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

The understanding of the relationship between excitation parameters andoscillation regimes is a classical topic concerning bowed stringinstruments. The paper aims to study the case of reed woodwinds and attemptsto find consequences on the ease of playing.In the minimum model of clarinet-like instruments, three parameters areconsidered: i) the mouth pressure, ii) the reed opening at rest, iii) thelength of the resonator \ assumed to be cylindrical. Recently asupplementary parameter was added: the loss parameter of the resonator(using the \textquotedblleft Raman model\textquotedblright, that considersresonator losses to be independent of frequency). This allowed explainingthe extinction of sound when the mouth pressure becomes very large. Thepresent paper presents an extension of the paper by Dalmont et al (JASA,2005), searching for a diagram of oscillation regimes with respect to thereed opening and the loss parameter. An alternative method is used, whichallows easier generalization and simplifies the calculation. The emphasis is done on the emergencebifurcation: for very strong losses, it can be inverse, similarly to theextinction one for weak losses. The main part of the calculations areanalytical, giving clear dependence of the parameters. An attempt to deducemusical consequences for the player is given

Analysis and optimisation of the tuning of the twelfths for a clarinet resonator

Even if the tuning between the first and second register of a clarinet has been optimized by instrument makers, the lowest twelfths remain slightly too large (inharmonicity). In this article, we study the problem from two different points of view. First, we systematically review various physical reasons why this inharmonicity may take place, and the effect of different bore perturbations inserted in cylindrical instruments. Applications to a real clarinet resonator and comparisons with impedance measurements are then presented. A commonly accepted idea is that the register hole is the dominant cause for this inharmonicity: it is natural to expect that opening this hole will raise the resonance frequencies of the instrument, except for the note for which the hole is at the pressure node. We show that the real situation is actually more complicated because other effects, such as open holes or bore taper and bell, introduce resonance shifts that are comparable but with opposite sign, so that a relatively good overall compensation takes place. The origin of the observed inharmonicity in playing frequencies is therefore different. In a second part, we use an elementary model of the clarinet in order to isolate the effect of the register hole: a perfect cylindrical tube without closed holes. Optimization techniques are then used to calculate an optimum location for the register hole; the result turns out to be close to the location chosen by clarinet makers. Finally, attempts are made numerically to improve the situation by introducing small perturbations in the higher part of the cylindrical resonator, but no satisfactory improvement is obtained.Comment: 28 June 2004 (submitted to Applied Acoustics

Complex resonance frequencies of a finite, circular radiating duct with an infinite flange

Radiation by solid or fluid bodies can be characterized by resonance modes. They are complex, as well as resonance frequencies, because of the energy loss due to radiation. For ducts, they can be computed from the knowledge of the radiation impedance matrix. For the case of a flanged duct of finite length radiating on one side in an infinite medium, the expression of this matrix was given by Zorumski, using a decomposition in duct modes. In order to calculate the resonance frequencies, the formulation used in Zorumski's theory must be modified as it is not valid for complex frequencies. The analytical development of the Green's function in free space used by Zorumski depends on the integrals of Bessel functions which become divergent for complex frequencies. This paper proposes first a development of the Green's function which is valid for all frequencies. Results are applied to the calculation of the complex resonance frequencies of a flanged duct, by using a formulation of the internal pressure based upon cascade impedance matrices. Several series of resonance modes are found, each series being shown to be related to a dominant duct mode. Influence of higher order duct modes and the results for several fluid densities is presented and discussed

Generic resonator models for real-time synthesis of reed and brass instruments

International audienceFrom accurate measurements of bore profiles of various reed and brass instruments, a common and simplified geometrical model made of three parts totalizing seven geometrical parameters is proposed. From this geometry, it is shown that a good approximation of the input impedance can be obtained by a combination of two lumped elements gathered in series and parallel with a distributed element. Each element is approximated and discretized in order to end up with costless digital filters representing the impedance impulse response. These filters require the order of twenty multiplication/additions per sample and their coefficients are analytically expressed as functions of the geometrical parameters. The choice of the geometry and the time discretization schemes are validated both through comparison with continuous models and through the estimation of the geometrical parameters via a global optimization procedure, using measured input impedance curves

Guides d'ondes inhomogènes couplés par un réseau périodique de perforations : modes basse fréquence et interférence dans un réseau de longueur finie

6 pages, 7 refsInternational audienceLa propagation dans deux guides d'ondes couplés périodiquement par des perforations est étudiée aux basses fréquences. Les outils classiques de propagation dans les milieux périodiques permettent d'établir analytiquement les expressions qui caractérisent une cellule élémentaire d'un réseau avec deux milieux de propagation différents (inhomogène) : la matrice de transfert, l'équation de dispersion et les modes propres du réseau. Ces résultats à caractère général sont ensuite utilisés pour décrire le cas où un des guides joue pour l'autre le rôle d'un traitement acoustique en paroi à réaction non locale. Dans le cas réciproque, on exprime également de façon analytique la matrice de transfert et la perte par insertion pour un réseau périodique inhomogène de longueur finie. A titre d'exemple, le cas de deux guides homogènes est étudié. Le modèle permet de proposer une interprétation de la perte par insertion d'un silencieux à tube perforé, basée le nombre de modes propres se propageant dans le réseau et leur interférence dans un réseau de longueur finie. Le caractère général des expressions obtenues devrait permettre d'explorer ultérieurement d'autres configurations, notamment pour des guides inhomogènes présentant un matériaux poreux et/ou un diaphragme dans les guides. Les applications concernent par exemple les silencieux automobiles ou les traitements acoustiques à réaction non locale (nacelles de turboréacteurs)

