ABSTRACT
Previously we used some basic geometric ideas-circles and triangles, rotations and reflectionsto examine pitch organization in music. In this chapter we explore more advanced applications of geometric ideas relating to chord progressions, consonance, dissonance, and the sound qualities of different keys and modes. Our focus will be the network of chords and tones, called the Tonnetz, devised by the mathematician Leonhard Euler in the eighteenth century. The Tonnetz was later used, in the context of just intonation, by the music theorists Arthur von Oettingen and Hugo Riemann in the 19th century. It was then applied, in the context of equal temperament, by a number of music theorists at the end of the 20th century. We will focus exclusively on the Tonnetz in this latter context of equal temperament. We shall see that this version of the Tonnetz provides a geometric picture of pitch relationships (such as consonance and dissonance), as well as some of the more commonly used progressions of triadic chords.
