Cross-set

Pajara[10] as a cross-set of Archy[5] and 2edo.

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" (in typeset math, $C_1\mathsf{b}\mathsf{y}{C}_2$) in more technical contexts or as shorthand (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 (and equave multiples if equave reduction is applied).

Examples

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

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.

Note that a tonality diamond considered as a scale contains intervals that aren't necessarily in the odd-limit. For example, 6/5 and 5/4 are 5-odd-limit, but their difference (5/4)/(6/5) = 25/24 is not in the 5-odd-limit.

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

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

Interleaving

A cross-set of a scale s and an 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] is an interleaving of 2L3s (generated by a somewhat sharp 3/2) by 600c.

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 of a scale s and an offset chord 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.