### 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

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
[128]

#### Digital Waveguide Plucked-String Model

When plucking a string, it is necessary to detect ``collisions''
between the plectrum and string. Also, more complete plucked-string
models will allow the string to ``buzz'' on the frets and ``slap''
against obstacles such as the fingerboard. For these reasons, it is
convenient to choose *displacement waves* for the waveguide
string model. The reflection and transmission filters for
displacement waves are the same as for velocity, namely,
and
.

As in the mass-string collision case, we obtain the one-filter scattering-junction implementation shown in Fig.9.23. The filter may now be digitized using the bilinear transform as previously (§9.3.1).

#### Incorporating Control Motion

Let denote the vertical position of the *mass*
in Fig.9.22. (We still assume .) We can think of
as the position of the *control point* on the
plectrum, *e.g.*, the position of the ``pinch-point'' holding the
plectrum while plucking the string. In a harpsichord, can be
considered the *jack* position [347].

Also denote by the *rest length* of the spring
in Fig.9.22, and let
denote the
position of the ``end'' of the spring while not in contact with the
string. Then the plectrum makes *contact* with the string when

*collision detection*equation.

Let the subscripts and each denote one side of the scattering system, as indicated in Fig.9.23. Then, for example, is the displacement of the string on the left (side ) of plucking point, and is on the right side of (but still located at point ). By continuity of the string, we have

When the spring engages the string () and begins to compress, the upward force on the string at the contact point is given by

^{10.15}For or the applied force is zero and the entire plucking system disappears to leave and , or equivalently, the force reflectance becomes and the transmittance becomes .

During contact, force equilibrium at the plucking point requires
(*cf.* §9.3.1)

where as usual (§6.1), with denoting the string tension. Using Ohm's laws for traveling-wave components (p. ), we have

Substituting and taking the Laplace transform yields

where, as first noted at Eq.(9.24) above,

This system is diagrammed in Fig.9.24. The manipulation of the
minus signs relative to Fig.9.23 makes it convenient for
restricting to positive values only (as shown in the
figure), corresponding to the plectrum engaging the string going up.
This uses the approximation
,
which is exact when
, *i.e.*, when the plectrum does not affect
the string displacement at the current time. It is therefore exact at
the time of collision and also applicable just after release.
Similarly,
can be used to trigger a release of the
string from the plectrum.

#### Successive Pluck Collision Detection

As discussed above, in a simple 1D plucking model, the plectrum comes
up and engages the string when
, and above some
maximum force the plectrum releases the string. At this point, it is
``above'' the string. To pluck again in the same direction, the
collision-detection must be disabled until we again have ,
requiring one bit of state to keep track of that.^{10.16} The harpsichord jack
plucks the string only in the ``up'' direction due to its asymmetric
behavior in the two directions [143]. If only
``up picks'' are supported, then engagement can be suppressed after a
release until comes back down below the envelope of string
vibration (*e.g.*,
). Note that
intermittent disengagements as a plucking cycle begins are normal;
there is often an audible ``buzzing'' or ``chattering'' when plucking
an already vibrating string.

When plucking up and down in alternation, as in the *tremolo*
technique (common on mandolins), the collision detection alternates
between and , and again a bit of state is needed to
keep track of which comparison to use.

#### Plectrum Damping

To include damping in the plectrum model, the load impedance goes back to Eq.(9.22):

#### Digitization of the Damped-Spring Plectrum

Applying the bilinear transformation (§7.3.2) to the reflectance in Eq.(9.23) (including damping) yields the following first-order digital force-reflectance filter:

The transmittance filter is again , and there is a one-filter form for the scattering junction as usual.

#### Feathering

Since the pluck model is linear, the parameters are not
signal-dependent. As a result, when the string and spring separate,
there is a discontinuous change in the reflection and transmission
coefficients. In practice, it is useful to ``feather'' the
switch-over from one model to the next [470]. In
this instance, one appealing choice is to introduce a *nonlinear
spring*, as is commonly used for piano-hammer models (see
§9.3.2 for details).

Let the nonlinear spring model take the form

The foregoing suggests a nonlinear tapering of the damping in addition to the tapering the stiffness as the spring compression approaches zero. One natural choice would be

In summary, the engagement and disengagement of the plucking system can be ``feathered'' by a nonlinear spring and damper in the plectrum model.

**Next Section:**

Stiff Piano Strings

**Previous Section:**

Piano Hammer Modeling