# 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.^{[1]}

## 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]\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.

## Music

### 4:5:6:7 cross-set scale

## References

- ↑ Narushima, T. (2017). Microtonality and the tuning systems of Erv Wilson. Routledge.