Interleaving

A scale is (k-)flought^{[idiosyncratic term]} (/flɔːt/, rhymes with bought) if it is made of k > 1 copies (called strands) of an n-note scale, where any two copies are interleaved so that any note of the first copy falls between two notes of the other copy, and vice versa. The set of offsets that separate the strands from a fixed strand is a chord called the polyoffset. To floughten a scale is to use said scale as the strand scale of a flought scale. 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.

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 s can be floughtened with the polyoffset chord Δ if and only if no nonunison 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:

• strand 12:14:16:18:21:24, offset 11:12
• strand 12:14:16:18:21:24, offset 12:13:22
• strand 12:14:16:18:21:24, polyoffset 8:10:11
  • User:Userminusone/Userminusone's_11_limit_15_tone_scale
• strand 12:14:16:18:21:24, polyoffset 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)
• strand Pyth[5], polyoffset 8:10:11
• strand Pyth[5], polyoffset 9:10:11
  • Note: detempered 2.3.5.11 Porcupine[15]; well-formed generator sequence GS(10/9, 11/10, 12/11)