Stiff-String Dispersion Filter Design

In the context of a digital waveguide string model, dispersion associated with stiff strings can be supplied by an allpass filter in the basic feedback loop. A modification of the method in [563] was suggested for designing allpass filters having a phase delay corresponding to the delay profile needed for a stiff string simulation [432, pp. 60,172]. The method of [563] was streamlined in [372]. In [80], piano strings were modeled using finite-difference techniques. An update on this approach appears in [46]. In [346], high quality stiff-string sounds were demonstrated using high-order allpass filters in a digital waveguide model. In [384], this work was extended by applying a least-squares allpass-design method [278] and a spectral Bark-warping technique [463] to the problem of calibrating an allpass filter of arbitrary order to recorded piano strings. They were able to correctly tune the first several tens of partials for any natural piano string with a total allpass order of 20 or less. Additionally, minimization of the $L^\infty$ norm [277] has been used to calibrate a series of allpass-filter sections [42,41], and a dynamically tunable method, based on Thiran's closed-form, maximally flat group-delay allpass filter design formulas (§J.2), has recently been proposed [371].

Perceptual studies regarding the audibility of inharmonicity in stringed instrument sounds [214] indicate that the just noticeable coefficient of inharmonicity is given approximately by

$$B_{\mbox{thresh}} = \exp(2.54\log(f_0)-24.6)$$

where $f_0$ is the fundamental frequency of the string vibration in Hertz, and $B$--the so-called coefficient of inharmonicity--affects the $n$th partial overtone tuning via

$$f_n = nf_0\sqrt{1+Bn^2}.$$

For a stiff string with Young's modulus $E$, radius $a$, length $L$, and tension $K$, the coefficient of inharmonicity, $B$, is predicted from theory [142, p. 65],[214] to be

$$B = \frac{\pi^2 E S I^2}{K L^2} = \frac{\pi^3 E a^4}{16 L^2 K},$$

where $S=\pi a^2$ is the string cross-sectional area, and $I=a/2$ is the radius of gyration of the string cross-section (see §E.4.5).

In general, when designing dispersion filters for vibrating string models, it is highly cost-effective to obtain an allpass filter which correctly tunes only the lowest-frequency partial overtones, where the number of partials correctly tuned is significantly less than the total number of partials present, as in [384].