Combination product set

This is an expert page. It either assumes experience with xen theory or involves fairly technical procedures.

A combination product set (CPS) is a scale generated by the following procedure:

1. A set of chosen intervals (usually odd harmonics) is the starting point.
2. All the combinations of some number of intervals from the set are obtained, and each combination of intervals is stacked together. The same number of intervals is used for every combination.
3. This results in a bunch of notes. One note is chosen as the tonic.
4. The resulting intervals relative to the tonic are octave-reduced.

CPSes were developed by Erv Wilson.

Example (1, 3, 5, 7 hexany)

1. In this example we choose four odd harmonics: 1, 3, 5, 7.
2. We get all combinations of 2 different odd harmonics: [1, 3], [1, 5], [1, 7], [3, 5], [3, 7], [5, 7].
3. For each combination of intervals, stack the intervals together: 3, 5, 7, 15, 21, 35.
4. Choose 3 as the tonic. (This choice is arbitrary.) Measure all the other notes relative to the chosen tonic: 1/1, 5/3, 7/3, 15/3 = 5/1, 21/3 = 7/1, 35/3.
5. Octave-reduce everything: 1/1, 5/3, 7/6, 5/4, 7/4, 35/24.

This results in the 6-note scale [1/1, 7/6, 5/4, 35/24, 5/3, 7/4, 2/1], hence "hexany".

The reason we have 6 notes is that there are 6 combinations of two elements from a set of foue elwments.