Introduction

According to prevalent theories of bow-string interaction [6,3], disturbances sent out by the stick-slip process along the string are fundamentally impulsive in nature. That is, the bow is normally either sticking or slipping against the string, and the main excitation events on the string occur when the slipping starts or ends, at which point there is a narrow acceleration pulse sent out in both directions along the string. (There is also sliding noise during slipping each period, but that can be dealt with separately.) Both the Helmholtz [1863] and Raman [1918] models of bowed string behavior consist only of sparse acceleration impulses on the string. Raman's theory, in fact, classifies the various motions according to how many impulses there are per period. Basic Helmholtz motion only consists of one impulse per period, while other modes, such as ``surface sounds'' generated by ``multiple slips,'' or ``multiple flybacks,'' consist of two or more acceleration impulses per period.

The implication of any ``sparse impulse model'' of bowed-string interaction is that it can be used to efficiently drive a commuted synthesis implementation for bowed strings [12,4]. The advantage of commuted synthesis is that a potentially enormous recursive digital filter representing the resonating body is avoided. When an impulse reaches the bridge, a body impulse response (BIR) is ``triggered'' at the amplitude of the impulse. The commuted synthesis implementation thus ``watches'' impulses arriving at the bridge in the bowed-string model, and instantiates a BIR playback into a separate string model on the arrival of each impulse. (BIR playbacks which overlap in time are summed.) A BIR playback may be implemented, for example, using a wavetable oscillator in ``one-shot'' mode. The variable playback rate normally available in such an oscillator can be used to modulate apparent ``body size'' [2, Mandolin.cpp]. The impulse-triggered BIR playback scheme can be classified as an efficient ``sparse-input FIR filter'' implementation of the body resonator. For simple Helmholtz motion, this model reduces to the original bowed-string commuted-synthesis model, except that we may now generate automatically impulse amplitude and timing information from the bow-string interaction model, and we can use physical bow force, position, and velocity signals as the control inputs. In this way, we obtain the reduced computational cost of commuted synthesis, at least during smooth playing, while allowing for fully general interaction between the bow and string.