An Introduction to the Combination Product Sets [CPS]
The Combination Product Sets are one of Wilson's more important achievements. These are just intonational structures without a predetermined central point of gravity or tonality. Wilson once commented on how the moon expeditions during the late 1960s provided a ripe vision of weightlessness that inspired him to investigate the possibility of analogous musical structures. It was also at this time of discovery that Wilson was working with Partch and without a doubt the CPS structures form a fitting complement to Partch’s diamond.
Combination Product sets [CPS] are sets where different factors of a set are combined into their product which results in a single number, or in this case a tone that functions interactively with the whole in a unique way. The properties of CPS structures can be predicted by Pascal's triangle [Mt. Meru].
At its core, the CPS represent a unique form of mathematics that improves and distinguishes itself from Euler's sets by bringing out substructures obscured by the latter. The difference is that Wilson includes 1 as a possible member of a combined set. While the resulting sets of the two can overlap, certain structures are not possible with Euler’s method, and the simplest and most musically significant of the uncentered structures are not articulated. Wilson focuses on these very structures, which he calls the Hexany and the Eikosany, and in so doing he provides us with some of the most innovative of musical structures. These are explained in the papers.
This page http://anaphoria.com/eikopapers.html contains an introduction on the Eikosany as found in Xenharmonikon IX as well as many of the patterns possible in the Eikosany that were used by this archivist in countless pieces and improvisations.