Cross-set scale

A cross-set scale (or simply cross-set) is a scale produced by taking every ordered pair in the Cartesian product of two scales, or of a scale with itself, and stacking both elements in each ordered pair. Cross-set scales may also be generalized to more than two initial scales.

If the second scale is the inverse of the first scale (e.g. a becomes 1/a), the result is a reciprocal cross-set (scale). If additionally the first scale is a sequence of odd harmonics starting from 1, the result is a tonality diamond.

The term cross-set goes back to Erv Wilson.

Notation

The notation "chord1 by chord2" has been proposed as shorthand in lists or tables. This is borrowed from previous use on the Xen Wiki where the second chord is an interval (example).

Example

The 4:5:6:7 cross-set scale is produced by multiplying every pair of intervals from the 4:5:6:7 tetrad (1/1 - 5/4 - 3/2 - 7/4), including an interval with itself, and octave-reducing as necessary. It contains 10 distinct intervals out of 16 combinations.

1/1 × 1/1 1/1	5/4 × 1/1 5/4	3/2 × 1/1 3/2	7/4 × 1/1 7/4
1/1 × 5/4 5/4	5/4 × 5/4 25/16	3/2 × 5/4 15/8	7/4 × 5/4 35/32
1/1 × 3/2 3/2	5/4 × 3/2 15/8	3/2 × 3/2 9/8	7/4 × 3/2 21/16
1/1 × 7/4 7/4	5/4 × 7/4 35/32	3/2 × 7/4 21/16	7/4 × 7/4 49/32

The starting scales do not need to be in just intonation; a cross-set scale could be constructed from any kind of scale.

Theory

In mathematical notation, the cross-set of scales A, B, ..., Z is (note that interval stacking has been written as addition):

[math]\displaystyle{ \begin{align*}\text{Cross-set}(A, B, ..., Z) &= A + B + \cdots + Z \\ &= \{ a + b + \cdots + z : (a, b, ..., z) \in A \times B \times \cdots \times Z\}.\end{align*} }[/math]

In combinatorics, this operation is called a sumset.

Subtlety: The cross-set of two chords is properly an unreduced chord, the sumset of two finite subsets of [math]\displaystyle{ \mathbb{R}, }[/math] whereas the cross-set of two scales with the same equave is best thought of as the sumset of two finite subsets of [math]\displaystyle{ \mathbb{R}/(\text{equave})\mathbb{Z}. }[/math] (While you can theoretically take a cross-set of scales with incommensurable equaves, that requires thinking of the scales as infinite albeit periodically repeating subsets of [math]\displaystyle{ \mathbb{R}. }[/math] In fact, the resulting cross-set is dense in [math]\displaystyle{ \mathbb{R}, }[/math] thus not properly a scale.)

Music

4:5:6:7 cross-set scale

Nick Vuci

• Prelude and Fugue
• Short ambient piece
• Improv

Frédéric Gagné

• Floating in Outer Space