Donald Casadonte
Dept. of Music, Ohio State Univ., 1899 College Rd., Columbus, OH 43210
Since the time of the ancient Greeks music theorists have dreamed of a ``musical space'' in which musical events reside. In this paper the complete phase space portrait of a piece of music, Bach's Two Part Invention No. 4 in D Minor is reconstructed using techniques of dynamical systems theory. The space consists of a central strange attractor surrounded by wandering orbits that continually ``revisit the attractor as the musical structure develops. This simple example is taken as a paradigm for more general types of music structure, in which stable melodic lines are topologically represented as orbitally stable subsets in phase space (most commonly as strange attractors), and the transitional or nonmelodic sections are topologically unstable or wandering subsets. Musical structure is generated as the musical ``trajectories'' contained in the different simultaneous musical lines of a composition interact via the stable and unstable regions of the musical phase space structure ((omega)-limit structure) defined by the composer. This (omega)-limit hypothesis is illustrated with several classical music examples. The implications of time-evolutionary pictures of the unfolding of musical structure and the topological process involved in the composer-shaping of melodic contours are discussed.