Nonlinear modes of clarinet-like musical instruments

The concept of nonlinear modes is applied in order to analyze the behavior of a model of woodwind reed instruments. Using a modal expansion of the impedance of the instrument, and by projecting the equation for the acoustic pressure on the normal modes of the air column, a system of second order ordinary differential equations is obtained. The equations are coupled through the nonlinear relation describing the volume flow of air through the reed channel in response to the pressure difference across the reed. The system is treated using an amplitude-phase formulation for nonlinear modes, where the frequency and damping functions, as well as the invariant manifolds in the phase space, are unknowns to be determined. The formulation gives, without explicit integration of the underlying ordinary differential equation, access to the transient, the limit cycle, its period and stability. The process is illustrated for a model reduced to three normal modes of the air column

Optimal control theory : a method for the design of wind instruments

It has been asserted previously by the author that optimal control theory can be a valuable framework for theoretical studies about the shape that a wind instrument should have in order to satisfy some optimization criterion, inside a fairly general class. The purpose of the present work is to develop this new approach with a look at a specific criterion to be optimized. In this setting, the Webster horn equation is regarded as a controlled dynamical equation in the space variable. Pressure is the state, the control being made of two parts: one variable part, the inside diameter of the duct and one constant part, the weights of the elementary time-harmonic components of the velocity potential. Then one looks for a control that optimizes a criterion related to the definition of an {oscillation regime} as the cooperation of several natural modes of vibration with the excitation, the {playing frequency} being the one that maximizes the total generation of energy, as exposed by A.H. Benade, following H. Bouasse. At the same time the relevance of this criterion is questioned with the simulation results.Comment: To appear in Acta Acustica united with Acustica, 201

Developing and evaluating a hybrid wind instrument

A hybrid wind instrument generates self-sustained sounds via a real-time interaction between a computed excitation model (such as the physical model of human lips interacting with a mouthpiece) and a real acoustic resonator. Attempts to produce a hybrid instrument have so far fallen short, in terms of both the accuracy and the variation in the sound produced. The principal reason for the failings of previous hybrid instruments is the actuator which, controlled by the excitation model, introduces a fluctuating component into the air flow injected into the resonator. In the present paper, the possibility of using a loudspeaker to supply the calculated excitation signal is evaluated. A theoretical study has facilitated the modeling of the loudspeaker-resonator system and the design of a feedback and feedforward filter to successfully compensate for the presence of the loudspeaker. The resulting self-sustained sounds are evaluated by a mapping of their sound descriptors to the input parameters of the physical model of the embouchure, both for sustained and attack sounds. Results are compared with simulations. The largely coherent functioning confirms the usefulness of the device in both musical and research contexts

Digital waveguide modeling for wind instruments: building a state-space representation based on the Webster-Lokshin model

This paper deals with digital waveguide modeling of wind instruments. It presents the application of state-space representations for the refined acoustic model of Webster-Lokshin. This acoustic model describes the propagation of longitudinal waves in axisymmetric acoustic pipes with a varying cross-section, visco-thermal losses at the walls, and without assuming planar or spherical waves. Moreover, three types of discontinuities of the shape can be taken into account (radius, slope and curvature). The purpose of this work is to build low-cost digital simulations in the time domain based on the Webster-Lokshin model. First, decomposing a resonator into independent elementary parts and isolating delay operators lead to a Kelly-Lochbaum network of input/output systems and delays. Second, for a systematic assembling of elements, their state-space representations are derived in discrete time. Then, standard tools of automatic control are used to reduce the complexity of digital simulations in the time domain. The method is applied to a real trombone, and results of simulations are presented and compared with measurements. This method seems to be a promising approach in term of modularity, complexity of calculation and accuracy, for any acoustic resonators based on tubes

How do clarinet players adjust the resonances of their vocal tracts for different playing effects

In a simple model, the reed of the clarinet is mechanically loaded by the series combination of the acoustical impedances of the instrument itself and of the player's vocal tract. Here we measure the complex impedance spectrum of players' tracts using an impedance head adapted to fit inside a clarinet mouthpiece. A direct current shunt with high acoustical resistance allows players to blow normally, so the players can simulate the tract condition under playing conditions. The reproducibility of the results suggest that the players' "muscle memory" is reliable for this task. Most players use a single, highly stable vocal tract configuration over most of the playing range, except for the altissimo register. However, this 'normal' configuration varies substantially among musicians. All musicians change the configuration, often drastically for "special effects'' such as glissandi and slurs: the tongue is lowered and the impedance magnitude reduced when the player intends to lower the pitch or to slur downwards, and vice versa

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