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, where any two copies of s are interleaved so that any note of the first copy falls 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.

Terminology

Given an E-equivalent scale, scale s, offsets between 0 and min({step sizes in s}) (equivalently, between max{len(s)-1-steps in s} and E) are called small. Small offsets are significant because it copies the underlying scale structure: if s = w(a1, a2, ..., an) then Fl(s; δ) = w(δ b1, δ b2, ..., δ bn) where bi = ai - δ.

Properties

• 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).

Condition for floughtenability

Let:

• s be a scale with equave P,
• [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, P).

Then the polyoffset chord Δ floughtens s if and only if no nonunison (positive) interval in Δ falls within

[math]\displaystyle{ \bigcup_{i=0}^{\mathrm{len}(s) - 1} [\min \mathcal{D}_k(s), \max \mathcal{D}_k(s)]. }[/math]

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)
  • User:Userminusone/Userminusone's_11_limit_15_tone_scale
• 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)