Fishing for complements with chord, scale, and rhythm nets

The aim of this paper is to argue that complementation is an operation similarly fundamental to music theory as transposition and inversion. We focus on studying the chromatic complement mapping that translates diatonic seventh chords into 8-note scales which can also be interpreted as rhythmic beat patterns. Such complements of diatonic seventh chords are of particular importance since they correspond to the scales popularized by the Jazz theorist Barry Harris, as well as to rhythms used in African drum music and Steve Reich's Clapping Music. Our approach enables a systematic study of these scales and rhythms using established theories of efficient voice leading and generalized diatonic scales and chords, in particular the theory of second-order maximally even sets. The main contributions of this research are (1) to explicate the correspondence between voice leadings and rhythmic transformations, (2) to systematize the family of Barry Harris scales, and (3) to describe classes of voice leadings between chords of different cardinality that are invariant under complementation.