# Combination product set

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

- A set
*S*of*n*positive real numbers is the starting point. - All the combinations of
*k*elements of the set are obtained, and their products taken. - These are combined into a set, and then all of the elements of that set are divided by one of them (which one is arbitrary; if a canonical choice is required, the smallest element could be used).
- The resulting elements are octave-reduced and sorted in ascending order, resulting in an octave period of a periodic scale (the usual sort of scale, in other words) which we may call CPS(
*S*,*k*).

This is sometimes called a *k*)*n* CPS, where the *n* denotes the size of the set *S*. There are special names for special cases: a 2)4 CPS is called a hexany; both 2)5 and 3)5 CPS are called dekanies; both 2)6 and 4)6 CPS are called pentadekanies, a 3)6 CPS is called an eikosany, etc. These are normally considered in connection with just intonation, so that the starting set is a set of positive rational numbers, but nothing prevents consideration of the more general case.

The idea can be further generalized so that the thing we start from is not a set but a multiset. A multiset is like a set, but the elements have multiplicities; there can be two, three or any number of a given kind of element. Submultisets of a multiset are also multisets, and the product over a multiset takes account of multiplicity. Hence the 2)4 hexany resulting from the multiset [1,1,3,5] first generates the two-element submultisets, which are [1, 1], [1, 3], [1, 5], [3, 5], and we obtain products 1, 3, 5, and 15. Reducing to an octave gives 5/4, 3/2, 15/8, 2 which can be transposed to 5/4, 4/3, 5/3, 2. In spite of being called a hexany it has only four notes.

CPS are closely related to Euler genera, since if we combine 0)*n*, 1)*n*, 2)*n* ... *n*)*n* before reducing to an octave, and then reduce, we get an Euler genus.

CPS were invented by Erv Wilson.