composition, music theory, mathematics
The point of departure for this paper is Kaija Saariaho’s Vers le blanc, the pitch structure of which consists of a continuous glissando between two chords over a duration of 15 minutes. About this work the composer writes, “The harmony is a continuous stream and cannot be heard as a series of changing chords. One only notices from time to time that the harmonic situation has changed.” What are these harmonic ‘situations’?
The solution offered in this paper is to consider this and other continuous transformations as trajectories through continuous spaces of chords, rhythms, tempos, etc. While the paper focuses on a number of specific examples, the theoretical ideas build on work by John Roeder (1987) and lay the groundwork for the later Science article on “Generalized Voice-Leading Spaces.”
Continuous Transformations, Music Theory Online 10.3, 2004.