- Direct Simulation of Reed Wind Instruments
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 frequency-domain (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 at a satisfactory model of a musical instrument, written as a time-space partial-differential-equation (PDE) system (for the resonator) coupled to ordinary differential equations (ODEs, the excitation element and a radiation boundary condition), one may proceed directly to a synthesis algorithm without invoking any notion of frequency, impedance, wave variables, or reflectance, or otherwise making any hypotheses about the dynamics of the air in the tube. Though one of course loses the powerful analysis perspective mentioned previously, the treatment of the resonator becomes independent of any particular bore profile, and the system as a whole is now much more amenable to interesting extensions involving, e.g., time-varying and nonlinear effects which do indeed play a role in wind instruments, and which are not easily approached using impedance or scattering concepts.
In the present case, concerned with audio synthesis (and thus efficiency), the model remains one-dimensional. Standard numerical techniques, and, in particular, finite-difference schemes, have been applied (infrequently) to acoustic tube modeling for some time, especially in the case of speech synthesis (see, e.g., the recent paper by van den Doel and Ascher 2008 and the much older but prescient and comprehensive treatment of Portnoff 1973). Finite-difference schemes have also been applied in multi-dimensional spaces, in the setting of acoustical analysis of wind instruments, though generally not directly for synthesis (e.g., Nederveen 1998; Noreland 2002).
In this article, a standard model of a reed wind instrument is presented first, followed by the development of a finite-difference time-domain algorithm, including some discussion of implementation details, such as the operation count and computability issues. Connections to toneholes are then introduced, followed by simulation results.
This article appeared, in a modified form, at a recent conference (Bilbao 2008), and also forms the basis for a section in a new text (Bilbao 2009). [End Page 43]
Click for larger view
View full resolution
A Standard Wind Instrument Model
A standard model of one-dimensional linear wave propagation in an acoustic tube (Morse and Ingard 1968) is given by the following set of equations:
Here, u(x, t) and p(x, t) are the volume velocity and pressure, respectively, at position x, and at time t, and subscripts...