Propagation of acoustic waves in two waveguides coupled by perforations. II. Analysis of periodic lattices of finite length

first revision, submitted to editor.International audienceThe paper deals with the generic problem of two waveguides coupled by perforations, which can be perforated tube mufflers without or with partitions, possibly with absorbing materials. Other examples are ducts with branched resonators of honeycomb cavities , which can be coupled or not, and splitter silencers. Assuming low frequencies, only one mode is considered in each guide. The propagation in the two waveguides can be very different, thanks e.g. to the presence of constrictions. The model is a discrete, periodic one, based upon 4th-order impedance matrices and their diagonalization. All the calculation is analytical, thanks to the partition of the matrices in 2nd-order matrices, and allows the treatment of a very wide types of problems. Several aspects are investigated: the local or non-local character of the reaction of one guide to the other; the definition of a coupling coefficient; the effect of finite size when a lattice with n cells in inserted into an infinite guide; the relationship between the Insertion Loss and the dispersion. The assumptions are as follows: linear acoustics, no mean flow, rigid wall. However the effect of the series impedance of the perforations, which is generally ignored, is taken into account, and is dis- cussed. When there are no losses, it is shown that, for symmetry reasons, the cutoff frequencies depend on either the series impedance or the shunt admittance , and are the eigenfrequencies of the cells of the lattice, with zero-pressure or zero-velocity at the ends of the cells

Statistical Estimation of Mechanical Parameters of Clarinet Reeds Using Experimental and Numerical Approaches

A set of 55 clarinet reeds is observed by holography, collecting 2 series of measurements made under 2 different moisture contents, from which the resonance frequencies of the 15 first modes are deduced. A statistical analysis of the results reveals good correlations, but also significant differences between both series. Within a given series, flexural modes are not strongly correlated. A Principal Component Analysis (PCA) shows that the measurements of each series can be described with 3 factors capturing more than $90\%$ of the variance: the first is linked with transverse modes, the second with flexural modes of high order and the third with the first flexural mode. A forth factor is necessary to take into account the individual sensitivity to moisture content. Numerical 3D simulations are conducted by Finite Element Method, based on a given reed shape and an orthotropic model. A sensitivity analysis revels that, besides the density, the theoretical frequencies depend mainly on 2 parameters: $E_L$ and $G_{LT}$. An approximate analytical formula is proposed to calculate the resonance frequencies as a function of these 2 parameters. The discrepancy between the observed frequencies and those calculated with the analytical formula suggests that the elastic moduli of the measured reeds are frequency dependent. A viscoelastic model is then developed, whose parameters are computed as a linear combination from 4 orthogonal components, using a standard least squares fitting procedure and leading to an objective characterization of the material properties of the cane \textit{Arundo donax}

On the cutoff frequency of clarinet-like instruments. Geometrical versus acoustical regularity

A characteristic of woodwind instruments is the cutoff frequency of their tone-hole lattice. Benade proposed a practical definition using the measurement of the input impedance, for which at least two frequency bands appear. The first one is a stop band, while the second one is a pass band. The value of this frequency, which is a global quantity, depends on the whole geometry of the instrument, but is rather independent of the fingering. This seems to justify the consideration of a woodwind with several open holes as a periodic lattice. However the holes on a clarinet are very irregular. The paper investigates the question of the acoustical regularity: an acoustically regular lattice of tone holes is defined as a lattice built with T-shaped cells of equal eigenfrequencies. Then the paper discusses the possibility of division of a real lattice into cells of equal eigenfrequencies. It is shown that it is not straightforward but possible, explaining the apparent paradox of Benade's theory. When considering the open holes from the input of the instrument to its output, the spacings between holes are enlarged together with their radii: this explains the relative constancy of the eigenfrequencies

Oscillation thresholds for "striking outwards" reeds coupled to a resonator

International audienceThis paper considers a "striking outwards" reed coupled to a resonator. This expression, due to Helmholtz, is not discussed here : it corresponds to the most common model of a lip-type valve, when the valve is assumed to be a one degree of freedom oscillator. The presented work is an extension of the works done by Wilson and Beavers (1974), Tarnopolsky (2000). The range of the playing frequencies is investigated. The first results are analytical : when no losses are present in the resonator, it is proven that the ratio between the threshold frequency and the reed resonance frequency is found to be necessarily within the interval between unity and the square root of 3. This is a musical sixth. Actually the interval is largely smaller, and this is in accordance with e.g. the results by Cullen et al.. The smallest blowing pressure is found to be directly related to the quality factor of the reed. Numerical results confirm these statements, and are discussed in comparison with previous ones by Cullen et al (2000)

The logical clarinet: numerical optimization of the geometry of woodwind instruments

The tone hole geometry of a clarinet is optimized numerically. The instrument is modeled as a network of one dimensional transmission line elements. For each (non-fork) fingering, we first calculate the resonance frequencies of the input impedance peaks, and compare them with the frequencies of a mathematically even chromatic scale (equal temperament). A least square algorithm is then used to minimize the differences and to derive the geometry of the instrument. Various situations are studied, with and without dedicated register hole and/or enlargement of the bore. With a dedicated register hole, the differences can remain less than 10 musical cents throughout the whole usual range of a clarinet. The positions, diameters and lengths of the chimneys vary regularly over the whole length of the instrument, in contrast with usual clarinets. Nevertheless, we recover one usual feature of instruments, namely that gradually larger tone holes occur when the distance to the reed increases. A fully chromatic prototype instrument has been built to check these calculations, and tested experimentally with an artificial blowing machine, providing good agreement with the numerical predictions