Interleaving

A scale is an interleaving if it is (1) made of multiple copies (called strands) of a smaller scale and (2) any two copies of the smaller scale are interleaved so that any note of the first copy falls strictly between two adjacent notes of the other copy.

An interleaving is defined by the choice of strand scale and the choice of the offset chord that copies of the strand are placed on. For example, the ternary scale blackdye is an interleaving since it has strand pyth[5] and offset chord 9:10 or 5:9. We express this fact as: "9:10 (or 10/9) interleaves pyth[5]."

Examples

The notation used is Interleave(strand; offset_chord).

Pental blackdye is an example (Interleave(pyth[5]; 9:10) = sLmLsLmLsL with L = 10/9). More generally: If w(x, y) (a sequence of steps x and y) is a binary scale, then the ternary scale w(z(x-z), z(y-z)) (the same sequence but with substitutions x -> z(x-z), y -> z(y-z)) is an interleaving, namely Interleave(w(x, y); z).

Pajara[10] = Interleave(2L3s with a somewhat sharp 3/2; [0c 600c]) is an interleaved scale.

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)
• 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)

Condition for interleaving

A scale that is a union of multiple copies of a smaller scale s is an interleaving if and only if no interval between any two notes of the offset chord falls between the smallest k-step of s and the largest k-step of s (inclusive) for any k, 1 ≤ k < size of s.

For example, 9:10 interleaves pyth[5] since 1/1 < 10/9 < 9/8 = smallest 1-step of pyth[5]. But 5:6:7 does not create an interleaving of pyth[5], since 7/6 falls between 9/8 = the smallest 1-step and 32/27 = the largest 1-step.