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

When designing a wind instrument such as a clarinet, it can be useful to be able to predict the playing frequencies. This paper presents an analytical method to deduce these playing frequencies using the input impedance curve. Specifically there are two control parameters that have a significant influence on the playing frequency, the blowing pressure and reed opening. Four effects are known to alter the playing frequency and are examined separately: the flow rate due to the reed motion, the reed dynamics, the inharmonicity of the resonator, and the temperature gradient within the clarinet. The resulting playing frequencies for the first register of a particular professional level clarinet are found using the analytical formulas presented in this paper. The analytical predictions are then compared to numerically simulated results to validate the prediction accuracy. The main conclusion is that in general the playing frequency decreases above the oscillation threshold because of inharmonicity, then increases above the beating reed regime threshold because of the decrease of the flow rate effect.

Passive models of viscothermal wave propagation in acoustic tubes.

A continued fraction expansion to the immittances defining viscothermal wave propagation in a cylindrical tube has been presented recently in this journal, intended as a starting point for time domain numerical method design. Though the approximation has the great benefit of passivity, or positive realness under truncation, its convergence is slow leading to approximations of high order in practice. Other passive structures, when combined with optimisation methods, can lead to good accuracy over a wide frequency range, and for relatively low order.

Interaction of reed and acoustic resonator in clarinetlike systems.

Sound emergence in clarinetlike instruments is investigated in terms of instability of the static regime. Various models of reed-bore coupling are considered, from the pioneering work of Wilson and Beavers ["Operating modes of the clarinet," J. Acoust. Soc. Am. 56, 653-658 (1974)] to more recent modeling including viscothermal bore losses and vena contracta at the reed inlet. The pressure threshold above which these models may oscillate as well as the frequency of oscillation at threshold are calculated. In addition to Wilson and Beavers' previous conclusions concerning the role of the reed damping in the selection of the register the instrument will play on, the influence of the reed motion induced flow is also emphasized, particularly its effect on playing frequencies, contributing to reduce discrepancies between Wilson and Beavers' experimental results and theory, despite discrepancies still remain concerning the pressure threshold. Finally, analytical approximations of the oscillating solution based on Fourier series expansion are obtained in the vicinity of the threshold of oscillation. This allows to emphasize the conditions which determine the nature of the bifurcation (direct or inverse) through which the note may emerge, with therefore important consequences on the musical playing performances.

Contribution to harmonic balance calculations of self-sustained periodic oscillations with focus on single-reed instruments.

The harmonic balance method (HBM) was originally developed for finding periodic solutions of electronical and mechanical systems under a periodic force, but has been adapted to self-sustained musical instruments. Unlike time-domain methods, this frequency-domain method does not capture transients and so is not adapted for sound synthesis. However, its independence of time makes it very useful for studying any periodic solution, whether stable or unstable, without care of particular initial conditions in time. A computer program for solving general problems involving nonlinearly coupled exciter and resonator, HARMBAL, has been developed based on the HBM. The method as well as convergence improvements and continuation facilities are thoroughly presented and discussed in the present paper. Applications of the method are demonstrated, especially on problems with severe difficulties of convergence: the Helmholtz motion (square signals) of single-reed instruments when no losses are taken into account, the reed being modeled as a simple spring.

An analytical prediction of the oscillation and extinction thresholds of a clarinet.

This paper investigates the dynamic range of the clarinet from the oscillation threshold to the extinction at high pressure level. The use of an elementary model for the reed-mouthpiece valve effect combined with a simplified model of the pipe assuming frequency independent losses (Raman's model) allows an analytical calculation of the oscillations and their stability analysis. The different thresholds are shown to depend on parameters related to embouchure parameters and to the absorption coefficient in the pipe. Their values determine the dynamic range of the fundamental oscillations and the bifurcation scheme at the extinction.

Real-time synthesis of clarinet-like instruments using digital impedance models.

A real-time synthesis model of wind instruments sounds, based upon a classical physical model, is presented. The physical model describes the nonlinear coupling between the resonator and the excitor through the Bernoulli equation. While most synthesis methods use wave variables and their sampled equivalent in order to describe the resonator of the instrument, the synthesis model presented here uses sampled versions of the physical variables all along the synthesis process, and hence constitutes a straightforward digital transposition of each part of the physical model. Moreover, the resolution scheme of the problem (i.e., the synthesis algorithm) is explicit and all the parameters of the algorithm are expressed analytically as functions of the physical and the control parameters.