Cross-set

A cross-set of two chords is a scale formed by

1. taking every element of the set of ordered pairs from the pitches of the chords
2. stacking all the intervals listed in the ordered pair
3. (and reducing by the equave, and centering the scale on one root note if necessary).

This operation can be iterated over more than two chords.

On this wiki, a cross-set of two chords may be denoted chord1 ⊗ chord2 (and the equave is usually understood).

Example

Partch tonality diamonds are cross-sets. Here is the 7-odd-limit tonality diamond as the octave-reduced cross-set of 4:5:6:7 and 1/4:1/5:1/6:1/7:

	4	5	6	7
1/4	(1/4, 4) 4/4 = 1/1	(1/4, 5) 5/4	(1/4, 6) 6/4 = 3/2	(1/4, 7) 7/4
1/5	(1/5, 4) 4/5 -> 8/5	(1/5, 5) 5/5 = 1/1	(1/5, 6) 6/5	(1/5, 7) 7/5
1/6	(1/6, 4) 4/6 -> 4/3	(1/6, 5) 5/6 -> 5/3	(1/6, 6) 6/6 = 1/1	(1/6, 7) 7/6
1/7	(1/7, 4) 4/7 -> 8/7	(1/7, 5) 5/7 -> 10/7	(1/7, 6) 6/7 -> 12/7	(1/7, 7) 7/7 = 1/1

Here -> denotes octave reduction. The resulting scale is [1/1, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4].

Tonality diamond

As stated above, odd-limit tonality diamonds are cross-sets. Harry Partch is well-known for developing this idea, in particular basing his compositions on the 11-odd-limit tonality diamond.

A generalized definition: Given a set {1, n1, n2, ..., nk} of odd harmonics (usually n-odd-limit), the corresponding tonality diamond is the 2/1-equivalent cross-set {1, n1, n2, ..., nk} ⊗ {1, 1/n1, 1/n2, ..., 1/nk}.

Interleaving

A cross-set s ⊗ (offset chord) is an interleaving if (1) the offset chord has multiple notes, and (2) any two copies of s (called the strand) 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 step sizes 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 cross-set s ⊗ (offset) 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.