### ASA 124th Meeting New Orleans 1992 October

## 4aMU4. Fractal waveforms and cellular dynamata.

**Ami Radunskaya
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*Math. Dept., Rice Univ., Houston, TX 77252
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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.