This is the homepage of the paper

Balázs Bank and László Sujbert, "Efficient Modeling Strategies for the Geometric Nonlinearities of Musical Instrument Strings," Proceedings of the Forum Acusticum, Budapest, Hungary, Aug. 2005.

Abstract

Sound synthesis algorithms modeling the linear behavior of strings are well developed. However, some musical instruments require the modeling of such nonlinear phenomena as the appearance of longitudinal string modes, phantom partials, or mode coupling and pitch glide due to tension variation. Accordingly, the effects of geometric nonlinearities in strings are gaining more and more interest in the sound synthesis community. These effects can be grouped into four different regimes, depending on the transverse slope and on the ratio of longitudinal and transverse fundamental frequencies. In some cases only the coupling from the transverse to the longitudinal polarization is significant, while in others both directions of coupling are important. Another question is whether the inertial effects of longitudinal modes have to be modeled or not. The four cases arising from the combinations of these factors are outlined in the paper. The most common string modeling approaches -- finite difference modeling, digital waveguides and modal models -- are investigated with respect to their ability to model the different effects of geometric nonlinearities. The paper proposes the combined use of different modeling approaches to reduce the computational cost required for modeling the aforementioned phenomena.

Sound examples

The following examples demonstrate the sound of the different regimes of nonlinearity by varying the parameters of a piano string. The example G1 forte tone with realistic parameters is here.

The examples demonstrated by the following wav files are also available as a ppt slide, where the different regimes of nonlinearity are plotted graphically.

Linear motion

Here the amplitude of the string excitation was set to 10 % of its real value. As a result, only the linear transverse components are present.

Phantom partials

This is an example with normal amplitude, but due to the large longitudinal/transverse fundamental frequency ratio the longitudinal modes are not excited. Thus, the tension is spatially uniform along the string, and the tension variation leads to double frequency terms in the longitudinal force, but it is to small to act back to the transverse vibration. Note that for this particular amplitude of vibration this is hardly audible.

Tension modulation

This is an example with five times higher amplitude. As before, due to the large longitudinal/transverse fundamental frequency ratio the longitudinal modes are not excited. Thus, the tension is spatially uniform along the string, but now the tension variation is large enough to influence the transverse vibration. The audible effect is a pitch glide showing the continous decrease of tension.

Longitudinal components

This is the G1 forte tone with realistic parameters, meaning that due to the longitudinal/transverse fundamental frequency ratio is in the order of 20, the longitudinal components are excited around and above resonance, leading to a spatially nonuniform tension. However, because of the moderate amplitude of vibration, the tension variation does not influence the transverse vibration significantly. The main perceptual effect is the metallic sound due to the longitudinal components.

Bidirectional coupling

The same as above but with five times higher excitation force. Here neither the tension is spatially unifrom along the string, nor the tension variation is negligible in comparison to the tension at rest. As a result, the transverse and longitudinal polarizations interact with each other, leading to the main perceptual effect of the metallic longitudinal components and to a pitch glide.

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