Some Non-Isomorphisms between Pitch and Time

Abstract

The idea that a single principle governs each and every musical parameter goes back at least to the Pythagoreans, who believed that melody, harmony and rhythm were all ruled by ratio and proportion. Perhaps ever since, western musicians and music theorists have sought universal principles of musical structure. At bottom of such claims is the belief that pitch and time are somehow isomorphic. Faith in the isomorphism of pitch and time has continued through the twentieth-century and to the present. I begin with a few examples. Isomorphism was an attractive idea to many twentieth-century composers. Messiaen (1956, 13) catalogued symmetrical pitch patterns in his modes of limited transposition and found analogous symmetries in his non-retrogradeable rhythms. Similarly, Boulez draws the distinction between smooth versus varieties of space (for pitches) and time (for durations): Pulsation is for striated time what temperament is for striated space; it has been shown that, depending on whether a partition is fixed or variable, defined space will be regular or irregular; similarly, that the pulsation of striated time will be regular or irregular, but system-