''M''<sub>''k''</sub>" and "smaller" are replaced with "''m''<sub>''k''</sub>" and "larger".
+For the forward direction, we wish to show that the interleaving condition is violated if ''m''<sub>''k''</sub> < ''M''<sub>''k''</sub> and δ ∈ [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>] for some ''k'', 1 &le; ''k'' &le; ''n'' &minus; 1, ''n'' = len(''S''). We assert that if this holds, then ''S'' has some pair of stacked ''k''-steps, say (''S''(''n''<sub>0</sub>),&nbsp;''S''(''n''<sub>0</sub>&nbsp;+&nbsp;''k''))&nbsp;(''S''(''n''<sub>0</sub>&nbsp;+&nbsp;''k''), ''S''(''n''<sub>0</sub>&nbsp;+&nbsp;2''k'')), whose sizes ''t''<sub>0</sub>, ''t''<sub>1</sub> are unequal and both contained in [''m''<sub>''k''</sub>, ''M''<sub>''k''</sub>] This is because such intervals [''t''<sub>0</sub>, ''t''<sub>1</sub>] or [''t''<sub>1</sub>, ''t''<sub>0</sub>] must cover [''m''<sub>''k''</sub>, ''M''<sub>k</sub>]. Indeed, if a circle of ''k''-steps in ''S'' has the ''k''-step ''M''<sub>''k''</sub>, that circle must also have a ''k''-step smaller than ''k''/gcd(''n'', ''k'') steps of ''n''/gcd(''n'', ''k'')-ed''E'', and by symmetry, the previous clause holds when "''M''<sub>''k''</sub>" and "smaller" are replaced with "''m''<sub>''k''</sub>" and "larger".
 Now assume a stacked pair ''t''<sub>0</sub>, ''t''<sub>1</sub> of unequal ''k''-steps in ''S''. Assume ''t''<sub>0</sub> < ''t''<sub>1</sub> and δ ∈ [''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'' &minus; 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>, each of them 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.