Cross-set

A cross-set of two chords is a scale consisting of all the notes that result when constructing a copy of one chord on each note of the other (and reducing by the equave).^{[adv 1]}

This operation can be iterated over more than two chords (and is commutative and associative; mathematical parlance calls this operation a sumset).

Conventions

On this wiki, a cross-set of two chords may be denoted "chord1 by chord2" (and the equave is usually understood). An n-fold iterated cross-set may be written chord^{by n}, and by convention chord^{by 0} is a singleton consisting only of the unison.

Examples

A cross-set of a scale and a dyad is two copies of that scale offset by that dyad. For example, blackwood[10] is the cross-set of 5edo and 5/4.

Partch tonality diamonds are cross-sets. Here is the 7-oddlimit 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, oddlimit tonality diamonds are cross-sets. Harry Partch is well-known for developing this idea, in particular basing his compositions on the 11-oddlimit tonality diamond.

A generalized definition: Given a set $\{1,n_1,n_2,...,n_k\}$ of odd harmonics (usually some oddlimit), the corresponding tonality diamond is the 2/1-equivalent cross-set

$\{1,n_1,n_2,...,n_k\}\mathsf{b}\mathsf{y}\{1,1/n_1,1/n_2,...,1/n_k\}.$

Tonality diamonds can be explored at https://tonalitydiamondapplet.nickvuci.com/.

Interleaving

A cross-set s by (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

Pental blackdye is an example (pyth[5] by 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 w(x, y) by z.

Pajara[10] = (2L3s with a somewhat sharp 3/2) by 600c is an interleaving.

Interleavings 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:

• 12:14:16:18:21:24 by 11:12
• 12:14:16:18:21:24 by 12:13:22
• 12:14:16:18:21:24 by 8:10:11
• 12:14:16:18:21:24 by 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)
• Pyth[5] by 8:10:11
• Pyth[5] by 9:10:11
  • Note: detempered 2.3.5.11 Porcupine[15]; well-formed generator sequence GS(10/9, 11/10, 12/11)
• 9/8-14/11-4/3-3/2-56/33-21/11-2/1 by 9/7

Condition for interleaving

A cross-set s by (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.

Footnotes

^{[adv 1]} Subtlety: The cross-set of two chords is most naturally an unreduced chord, the sumset of two finite subsets of pitch space $\mathbb{ℝ},$ whereas the cross-set of two scales with the same equave is best thought of as the sumset of two finite subsets of pitch class space, $\mathbb{ℝ}/(\mathsf{e}\mathsf{q}\mathsf{u}\mathsf{a}\mathsf{v}\mathsf{e})\mathbb{\mathbb{Z}}.$ To make a chord into a scale we copy notes into every equave or equivalently equave-reduce; some type conversions are left implicit in this discussion.