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 <i>A</i> and <i>B</i> of finite cardinality <i>n</i> (<i>n</i>-sets) are in parsimonious relation if there exists a (<i>n</i> – 1)-set <i>C</i> that is included in both <i>A</i> and <i>B</i>. A sequence of <i>n</i>-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 <i>p</i>-set invariant, there exists a bipartition of its <i>n</i>-subsets into two non-redundant parsimonious sequences that are related by the involution and have only the invariant <i>n</i>-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 <i>n</i>-chords of a <i>p</i>-tonic scale in various musical temperaments enables the composer to design chord progressions and structure compositions from the intrinsic properties of the starting scale.