Math. Dept., Rice Univ., Houston, TX 77252
Chaotic dynamical systems have intrigued artists since the computer produced images of their complex trajectories. For the composer, the structures inherent in these systems and their time evolution provide a framework for musical expression that reflects a contemporary interpretation of reality; for the sound-designer, the same structures can be used to synthesize rich, organic timbres. These two applications are explored in this paper. The map R[sub (alpha)](x):x->x+(alpha) (mod 1) is iterated to produce a fractal waveform for each irrational number (alpha). These quasiperiodic waveforms have complex overtones, and exhibit a quasi-self-similarity reflecting the number-theoretic properties of (alpha). This audification of a system is often a biproduct of the musician's explorations of mathematical structures, and on the compositional level methods of hearing and performing a dynamical system become a primary focus. Performance gestures can be mapped to the space of initial conditions and the parameter space of the system. The time series from the resulting system, along with averaged data (Lyapunov exponents, for example), are mapped to musical parameter space. Several such mappings are illustrated using a toral lattice of two-dimensional twisted logistic maps.