Interleaving
A scale is (k-)flought (/flɔːt/, rhymes with bought) if it is made of k > 1 copies (called strands) of an n-note periodic scale s, and any two copies of s are interleaved so that any note of the first copy falls strictly between two notes of the other copy. The set of offsets that separate the strands from a fixed strand is a chord called the polyoffset, which is determined up to inversion and equave-equivalence. A flought scale is thus a cross-set with a little additional structure. One can floughten a scale s with a certain polyoffset Δ (or: "Δ floughtens s" or "s is floughtenable with Δ") if s is the strand scale of a flought scale with polyoffset Δ. Such a scale is denoted Fl(s; Δ). The concept of flought scales is a generalization of dipentatonic scales and (even-length) generator-offset scales.
Blackdye, Zil[14], and bicycle are examples of flought scales, because they each have two interleaved strands, respectively Pyth[5], Zarlino, and 8:9:10:11:13:14. The terminology, however, is intended to cover any number of strands and any choice of strand scale.
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]\displaystyle{ \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]\displaystyle{ [\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.
- 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]\displaystyle{ w(a_1, a_2, ..., a_n) }[/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]\displaystyle{ w(\delta b_1, \delta b_2, ..., \delta b_n) }[/math] where [math]\displaystyle{ 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
Some flought scales
Flought scales can easily be built from a harmonic series mode as the strand: for example, if n::2n is the strand, then (2n + 1)/2n always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples:
- Fl(12:14:16:18:21:24; 11:12)
- Fl(12:14:16:18:21:24; 12:13:22)
- Fl(12:14:16:18:21:24; 8:10:11)
- Fl(12:14:16:18:21:24; 9:10:11)
- Note: detempered 11-limit Porcupine[15]; well-formed generator sequence GS(10/9, 11/10, 12/11, 10/9, 11/10, 12/11, 10/9, 11/10, 189/176)
- Fl(Pyth[5]; 8:10:11)
- Fl(Pyth[5]; 9:10:11)
- Note: detempered 2.3.5.11 Porcupine[15]; well-formed generator sequence GS(10/9, 11/10, 12/11)
- Fl(9/8-14/11-4/3-3/2-56/33-21/11-2/1; 9/7)