Interleaving: Difference between revisions
→Properties: Correction: If the polyoffset \Delta is chiral, then Fl(s; E-\Delta) != Fl(s; \Delta) in general. |
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== Properties == | == Properties == | ||
# The following is a necessary and sufficient condition for floughtenability. Let ''s'' be a scale with equave ''P'', <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, ''P''). 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 ''P'', <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, ''P''). 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; | # 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). Nor does shifting any individual offset by equaves change the associated flought scale. | ||
# 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 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 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. | ||
# 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). | # 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). | ||