Abstract
A reed instrument model with N acoustical modes can be described as a 2N dimensional autonomous nonlinear dynamical system. Here, a simplified model of a reed-like instrument having two quasi-harmonic resonances, represented by a four dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams of equilibria and periodic solutions are explored with respect to the blowing mouth pressure, with focus on amplitude and frequency evolutions along the different solution branches. Equilibria and periodic regimes are connected through Hopf bifurcations, which are found to be direct or inverse depending on the physical parameters values. Emerging periodic regimes mainly supported by either the first acoustic resonance (first register) or the second acoustic resonance (second register) are successfully identified by the model. An additional periodic branch is also found to emerge from the branch of the second register through a period-doubling bifurcation. The evolution of the oscillation frequency along each branch of the periodic regimes is also predicted by the continuation method. Stability along each branch is computed as well. Some of the results are interpreted in terms of the ease of playing of the reed instrument. The effect of the inharmonicity between the first two impedance peaks is observed both when the amplitude of the first is greater than the second, as well as the inverse case. In both cases, the blowing pressure that results in periodic oscillations has a lowest value when the two resonances are harmonic, a theoretical illustration of the Bouasse-Benade prescription.
Highlights
An important goal of the acoustics of wind instruments is to understand key components of intonation and the ease of playing
Equilibria and periodic regimes are connected through Hopf bifurcations, which are found to be direct or inverse depending on the physical parameters values
Bifurcation diagrams of a basic reed instrument modeled by two quasi-harmonic resonances have been computed by using a continuation method (AUTO software), where the mouth pressure is the control parameter
Summary
An important goal of the acoustics of wind instruments is to understand key components of intonation and the ease of playing. It is often commented that the flaring bore of brass instruments are designed such that the input impedance are, as close as possible, harmonically related While this alignment is said to be important for intonation, it is likely to determine the oscillation threshold and improve the ease of playing. In order to illustrate this prescription, a horn was designed to provide an air column whose resonance frequencies (frequencies of maximum input impedance) were chosen to avoid all possible integer relations between them, called “tacet horn” in [7]. The purpose of this instrument is to deliberately make the conditions for oscillation unfavorable.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
