aLaboratoire d'Acoustique de l'Université du Mans - UMR 6613 CNRS
bLaboratoire Manceau de Mathématiques - Le Mans Université
cAix Marseille Univ, CNRS, Centrale Marseille, LMA, UMR 7031, Marseille
Abstract :
Reed musical instruments can be described in terms of conceptually
separate linear and nonlinear mechanisms: a localized nonlinear
element (the valve effect due to the reed) excites a linear, passive
acoustical multimode element (the musical instrument usually
represented in the frequency domain by its input impedance). The
linear element in turn influences the operation of the nonlinear
element. The reed musical instruments are self-sustained oscillators.
They generate an oscillating acoustical pressure (the note played)
from a static overpressure in the player’s mouth (the blowing
pressure).
A reed instrument having N acoustical modes can be described as a
2N dimensional autonomous nonlinear dynamical system. A reed-like
instrument having two quasi-harmonic resonances, represented by a
4 dimensional dynamical system, is studied using the continuation
and bifurcation software AUTO. Bifurcation diagrams are explored
with respect to the blowing pressure, with focus on amplitude and
frequency evolutions along the different solution branches. Some of
the results are interpreted in terms of the ease of playing of the reed
instrument. They can be interpreted as a theoretical illustration of the
Bouasse-Benade prescription.