Scattering Parameters for the Keefe Clarinet Tonehole Model

The clarinet tonehole model developed by Keefe [1981] is parametrized as the cascade of a series reactance, a shunt complex impedance, and another series reactance. The transmission matrix description of this two-port tonehole model is given by the product of the transmission matrices for each of the three impedances. For implementation in a digital waveguide model, these ”lumped” parameters of the Keefe tonehole model must be converted to traveling-wave scattering parameters. Such formulations have recently appeared in the literature [Välimäki et al., 1993] based on a three-port digital waveguide junction loaded by an inertance as described in Fletcher and Rossing [1991]. The scattering parameters of any high quality tonehole model are frequency dependent and therefore require a filter-design problem to be solved. This paper investigates a ”four-filter” form for the Keefe tonehole scattering junction, as well as an improved one-multiply, one-filter three-port digital waveguide junction implementation. 1 THE TWO-PORT TONEHOLE MODEL Keefe [1981, 1990] presents a model of a single woodwind tonehole unit in terms of a symmetric T section transmission matrix of series and shunt impedance parameters, as shown in Fig. 1. The series impedance