Plucked-String Models: From the Karplus-Strong Algorithm to Digital Waveguides and Beyond

Matti Karjalainen, Vesa Välimäki, and Tero Tolonen

Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing, P.O. Box 3000, FIN-02015 HUT, Espoo, Finland, URL:

The emergence of what is called physical modeling and model-based sound synthesis is closely related to the development of computational simulations of plucked string instruments. Historically, the first physical approaches (Hiller and Ruiz 1971a, 1971b; McIntyre and Woodhouse 1979; McIntyre, Schumacher, and Woodhouse 1983) were followed by the Karplus-Strong (KS) algorithm (Karplus and Strong 1983). The KS algorithm was discovered as a simple computational technique that seemingly had nothing to do with physics. Soon thereafter, Julius Smith and David Jaffe showed a deeper understanding of its relation to the physics of the plucked string (Smith 1983; Jaffe and Smith 1983).

Later, Julius Smith generalized the underlying ideas of the KS algorithm by introducing the theory of digital waveguides (Smith 1987). Digital waveguides are physically relevant abstractions yet computationally efficient models, not only for plucked strings, but for a variety of one-, two-, and three-dimensional acoustic systems (Van Duyne and Smith 1993; Savioja, Rinne, and Takala 1994; Van Duyne, Pierce, and Smith 1994). Further investigations embodied these ideas in more detailed synthesis principles and implementations, resulting in high-quality and realistic syntheses of plucked string instruments (Sullivan 1990; Karjalainen and Laine 1991; Smith 1993; Karjalainen, Välimäki, and Jánosy 1993; Välimäki, Huopaniemi, Karjalainen, and Jánosy 1996). A recent overview of research in this field is given by Smith (1996).

The equivalence of Karplus-Strong and digital waveguide formulations in sound synthesis was al-ready known when the waveguide theory appeared (Smith 1987, 1992, 1997); however, the relation has never been explicated in full detail. The first aim of this article is to show how the more "physical" waveguide model of a plucked string can be reduced to an extended form of the Karplus-Strong type that we call the single delay-loop (SDL) model. For a linear and time-invariant (LTI) case, this reduction is relatively straightforward, and results in a computationally more efficient digital filter structure. (Note that the historical order of the KS algorithm and digital waveguides is the reverse of their logical order, since the generalization was not developed until after the KS algorithms was designed. This article's title reflects the historical evolution: the "beyond" refers to recent generalizations and extensions of both concepts.)

The second aim of this article is to discuss further extensions to the basic SDL models that make them capable of simulating plucking styles, beats in string vibration, sympathetic vibrations, and resonant strings. Such techniques have already been proposed and studied (Jaffe and Smith 1983; Smith 1993; Karjalainen, Välimäki, and Jánosy 1993). Here we discuss them in the context of our recent implementations of plucked-string models.

Keywords: physical modeling of musical instruments, sound synthesis, musical acoustics, audio signal processing, vibrating string

This article was published in Computer Music Journal, vol. 22, no. 3, pp. 17-32, Fall 1998.

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Last modified: October 19, 1998
Author: Vesa Välimäki