Describing minimum bow force using Impulse Pattern Formulation (IPF) – an empirical validation
Simon Linke, Robert Mores, Rolf Bader · Proceedings of meetings on acoustics · 2022
With bowed string instruments, the minimum bow force necessary to produce stable Helmholtz motion has often been discussed over the last decades. If the bowing force is too small or the bowing velocity too large, no stable tone is produced, and bifurcations or noise occurs. The Impulse Pattern Formulation (IPF) is a top-down method proposed previously (Bader, 2013), which explains such transitions between regular periodicity, bifurcation scenarios, and noise. The recursive equation is based on the idea that impulses, produced at a generator within a musical instrument, travel through the instrument, are reflected, exponentially dampened, and finally trigger or interact with succeeding impulses at the generator. Real bowing is measured on an experimental pendulum for self-organized bowing. Bowing force and velocity are recorded during the transitions from bifurcation to stable tone production. Then the IPF is used under a Simulated Annealing paradigm to reproduce this behavior, using the measured bowing pressure and velocity as input parameters. Compared to previous models, the IPF predicts minimum bow force with much higher precision, which is caused by bowing being a nonlinear dynamical system, including, e.g., hysteresis. In contrast, previous works mainly used analytical equations, therefore not considering the self-organizing nature of the process.