Nonlinear Dynamics in Physical Models: Simple Feedback-Loop Systems and Properties

Abstract

This work establishes a new approach to the functioning of musical instruments that is oriented toward sound synthesis by computer, and its application to musical creation and composition. In this work, we only consider musical instruments that produce sustained sound. Our approach, which relies on the theory of nonlinear dynamical systems, provides theoretical results regarding instruments, their models, a class of equations with delay, and sound synthesis itself. Experimental and practical results open new sonic possibilities in terms of sound material and the control of sound synthesis, both of which are important to performers and composers of contemporary music. Out of these results, new possibilities also arise for the control of chaotic sounds and the control of the proportion of nonperiodic components introduced in sound by chaotic behavior. Therefore, new artificial instruments can be designed that fulfill the fundamental properties of a musical instrument: richness of the sonic space, expressivity, flexibility, predictability, and ease of control of sonic results. A model of brass instruments simulated on a workstation and played in real time which exemplifies these remarkable properties is presented in a second article in this issue, entitled “Nonlinear Dynamics in Physical Models: From Basic Models to True Musical-Instrument Models.” The notion of a “model” is essential for a better comprehension and use of the properties of sound analysis and synthesis methods. Many physical models of musical instruments have been proposed and studied by various authors (for an overview, see Smith’s [1996] work, for instance). The approach described in this article is based on some advantages and difficulties specific to physical models. From a fundamental point of view, it appears that the complexity of physical models comes in part from their nonlinear nature. Therefore, their study should rely on the increasingly rich theory of nonlinear dynamical systems. However, developers should not merely build models and deliver them to musicians; rather, they should help musicians understand the models by conceiving abstractions of the models and offering useful explanations. Above all, it is necessary for the user to understand the structure of the space of instrumental sounds. In particular, this comprehension is indispensable for elaborating the control of synthesis models that are at the same time efficient and musically pertinent. To fulfill the requirements mentioned above, we have developed models as archetypes, i.e., models that retain the essence of the behavior of a class of instruments while disregarding all details that are not useful for understanding what is typical of that class. The existence of delayed-feedback loops in the equations of the models is a characteristic