Commuted Piano Synthesis

While the piano-hammer is a nonlinear element, it is nevertheless possible to synthesize a piano using commuted synthesis with high fidelity and low computational cost [475,528]. The key observation is to note that the interaction between the hammer and string is essentially discrete at only one or a few time instants per hammer strike. That is, the hammer-string interaction can be modeled as one or a few discrete impulses which are filtered in a velocity-dependent way.

When the hammer first strikes the string at rest, it ``sees'' the wave impedance of an infinitely long string. As a result, the collision pulse can be well modeled as a filtered impulse.^6.2 This disturbance must propagate to the agraffe and back before any further interaction can take place. Since what comes back is a slightly filtered version of the original interaction pulse, it is also representable as a filtered impulse (just filtered a little more). This secondary pulse passes over the hammer. If the hammer has fallen away from contact with the string by then, there is no further interaction, and the original force pulse is the complete excitation. In cases in which the hammer is still in contact with the string, a new interaction pulse is issued which travels outward from the striking point in a manner similar to the original force pulse. In real pianos, there are typically only a few interaction pulses before the hammer falls away, except at very high pitches.

The nature of the hammer-string interaction is predictable to first order. To a large extent, for example, the number of interaction impulses is determined by which string is being struck. Thus, given key number and hammer velocity, together with current string velocity, one can predict rather well the amplitude and timing of all interaction impulses. There is a slight amount of unpredictability which we neglect having to do with the fact that when the hammer strikes an already vibrating string, the entire history of string vibration influences the exact details of the hammer-string interaction; however, this is a second-order effect which may not even be desirable in the simulation. A pianist has observed that sometimes a note is unexpectedly bright, corresponding perhaps to striking the string while it is rapidly approaching the hammer, and as a result, he would have to play a little brighter thereafter to keep the dynamic level smooth. Perhaps we can actually improve the piano by remov