Approximate Set Theory

This article describes an approximate set theory modeling intuitions shared by musicians such as Cowell, Schoenberg, Messiaen, and Persichetti. The author considers five approximation strategies, showing that in each case the result resembles an exact seven-tone set theory. Since most seven-tone sets are interval cycles, approximate twelve-tone sets are approximately cyclic as well. The theory explains how to highlight this cyclic structure using voicings, modeled by intervals in the intrinsic scale formed from a chord's own notes. This connection to voicing is what gives approximate chord categories much of their significance. The approach is most useful for chords with five or fewer notes and works tolerably for hexachords, but it breaks down with larger collections. This is not a failure of the model but a reflection of the fact that quality space contracts as cardinality increases.