An Analytical Prediction of the Bifurcation Scheme of a Clarinet-Like Instrument: Effects of Resonator Losses

Abstract

The understanding of the relationship between excitation parameters and oscillation regimes is a classical topic concerning bowed string instruments. The paper aims to study the case of reed woodwinds and attempts to find consequences on the ease of playing.In the minimum model of clarinet-like instruments, three parameters are considered: i) the mouth pressure, ii) the reed opening at rest, iii) the length of the resonator assumed to be cylindrical. Recently a supplementary parameter was added: the loss parameter of the resonator (using the "Raman model", that considers resonator losses to be independent of frequency). This allowed explaining the extinction of sound when the mouth pressure becomes very large. The present paper presents an extension of the paper by Dalmont et al (JASA, 2005), searching for a diagram of oscillation regimes with respect to the reed opening and the loss parameter. An alternative method is used, which allows easier generalization and simplifies the calculation. The emphasis is done on the emergence bifurcation: for very strong losses, it can be inverse, similarly to the extinction one for weak losses. The main part of the calculations are analytical, giving clear dependence of the parameters. An attempt to deduce musical consequences for the player is * Tel 33 491164381, Fax 33 491228248, kergomard@lma.cnrs-mrs.fr given.

Idealized digital models for conical reed instruments, with focus on the internal pressure waveform

“…2, and with a supplementary parameter, corresponding to frequency-independent losses in the resonator. 3,4,5 Comparison with experiments shows a good agreement for the bifurcation scheme. 6 Therefore, thanks to these extremely simplified models, basic features of the sound production can be understood, while refined details of the waveform, that are important for the high frequencies and the external sound perception, cannot be predicted.…”

Assessing the sound of a woodwind instrument that cannot be played

“…The understanding of woodwind acoustics progressed through analytical expressions for lossless and then lossy systems [17,18,19], linear system calculations [20], analysis of the reed/mouthpiece system [e.g. 15,21,22,23], impedance of the bell [24,25], non-linear treatment of the reed generator [26] and other factors; an excellent recent treatment appears in Chaigne and Kergomard [27]. In 1979, Plitnik and Strong [28] first applied the computer modelling method to the whole instrument.…”

“…Nederveen [30] has added valuable insight into the elements of the modelling equations and a number of experimental measurements. Research on simulating clarinet and saxophone sounds dynamically using digital formulations of the air column and reed/mouthpiece system in the time domain are also reaching an interesting stage [23,31,32,34].…”

Predicting playing frequencies for clarinets: A comparison between numerical simulations and simplified analytical formulas

“…The second case represents the reed beating against the reed table. According to past studies on clarinet-like instruments, negative flow rate does not occur in the steady-state regime [6,7]. Equation 1 is the first the 3 equations and represents the so-called "nonlinear characteristic".…”