The Stiff String
Stiffness in a vibrating string introduces a restoring force proportional to the bending angle of the string. As discussed further in §C.6, the usual stiffness term added to the wave equation for the ideal string yields

Stiff-string models are commonly used in piano synthesis. In §9.4, further details of string models used in piano synthesis are described (§9.4.1).
Experiments with modified recordings of acoustic classical guitars indicate that overtone inharmonicity due to string-stiffness is generally not audible in nylon-string guitars, although just-noticeable-differences are possible for the 6th (lowest) string [225]. Such experiments may be carried out by retuning the partial overtones in a recorded sound sample so that they become exact harmonics. Such retuning is straightforward using sinusoidal modeling techniques [359,456].
Stiff String Synthesis Models
An ideal stiff-string synthesis model is drawn in
Fig. 6.13 [10]. See
§C.6 for a detailed derivation. The delay-line length
is the number of samples in
periods at frequency
, where
is the number of the highest partial supported (normally the last
one before
). This is the counterpart of
Fig. 6.12 which depicted ideal-string damping which
was lumped at a single point in the delay-line loop. For the
ideal stiff string, however, (no damping), it is dispersion
filtering that is lumped at a single point of the loop. Dispersion
can be lumped like damping because it, too, is a linear,
time-invariant (LTI) filtering of a propagating wave. Because it is
LTI, dispersion-filtering commutes with other LTI systems in
series, such as delay elements. The allpass filter in
Fig.C.9 corresponds to filter
in Fig.9.2 for
the Extended Karplus-Strong algorithm. In practice, losses are also
included for realistic string behavior (filter
in
Fig.9.2).
Allpass filters were introduced in §2.8, and a fairly comprehensive summary is given in Book II of this series [449, Appendix C].7.8The general transfer function for an allpass filter is given (in the real, single-input, single-output case) by












Section 6.11 below discusses some
methods for designing stiffness allpass filters from
measurements of stiff vibrating strings, and
§9.4.1 gives further details for the case of piano
string modeling.
Next Section:
The Externally Excited String
Previous Section:
Frequency-Dependent Damping