Minimal blowing pressure allowing periodic oscillations in a model of bass brass instruments

Abstract

In this study, an acoustic resonator – a bass brass instrument – with multiple resonances coupled to an exciter – the player’s lips – with one resonance is modelled by a multidimensional dynamical system, and studied using a continuation and bifurcation software. Bifurcation diagrams are explored with respect to the blowing pressure, in particular with focus on the minimal blowing pressure allowing stable periodic oscillations and the associated frequency. The behaviour of the instrument is first studied close to a (non oscillating) equilibrium using linear stability analysis. This allows to determine the conditions at which an equilibrium destabilises and as such where oscillating regimes can emerge (corresponding to a sound production). This approach is useful to characterise the ease of playing of a brass instrument, which is assumed here to be related – as a first approximation – to the linear threshold pressure. In particular, the lower the threshold pressure, the lower the physical effort the player has to make to play a note [The Science of Brass Instruments. Springer-Verlag, 2021]. Cases are highlighted where periodic solutions in the bifurcation diagrams are reached for blowing pressures below the value given by the linear stability analysis. Thus, bifurcation diagrams allow a more in-depth analysis. Particular attention is devoted to the first playing regime of bass brass instruments (the pedal note and the ghost note of a tuba in particular), whose behaviour qualitatively differs from a trombone to a euphonium for instance.

Diversity of ghost notes in tubas, euphoniums and saxhorns

“…It has already been thoroughly described in [McIntyre et al, 1983, Hirschberg, 1995, Velut et al, 2017, Mattéoli et al, 2021. Brass instruments as a whole can be described through both linear and nonlinear mechanisms.…”

“…frequency and the quality factor of the lips, respectively, µ is the lips' mass per unit area, and H(ω ) is the lip-opening height at rest. Based on in vivo measurements of H with respect to ω from [Elliott and Bowsher, 1982], H is assumed to be proportional to the inverse of the lips' resonance frequency as in [Mattéoli et al, 2021]. In this respect, we write H(ω ) = 2πk/ω , k being given in table 2 along with the other parameters of the model.…”

“…Continuation methods rely on the assumption of a smooth nonlinear vector function F , which needs to be at least C 1 . Consequently, equation ( 2) is regularised in the exact same way as in [Mattéoli et al, 2021, Colinot et al, 2019:…”

“…In the following, the system of equations ( 9) with u(X) given by equation ( 11) are processed numerically in a dimensionless form, which is detailed in appendix B of [Mattéoli et al, 2021]. Typical bifurcation diagrams of the system will be shown in section 3.…”

“…The oscillation regimes and ease of playing of bass brass instruments have been investigated in [Mattéoli et al, 2021]: a bifurcation analysis showed that a simple mathematical model accurately describes all the natural notes of a bass trombone and a euphonium in terms of frequency. However, a main difference was highlighted in [Velut et al, 2017, Mattéoli et al, 2021, between instruments with predominantly-cylindrical bore profile such as the trombone, and instruments with predominantly-expanding bore profile such as the euphonium or bass saxhorn. Indeed, the latter exhibit an extra regime located between the pedal note and the second regime, referred to as "ghost note".…”