Flute-like musical instruments: A toy model investigated through numerical continuation

Abstract

Self-sustained musical instruments (bowed string, woodwind and brass instruments) can be modeled by nonlinear lumped dynamical systems. Among these instruments, flutes and flue organ pipes present the particularity to be modeled as a delay dynamical system. In this paper, such a system, a toy model of flute-like instruments, is studied using numerical continuation. Equilibrium and periodic solutions are explored with respect to the blowing pressure, with focus on amplitude and frequency evolutions along the different solution branches, as well as "jumps" between periodic solution branches. The influence of a second model parameter (namely the inharmonicity) on the behaviour of the system is addressed. It is shown that harmonicity plays a key role in the presence of hysteresis or quasi-periodic regime. Throughout the paper, experimental results on a real instrument are presented to illustrate various phenomena, and allow some qualitative comparisons with numerical results.

Calculation of the Steady-State Oscillations of a Flute Model Using the Orthogonal Collocation Method

“…Especially, it highlights that the methods used here predict precisely the saturation of the oscillation amplitude (figure 9), a commonly observed behaviour in experiments and simulations [1,3,5,43], which is not explained or predicted by the often-used linear analysis of the model [3]. In the same way, if the strong dependance of the frequency on the jet velocity U j , highlighted in figure 10, is a well-known behaviour of both models and real instruments (see for example [1,3,22,43,44,45]), a linear analysis of the model only gives a rough estimation of the frequency evolution, and does not distinguinsh between stable and unstable parts of the branch. As highlighted in figure 10, the bifurcation diagram not only predicts precisely the frequency evolution along the branch, but also the stabilisation of the frequency slightly above the resonance frequency (observed experimentally for example in [3]).…”

“…Such a bifurcation can lead to the birth of a quasiperiodic regime, which would be called a multiphonic sound in a musical context, and whose generation mechanism in flute-like instruments is described more precisely in [22].…”

“…A first attempt to study flute-like instruments through numerical continuation, using a toymodel (mathematically a more simple non-neutral delayed dynamical system), has highlighted that such an approach provides new kind of information, explaining behaviours which can be related to some experimental phenomena [22]. It thus suggests to study the state-of-the-art model of flute-like instruments in the same way, and therefore to introduce numerical tools adapted to neutral systems.…”

Evidence of biphonation and source–filter interactions in the bugles of male North American wapiti (Cervus canadensis)

“…Whistle elements within bugles may also function to encode quality if whistle pitch correlates with the caller's size or strength and is held reliable by a physical constraint or production cost. For example, F0 is expected to increase directly with flow velocity in both vortex and flute-like whistles (Terrien et al, 2013). As a consequence, male wapitis that are capable of producing higher G0 may advertise stronger muscles or higher lung capacities to receivers, allowing G0 to function as an index of physical quality, condition or motivational state in inter-and intra-selection contexts.…”

Regime Change Thresholds in Recorder-Like Instruments: Influence of the Mouth Pressure Dynamics

“…Thus, aj ump from the first register to the second register (synchronised on the first and the second resonance modes, respectively)i sc haracterised by af requencyl eap approximately an octave higher (see for example [1]). Register change is known to be accompanied by hysteresis (see for example [1,2,3,4]): the blowing pressure at which the jump between twor egisters occurs (the socalled regime change threshold)i sh igher for rising pressures than for diminishing pressures. As it is related to the selection, through the control of the blowing pressure, of the note played by the instrument, the phenomenon of register change is particularly important for recorder players.…”