Navigating a Chaotic System

Musical signals are acoustic signals produced by well-trained musicians who have a fine sense of tuning their physical movements with respect to their instruments. These musicians often display a good ability to achieve desired quality of sounds. Orderly movements on their instruments with intended irregularities in their physical control are guided by their ears. Ears are fine judges of sounds that musicians are producing. To be a performer she/he is a listener. As a listener she/he remembers and predicts correlations between sounds and actions. And making the correlation is what distinguishes music, even contemporary or modern music, from sounds generated by cats walking on a piano keyboard.

At the same time when we refer to musical signals we refer to a flexible semiotic function of a broad range of sounds. When we walk through an alley in our neighborhood on Sunday afternoon we hear the five-year-olds fingering through a simple Bayer tune. The next moment we hear the graduate music student sweeping through piano etudes by Chopin. These are musical signals that we distinguish one from another. Given our familiarity of social context we accept the differences of these sounds and attribute their differences to our notion of musical signals.

With a computer we can model the familiar musical instruments such as piano, clarinet, and more. With a computer we can also model unfamiliar instruments of which sounds are unknown to us. For simulating instruments such as the piano the goal of the project is to achieve a quality of sounds as real as the sounds from a real instrument. On the other hand unfamiliar instruments are modeled after abstract numerical descriptions. In this case the goal of the project is to realize the range of expression of inherent properties of numerical models through acoustic information space.

We present one of these unfamiliar instruments, a chaotic oscillator. The system dynamics of this oscillator is described by a set of three ordinary differential equations with a nonlinear function. The solutions from these equations range from simple periodic signals to chaotic signals. Different signals produce different auditory characteristics. We apply parameter variation technique to achieve a variety of signal characteristics. In a computer simulation of this oscillator we control the values of six parameters {R, R0, C1, C2, BP2, Gb} through a graphic interface called the manifold interface. The manifold interface was developed for intuitive navigation of a control space of arbitrary dimensions.

High-dimensional control space is represented as a graphical surface we call a manifold surface. The manifold provides visual cues to regions in control space and assists our interactive exploration of the signals generated from those regions. Complex navigation tasks are encountered as intuitive gestures providing immediate visual and auditory feedback. We provide a brief movie of a composer using the manifold interface in the CAVE to control the simulated oscillator. In this implementation the phase portrait of the signal is displayed in the visual space with the manifold surfaces. Both sound and image are computed and displayed in real time.

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