Parsimonious graphs: A study in parsimony, contextual transformations, andmodes of limited transposition

Connections between parsimonious structures and modes of limited transporta tion 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 gr aphs are constructed. Each mode graph consists of several components, and t he vertices in each of these components represent triads or seventh chords embedded in a particular mode of limited transposition. Two parsimonious me thods of modulating between modes of limited transposition are explored, on e 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 fo r seventh chords. In both tori, contextual transformations are evident in t heir structures, and the torus for triads is equivalent to the toroidal ver sion of the Oettingen/Riemann Tonnetz. Coupling techniques of modulation le ad to the graphs known as Cube Dance and Power Towers. Analytical implicati ons of patterns of chord sequences embedded in parsimonious graphs are also discussed.