Cardinality Transformations in Diatonic Space

Abstract

This paper introduces a system in which parsimonious and continuous transformations occur seamlessly between triads and tetrachords. Such fluidity is abundant in common practice music, but unprecedented in theoretical literature, largely because there has been no consistent way to approach transformations independent of cardinality. Neo-Riemannian theory elegantly unites harmonic change and voice-leading efficiency, but deals exclusively with set class [037] in a 12-gamut pcset space. Attempts to extend the neo-Riemannian approach to tetrachords in 12-gamut space often fall short; the elegant characteristics of the triadic theory do not carry over. However, when a scalar context arbitrates the parsimoniousness of transformations, triads and tetrachords can be treated in a consistent manner. Within this consistently modeled space, cardinality itself can be transformed. In this paper, we see that filtered point-symmetry is an essential tool for working through the iterated maximally even sets that establish scalar contexts. To understand cardinality transformations, we also extend filtered point-symmetry to model partially symmetric distributions and relatively even sets.