Influence of the basins of attractions in the register jumps of the clarinet

TL;DR

This work addresses the predictability of register transitions on the clarinet, focusing on how basins of attraction determine whether opening the register hole leads to the first or second register ($R_1$ or $R_2$) under multistability. It introduces a waveguide-based model with digital resonators, viscothermal losses, forward/backward pressure waves, nonlinear reed-flow, and localized nonlinear losses in the register hole, then performs blowing-pressure ramps and repeated hole openings to map stability regions and transition probabilities. The key findings show that the open-hole regime allows coexistence of $R_2$ and the equilibrium $R_0$ over a range of blowing pressures, with the probability of landing in $R_2$ following a sigmoid as a function of $\gamma$ and shaped by the geometry of the basins of attraction; phase-tipping emerges as the opening timing interacts with the current phase on the $R_1$ limit cycle. These results offer practical insights for instrument makers and players regarding the playability of twelfths, highlighting the role of basin geometry and perturbation robustness in the reliability of register jumps.

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 mouth, 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 each blowing pressure value. The evolution of the probability of reaching the second register is computed, and its relationship with the structure of the basin of attraction of the second register is analyzed.

Figures (4)

• Figure 1: Definition of the digital resonators studied.
• Figure 2: Evolution of the amplitude of the acoustic pressure into the mouthpiece when the blowing pressure $\gamma$ increases linearly , for a constant embouchure $\zeta=0.3$. Two crescendi are represented: one for the hole closed (blue curve), and one for the hole open (yellow curve). Colored surfaces in the background show the ranges of $\gamma$ where the second register is reached with a given probability when the hole is opened. Green: $100~\%$. Blue: between $0~\%$ and $100~\%$. Red: $0~\%$.
• Figure 3: Evolution of the proportion of the second register (R2) when opening the register hole, for $\gamma\in[1.334,1.339]$. The red, yellow, purple and green curves show the evolution of the proportion of R2 when random perturbations are added to $\hat{p}^+_{in}$ when the register hole is opened.
• Figure 4: Positions on the limit cycle of the first register leading, when the hole is opened, to the second register (in red), and to the equilibrium (in black). Limit cycles are represented in the $(\hat{p}_1, \dot{\hat{p} }_1)$ space. Three different values of $\gamma$ are displayed on each row. On the second row, a random perturbation of $\pm 0.1$ is added to $\hat{p}_{in}^+$ when the hole is opened; on the third row the amplitude is $0.33$.