Stiff String Synthesis Models
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 [449, Appendix C].7.8The general transfer function for an allpass filter is given (in the real, single-input, single-output case) by
Terminated String Impedance