Cross-set scale

Revision as of 18:20, 24 July 2023 by Inthar (talk | contribs) (Cited Narushima's book which apparently attributes the term "cross-set" to Wilson.)

A cross-set scale is a scale generated by taking every ordered pair in the Cartesian product of two or more scales, or of a scale with itself, and stacking all elements in each ordered pair. In mathematical notation, the cross-set of scales A, B, ..., Z is (note that stacking has been written as addition):

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

In combinatorics, this operation is called a sumset.

The term cross-set goes back to Erv Wilson.[1]

Examples

The 4:5:6:7 cross-set scale is generated 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 pitches 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.

Music

4:5:6:7 cross-set tuning

Nick Vuci
Frédéric Gagné

See also

References

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