Interleaving: Difference between revisions

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

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.

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. The same holds when we reverse the roles of ''S''1 and ''S''2 with offset ''E'' − δ; 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'']. 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 stacked ''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".

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 Let ''S''<sub>1</sub>, ''S''<sub>2</sub> denote the two copies of ''S'', where ''F''(0) = ''S''<sub>1</sub>(0) = 1/1, ''S''<sub>2</sub>(0) = δ and the scale ''F'' is the union of ''S''<sub>1</sub> and ''S''<sub>2</sub>. Let <math>m_k = \min \mathcal{D}_k(S)</math> and <math>M_k = \max \mathcal{D}_k(S).</math>
-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.
+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>. The same holds when we reverse the roles of ''S''<sub>1</sub> and ''S''<sub>2</sub> with offset ''E'' &minus; δ; 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>]. 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 stacked ''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".