#### Efficient Waveguide Synthesis of Nonlinear Piano Strings

The following nonlinear waveguide piano-synthesis model is under current consideration [427]:

- Initial striking force determines the starting regime (1, 2, or 3 above).
- The string model is simplified as it decays.

*e.g.*, fortissimo for the first three octaves of the piano keyboard [26]), the string model starts in regime 2.

Because the energy stored in a piano string decays monotonically after
a hammer strike (neglecting coupling from other strings), we may
*switch to progressively simpler models* as the energy of
vibration falls below corresponding thresholds. Since the purpose of
model-switching is merely to save computation, it need not happen
immediately, so it may be triggered by a string-energy estimate based
on observing the string at a *single point* over the past
period or so of vibration. Perhaps most simply, the model-regime
classifier can be based on the maximum magnitude of the *bridge
force* over at least one period. If the regime 2 model includes an
instantaneous string-length (tension) estimate, one may simply compare
that to a threshold to determine when the simpler model can be used.
If the longitudinal components and/or phantom partials are not
completely inaudible when the model switches, then standard
cross-fading techniques should be applied so that inharmonic partials
are faded out rather than abruptly and audibly cut off.

To obtain a realistic initial shock noise in the tone for regime 1 (and for any model that does not compute it automatically), the appropriate shock pulse, computed as a function of hammer striking force (or velocity, displacement, etc.), can be summed into the longitudinal waveguide during the hammer contact period.

The longitudinal bridge force may be generated from the estimated string length (§9.1.6). This force should be exerted on the bridge in the direction coaxial with the string at the bridge (a direction available from the two transverse displacements one sample away from the bridge).

Phantom partials may be generated in the longitudinal waveguide as explicit intermodulation products based on the transverse-wave overtones known to be most contributing; for example, the Goetzel algorithm [451] could be used to track relevant partial amplitudes for this purpose. Such explicit synthesis of phantom partials, however, makes modal synthesis more compelling for the longitudinal component [30]; in a modal synthesis model, the longitudinal attack pulse can be replaced by a one-shot (per hammer strike) table playback, scaled and perhaps filtered as a function of the hammer striking velocity.

On the high end for regime 2 modeling, a full nonlinear coupling may be implemented along the three waveguides (two transverse and one longitudinal). At this level of complexity, a wide variety of finite-difference schemes should also be considered (§7.3.1) [53,555].

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Checking the Approximations

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Regimes of Piano-String Vibration