# 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.

## 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 × 1/11/1 5/4 × 1/15/4 3/2 × 1/13/2 7/4 × 1/17/4 1/1 × 5/45/4 5/4 × 5/425/16 3/2 × 5/415/8 7/4 × 5/435/32 1/1 × 3/23/2 5/4 × 3/215/8 3/2 × 3/29/8 7/4 × 3/221/16 1/1 × 7/47/4 5/4 × 7/435/32 3/2 × 7/421/16 7/4 × 7/449/32

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):

\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*}

In combinatorics, this operation is called a sumset.

Nick Vuci
Frédéric Gagné