### 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

*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

^{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

*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

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Stiff Piano Strings

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Piano Hammer Modeling