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

TL;DR

This work addresses how opening the register hole in clarinet-like systems induces transitions between registers, revealing multistability and phase-tipping phenomena. It combines an artificial player experiment with a minimal waveguide model that includes localized nonlinear losses at the register hole, enabling detailed exploration of the control-parameter space. The results show that stable second-register play after hole opening occurs in a narrower pressure range than the region in which the second register is intrinsically stable, with transition outcomes strongly depending on the phase of the first-register limit cycle and on noise, leading to probabilistic switching and long transient states. The findings advance understanding of register transitions in musical acoustics and demonstrate how basin structure and rate effects govern phase-sensitive tipping in a simplified clarinet model, with implications for performance and instrument design.

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.

Figures (10)

• Figure 1: Definition of the digital resonator studied.
• Figure 2: Experimental protocol for the blowing pressure ramps. The graph shows for the closed hole, the evolution of the blowing pressure $P_\mathrm{blow}$ (in blue) and the amplitude of the external acoustic pressure recorded by the microphone $\|P_\mathrm{out}\|_2$ (in red), with respect to time. The three threshold values $P_\mathrm{osc}^{(c)}$, $P_\mathrm{ext}^{(c)}$, $P_\mathrm{inv}^{(c)}$ are represented.
• Figure 3: Experimental protocol for the hole openings procedure. The four threshold values $P^\mathrm{I}, P^\mathrm{II}, P^\mathrm{III}$ and $P^\mathrm{IV}$, are determined iteratively by opening the hole five times for a selected blowing pressure, $P^\odot$. $P^\odot$ is then increased until $P^\mathrm{IV}$ is found.
• Figure 4: Example of a cartography of the control parameter space $(\gamma, \zeta)$ by Latin Hypercube sampling (a). The time evolution of $\gamma$ (b), $\zeta$ (c), the amplitude of the acoustic pressure $\|p_{in}\|_2$ (d) and the playing frequency (e) is also shown for two different target control parameters. In this example, the hole is closed.
• Figure 5: Evolution of the amplitude of the external acoustic pressure $P_\mathrm{out}$ when the scaled blowing pressure $\hat{\gamma}$ increases (a) or decreases (b) monotonically, for a constant embouchure, and for the hole closed and open. The measured values of $P_\mathrm{blow}$ are displayed at the top of the graphs. Five measurements are represented for each case. The color of the curves is indexed on the playing frequency. The blue and yellow dots on the x-axis show the values of $\hat{\gamma}_\mathrm{inv}^{(c)}$ and $\hat{\gamma}_\mathrm{inv}^{(o)}$ respectively. Colored surfaces in the background show the ranges of $\hat{\gamma}_\mathrm{blow}$ where the second register is reached with a given probability when the hole is opened. Green: $100~\%$. orange: between $0~\%$ and $100~\%$.