Pluck Modeling

The piano-hammer model of the previous section can also be configured
as a *plectrum* by making the mass and damping small or zero, and
by releasing the string when the contact force exceeds some threshold
. That is, to a first approximation, a plectrum can be modeled
as a *spring* (linear or nonlinear) that disengages when either
it is far from the string or a maximum spring-force is exceeded. To
avoid discontinuities when the plectrum and string engage/disengage,
it is good to taper both the damping and spring-constant to zero at
the point of contact (as shown
below).

Starting with the piano-hammer impedance of Eq.
(9.19) and setting
the mass
to infinity (the plectrum holder is immovable), we define
the *plectrum impedance* as

The force-wave reflectance of impedance
in Eq.
(9.22), as
seen from the string, may be computed exactly as in
§9.3.1:

If the spring damping is much greater than twice the string wave impedance ( ), then the plectrum looks like a rigid termination to the string (force reflectance ), which makes physical sense.

Again following §9.3.1, the transmittance for force waves is given by

and for velocity and displacement waves, the reflectance and transmittance are respectively and .

If the damping
is set to zero, *i.e.*, if the plectrum is to be modeled
as a simple linear spring, then the impedance becomes
,
and the force-wave reflectance becomes
[129]

- Digital Waveguide Plucked-String Model
- Incorporating Control Motion
- Successive Pluck Collision Detection
- Plectrum Damping
- Digitization of the Damped-Spring Plectrum
- Feathering

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University