Models of musical string vibration

Musical string vibration has been the subject of scientific study for centuries. Recent increases in computational power have allowed the exploration of increasingly detailed features of perceptual significance through simulation approaches. The starting point for any simulation is a well-defined model, usually framed as a system of differential equations, with parameters determined by measurement and experiment. This review article is intended to take the reader through models of string vibration progressively, beginning with well-known and well-studied linear models, and then introducing new features that form the basis for the modern study of realistic musical string vibration. These include, first, nonlinear excitation mechanisms, such as the hammer-string and bow-string interaction, and then the collision mechanism, both for pointwise obstructions and over a distributed region. Finally, the linear model of string vibration is generalized to include geometric nonlinear effects, leading to typical nonlinear behaviour such as pitch glides and the appearance of so-called phantom partials due to nonlinear mixing of modes. The article concludes with a general overview of numerical simulation techniques for string vibration.