Light strikes lead to bell-like inharmonic sounds; at higher strike amplitudes, such instruments exhibit dramatic nonlinear behaviour, involving changes in pitch, a rapid increase in high-frequency energy, subharmonic generation, and even chaotic noise [1].
Linear plate and (when curved) shell models are completely insufficient to give even a rough approximation to these sounds…a nonlinear model is really necessary!
Linear shell model:
Nonlinear shell model:
…and a repeated striking gesture:
A gong:
Through the NESS project, we have concentrated on thin nonlinear flat plates and spherical curved shells [2], but it is clear that there is much more work to be done, specifically in the extension to more realistic settings. Features of interest are:
- curved shells of variable thickness
- supporting conditions at center
- loss modelling
The first feature has a great impact on the resulting sound, and is indeed a key design attribute for such instruments—gongs, for instance, often have a raised central dome, and a curved thick rim [3]. The second allows for distinctions among various types of cymbals: high-hats are clamped, ride cymbals are able to pivot, and orchestral crash cymbals are unsupported. Loss modelling in such structures is far more important than in the linear case, as it is a determining perceptual feature in the over-all migration of energy in the sound spectrum of such instruments, particularly in cymbal crashes and swells of gongs, and is also crucial, numerically, in preventing aliasing. A fourth feature, the full embedding of such structures in 3D for spatialized synthesis, has been dealt with in great detail by Alberto Torin.
References
[1] T. Rossing and N. Fletcher. Nonlinear Vibrations in Plates and Gongs, J. of the Acoustical Soc. of Am., 73(1):345—351, 1983.
[2] S. Bilbao. Percussion Synthesis Based on Models of Nonlinear Shell Vibration, IEEE Trans. on Audio Speech and Language Processing, 18(4):872—880, 2010.
[3] N. Fletcher and T. Rossing. The Physics of Musical Instruments, Springer-Verlag, New York, 1991.
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