Systems theory in musical acoustics

Abstract

Recent digital synthesis techniques for woodwind bores and strings (bowed, plucked, and the like) have been devised using long delay lines and sparsely distributed filter elements to efficiently simulate solutions to the one‐dimensional wave equation. In the simplest case, a string or bore can be modeled using a single delay line and a single, low‐order digital filter. This presentation describes techniques used to simulate specific physical phenomena in the context of these models. As is well known, in the lossless case, solutions to the 1‐D wave equation can be expressed in terms of left‐going and right‐going traveling waves, and these are efficiently simulated using pure delay lines plus, perhaps, a sign inversion at a termination. Abutting two waveguides of differing characteristic impedance gives rise to a scattering junction at the interface, leading to the ladder and lattice digital filter structures used extensively in the field of signal processing. When there are losses, or when string stiffness is important, there is linear filtering along the waveguide. Because linear, time‐invariant filters commute, the distributed losses and dispersion can be lumped at convenient places in the waveguide network without changing an input‐output transfer function. Once they are thus consolidated, it is usually possible, especially in audio applications, to find a very accurate approximation using a low‐order lumped digital filter in place of the distributed filtering. On top of this physically accurate yet efficient model, it is straightforward to introduce nonlinear extensions such as (1) bow‐string slippage due to absolute string displacement, (2) pitch decay due to the gradual decline of the average string tension, (3) wave‐front sharpening due to increased speed of sound at higher air pressures, (4) closure forces on double reeds and vocal folds due to the Bernoulli effect, (5) heat conduction losses, and (6) spontaneous generation of turbulence due to high volume velocity through a narrow aperture.