Geometric and Algebraic Aspects of Neo-Riemannian and Pitch-Class Set Theory

Abstract

In this article, a mathematical method based on lineal algebra is proposed to represent and compose harmonic progressions. First, in relation to chords and sets of notes, the mathematical concepts of the Neo-Riemannian and Pitch-Class-Set Theory are leveraged to obtain the respective equations. Second, with the formula of the distance between two points, the notes are interpreted as points in a Cartesian plane with their two-dimensional vectors. Third, these points serve as a basis for constructing polygons from the pitch-class sets. Fourth, as with notes, the triadic chords are defined as three-dimensional and four-dimensional (i.e., with four variables: x, y, z and t) vectors in seventh chords. Finally, linear transformations are introduced as a mathematical tool to express and generate harmonic progressions.