Page 22 - Spring2020

Synthesis of Musical Instrument Sounds
the excitation mechanism in woodwind instruments (Wolfe, 2018). Time-domain simulations of musical instruments are facilitated by solving the (partial) differential equations that describe their oscillations. Several time-stepping algorithms exist for the numerical treatment of such equations in the time domain. Most of the relevant algorithms found in the musical acoustics and sound-computing literature are based on finite differences or closely related methods (Välimäki et al., 2006). When nonlinear phenomena need to be modeled, stability can be shown for some specific cases or under spe- cific assumptions. However, this is not always the case, such as when dealing with systems where the underlying physi- cal quantities are nonanalytic functions of the phase space variables (Chatziioannou and van Walstijn, 2015). As such, numerical analysis techniques are still under investigation in the simulation community (Bilbao et al., 2015).
Regarding the actions of the player, numerical simulations are still rather limited because most physics-based models usually neglect the fact that the instrument oscillations can be excited in different manners during a virtuosic perfor- mance (Wolfe et al., 2015). To the contrary, idealized initial and boundary conditions are often employed in numerical simulations, which do not reflect the control of the player on the instrument. Studying the continuous interaction between the player and the instrument (inherently present
in woodwind instruments) may drive forward the formu- lation of physical models, enabling the modeling of a wide variety of playing gestures. We will use the case of single-reed woodwind instruments (e.g., clarinets and saxophones) as an example of player-instrument interactions.
During expressive woodwind performance, musicians use vari- ous articulation techniques, mostly involving different kinds of interaction between the players’ tongues and the vibrating reed (Scavone, 1996). To capture such interactions, an additional term has been included in the equations that model the sin- gle-reed excitation mechanism corresponding to tongue-reed interaction (Chatziioannou et al., 2019). This nonlinear term is added to two further nonlinearities that take place at the driving end of the instrument, namely, the collision between the reed and the mouthpiece (Dalmont et al., 2003) and the flow that enters the mouthpiece through the reed (Wolfe, 2018). Thus the force acting on the oscillating reed can be written as tongue + lay + ∆, where tongue is the force of the player’s tongue acting on the reed, lay is the collision force between reed and mouthpiece, and ∆ is the force due to the pressure difference across the reed. This excitation model may be coupled to a linear model that describes wave propagation inside a cylindri- cal tube, which is an accurate approximation in the case of the relatively short bores of woodwinds. The coupled model may be solved numerically using, for example, the finite-difference
Figure 1. A: artificial blowing machine for single-reed woodwind instruments. B: sketch of the blowing machine (view from above). C: examples of measured and synthesized mouthpiece pressure for staccato (an individual note separated from its neighboring notes by silence) and portato (the notes are generally sustained, using tonguing to achieve some separation) articulation.
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22 | Acoustics Today | Spring 2020