Parsimonious sequences of pitch-class sets: bipartition through inversion and its applications to music composition

The mathematical concept of parsimony has powerful applications in music 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 two successive sets are in parsimonious relation. Given an involution with zero, one or two fixed points (inversion) that leaves a p-set invariant, there exists a bipartition of its n-subsets into two non-redundant parsimonious sequences that are related by the involution and have only the invariant n-subsets in common. The corresponding algorithm is described. The properties of recombinations between the two complementary sequences at various crossover points are characterized. The application of these results to the n-chords of a p-tonic scale in various musical temperaments enables the composer to design chord progressions and structure compositions from the intrinsic properties of the starting scale.