Cross-set scale

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A cross-set scale is a scale generated by taking the Cartesian product of two or more scales, or of a scale with itself.

As reported by Nick Vuci, "the term Cross-Set as applied to scales is apparently the invention of Praveen Venkataramana".

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