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

TL;DR

This paper models a bass brass instrument as a multidimensional dynamical system to analyze the minimal blowing pressure needed for stable oscillations, providing insights into ease of playing and specific regimes like pedal and ghost notes.

Contribution

It introduces a detailed dynamical model of a bass brass instrument and uses bifurcation analysis to identify minimal blowing pressures for oscillations, extending understanding of playability.

Findings

Bifurcation diagrams reveal minimal blowing pressures for oscillations.

Linear stability analysis predicts thresholds close to actual oscillation onset.

Different behaviors are observed for pedal and ghost notes in bass 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 [Campbell et al., 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.