The creation of a single force-pulse for a given hammer-string collision velocity (a specific ``dynamic level'') is shown in Fig.9.32. The filter input is an impulse, and the output is the desired hammer-string force pulse. As increases, the output pulse increases in amplitude and decreases in width, which means the filter is nonlinear. In other words, the force pulse gets ``brighter'' as its amplitude (dynamic level) increases. In a real piano, this brightness increase is caused by the nonlinear felt-compression in the piano hammer. Recall from §9.3.2 that piano-hammer felt is typically modeled as a nonlinear spring described by , where is felt compression. Here, the brightness is increased by shrinking the duration of the filter impulse response as increases. The key property enabling commuted synthesis is that, when is constant, the filter operates as a normal LTI filter. In this way, the entire piano has been ``linearized'' with respect to a given collision velocity .
Multiple Force-Pulse Synthesis
Finite Difference Implementation