Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition

Abstract

Connections between parsimonious structures and modes of limited transposition from three set classes are explored. A graph-theoretic approach proves useful in illustrating the symmetries inherent in parsimonious structures and modes of limited transposition. Four parsimonious graphs called mode graphs are constructed. Each mode graph consists of several components, and the vertices in each of these components represent triads or seventh chords embedded in a particular mode of limited transposition. Two parsimonious methods of modulating between modes of limited transposition are explored, one by bridging and the other by coupling components of mode graphs. Bridging techniques of modulation lead to two tori, one for triads and the other for seventh chords. In both tori, contextual transformations are evident in their structures, and the torus for triads is equivalent to the toroidal version of the Oettingefliemann Tonnetz. Coupling techniques of modulation lead to the graphs known as Cube Dance and Power Towers. Analytical implications of patterns of chord sequences embedded in parsimonious graphs are also discussed.