The general technique of commuted synthesis was introduced in §8.7. This method is enabled by the linearity and time invariance of both the vibrating string and its acoustic resonator, allowing them to be interchanged, in principle, without altering the overall transfer function [#!JOSFP!#].
While the piano-hammer is a strongly nonlinear element (§9.3.2), it is nevertheless possible to synthesize a piano using commuted synthesis with high fidelity and low computational cost given the approximation that the string itself is linear and time-invariant (§9.4.2) [#!JOSSVDCP!#,#!SVDJOSCP!#,#!BankEtAl10!#]. The key observation is that the interaction between the hammer and string is essentially discrete (after deconvolution) at only one or a few time instants per hammer strike. The deconvolution needed is a function of the hammer-string collision velocity . As a result, the hammer-string interaction can be modeled as one or a few discrete impulses that are filtered in a -dependent way.
Figure 9.31 illustrates a typical series of interaction force-pulses at the contact point between a piano-hammer and string. The vertical lines indicate the locations and amplitudes of three single-sample impulses passed through three single-pulse filters to produce the overlapping pulses shown.