Iterated maps for clarinet-like systems

The dynamical equations of clarinet-like systems are known to be reducible to a non-linear iterated map within reasonable approximations. This leads to time oscillations that are represented by square signals, analogous to the Raman regime for string instruments. In this article, we study in more detail the properties of the corresponding non-linear iterations, with emphasis on the geometrical constructions that can be used to classify the various solutions (for instance with or without reed beating) as well as on the periodicity windows that occur within the chaotic region. In particular, we find a regime where period tripling occurs and examine the conditions for intermittency. We also show that, while the direct observation of the iteration function does not reveal much on the oscillation regime of the instrument, the graph of the high order iterates directly gives visible information on the oscillation regime (characterization of the number of period doubligs, chaotic behaviour, etc.)

Oscillation threshold of a clarinet model: a numerical continuation approach

This paper focuses on the oscillation threshold of single reed instruments. Several characteristics such as blowing pressure at threshold, regime selection, and playing frequency are known to change radically when taking into account the reed dynamics and the flow induced by the reed motion. Previous works have shown interesting tendencies, using analytical expressions with simplified models. In the present study, a more elaborated physical model is considered. The influence of several parameters, depending on the reed properties, the design of the instrument or the control operated by the player, are studied. Previous results on the influence of the reed resonance frequency are confirmed. New results concerning the simultaneous influence of two model parameters on oscillation threshold, regime selection and playing frequency are presented and discussed. The authors use a numerical continuation approach. Numerical continuation consists in following a given solution of a set of equations when a parameter varies. Considering the instrument as a dynamical system, the oscillation threshold problem is formulated as a path following of Hopf bifurcations, generalizing the usual approach of the characteristic equation, as used in previous works. The proposed numerical approach proves to be useful for the study of musical instruments. It is complementary to analytical analysis and direct time-domain or frequency-domain simulations since it allows to derive information that is hardly reachable through simulation, without the approximations needed for analytical approach

De la capacité d'une bouche artificielle a reproduire le geste d'un instrumentiste

National audienceCe travail présente une étude sur la capacité d'une bouche artificielle aspirante à reproduire le jeu d'une clarinette simplifiée. Le système étudié est composé d'un bec instrumenté connecté à un cylindre. L'instrument est joué d'une part par un musicien, d'autre part à l'aide de la bouche artificielle aspirante. Le bec permet la mesure de la pression dans la bouche du musicien, la pression acoustique dans le bec et le déplacement de l'extrémité de l'anche. Le geste instrumental est caractérisé par les paramètres de jeu et les paramètres d'anche estimés par l'analyse de la partie stationnaire des signaux mesurés. Les paramètres de jeu considérés dans ce travail sont la pression moyenne dans la bouche et la valeur maximale du déplacement de l'anche. Les paramètres d'anche estimés à l'aide d'une méthode inverse sont la raideur non linéaire, l'amortissement équivalent de l'anche et l'ouverture au repos. La bouche artificielle est réglée de façon à obtenir des valeurs de paramètres de jeu proches de ceux obtenus avec un musicien pour une nuance et une embouchure données. L'analyse et la comparaison d'un jeu de paramètres d'anche permet de quantifier la performance de la bouche artificielle à reproduire une situation de jeu. La gamme de fonctionnement de la bouche artificielle est ainsi définie pour pouvoir étudier à terme les caractéristiques d'anches considérées comme différentes

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

International audienceThe harmonic balance method ͑HBM͒ 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 signals͒ of single-reed instruments when no losses are taken into account, the reed being modeled as a simple spring

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

International audienceTwo models for the generation of self-oscillations of reed conical woodwinds are presented. They use the fewest parameters (of either the resonator or the ex-citer), whose influence can be quickly explored. The formulation extends iterated maps obtained for loss-less cylindrical pipes without reed dynamics. It uses spherical wave variables in idealized resonators, with one parameter more than for cylinders: the missing length of the cone. The mouthpiece volume equals that of the missing part of the cone, and is implemented as either a cylindrical pipe (first model) or a lumped element (second model). Only the first model adds a length parameter for the mouthpiece and leads to the solving of an implicit equation. For the second model, any shape of nonlinear characteristic can be directly considered. The complex characteristics impedance for spherical waves requires sampling times smaller than a round trip in the resonator. The convergence of the two models is shown when the length of the cylindrical mouthpiece tends to zero. The waveform is in semi-quantitative agreement with experiment. It is concluded that the oscillations of the positive episode of the mouthpiece pressure are related to the length of the missing part, not to the reed dynamics

UNE SÉRIE INFINIE DE RÉSONATEURS PRÉSENTANT DES FRÉQUENCES DE RÉSONANCES HARMONIQUES

We present a family of resonators which have harmonic resonance frequencies but in which some resonances are missing. Each resonator is composed by a succession of N cylinders of the same length. The cross-section area of each cylinder is equal to i(i+1)/2 times the cross-section area of the first cylinder (for i=1 to N) and the last cylinder in open. All the resonance frequencies are harmonic with F1 the lowest resonance frequency. However all multiples of (N+1)F1 correspond to antiresonance frequencies

Influence de la perce sur l'inharmonicité des fréquences de résonance du hautbois

In this paper it is shown theoretically how a local change on the conicity of the main bore can reduce the anharmonicity of the resonance frequencies. A comparative study of three different oboes has been made. The measurement of the respective main bore explains partially the differences of the anharmonicity

Mesure de matrices de transfert de systèmes acoustiques simples

The purpose of the present paper is to measure the transfer matrix of acoustical, reciprocal elements of wave guides by using two microphones and several terminal tubes, by mean of the least mean square method. We apply this method for the case of a lateral chimney. The results are satisfactory, but the accuracy is not large enough to determine the anti symmetrical field in the chimney