Interleaving: Difference between revisions
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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>\bigcup_{i=0}^{\mathrm{len}(s) - 1} [\min \mathcal{D}_k(s), \max \mathcal{D}_k(s)].</math> | * 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>\bigcup_{i=0}^{\mathrm{len}(s) - 1} [\min \mathcal{D}_k(s), \max \mathcal{D}_k(s)].</math> | ||
* For any periodic scale s with equave E, if Δ is a polyoffset and Fl(s; Δ) exists, then Fl(s; Δ) = Fl(s; E - Δ | * For any periodic scale s with equave E, if Δ is a polyoffset and Fl(s; Δ) exists, then Fl(s; Δ) = Fl(s; E - Δ) where E - Δ = {E - δ : δ ∈ Δ}. Nor does shifting any individual note in Δ 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 an [[AGS]]. 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 [[AGS]](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 an [[AGS]]. 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 [[AGS]](32/25 125/96 32/25 5/4). | ||