Bow control and playability of a two-polarisation time domain physical model of a bowed string

Many studies have explored bowed string control, first in theory, and later through experiments and simulations. Bowing control is most commonly described in terms of 3 parameters: downward bow force, bow position, and transverse bow velocity. The correlation between these parameters and the production of a note is generally referred to as the playability of the instrument. The musically-useful region within this parameter space where Helmholtz motion is achieved has been extensively researched. The coordination of these player-controlled parameters is crucial for the existence of steady-state oscillations, but also for their behaviour during note onset. Time domain models of bowed strings require the dynamic input of these physical control parameters. A detailed exploration of this parameter space is therefore of importance for playable physical models. In this work, a two-polarisation bowed string is coupled to a fully dynamic nonlinear lumped bow model. A finite difference scheme is used to solve the discretised equations of motion in the time domain. The bow-controlled input to the model is based on a two-dimensional force vector applied to the bow itself, coupling the two string polarisations. One force component is applied downwards, and the other orthogonally across the string axis, in contrast to the common convention of imposing a downward bow force and transverse bow velocity. The new model has the potential to allow for more realistic and dynamic gesture control. This study explores the parameter space implied by the model, relating it to the playability of the virtual instrument. Analysis of the steady state bow-string dynamics leads to classification of the playing regime, and a graphical representation inspired by the well known Schelleng diagram. The resulting playability-force diagrams may be used as an aid to control of the sound synthesis algorithm.