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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
.