Experimental analysis of non-periodic sound regimes in flute-like musical instruments.

Self-sustained musical instruments are complex nonlinear dynamical systems that are known to produce a wealth of dynamical regimes. This includes different kinds of non-periodic sounds, which are either played on purpose or avoided depending on the cultural and musical context. We investigate non-periodic sounds produced by two types of flute-like instruments, namely, an alto recorder and traditional pan-like flutes from Central Chile. We adopt a nonlinear dynamics point of view to characterize the multiphonics produced by the alto recorder and the sonidos rajados produced by the Chilean flutes. Our results unveil the common quasiperiodic nature of the two types of sound regimes and suggest that they result from a similar physical sound production mechanism. This paves the way for a better control of non-periodic sound regimes by the instrument makers.

Woodwind instrument design optimization based on impedance characteristics with geometric constraints.

Computational optimization algorithms coupled with acoustic models of wind instruments provide instrument makers with an opportunity to explore new designs. Specifically, they enable the automatic discovery of geometries exhibiting desired resonance characteristics. In this paper, the design optimization of woodwind instruments with complex geometrical features (e.g., non-cylindrical bore profile and side holes with various radii and chimney heights) is investigated. Optimal geometric designs are searched so that their acoustic input impedance has peaks with specific target frequencies and amplitudes. However, woodwind instruments exhibit complex input impedance whose features, such as resonances, might have non-smooth evolution with respect to design variables, thus hampering gradient-based optimization. For this reason, this paper introduces new formulations of the impedance characteristics (resonance frequencies and amplitudes) using a regularized unwrapped angle of the reflection function. The approach is applied to an illustrative instrument subjected to geometric constraints similar to the ones encountered by manufacturers (a key-less pentatonic clarinet with two-registers). Three optimization problems are considered, demonstrating a strategy to simultaneously adjust several impedance characteristics on all fingerings.

Numerical continuation of a physical model of brass instruments: Application to trumpet comparisons.

The system formed by a trumpet player and his/her instrument can be seen as a non-linear dynamic system and modeled by physical equations. Numerical tools can then be used to study these models and clarify the influence of the model parameters. The acoustic input impedance, for instance, is strongly dependent on the geometry of the air column and is therefore of primary interest for a musical instrument maker. In this study, a method of continuation of periodic solutions based on the combination of the Harmonic Balance Method (HBM) and the Asymptotic Numerical Method (ANM) is applied to a physical model of brass instruments. It allows the study of the evolution of the system where one parameter of the model (static mouth pressure) varies. This method is used to compare different B♭ trumpets on the basis of two descriptors (hysteresis behavior and dynamic range) computed from the continuation outputs. Results show that this methodology enables the differentiation of instruments in the space of the calculated descriptors. Calculations for different values of the lip parameters are also performed to confirm that the obtained categorization is independent of variations of lip parameters.

Multiple two-step oscillation regimes produced by the alto saxophone.

A saxophone mouthpiece fitted with sensors is used to observe the oscillation of a saxophone reed, as well as the internal acoustic pressure, allowing to identify qualitatively different oscillating regimes. In addition to the standard two-step regime, where the reed channel successively opens and closes once during an oscillation cycle, the experimental results show regimes featuring two closures of the reed channel per cycle, as well as inverted regimes, where the reed closure episode is longer than the open episode. These regimes are well-known on bowed string instruments and some were already described on the Uilleann pipes. A simple saxophone model using measured input impedance is studied with the harmonic balance method, and is shown to reproduce the same two-step regimes. The experiment shows qualitative agreement with the simulation: in both cases, the various regimes appear in the same order as the blowing pressure is increased. Similar results are obtained with other values of the reed opening control parameter, as well as another fingering.

Measurements and time-domain simulations of multiphonics in the trombone.

Multiphonic sounds of brass instruments are studied in this article. They are produced by playing a note on a brass instrument while simultaneously singing another note in the mouthpiece. This results in a peculiar sound, heard as a chord or a cluster of more than two notes in most cases. This effect is used in different artistic contexts. Measurements of the mouth pressure, the pressure inside the mouthpiece, and the radiated sound are recorded while a trombone player performs a multiphonic, first by playing an F3 and singing a C4, then playing an F3 and singing a note with a decreasing pitch. Results highlight the quasi-periodic nature of the multiphonic sound and the appearance of combination tones due to intermodulation between the played and the sung sounds. To assess the ability of a given brass instrument physical model to reproduce the measured phenomenon, time-domain simulations of multiphonics are carried out. A trombone model consisting in an exciter and a resonator nonlinearly coupled is forced while self-oscillating to reproduce simultaneous singing and playing. Comparison between simulated and measured signals is discussed. Spectral content of the simulated pressure match very well with the measured one, at the cost of a high forcing pressure.

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.

Oscillation regimes produced by an alto saxophone: Influence of the control parameters and the bore inharmonicity.

The aim of this work is to highlight experimentally how inharmonicity of the bore resonance frequencies of an alto saxophone influences the nature of the oscillation regimes. A variable volume branching from the neck of an alto sax at an appropriate position allows one to change the frequency of the first resonance independently from the second. A blowing machine with artificial lips is used to make the saxophone play while controlling independently the control parameters: the blowing pressure and an embouchure parameter. Values of these parameters are estimated experimentally through the measurement of the nonlinear characteristics linking the mean air flow blown into the instrument to the static pressure difference across the reed. Experiments with different values of the control parameters as well as of the inharmonicity produce different kinds of oscillation regimes. These regimes are categorized through the analysis of the pressure signal inside the mouthpiece. The resulting maps demonstrate that the emergence of quasi-periodic regimes, and their extent, depend on the level of inharmonicity, but also on the values of the control parameters. Periodic regimes playable by choosing appropriate values of the control parameters also differ according to the level of inharmonicity, a higher inharmonicity facilitating the emergence of the third register.

Response of an artificially blown clarinet to different blowing pressure profiles.

Using an artificial mouth with an accurate pressure control, the onset of the pressure oscillations inside the mouthpiece of a simplified clarinet is studied experimentally. Two time profiles are used for the blowing pressure: in a first set of experiments the pressure is increased at constant rates, then decreased at the same rate. In a second set of experiments the pressure rises at a constant rate and is then kept constant for an arbitrary period of time. In both cases the experiments are repeated for different increase rates. Numerical simulations using a simplified clarinet model blown with a constantly increasing mouth pressure are compared to the oscillating pressure obtained inside the mouthpiece. Both show that the beginning of the oscillations appears at a higher pressure values than the theoretical static threshold pressure, a manifestation of bifurcation delay. Experiments performed using an interrupted increase in mouth pressure show that the beginning of the oscillation occurs close to the stop in the increase of the pressure. Experimental results also highlight that the speed of the onset transient of the sound is roughly the same, independently of the duration of the increase phase of the blowing pressure.

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.

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.

Quasistatic nonlinear characteristics of double-reed instruments.

This article proposes a characterization of the double reed in quasistatic regimes. The nonlinear relation between the pressure drop, deltap, in the double reed and the volume flow crossing it, q, is measured for slow variations of these variables. The volume flow is determined from the pressure drop in a diaphragm replacing the instrument's bore. Measurements are compared to other experimental results on reed instrument exciters and to physical models, revealing that clarinet, oboe, and bassoon quasistatic behavior relies on similar working principles. Differences in the experimental results are interpreted in terms of pressure recovery due to the conical diffuser role of the downstream part of double-reed mouthpieces (the staple).

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.