The first attempt to generate musical sounds by solving the equations of vibrating strings by means of finite difference methods (FDM) was made by Hiller and Ruiz [J. Audio Eng. Soc. 19, 462472 (1971)]. It is shown here how this numerical approach and the underlying physical model can be improved in order to simulate the motion of the piano string with a high degree of realism. Starting from the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, as well as the force acting on the bridge, are given by the same scheme. The performance of the model is illustrated by a few examples of simulated string waveforms. A brief discussion of the aspects of numerical stability and dispersion with reference to the proper choice of sampling parameters is also included.
The vibration levels in four traditional stringed instruments during playing have been investigated, including the double bass, violin, guitar, and the piano. The vibration levels, which were measured at several positions and at different dynamic levels, were evaluated with respect to reported thresholds for detection of vibrotactile stimuli. The results show that the vibration levels are well above threshold for almost all positions on the instruments in normal playing. It is concluded that the perceived vibrations may be of some assistance with regard to intonation in ensemble playing, in particular for the bass instruments. The finger forces exerted when playing the bowed strings, as well as the touch forces in piano playing were studied briefly. It was concluded that the kinesthetic forces perceived in playing may assist the timing in performance.
A physical model of the piano string, using finite difference methods, has recently been developed. [Chaigne and Askenfelt, J. Acoust. Soc. Am. 95, 1112-1118 (1994)]. The model is based on the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, from which a numerical finite difference scheme is derived. In the present study, the performance of the model is evaluated by systematic comparisons between measured and simulated piano tones. After a w•rification of the accuracy of the method, the model is used as a tool for systematically exploring the influence of string stiffness, relative striking position, and hammer-string mass ratio on stri.ng waveforms and spectra.
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