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::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:

• 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)
  • User:Userminusone/Userminusone's_11_limit_15_tone_scale
• 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

1. 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}.
2. 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.
3. 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].
4. 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

If the polyoffset has more than two notes, the interleaving condition only needs to hold for 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 S₁, S₂ denote the two copies of S separated by δ, where S₁(0) = 0 (the unison), S₂(0) = δ. Assume that the scale F is the union of S₁ and S₂, 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, S₁(k) falls between adjacent notes of S₂. The same holds when we reverse the roles of S₁ and S₂ 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(n₀), S(n₀ + k)) (S(n₀ + k), S(n₀ + 2k)), whose sizes t₀, t₁ are unequal and both contained in [m_k, M_k]. Moreover, such intervals [t₀, t₁] or [t₁, t₀], taken over all non-edE 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)-edE, and by symmetry, the previous clause also holds when "M_k" and "smaller" are replaced with "m_k" and "larger".

The covering of [m_k, M_k] constructed above grants us a stacked pair t₀, t₁ of unequal k-steps in S such that δ ∈ [t₀, t₁] ⊆ [m_k, M_k]. Assume t₀ < t₁. (If t₀ > t₁, take equave complements and use the offset E − δ.) Then the corresponding occurrence of the k-step t₀ in S₂ is shifted into the closed interval I corresponding to the k-step t₁ in S₁. But we then have k + 1 notes of S₂ within I. Assuming none of these notes coincide with a note of S₁ (otherwise, interleaving would be violated), each of the k + 1 notes must fall within one of the k scale steps subtended by t₀ in S₁. By the pigeonhole principle, at least one of these steps in S₁ must contain two consecutive notes of S₂ 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 2n-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.