Interleaving: Difference between revisions

Line 25:

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

Suppose δ > 0 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 <math>\bigcup_{k=1}^{n-1} [m_k, M_k]</math> is invariant under taking equave complements, neither is ''E'' − δ within any [''m''''k'', ''M''''k'']. The reverse implication follows.

Suppose δ > 0 is not in any closed 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 <math>\bigcup_{k=1}^{n-1} [m_k, M_k]</math> is invariant under taking equave complements, neither is ''E'' − δ within any [''m''''k'', ''M''''k'']. The reverse implication follows.

For the forward implication, 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. We first observe that if ''m''''k'' < ''M''''k'', 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'']. Moreover, such intervals [''t''0, ''t''1] or [''t''1, ''t''0], taken over all non-ed''E'' subsets comprised of stacked ''k''-steps in ''S'', must cover [''m''''k'', ''M''k]. Indeed, if such a subset in ''S'' has the ''k''-step ''M''''k'', that subset 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 also holds when "''M''''k''" and "smaller" are replaced with "''m''''k''" and "larger".

For the forward implication, 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. We first observe that if ''m''''k'' < ''M''''k'', 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'']. Moreover, such closed intervals [''t''0, ''t''1] or [''t''1, ''t''0], taken over all non-ed''E'' subsets comprised of stacked ''k''-steps in ''S'', must cover [''m''''k'', ''M''k]. Indeed, if such a subset in ''S'' has the ''k''-step ''M''''k'', that subset 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 also holds when "''M''''k''" and "smaller" are replaced with "''m''''k''" and "larger".

The covering of [''m''''k'', ''M''k] constructed above grants us a stacked pair ''t''0, ''t''1 of unequal ''k''-steps in ''S'' such that δ ∈ [''t''0, ''t''1] ⊆ [''m''''k'', ''M''k]. Assume ''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 (otherwise, interleaving would be violated), each of the ''k'' + 1 notes 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.}}

@@ Line 25: / Line 25: @@
 Let ''S''<sub>1</sub>, ''S''<sub>2</sub> denote the two copies of ''S'' separated by δ, where ''S''<sub>1</sub>(0) = '''0''' (the unison), ''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) = '''0'''. Let <math>m_k = \min \mathcal{D}_k(S)</math> and <math>M_k = \max \mathcal{D}_k(S).</math>
-Suppose δ > 0 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 <math>\bigcup_{k=1}^{n-1} [m_k, M_k]</math> 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.
+Suppose δ > 0 is not in any closed 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 <math>\bigcup_{k=1}^{n-1} [m_k, M_k]</math> 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.
-For the forward implication, 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. We first observe that if ''m''<sub>''k''</sub> < ''M''<sub>''k''</sub>, 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>]. Moreover, such intervals [''t''<sub>0</sub>, ''t''<sub>1</sub>] or [''t''<sub>1</sub>, ''t''<sub>0</sub>], taken over all non-ed''E'' subsets comprised of stacked ''k''-steps in ''S'', must cover [''m''<sub>''k''</sub>, ''M''<sub>k</sub>]. Indeed, if such a subset in ''S'' has the ''k''-step ''M''<sub>''k''</sub>, that subset 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 also holds when "''M''<sub>''k''</sub>" and "smaller" are replaced with "''m''<sub>''k''</sub>" and "larger".
+For the forward implication, 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. We first observe that if ''m''<sub>''k''</sub> < ''M''<sub>''k''</sub>, 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>]. Moreover, such closed intervals [''t''<sub>0</sub>, ''t''<sub>1</sub>] or [''t''<sub>1</sub>, ''t''<sub>0</sub>], taken over all non-ed''E'' subsets comprised of stacked ''k''-steps in ''S'', must cover [''m''<sub>''k''</sub>, ''M''<sub>k</sub>]. Indeed, if such a subset in ''S'' has the ''k''-step ''M''<sub>''k''</sub>, that subset 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 also holds when "''M''<sub>''k''</sub>" and "smaller" are replaced with "''m''<sub>''k''</sub>" and "larger".
 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'' &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> (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.}}