Physical Models of Musical Instruments

The field of musical acoustics [16,61,78] is a branch of physics devoted to the improved understanding of musical instruments. This understanding typically takes the form of a mathematical model verified by experiment. For the most part, traditional musical instruments are pretty well understood, and their equations of motion can largely be formulated based on classical Newtonian mechanics [68,130]. For wind instruments, however, there are many details that have yet to be pinned down with sufficient accuracy to enable a straightforward computational model to be based upon pure theory. Fortunately, musically motivated scientific investigations (such as [47]) are ongoing, and we can expect this situation to improve over time.

A mathematical model of a musical instrument normally takes the form of a set of differential equations describing the various components of the instrument, such as a vibrating string, air column, or resonator. The components are described by the equations in the sense that their motion is always observed to obey the differential equations. As a result, given the initial conditions (initial positions and velocities), boundary conditions (constraints describing connections to other components) and knowing all external forces over time (``time-varying boundary conditions''), it is possible to simulate mathematically the motion of the real system with great accuracy.

Computational models of musical instruments may be obtained by discretizing the differential equations to obtain difference equations. This approximation step in going from a differential equation to a difference equation is sometimes called a finite difference approximation (FDA), since derivatives are replaced by finite differences defined on a discrete space-time grid [169]. An example will be given for strings in the next section. The difference equations can be ``integrated'' numerically to obtain estimates of the acoustic field in response to external forces and boundary conditions. Numerical integration of FDAs is perhaps the most general path to finite difference schemes yielding accurate computational models of acoustic systems (see, e.g., [32,49]).

While finite-difference methods are quite general, they are computationally more expensive than what is really necessary to achieve excellent real-time musical instruments based on physics. The following sections, organized by instrument family, will discuss various avenues of complexity reduction based on perceptual equivalents from psychoacoustics and computational equivalents from the field of signal processing.