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Page 00000001 Nonlinear Commuted Synthesis of Bowed Strings Julius O. Smith III CCRMA, Music Department, Stanford University josOccrma.stanford.edu, http://www-ccrma.stanford.edu/-jos Abstract A commuted-synthesis model for bowed strings is driven by a separate nonlinear model of bowedstring dynamics. This gives the desirable combination of a full range of complex bow-string interaction behavior together with an efficiently implemented body resonator. A "single-hair bow" may control a pulsed-noise version which provide the effects of multiple bow hairs. The pulsed noise may also include qualitatively the impulse responses of commuted high-frequency body modes. 1 Introduction According to prevalent theories of bow-string interaction [McIntyre and Woodhouse 1979, Guettler 1992], 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  and Raman  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 [Smith 1993, Jaffe and Smith 1995]. 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" [Cook 1996, 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. 2 Nonlinear Commuted Model The basic idea of commuted synthesis is to interchange the order of implementation of the string and the body resonator, as depicted in Fig. 1. The bowed string synthesizer of the present paper is shown in Fig. 2. The bottom half is Fig. 1c, with an external trigger input, and some further details regarding pulsed noise generation. The top half of Fig. 2 provides an explicit model of bow-string dynamics. The "Impulse Prioritizer" measures the timing and amplitude of the largest impulses in the string waveform at the bridge and passes on the most important ones subject to complexity constraints. The second string which is driven by the BIR oscillators may be a digital waveguide model driven at the bowing point,
Page 00000002 a) Output Amplitude(n) Impulse en ~ )xn Frequency(n) - Train String Resonator b)Y Output Amplitude(n) Ilmpulse e(n) a(n) x(n) Flclecv)~ra ~ -Resonatr toring Frqenyn Train - c) 4 Output Ampliude(n)- Impulse-Responaseh a on) xin) Frequency(n) TrrainSrn Figure 1: a) Simplified bowed string model, including only amplitude, pitch, and vibrato control capability. b) Equivalent diagram with resonator and string commuted. c) Equivalent diagram in which the resonator impulse response is played into the string each pitch perio d. or it may consist of an equivalent feedforward comb filter followed by a filtered delay loop. However, the advantage of a a full waveguide model of the string [Smith 1986] is that the time-varying, nonlinear, partial termination of the string by the bow can be more conveniently implemented. The Stick/Slip Bit can be used to switch between two models of partial string termination by the bow. For more accurate control of string damping by the bow, the contact force, relative velocity, position along the string, and bow angle can all be used to determine the frequency-dep endent scattering junction created by the bow on the string [Smith 1986]. It was found empirically that significant damping of the string by the bow is necessary for obtaining robust Helmholtz motion; otherwise, excessive ringing of the string segment between the bow and nut tends to cause slipping at times disruptive to the Helmholtz motion. Intuitively, one of the two "IHelmholtz corners" sent in opposite directions along the string on each slip/stick impulse must be "filtered out" by the bow, while the other is "amplified" by the stick/slip process. Graphical animation of the bowed string motion was found to be very helpful for determining qualitative factors such as this. 3 Friction Impulse Detection The output of the bow-string simulation must be converted to discrete trigger events, with each trigger initiating playback of the body impulse response (fIR). Ideally, we would like a means of "lthinning" the impulses coming from the bridge so as to keep the most important ones and neglect the least important ones to the degree necessary to meet computational re Filtered Noise-Burst Generator Fig ure 2: Commuted bowed string synthesis model driven by a separate bow-string model exhibiting full nonlinear dynamic behavior. source restrictions. There are several alternative impulse thinning schemes. Perhaps the simplest is to set an impulse amplitude threshold, such as ten percent of the expected main impulse amplitude, such that any impulse over the threshold in magnitude is passed on as a trigger, and anything smaller is suppressed. When the threshold is crossed by the absolute value of the bridge acceleration waveform in an upward direction, the next local maximnum is taken to determine the impulse amplitude and timing. No further impulses are accepted until the bridge acceleration falls below the threshold. As a further refinement, the samples on either side of the local maximum can be used to quadratically interpolate the peak, as is typically done for spectral peaks; alternatively, or in addition, the bow-string simulation can be run at a higher sampling rate than the commuted synthesis unit in order to further improve the impulse timing accuracy. The simple threshold method does not introduce latency, which is important in the real-time case, but it does not enable optimal impulse detection methods and there is no direct control over complexity (it is not easily known in advance what threshold will thin the impulse stream to the necessary extent). An indirect control over complexity is obtained by setting the threshold dynamically as a function of the number of overlapping fIRs. In this way, the threshold can be lifted to increase the thinning when the complexity becomes too great. An advantage of this thinning algorithm is that it doesn't matter what the source of complexity is. For example, impulses may be thinned because the pitch went higher causing more fIR overlap, or because other voices came in reducing the available number of fIR oscillators, or because the end user changed a preference specifying an upper limit on computing resources to be devoted to sound synthesis on a general purpose computer.
