Parsimonious sequences of finite sets and their applications to chord progressions and music composition

Abstract

Parsimony is a broad concept with applications in music theory and composition. Two sets A and B of finite cardinality n (n-sets) are in parsimonious relation if there exists a (n–1)-set C that is included in both A and B. A sequence of n-sets is parsimonious if any two successive sets in the sequence are in parsimonious relation. This work demonstrates that for any set of finite cardinality p, there exist sequences of its n-subsets (n ≤ p) that are circular, non-redundant, exhaustive and parsimonious, and it describes the corresponding algorithm. The image of such a sequence by a bijection and the retrograde sequence keep the same four properties. The consequences of these results for the pitch class (pc) subsets of cardinality n (or n-chords) of a pc set of finite cardinality p (p-tonic scale) are derived and discussed in the context of music harmony, microtonality and composition.