Playability of self-sustained musical instrument models: statistical approaches

Self-sustained musical instruments, such as wind or bowed string instruments, are complex nonlinear systems. They admit a wide variety of regimes, which sometimes coexist for certain values of the control parameters. This phenomenon is known as multistability. With fixed parameters, the selection of a regime and the shape of the transient depend not only on the values of the control parameters, but also on the initial conditions. In this article, we focus on the statistical influence of initial conditions on regime selection and transient duration. An existing sample-based method called basin stability is presented to calculate the probability of occurrence of each regime. A second sample-based method is proposed for the calculation of the probability density function of transient durations. Additionally, a study taking into account specific control scenarios is presented to highlight the influence of the distribution of initial conditions considered for the statistical methods. These methods are presented on a Van der Pol oscillator seen as a prototypical musical instrument model. They are then applied to a physical model of trumpet, to demonstrate their potential for a high dimensional self-oscillating musical instrument. Finally, their interest regarding questions of playability is discussed.