Page 00000003 A more direct impulse thinning scheme which introduces one period of latency delay is as follows: The most recent period of the bridge signal is kept in a circular buffer at all times. Let Ne denote the maximum number of stick-slip events allowed per period P. To restrict behavior to basic Helmholtz motion, Ne can be set to 1. To allow second-order Raman motion, Ne = 2 would be appropriate, and so forth. At each time step, the largest Ne peaks in the last period are defined as the impulses to send out. Since there is one period of latency, it is always the case that the emitted impulses are the most important ones within the past period. Having a period of "look ahead" enables use of more sophisticated peak detection schemes than the simple local-maximumafter-threshold-crossing method. A variation on the threshold method which does not need threshold adaption for complexity control is analogous to voice allocation in polyphonic synthesizers: When an impulse crosses a nominal threshold level, the next local maximum triggers a BIR playback unless (1) all playback units are busy and (2) the desired playback amplitude is smaller than that of all of the playing BIRs. When all BIR units are busy but one of them is deemed less important than the desired new BIR, the least important BIR is preempted, interrupting its playback and restarting it at the desired amplitude for the new BIR playback. 4 Pulsed Noise A stick-slip event never involves only one bow hair, and during the slipping interval, or string "flyback," there is a soft noise burst which is audible, especially at close range. It is well known that pulsed noise is an important feature of high quality bowed-string synthesis as well as other instruments [Chafe 1990]. The Stick/Slip Bit provided by the bow-string contact model (see Fig. 2) indicates when sliding noise is appropriate. As in the case of the time-varying string-damping discussed above, more refined noisegeneration models can be devised based on the bow force, differential velocity, and position information available from the bow-string simulator, as well as an external "bow angle" control. When the resonating body transfer function is factored [Karjalainen and Smith 1996] into slowly decaying modes (implemented parametrically using recursive filters and not necessarily commuted) and rapidly decaying modes (which are commuted and used in nonparametric form as impulse response data), the commuted nonparametric impulse response is qualitatively a short, high-frequency noise Frictional Force at Bow 0.5 -0.5-- 0 200 400 600 800 100 0 U. - E a 0 S-0.2' 0 0.1 0 p -0.1 0 Displacement at Bridge 200 400 600 Body Output Sound Pressure 800 SI I 1000 1000 200 400 600 800 Time (samples) Figure 3: Output of the bow-string model before extracting bridge imimpulses. Top: Frictional force between bow and string. Middle: String displacement 1/2 sample from the bridge. Bottom: Sound pressure radiated from simulated body filter. Bowing parameters (fixed): speed 15 cm/sec, force 20 grams, position 3 cm from bridge. A two-pole, two-zero bridge-filter for a digital waveguide string model was calibrated to measurements of violin pizzicato waveforms. A torsional-wave loss coefficient of 0.9 was implemented at the bow at all times. burst, since it consists of the impulse responses of thousands of high-frequency, highly damped modes. In principle, this "damped-modes-noiseburst" should be convolved with the noise arising from the slipping bow. In other words, the string excitation for each stick-slip event can be modeled as a filtered noise burst which includes both the highly damped resonator modes and the bow noise. 5 Simulation Results Figure 3 displays waveforms generated by the bowstring model given a constant bow force, velocity, and position. The frictional force applied to the string by the bow can be seen to diminish as the oscillation develops. The string displacement near the bridge clearly exhibits the single main impulse once per period associated with canonical Helmholtz bowed-string motion; there are also many secondary impulses associated with the ringing of the piece of the string between the bridge and the bow. The complexity control will determine whether these secondary impulses are included or suppressed. Figure 4 shows an overlay of the first 40 periods of oscillation of the bowed string, with each string
