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

Examples

The notation used is Interleave(strand; polyoffset_chord).

If w(x, y) is a binary scale, then the ternary scale w(z(x-z), z(y-z)) is an interleaving Interleave(w(x, y); z). Blackdye and Pajara[10] are examples of this with strand Pyth[5].

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)