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 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, Vilimiki, and Janosy 1993). Here we discuss them in the context of our recent implementations of plucked-string models.

https://www.jstor.org/stable/pdfplus/3681155.pdf

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

Recent research in physical modeling of musical instruments for purposes of sound synthesis is reviewed. Recent references, results, and outstanding problems are highlighted for models of strings, winds, brasses, percussion, and acoustic spaces. Emphasis is placed on digital waveguide models and the musical acoustics research on which they are based.

/pdf/physical-modeling-synthesis-update-1e4o3kdlt6.pdf

Physical Modeling Synthesis Update

As machines become more and more portable, and part of our everyday life, it becomes apparent that developing interactive and ubiquitous systems is an important aspect of new music applications created by the research community. We are interested in developing a robust layer for the automatic annotation of audio signals, to be used in various applications, from music search engines to interactive installations, and in various contexts, from embedded devices to audio content servers. We propose adaptations of existing signal processing techniques to a real time context. Amongst these annotation techniques, we concentrate on low and mid-level tasks such as onset detection, pitch tracking, tempo extraction and note modelling. We present a framework to extract these annotations and evaluate the performances of different algorithms. The first task is to detect onsets and offsets in audio streams within short latencies. The segmentation of audio streams into temporal objects enables various manipulation and analysis of metrical structure. Evaluation of different algorithms and their adaptation to real time are described. We then tackle the problem of fundamental frequency estimation, again trying to reduce both the delay and the computational cost. Different algorithms are implemented for real time and experimented on monophonic recordings and complex signals. Spectral analysis can be used to label the temporal segments; the estimation of higher level descriptions is approached. Techniques for modelling of note objects and localisation of beats are implemented and discussed. Applications of our framework include live and interactive music installations, and more generally tools for the composers and sound engineers. Speed optimisations may bring a significant improvement to various automated tasks, such as automatic classification and recommendation systems. We describe the design of our software solution, for our research purposes and in view of its integration within other systems.

/pdf/automatic-annotation-of-musical-audio-for-interactive-2t6jfn3vud.pdf

Automatic annotation of musical audio for interactive applications

A model of transverse piano string vibration, second order in time, which models frequency-dependent loss and dispersion effects is presented here. This model has many desirable properties, in particular that it can be written as a well-posed initial-boundary value problem (permitting stable finite difference schemes) and that it may be directly related to a digital waveguide model, a digital filter-based algorithm which can be used for musical sound synthesis. Techniques for the extraction of model parameters from experimental data over the full range of the grand piano are discussed, as is the link between the model parameters and the filter responses in a digital waveguide. Simulations are performed. Finally, the waveguide model is extended to the case of several coupled strings.

/pdf/the-simulation-of-piano-string-vibration-from-physical-xp8nscvivp.pdf

The simulation of piano string vibration: from physical models to finite difference schemes and digital waveguides.

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.

/pdf/modeling-of-tension-modulation-nonlinearity-in-plucked-1hbgej4car.pdf

Modeling of tension modulation nonlinearity in plucked strings

Efficient Synthesis of Stringed Musical Instruments

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Efficient Synthesis of Stringed Musical Instruments