��� The synthesis of sound based on physical models of wind instruments has traditionally been carried out in a variety of ways. Digital waveguides (Smith 1986; Scavone 1997; van Walstijn and Campbell 2003; van Walstijn 2007) have been extensively explored, especially in the special cases of cylindrical and conical tubes, in which case they yield an extreme efficiency advantage. A related scattering method, wave digital filtering (Fettweis 1986), is also used to connect waveguide tube models with lumped elements such as an excitation mechanism (Bilbao, Bensa, and Kronland-Martinet 2003) or toneholes (van Walstijn and Scavone 2000). Another body of techniques, closely related to digital waveguides, and based around impedance descriptions, has been developed recently (Guillemain 2004). Other techniques, employing finite-difference approximations to the reed model (as opposed to wave- and scattering-based methods) bear a closer resemblance to the direct simulation methods to be discusssed here (Avanzini and Rocchesso 2002; van Walstijn and Avanzini 2007; Avanzini and van Walstijn 2004). Most of these methods owe a great deal to the much earlier treatment of self-sustained musical oscillators by McIntyre, Schumacher and Woodhouse (1983). All of these methods rely, to some degree, on simplified descriptions of the resonator (tube). For example, digital waveguides make use of a traveling wave decomposition, accompanied by frequencydomain (impedance or reflectance) characterizations of lumped elements or phenomena such as bell radiation and tone holes. Other methods make use of the Green’s function or impulse response of the tube directly (McIntyre, Schumacher and Woodhouse 1983). These methods are, in the end, implemented in the time domain, but the notion of the spatial extent of the tube is suppressed: The system is viewed in an input–output sense. When it comes to sound synthesis, however, it is not clear that it is necessary to do so; once one has arrived
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