Interleaving: Difference between revisions

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* Flought(9/8-14/11-4/3-3/2-56/33-21/11-2/1; 9/7)

== 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 dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within the open interval (0, ''E''). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) dyad 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 dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within the open interval (0, E). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) dyad 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 Flought(''S''; δ) exists, then Flought(''S''; δ) = Flought(''S''; E - δ) = Flought(''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 the open interval (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(a_1, a_2, ..., a_n)</math> then Flought(''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>.

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 * Flought(9/8-14/11-4/3-3/2-56/33-21/11-2/1; 9/7)
 == 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 dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within the open interval (0, ''E''). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) dyad 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 dyads of ''S'', and Δ be a chord such that every dyad of Δ falls within the open interval (0, E). Then the polyoffset chord Δ floughtens ''S'' if and only if no nonunison (positive) dyad 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 Flought(''S''; δ) exists, then Flought(''S''; δ) = Flought(''S''; E - δ) = Flought(''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 the open interval (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(a_1, a_2, ..., a_n)</math> then Flought(''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>.