The practical application of the method involves four elementary actions, which can be chained in different ways to progressively approximate a measured input impedance. It consists in adjusting the frequency and damping of one mode at a time while taking into account the presence of all other modes in the basis. To bypass the limitation of usual peak-picking approaches, which are valid only for well separated resonances, the present method is based on a semi-local optimization problem. The method operates as a peak-picking procedure, which makes it particularly intuitive for users who are not experts in modal analysis. The paper presents a method to obtain the modal expansion of the measured input impedance of a brass instrument. 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. Thus, bifurcation diagrams allow a more in-depth analysis. Cases are highlighted where periodic solutions in the bifurcation diagrams are reached for blowing pressures below the value given by the linear stability analysis. In particular, the lower the threshold pressure, the lower the physical effort the player has to make to play a note. 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. This allows to determine the conditions at which an equilibrium destabilises and as such where oscillating regimes can emerge (corresponding to a sound production). The behaviour of the instrument is first studied close to a (non oscillating) equilibrium using linear stability analysis. 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. 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. #Ghostnote serial number professionalThe numerical results provided in terms of frequency intervals between pedal note and ghost note are compared with frequency intervals experimentally inferred from recordings of seven different types of tuba, each of them being played by two professional tuba players. #Ghostnote serial number softwareHere, we adopt a dynamical systems point of view and perform a bifurcation analysis using a software of numerical continuation. This study shows that an elementary brass model describing the player coupled to the instrument is capable of bringing both the ghost and the pedal note to light. References about this note are very scarce, and it is not commonly known among tuba players. However, if the interval between the pedal note and the second regime remains close to an octave regardless of the instrument, the interval between the pedal note and the ghost note vary from a minor third to a perfect fourth. It stands between the pedal note – the lowest natural note playable, or first regime – and the instrument’s second regime. The ghost note is a natural note which can be played exclusively on bass brass instruments with a predominantly-expanding bore profile such as tubas, euphoniums or saxhorns.
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