### 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].^{7.8}The general transfer function for an allpass filter is given (in the
real, single-input, single-output case) by

*flip*of . For example, if , we have . Thus, is obtained from by simply reversing the order of the coefficients (and conjugating them if they are complex, but normally they are real in practice). For an allpass filter simulating stiffness, we would normally have , since the filter is already in series with a delay line.

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:**

Equivalent Forms

**Previous Section:**

Terminated String Impedance