When the piano hammer strikes the string(s), a pulse travels to the agraffe and back. As long as the hammer remains in contact with the string, there will be subsequent pulses and reflections.
The piano hammer is typically “thrown away” from the string by one of the reflected pulses.
A plot depicting a theoretical hammer-string interaction force is shown in Fig. 8. This plot corresponds to three pulse reflections.
Figure 8:
A theoretical hammer force characteristic.
The actual number of pulse reflections that occur is dependent on the string being hit and the initial velocity of the hammer.
The hammer-string interaction is inherently non-linear, in large part because of the compression characteristics of the hammer felt.
However, because the pulse-hammer interactions occur at discrete times, it is possible to consider them as distinct events that overlap to produce a complete hammer-string characteristic.
Each single pulse can be modeled as a lowpass filtered impulse signal. This suggests a hammer-excited piano synthesis system as diagrammed in Fig. 9.
Figure 9:
A hammer-excited piano synthesis block diagram implementation.
In the system of Fig. 9, an input velocity control triggers an impulse signal. The tapped delay-line is used to produce three subsequent time-delayed impulses, each of which drive a particular lowpass filter. The resultant linear summation of time-delayed lowpass filter impulse responses then drives the string model and its soundboard response filter.
Hammer-string characteristics can be evaluated for various strings and input velocities and appropriate lowpass filter responses designed. The input velocity can then be used as a control parameter to select the appropriate filter characteristics.
Finally, because the implementation described by Fig. 9 is linear, the piano soundboard impulse response can be “commuted” through the system and used to trigger the hammer-string filters as shown in Fig. 10.
Figure 10:
A commuted piano hammer block-diagram implementation.