Learned Bow Control on a Measured Bowed-String Model: a Revised Minimum-Bow-Force Law, a Recurrent Controller, and the Domain of a Supervision Ceiling

Abstract

A finite-difference bowed-string model with implicitly resolved Stribeck friction is presented, with a regime diagnostic, the Schelleng bow-force limits on four strings, and a comparison of learned bow controllers. Implicit resolution is necessary, and quantitatively so: a lagged contact force cannot capture the string on a discrete grid, so no stick phase forms at any bow force. With friction, impedance and quality factor taken from published measurement rather than fitted, all four strings return a stick fraction of 89.1% against an ideal 90%. Schelleng's maximum bow force is recovered on every string. The minimum is not: it follows $Z v_b β^{-1}$ rather than the predicted $Z^2 v_b β^{-2}$, reducing both squared dependences to first powers. Six controllers at matched capacity, over four strings and twenty seeds each, place a gated recurrent network ahead of a feedforward one, by most under a mid-stroke disturbance. The feedforward network completes more strokes only from a start the model's own playability map places outside the Helmholtz region. A minimal gated variant fails because gates computed from the input alone cannot clear a latched state. Training loss selects neither the capacity nor the context length, and no learned controller improves on the lookup rule that generated its labels. That bound has a domain. Regressing the controller's score on the rule's gives a slope of 0.32, more than ten standard errors below unity, so the controller overtakes the rule where the rule fails and is bounded by it where it holds. Under a rigid finger stop the plant is provably invariant, so transfer loss between pitches belongs to the controller alone and is traced to one feature. A regime classifier without a stick test labels small-amplitude periodic slipping as Helmholtz motion, and a harmonicity measure rates a string the bow never grips above Helmholtz motion.