Approximate Set Theory

This arti cle describes an approx i mate set the ory mod el ing intu i tions shared by musi cians such as Cowell, Schoenberg, Messiaen, and Persichetti. The author con sid ers five approx i ma tion strat e gies, show ing that in each case the result resem bles an exact seven-tone set the ory. Since most seven-tone sets are inter val cycles, approx i mate twelve-tone sets are approx i ma tely cyclic as well. The the ory explains how to high light this cyclic struc ture using voicings, mod eled by inter vals in the intrinsic scale formed from a chord’s own notes. This con nec tion to voic ing is what gives approx i mate chord categories much of their sig nif i cance. The approach is most use ful for chords with five or fewer notes and works tol er a bly for hexachords, but it breaks down with larger col lec tions. This is not a fail ure of the model but a reflec tion of the fact that qual ity space con tracts as car di nal ity increases.