Force-Pulse Synthesis

Figure 9.32: Creating a single hammer-string interaction force-pulse as the impulse response of a filter. The filter depends on the hammer-string collision velocity $ v_c$, but it is LTI while $ v_c$ is fixed.

The creation of a single force-pulse for a given hammer-string collision velocity $ v_c$ (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 $ v_c$ 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 $ f(x)=k\,x^p$, where $ x$ is felt compression. Here, the brightness is increased by shrinking the duration of the filter impulse response as $ v_c$ increases. The key property enabling commuted synthesis is that, when $ v_c$ 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 $ v_c$.

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Finite Difference Implementation