Interval Preservation in Group- and Graph-Theoretical Music Theories: A Comparative Study

Abstract

Interval preservation—wherein intervals remain unchanged among varying musical objects—is among the most basic means of manifesting coherence in musical structures. Music theorists since (1960) seminal publication of “Twelve-Tone Invariants as Compositional Determinants” have examined and generalized situations in which interval preservation obtains. In the course of this investigation, two theoretical contexts have developed: the group-theoretical, as in (1987) Generalized Interval Systems; and the graph-theoretical, as in (1991) K-net theory. Whereas the two approaches are integrally related— the latter’s being particularly indebted to the former—they have also essential differences, particularly in regard to the way in which they describe interval preservation. Nevertheless, this point has escaped significant attention in the literature. The present study completes the comparison of these two methods, and, in doing so, reveals further-reaching implications of the theory of interval preservation to recent models of voice-leading and chord spaces (Cohn 2003, Straus 2005, Tymoczko 2005, among others), specifically where the incorporated chords have differing cardinalities and/or symmetrical properties.