An Algebraic Approach to Time-Span Reduction

In this chapter, we present an algebraic framework in which a set of simple, intuitive operations applicable to music can be flexibly combined to realize a target application and generate music. We formalize the concept of time-span tree introduced by Lerdahl and Jackendoff (1983) in their Generative Theory of Tonal Music (GTTM) and define the distance between time-span trees, on the hypothesis that this might coincide with the psychological resemblance between melodies heard by human listeners. To confirm the feasibility of the proposed framework, we conduct an experiment to determine whether the distance calculated on the basis of the framework reflects cognitive distance in human listeners. To demonstrate that the algebraic framework is computationally tractable, we present the implementation of a musical morphing system that, given two original melodies, generates an intermediate melody at any internally dividing point between them (i.e., at any ratio).