Register jumps on the clarinet: numerical and in-vitro investigation into basins of attraction and phase-tipping

Abstract

When playing the clarinet, opening the register hole allows for a transition from the first to the second register, producing a twelfth interval. On an artificial player system, the blowing pressure range where the second register remains stable can be determined by gradually varying the blowing pressure while keeping the register hole open. However, when the register hole is opened while the instrument is already producing the first register, the range of blowing pressures that lead to a stable second register is narrower than the full stability zone of the second register. This phenomenon is investigated numerically by performing multiple hole openings at different times, for various values of the blowing pressure and the embouchure parameter. In some narrow regions of the control parameters space, the success of a register transition depends on the phase at which the hole is opened. This illustrates an instance of phase-tipping, where the limit cycle of the closed-hole regime may intersect multiple basins of attraction associated with the open-hole regimes. Furthermore, to assess the robustness of the basins of attraction, random noise is introduced to the control parameters before the register hole is opened. Results indicate that the equilibrium regime is more robust to noise than the other oscillating regimes. Finally, long-lasting transient quasiperiodics are investigated. The phase at which the hole is opened influences both the transient duration and the resulting stable regime.