Sequences use familiar harmonies but combine them with a grammar quite separate from functional harmony. Let us assume that we are working in diatonic pitch-class space, and suppose each sequential unit is to be made of up exactly two chords. Building on preliminary work from 1996, John Clough explored the application and adaptation of neo-Riemannian theory to such a context. Using the nomenclature of Hook's uniform triadic transformations, Clough proposed that the binary opposition of major and minor triads that underlies so much neo-Riemannian theory could be extended to chord types found in such sequences. Instead of dichotomizing harmonic triads based on transposition class, chord classes in sequences can be based on chordal inversion, outer voice intervals, or other configural distinctions. Using this new conception of dualism, Clough went on to define transformations analogous to the neo-Riemannian Schritte and Wechsel to describe progressions within the diatonic space traversed by particular harmonic sequences.

The present study, building upon earlier work, investigates the sequence types explored by Clough from the perspective of parsimony. What types of diatonic sequences are possible under the constraint of parsimony? Is there some way to classify sequences based on their level of voice-leading parsimony? To answer these questions, of course, there must be a clear idea of what parsimony means in this context. To this end, we will identify two basic types of parsimony in such a sequence: chord-to-chord parsimony, as measured by the voice-leading connections between any two adjacent chords in the sequence, and unit-to-unit parsimony, determined by the relationship between corresponding chords in adjacent sequential units. Limiting the scope of our investigation to sequences made up entirely of triadic harmonies, we will explore the meaning and implication of both types of parsimony. In order to undertake this exploration in such a restricted context, the sense of the term “parsimony” must first be made precise in the general setting.

Part I. Toward a Definition of Parsimony

The general definition of parsimonious voice leading has been developed through a series of recent writings by Richard Cohn.