Cross-set scale

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.

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

• Prelude and Fugue
• Short ambient piece
• Improv

Frédéric Gagné

• Floating in Outer Space

See also

• Tonality diamond