Fuzzy Family Ties

Abstract

Melodic contour is one of a melody's defining characteristics. Music theorists such as Michael Friedmann, Robert Morris, Elizabeth West Marvin and Paul Laprade, and Ian Quinn have developed models for evaluating similarities between contours, but only a few compare similarities between pairs of contours with different lengths, and fewer still can measure shared characteristics among an entire family of contours. This article introduces a new method for evaluating familial similarities between related contours, even if the contours have different cardinalities. The model extends theories of contour transformation by using fuzzy set theory and probability, measuring a contour's degree of familial membership by examining the contour's transformational pathway and calculating the probability that each move in the pathway is shared by other family members. Through the potential of differing alignments along these pathways, the model allows for the possibility that pathways may be omitted or inserted within a contour that exhibits familial resemblance, despite its different cardinality. The analytical utility of the model is then demonstrated through an analysis of melodic possibility in phased portions of Steve Reich's The Desert Music. Integrating variable cardinality into contour similarity relations in this way more adequately accounts for familial relationships between contours and can provide new and valuable insights into one of music's most fundamental elements.