# Interleaved scale

A scale is (*k*-)**interleaved** or (*k*-)**flought scales** (/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. An interleaved scale is thus a cross-set with a little additional structure. One can **interleave** a scale *s* by a certain polyoffset Δ (or: "Δ *interleaves* *s*" or "*s* is *interleavable* by Δ") if *s* is the strand scale of an interleaved scale with polyoffset Δ. Such a scale is denoted Interleave(*s*; Δ). The concept of interleaved scales is a generalization of bipentatonic scales and (even-length) generator-offset scales.

Blackdye, Zil[14], and bicycle are examples of interleaved 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.

## Some interleaved scales

Interleaved scales can easily be built from a harmonic series mode as the strand: for example, if *n*::2*n* is the strand, then (2*n* + 1)/*2n* always works as the offset (e.g. strand 5:6:7:8:9:10, offset 10:11). Here are some other examples:

- Interleave(12:14:16:18:21:24; 11:12)
- Interleave(12:14:16:18:21:24; 12:13:22)
- Interleave(12:14:16:18:21:24; 8:10:11)
- Interleave(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)

- Interleave(Pyth[5]; 8:10:11)
- Interleave(Pyth[5]; 9:10:11)
- Note: detempered 2.3.5.11 Porcupine[15]; well-formed generator sequence GS(10/9, 11/10, 12/11)

- Interleave(9/8-14/11-4/3-3/2-56/33-21/11-2/1; 9/7)

## Properties

- The following is a necessary and sufficient condition for interleavability. Let
*S*be a scale with equave E, [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 the open interval (0, E). Then the polyoffset chord Δ interleaves*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 Interleave(*S*; δ) exists, then Interleave(*S*; δ) = Interleave(*S*; E - δ) = Interleave(*S*; δ + E). Thus, taking the equave complement of an offset in a polyoffset does not change the interleaved scale, nor does shifting any individual offset by equaves. - Given an E-equivalent scale
*S*, offsets δ within the open interval (0, min({step sizes in*S*})) are called*small*in the context of interleaving*S*. Small offsets are significant because the resulting interleaved scale has a structure that closely mimics the underlying scale structure: if*S*is a circular word [math]w(a_1, a_2, ..., a_n)[/math] then Interleave(*s*; δ) uses the same circular word but with δ followed by the difference between δ and every step size in w, namely [math]w(\delta b_1, \delta b_2, ..., \delta b_n)[/math] where [math]b_i = a_i - \delta[/math]. - An interleaved 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 Interleave(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 offset constraints**

*pairs*of distinct strands, and hence the above property only needs to hold for pairs of notes in the polyoffset. This reduces the proof to the case of one offset δ.

Let *S*_{1}, *S*_{2} denote the two copies of *S* separated by δ, where *S*_{1}(0) = **0** (the unison), *S*_{2}(0) = δ. Assume that the scale *F* is the union of *S*_{1} and *S*_{2}, and *F*(0) = **0**. Let [math]m_k = \min \mathcal{D}_k(S)[/math] and [math]M_k = \max \mathcal{D}_k(S).[/math]

Suppose δ > 0 is not in any intervals [*m*_{k}, *M*_{k}], 1 ≤ *k* ≤ *n* − 1, *n* = len(*S*). Then for any *k*, *S*_{1}(*k*) falls between adjacent notes of *S*_{2}. The same holds when we reverse the roles of *S*_{1} and *S*_{2} and use the offset *E* − δ; since the union [math]\bigcup_{k=1}^{n-1} [m_k, M_k][/math] is invariant under taking equave complements, neither is *E* − δ within any [*m*_{k}, *M*_{k}]. The reverse implication follows.

For the forward implication, we wish to show that the interleaving condition is violated if *m*_{k} < *M*_{k} and δ ∈ [*m*_{k}, *M*_{k}] for some *k*, 1 ≤ *k* ≤ *n* − 1. We first observe that if *m*_{k} < *M*_{k}, then *S* has some pair of stacked *k*-steps, say (*S*(*n*_{0}), *S*(*n*_{0} + *k*)) (*S*(*n*_{0} + *k*), *S*(*n*_{0} + 2*k*)), whose sizes *t*_{0}, *t*_{1} are unequal and both contained in [*m*_{k}, *M*_{k}]. Moreover, such intervals [*t*_{0}, *t*_{1}] or [*t*_{1}, *t*_{0}], taken over all non-ed*E* circles of *k*-steps in *S*, must cover [*m*_{k}, *M*_{k}]. Indeed, if a circle of stacked *k*-steps in *S* has the *k*-step *M*_{k}, that circle must also have a *k*-step smaller than *k*/gcd(*n*, *k*) steps of *n*/gcd(*n*, *k*)-ed*E*, and by symmetry, the previous clause also holds when "*M*_{k}" and "smaller" are replaced with "*m*_{k}" and "larger".

*m*

_{k},

*M*

_{k}] constructed above grants us a stacked pair

*t*

_{0},

*t*

_{1}of unequal

*k*-steps in

*S*such that δ ∈ [

*t*

_{0},

*t*

_{1}] ⊆ [

*m*

_{k},

*M*

_{k}]. Assume

*t*

_{0}<

*t*

_{1}. (If

*t*

_{0}>

*t*

_{1}, take equave complements and use the offset

*E*− δ.) Then the corresponding occurrence of the

*k*-step

*t*

_{0}in

*S*

_{2}is shifted into the closed interval

*I*corresponding to the

*k*-step

*t*

_{1}in

*S*

_{1}. But we then have

*k*+ 1 notes of

*S*

_{2}within

*I*. Assuming none of these notes coincide with a note of

*S*

_{1}(otherwise, interleaving would be violated), each of the

*k*+ 1 notes must fall within one of the

*k*scale steps subtended by

*t*

_{0}in

*S*

_{1}. By the pigeonhole principle, at least one of these steps in

*S*

_{1}must contain two consecutive notes of

*S*

_{2}in its interior, breaking the interleaving condition as desired. [math]\square[/math]

## Generalizations

### Co-interleavability (?)

Periodic scales [math]S, T : \mathbb{Z} \to \mathbb{R}[/math] of the same length and equave are *co-interleavable* (the *co-* is not meant to suggest any kind of duality) if there exists [math]\delta\in\mathbb{R}[/math] such that *S* and *T* + δ are interleaved. Note that though a given 2*n*-note scale being a co-interleaved result of some pair of scales may be trivial, a given pair of scales being co-interleavable is less so: for example, **MMMM** and **Lsss** are not co-interleavable when **s** is too small.

A *contrainterleaved* scale is a co-interleaved pair of the two chiralities of a chiral scale.