Finite Difference Implementation
Digitizing
via the centered second-order difference
[Eq.
(7.5)]

![$\displaystyle \underline{X}_{n+1} \eqsp \left[2\mathbf{I}+ \mathbf{M}^{-1}\mathbf{A}\right]\underline{X}_n - \underline{X}_{n-1} + B\uv_n,
$](http://www.dsprelated.com/josimages_new/pasp/img2296.png)

Note that requiring three adjacent spatial string samples to be in
contact with the piano hammer during the attack (which helps to
suppress aliasing of spatial frequencies on the string during the
attack) implies a sampling rate in the vicinity of 6 megahertz
[265]. Thus, the model is expensive to compute!
However, results to date show a high degree of accuracy, as desired.
In particular, the stretching of the partial overtones in the
stiff-string model of
Fig. has
been measured to be highly accurate despite using only three spring
attachment points on one side of each mass disk [265].
See [53] for alternative finite-difference formulations that better preserve physical energy and have other nice properties worth considering.
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Nonlinear Piano-String Equations of Motion in State-Space Form