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 [568] was suggested for designing allpass filters having a phase delay corresponding to the delay profile needed for a stiff string simulation [437, pp. 60,172]. The method of [568] was streamlined in [377]. In [80], piano strings were modeled using finite-difference techniques. An update on this approach appears in [45]. In [350], high quality stiff-string sounds were demonstrated using high-order allpass filters in a digital waveguide model. In [389], this work was extended by applying a least-squares allpass-design method [280] and a spectral Bark-warping technique [467] 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 norm [279] has been used to calibrate a series of allpass-filter sections [41,40], and a dynamically tunable method, based on Thiran's closed-form, maximally flat group-delay allpass filter design formulas (§K.2), has recently been proposed [376].

Perceptual studies regarding the audibility of inharmonicity in stringed instrument sounds [216] indicate that the just noticeable coefficient of inharmonicity is given approximately by