Combination product set: Difference between revisions

A '''combination product set''' is a [[scale|scale]] generated by the following means:

<ol><li>A set S of n positive real numbers is the starting point.</li><li>All the combinations of k elements of the set are obtained, and their products taken.</li><li>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).</li><li>The resulting elements are reduced to an octave and sorted in ascending order, resulting in an octave period of a [[Periodic_scale|periodic scale]] (the usual sort of scale, in other words) which we may call Cps(S, k).</li></ol>

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|hexany]]; both 2)5 and 3)5 cps are called [[dekanies|dekanies]]; both 2)6 and 4)6 cps are called [[pentadekanies|pentadekanies]], and a 3)6 cps an [[eikosany|eikosany]]. 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 [http://en.wikipedia.org/wiki/Multiset multiset]. A multiset is like a set, but the elements have multiplicites; 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|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.

CPSs were invented by [[Erv Wilson|Erv Wilson]].