An ideal stiff-string synthesis model is drawn in Fig. 6.13 . 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 [452, Appendix C].7.8The general transfer function for an allpass filter is given (in the real, single-input, single-output case) by
where is an integer pure-delay in samples (all delay lines are allpass filters),
We may think of as the 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.