Combination product set

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

1. A list of chosen intervals (usually odd harmonics) is the starting point.
2. All the combinations of some number of intervals from the list are obtained. The same number of intervals is used for every combination.
3. Each of the above combinations of intervals is stacked together into one interval.
4. This results in a list of notes. One note is chosen as the tonic.
5. 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 just amounts to choosing a mode of the final scale.)
5. 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.
6. 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".

Types of CPSes

Common sizes for CPSes have specific names:

1. Hexany: Choose 2 out of a list of 4 intervals
   • Stellated hexany: A hexany combined with combinations of 1 and combinations of 3
2. Dekany: Choose 2 (or 3) out of a list of 5 intervals
3. Pentadekany: Choose 2 (or 4) out of a list of 6 intervals
4. Eikosany: Choose 3 out of a list of 6 intervals