Matti Karjalainen

Plucked-string models: From the Karplus-Strong algorithm to digital waveguides and beyond

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, Valimaki, and Janosy 1993; Vilimaki, Huopaniemi, Karjalainen, and Jainosy 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 already 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) mod l. 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 algorithm was designed. This articles 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, Vilimiki, and Janosy 1993). Here we discuss them in the context of our recent implementations of plucked-string models.

Modeling of tension modulation nonlinearity in plucked strings

A nonlinear discrete-time model that simulates a vibrating string exhibiting tension modulation nonlinearity is developed. The tension modulation phenomenon is caused by string elongation during transversal vibration. Fundamental frequency variation and coupling of harmonic modes are among the perceptually most important effects of this nonlinearity. The proposed model extends the linear bidirectional digital waveguide model of a string. It is also formulated as a computationally more efficient single-delay-loop structure. A method of reducing the computational load of the string elongation approximation is described, and a technique of obtaining the tension modulation parameter from recorded plucked string instrument tones is presented. The performance of the model is demonstrated with analysis/synthesis experiments and with examples of synthetic tones.

Acoustical analysis and model-based sound synthesis of the kantele

The five-string Finnish kantele is a traditional folk music instrument that has unique structural features, resulting in a sound of bright and reverberant timbre. This article presents an analysis of the sound generation principles in the kantele, based on measurements and analytical formulation. The most characteristic features of the unique timbre are caused by the bridgeless string termination around a tuning pin at one end and the knotted termination around a supporting bar at the other end. These result in prominent second-order nonlinearity and strong beating of harmonics, respectively. A computational model of the instrument is also formulated and the algorithm is made efficient for real-time synthesis to simulate these features of the instrument timbre.

Plucked-string synthesis algorithms with tension modulation nonlinearity

Digital waveguide modeling of a nonlinear vibrating string is investigated when the nonlinearity is essentially caused by tension modulation. We derive synthesis models where the nonlinearity is implemented with a time-varying fractional delay filter. Also, conversion from a dual-delay-line physical model into a single-delay-loop model is explained. Realistic synthetic tones with nonlinear effects are obtained by introducing minor amendments to a linear string synthesis algorithm. It is shown how synthetic plucked-string tones are modified as a consequence of tension modulation.

Analysis, modeling, and real-time sound synthesis of the kantele, a traditional Finnish string instrument

The kantele is a unique old Finnish string instrument that has been found to have interesting acoustical principles of sound production, including nonlinear behavior and dual-mode vibrations of the strings. The main interest was in the most traditional five-string kantele. It is a musical instrument tuned to the pentatonic scale. The authors describe results from a study where the acoustics of the instrument were first analyzed, its sound generation principles were then modeled computationally, and finally a real-time sound synthesis algorithm was developed for a floating-point signal processor. The simplified model of the kantele, when properly adjusted, leads to synthetic sounds that well resemble the original instrument.<<ETX>>