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Multisymplectic Pseudo-Spectral Finite Difference Methods for Physical Models of Musical Instruments

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Sound - Perception - Performance

Part of the book series: Current Research in Systematic Musicology ((CRSM,volume 1))

Abstract

Physical Modeling (PM) of musical instruments has gained rising interest over the last decade. This is mainly due to the rising processing capabilities of standard computers making it possible to calculate PM solutions of complete instrument geometries in reasonable computational time or real-time using specialised hardware. Several theoretical advances of discrete solution methods developed over the last 100 years are being explored for PM of musical instruments for the first time. In this work some basic properties of two methods for PM of musical instruments a) Pseudo-Spectral Methods and b) geometry preserving methods known as symplectic (for ODEs) or multisymplectic (for PDEs) are presented and implemented. Two simplified models of musical instruments are presented as a proof of concept.

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Notes

  1. 1.

    COMSOL, ANSYS and many others.

  2. 2.

    For the wave equation all space variables and the time variable is discretised.

  3. 3.

    Moore and Reich (2003a) show that in many cases the reversibility of an algorithm is far more important than an exact energy conservation.

  4. 4.

    To this point many results for multisymplectic integrators are only shown numerically. Some properties have yet to be generalised (Moore 2009; Moore and Reich 2003b).

  5. 5.

    If this is physically correct is a question that can not be answered here. The analytical wave equation has no speed limit for transported information. Whereas discrete approximations have a speed limit set by the discretisation step width in space and time. This is the stability condition expressed by Courant et al. (1928).

  6. 6.

    The Velocity Verlet method is similar to the Leapfrog algorithm.

  7. 7.

    In MATLAB the build-in fft()-function is used. For the C++ implementation the fftw-library (Frigo and Johnson 1997, 1998), the CUFFT-library (NVIDIA CUDA 2013) and a template based Fast Fourier Transform is used.

  8. 8.

    Mean length between two ears.

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Authors and Affiliations

  1. Institute of Musicology, University of Hamburg, Neue Rabenstr. 13, 20354, Hamburg, Germany

    Florian Pfeifle

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  1. Florian Pfeifle
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Correspondence to Florian Pfeifle .

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  1. Institute of Musicology, University of Hamburg, Neue Rabenstr. 13, Hamburg, 20354, Germany

    Rolf Bader

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Pfeifle, F. (2013). Multisymplectic Pseudo-Spectral Finite Difference Methods for Physical Models of Musical Instruments. In: Bader, R. (eds) Sound - Perception - Performance. Current Research in Systematic Musicology, vol 1. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00107-4_15

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  • DOI: https://doi.org/10.1007/978-3-319-00107-4_15

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