Page 00000004 Overlay of first 40 periods of oscillation Figure 4: Snapshots of string state for first 40 periods of oscillation. snapshot taken slightly later than one period after the previous, and the first snapshot being taken at time zero. The bow is at the sharp upper corner on the left. Note that the vertical scale is highly magnified relative to the horizontal scale. There is also some distortion in the string shape resulting from the lumping of the string losses at the bridge and bowing point, as is typical in waveguide string modeling. 6 Conclusions The commuted bowed-string synthesis model was extended to incorporate driving information from a nonlinear model of bowed-string dynamics. The formulation allows a simplified "single-hair bow" to control a pulsed-noise driven commuted synthesis model, thereby simulating a full-width bow in the final sound quality. Commuting only the fastest decaying (high frequency) body modes results in a short, damped impulse response which can be regarded as a component of the pulsed noise. In summary, driving a commuted-synthesis model for bowed strings from a nonlinear model of bowed-string dynamics gives the desirable combination of a full range of complex bowstring interaction behavior together with an reducecomplexity body resonator. References [Chafe 1990] Chafe, C. 1990. "Pulsed Noise in SelfSustained Oscillations of Musical Instruments." In: Proc. Int. Conf. Acoustics, Speech, and Signal Processing, Albuquerque. New York: IEEE Press. Available as CCRMA Technical Report STAN-M-65, Music Dept., Stanford University. [Cook 1996] Cook, P. R. 1996. "Synthesis ToolKit in C++, Version 1.0." In: SIGGRAPH Proceedings. Assoc. Comp. Mach. See http://www.cs.princeton.edu/ prc/NewWork.html for a copy of this paper and the software. (All simulations for this paper were done using the STK and Matlab.) [Guettler 1992] Guettler, K. 1992. "The Bowed String Computer Simulated - Some Characteristic Features of the Attack." Catgut Acoustical Soc. Journal, 2(2):22-26. Series II. [Jaffe and Smith 1995] Jaffe, D. A., and J. O. Smith. 1995. "Performance Expression in Commuted Waveguide Synthesis of Bowed Strings." Pages 343-346 of: Proc. 1995 Int. Computer Music Conf., Banff. Computer Music Association (CMA). [Karjalainen and Smith 1996] Karjalainen, M., and J. 0. Smith. 1996. "Body Modeling Techniques for String Instrument Synthesis." In: Proc. 1996 Int. Computer Music Conf., Hong Kong. CMA. [McIntyre and Woodhouse 1979] McIntyre, M. E., and J. Woodhouse. 1979. "On the Fundamentals of Bowed String Dynamics." Acustica, 43(2):93-108. [Pickering 1991] Pickering, N. C. 1991. The Bowed String. Mattituck NY: Amereon, Ltd. Also available from Bowed Instruments, 23 Culver Hill, Southampton, NY 11968. [Pitteroff 1993] Pitteroff, R. 1993. "Modelling of the bowed string taking into account the width of the bow." Pages 407-410 of: Proc. Stockholm Musical Acoustics Conference (SMAC-93). Stockholm: Royal Swedish Academy of Music. [Raman 1918] Raman, C. V. 1918. "On the Mechanical Theory of Vibrations of Bowed Strings, etc." Indian Assoc. Cult. Sci. Bull., 15:1-158. [Smith 1990] Smith, J. H. 1990. Stick-Slip Vibration and its Constitutive Laws. Ph.D. thesis, Cambridge Univ. [Smith 1986] Smith, J. 0. 1986. "Efficient Simulation of the Reed-Bore and Bow-String Mechanisms." Pages 275-280 of: Proc. 1986 Int. Computer Music Conf., The Hague. CMA. [Smith 1993] Smith, J. 0. 1993. "Efficient Synthesis of Stringed Musical Instruments." Pages 64-71 of: Proc. 1993 Int. Computer Music Conf., Tokyo. CMA. [Smith 1996] Smith, J. 0. 1996. "Physical Modeling Synthesis Update." Computer Music J., 20(2):44-56. Available online at http://www-ccrma.stanford.edu/~jos/. [von Helmholtz 1863] von Helmholtz, H. L. F. 1863. On the Sensations of Tone as a Physiological Basis for the Theory of Music. New York: Dover. English translation by A. J. Ellis. 1954.