TitleNonlinear Oscillator Models of Musical Instrument Excitation
Publication TypeConference Paper
Year of Publication1992
AuthorsRodet, X
Conference NameInternational Computer Music Conference
PublisherInternational Computer Music Association
Conference LocationSan Jose, CA

Many physical models of sustained musical instruments have been proposed (strings, brass, reeds, flutes and voice [ACROE 90]). Mathematically, these models are dynamic systems [Khalil 92] which can be described by autonomous retarded functional difference and differential equations such as: (1), x'(t)=f(x(t), x(t-T)), or (2), x(t)=f(h[[circlemultiply]]x(t-T)), where f is a nonlinear function, T some delay, h a convolution kernel and [[circlemultiply]] the convolution operator. As an example, the system x' = v and v' = av-bv3-kx has been proposed by Lord Rayleigh [Abraham 82] as a model of clarinet oscillation but we will see below that another model is now preferred.A standard model of speech production [Ishizaka 72] uses a two-mass model of vocal cords consisting of two coupled Nonlinear Oscillators (NLO) representing a vibrating vocal cord. A one mass model can be described by a nonlinear system: x' = v, v' = g(x,u,v) and u' = (c+ax)[f(x,u,v)-h[[circlemultiply]]u], which has the form of the equations (1) and (2) above without the x(t-T) delayed term. But for strings, reed-woodwinds or brass, the delay term plays an essential role. In some clarinet models the instrument itself is represented by a delay line and the non-linear excitation is represented by a time-varying pressure- or velocity-controlled reflection coefficient. Similarly, a very accurate model of clarinet excitation has been designed at IRCAM. The reed is an NLO driven by mouth pressure and bore pressure. In violin models also (e.g. at CCRMA, ACROE and IRCAM) the string is set into oscillation by the bow and the combination is an NLO. We have also implemented a model for the lips of the trumpet player [Rodet in ACROE 90], where lips are represented by an NLO with two degrees of freedom, moved by pressure from the mouth and mouth piece.The delay found in the previous instrumental models comes from the instrument itself which is relatively easy to measure or estimate and can be modeled rather accurately as a linear system, by using, for instance, a state-space representation[Depalle 92]. One of the other key points for music synthesis is modeling the excitation process. This is why we present here research on non linear oscillators coupled to passive linear systems as a general model of a large class of musical instruments [McIntyre 83].