Dispersion Filter-DesignIn 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. Methods for designing dispersion allpass filters were summarized in §6.11.3. In this section, we are mainly concerned with how to specify the desired dispersion allpass filter for piano strings.
Perceptual studies regarding the audibility of inharmonicity in stringed instrument sounds  indicate that the just noticeable coefficient of inharmonicity is given approximately by
where is the fundamental frequency of the string vibration in hertz, and --the so-called coefficient of inharmonicity--affects the th partial overtone tuning via
For a stiff string with Young's modulus , radius , length , and tension , the coefficient of inharmonicity is predicted from theory [144, p. 65], to be
where is the string cross-sectional area, and is the radius of gyration of the string cross-section (see §B.4.9). 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 . Application of the method of  to piano-string dispersion-filter design is reported in . A Faust implementation of a closed-form expression  for dispersion allpass coefficients as a function of inharmonicity coefficient may be found in the function piano_dispersion_filter within effect.lib.
Nonlinear Piano-String Synthesis