Normalizing Musical Contour Theory

Abstract

The advent of the numerical representation of contour was a watershed in the development of musical contour theory. Both Michael L. Friedmann and Robert D. Morris pioneered the practice by mapping pitches in ascending registral order onto the subset of nonnegative integers from 0 to n − 1, where n equals cardinality. This notation identifies pitches solely by their relative height, thereby eschewing any reference to specific interval size and effectively transforming pitches in pitch space into contour pitches in contour space. The variable end-point mechanism this procedure entails, however, often yields counterintuitive and inconsistent results when comparing contour segments (csegs) of different cardinalities. In some instances, it falsely implies that an expansion or contraction of contour space has transpired. In other cases, it grossly misrepresents the phenomenological disposition of certain members of a given cseg yet remains perfectly true to that of others. This article addresses the analytical pitfalls of the integer-based contour labeling system by instead adopting a normalized scheme that maps pitches onto evenly distributed subsets of the real numbers from 0 to 1 inclusive. This not only systematically eliminates the distortions and inconsistencies that crop up with respect to mixed cardinality csegs but also provides a considerably more nuanced metric for intervallic distances in contour space. Using analytical case studies juxtaposing the two notational systems, this article demonstrates how the normalized representation of contour both enhances and extends the analytical capabilities of musical contour theory by more effectively modeling the transformational implications embedded therein.