On the oscillations of musical instruments

Abstract

The time-domain description of musical and other nonlinear oscillators complements the more commonly used frequency-domain description, and is advantageous for some purposes. It is especially advantageous when studying large-amplitude oscillations, for which nonlinearity may be severe. It gives direct insight into the physical reasons for the variation of waveform as playing conditions vary, and into certain phenomena which may seem counter-intuitive from the frequency-domain viewpoint, such as the musically undesirable flattening in the pitch of a bowed string when the bow is pressed too hard onto the string. It is easy to set up efficient time-domain simulations on a small computer, a fact that has been surprisingly little exploited in musical acoustics. The simplest relevant model is described here. It demonstrates some of the basic nonlinear behavior of the clarinet, violin, and flute families with very little programming effort. Remarkably, a single set of model equations has relevance to all three cases, at a certain level of idealization, with appropriate choices of parameter values and of linear and nonlinear characteristics. For the flute family, this simplest model gives waveforms and phase relations closely resembling those observed at resonance in the organ-pipe experiments of Coltman [J. Acoust. Soc. Am. 60, 725-733 (1976)], including the triangular pressure and velocity waveforms. It can be shown (again using a time-domain approach) that the triangular waveform is a universal limiting form, independent of detailed acoustic 'loss mechanisms provided losses are small.

The BRASS Project, from Physical Models to Virtual Musical Instruments: Playability Issues

“…The physical models used in the Brass project rely on a formulation of the physical functionning principles in term of nonlinear delay differential equations (popularized for sound synthesis of self-sustained musical instruments by [1]). The trumpet and the trombone models have been mainly developped during the Phd thesis of Christophe Vergez (see [2], [3] for a general description, and [4], [5] for precise aspects of the models).…”

A generalized frequency detuning method for multidegree-of-freedom oscillators with nonlinear stiffness

“…option N2, (27), must be applied). From β 1 it can be seen that the resonant terms are [1,2], [1,4], [1,5], [1,9], [1,12] and [1,15] for mode 1 and [2, 1], [2,6], [2,7], [2,11], [2,18] and [2,19] for mode 2. Applying option N2 to these terms gives the transformed equations of motionü…”

“…Note the right hand side of the second equation arises from the [2,1] and [2,11] resonance terms. The first of these equations, (50), can be solved to give a relationship between forcing frequency and response amplitude U 1…”

“…In the case where there is more than a single-degree-of-freedom, analysis of this type becomes significantly more complex, because for each primary resonance, there can be multiple secondary resonances [8,11]. For example, in the case of musical instruments, an integer (or near integer) relationship between primary and secondary resonances is deliberately exploited to give an instrument its characteristic sound quality [1].…”

“…In terms of practical motivation, resonances between primary and/or secondary resonant frequencies are important for a wide range of physical applications -see for example [1][2][3][4][5][6][7]. Typically models of the these type of systems are in the form of weakly nonlinear oscillators.…”