Interleaving: Difference between revisions

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The term ''flought'' was coined by Inthar by evolving the Old English past participle ''(ġe)flohten'' of the verb ''fleohtan'' 'to weave; to plait' into a hypothetical Modern English word. It is cognate to the Modern English words ''plait'' and ''plexus''.

== Properties ==

# The following is a necessary and sufficient condition for floughtenability. Let ''s'' be a scale with equave ''E'', <math>\mathcal{D}_k(s)</math> be the set of all ''k''-step intervals of ''s'', and Δ be a chord such that every interval of Δ falls within (0, ''E''). Then the polyoffset chord Δ floughtens ''s'' if and only if no nonunison (positive) interval in Δ falls within <math> [\min \mathcal{D}_k(s), \max \mathcal{D}_k(s)]</math> for any ''k'' = 0, ... len(''s'') - 1.

# The following is a necessary and sufficient condition for floughtenability. Let ''S'' be a scale with equave ''E'', <math>\mathcal{D}_k(s)</math> be the set of all ''k''-step intervals of ''S'', and Δ be a chord such that every interval of Δ falls within (0, ''E''). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) interval in Δ falls within <math> [\min \mathcal{D}_k(S), \max \mathcal{D}_k(S)]</math> for any ''k'' = 0, ... len(''S'') - 1.

# For any periodic scale s with equave E, if δ is an offset and Fl(s; δ) exists, then Fl(s; δ) = Fl(s; E - δ) = Fl(s; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the flought scale, nor does shifting any individual offset by equaves.

# For any periodic scale ''S'' with equave E, if δ is an offset and Fl(''S''; δ) exists, then Fl(''S''; δ) = Fl(''S''; E - δ) = Fl(''S''; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the flought scale, nor does shifting any individual offset by equaves.

# Given an E-equivalent scale s, offsets δ within (0, min({step sizes in s})) are called ''small'' in the context of floughtening s. Small offsets are significant because the resulting flought scale has a structure that closely mimics the underlying scale structure: if s is a circular word w(a1, a2, ..., an) then Fl(s; δ) uses the same circular word but with δ followed by the difference between δ and every step size in w, namely w(~~δ b1~~, ~~δ b2~~, ..., ~~δ bn~~) where bi = ai - δ.

# Given an E-equivalent scale ''S'', offsets δ within (0, min({step sizes in ''S''})) are called ''small'' in the context of floughtening s. Small offsets are significant because the resulting flought scale has a structure that closely mimics the underlying scale structure: if s is a circular word <math>w(a1, a2, ..., an)</math> then Fl(''s''; δ) uses the same circular word but with δ followed by the difference between δ and every step size in w, namely <math>w(\delta b_1, \delta b_2, ..., \delta b_n)</math> where <math>b_i = a_i - \delta</math>.

# 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 ===

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.

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''. By symmetry, the previous statement 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.

== Some flought scales ==

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 The term ''flought'' was coined by Inthar by evolving the Old English past participle ''(ġe)flohten'' of the verb ''fleohtan'' 'to weave; to plait' into a hypothetical Modern English word. It is cognate to the Modern English words ''plait'' and ''plexus''.
 == Properties ==
-# The following is a necessary and sufficient condition for floughtenability. Let ''s'' be a scale with equave ''E'', <math>\mathcal{D}_k(s)</math> be the set of all ''k''-step intervals of ''s'', and Δ be a chord such that every interval of Δ falls within (0, ''E''). Then the polyoffset chord Δ floughtens ''s'' if and only if no nonunison (positive) interval in Δ falls within <math> [\min \mathcal{D}_k(s), \max \mathcal{D}_k(s)]</math> for any ''k'' = 0, ... len(''s'') - 1.
+# The following is a necessary and sufficient condition for floughtenability. Let ''S'' be a scale with equave ''E'', <math>\mathcal{D}_k(s)</math> be the set of all ''k''-step intervals of ''S'', and Δ be a chord such that every interval of Δ falls within (0, ''E''). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) interval in Δ falls within <math> [\min \mathcal{D}_k(S), \max \mathcal{D}_k(S)]</math> for any ''k'' = 0, ... len(''S'') - 1.
-# For any periodic scale s with equave E, if δ is an offset and Fl(s; δ) exists, then Fl(s; δ) = Fl(s; E - δ) = Fl(s; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the flought scale, nor does shifting any individual offset by equaves.
+# For any periodic scale ''S'' with equave E, if δ is an offset and Fl(''S''; δ) exists, then Fl(''S''; δ) = Fl(''S''; E - δ) = Fl(''S''; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the flought scale, nor does shifting any individual offset by equaves.
-# Given an E-equivalent scale s, offsets δ within (0, min({step sizes in s})) are called ''small'' in the context of floughtening s. Small offsets are significant because the resulting flought scale has a structure that closely mimics the underlying scale structure: if s is a circular word w(a1, a2, ..., an) then Fl(s; δ) uses the same circular word but with δ followed by the difference between δ and every step size in w, namely w(δ b1, δ b2, ..., δ bn) where bi = ai - δ.
+# Given an E-equivalent scale ''S'', offsets δ within (0, min({step sizes in ''S''})) are called ''small'' in the context of floughtening s. Small offsets are significant because the resulting flought scale has a structure that closely mimics the underlying scale structure: if s is a circular word <math>w(a1, a2, ..., an)</math> then Fl(''s''; δ) uses the same circular word but with δ followed by the difference between δ and every step size in w, namely <math>w(\delta b_1, \delta b_2, ..., \delta b_n)</math> where <math>b_i = a_i - \delta</math>.
 # 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 ===
+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.
+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''. By symmetry, the previous statement 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.
 == Some flought scales ==