Brian Robinson
Cornell University
The present study begins by surveying the intrinsic vagueness of many musical parameters, including timbre, texture, dynamics, and duration, for which our traditional notation allows only limited precision. The problem of intrinsic vagueness extends to parameter which we can notate with a higher degree of precision, such as pitch; the concept of quartal harmony provides an example of a familiar notion of intervallic structure that resists precise formulation. That is, although precise definitions of quartal harmony are possible, they prove overly restrictive in relation to actual musical practice. Analysis can proceed more fruitfully from evaluating any given pitch set as possessing some degree of quartal character (e.g. as extremely quartal, strongly quartal, loosely quartal, weakly quartal, or negligibly quartal, rather than as either (absolutely) quartal or (absolutely) not quartal.
Fuzzy-set theory provides a formal methodology for describing sets which are based on graded membership. An introduction of the basic concepts of fuzzy-set theory leads to the definition of three broad classes od sonorities. One of these if the set of quartal/quintal sonorities, labeled as the set Q; for the sake of comparison, the sets of secundal/septal sonorities (S) and tertial/sextal sonorities (T) are also defined. A simple algorithm uses the ordinal proximity of pitches in a sonority to weight the terms of its interval-class vector; these weighted terms can be used to assign the sonority grades of membership in the sets Q, S, and T. This procedure is then used to evaluate the quartal character of pitch sets in examples from the music of Bartók, Messiaen, and Ligeti.