Page  110 ï~~General Weirdness with the Karplus-Strong String Tim Stilson Center for Computer Research in Music and Acoustics (CCRMA) Stanford University stilti@ccrma.stanford.edu ABSTRACT: A Karplus-Strong string is length-modulated by a sine wave whose frequency is very close to the fundamental frequency of the string. The resulting waveform is quite complex and sonically intriguing. The addition of an instability at 4allows the system to regenerate, giving a sound that can go for long periods of time before repeating. The string is analyzed in comparison to FM and time-domain pitch shifting, both of which have similar architectures and actions. The introduction of non-linearities into digital]mod. index waveguide models has been shown to add very in- Af teresting and satisfying complexity to the resulting sound (Van Duyne, et. al. 94). A version of the Extended Karplus-Strong string (Jaffe and Smith 83) f-q f"-"l is presented which achieves its effect not through a nonlinearity in the string loop, but instead as a dis- excitation in!-1,- s ruption of the time- invariance of the loop. Audiofrequency modulation of a string pick-up point was explored by Van Duyne (Van Duyne and Smith 92), when the string length is modulated rather than the pickup position, certain effects are similar (an FM- Figure 1: "WeirdString", (string loop in bold) like effect is seen in many cases). However, when the delay-line length is modulated at a frequency near to the fundamental frequency of the string, a more complex interaction occurs, and a self-regenerating sound can be produced which is exceedingly complex, yet quite intriguing. How it Acts: When the modulation index is zero, the system acts just like a normal string model. As the modulation index is increased, the effect appears and increases in intensity until, depending on the string length and ALf, the sound either kills itself or the regeneration becomes overpowering (see the Regeneration section). Moving the modulating frequency away from the string's frequency reduces the effect to FM, and moving it too close to the string's frequency causes the effect to blow up even at low modulation indices. Finally, everything sounds best at very low frequencies (freq < 30 Hz, preferably 10 Hz) because of the patterns that become audible far above the fundamental frequency. Doppler Effects: Because the string length is being modulated at nearly the same frequency as the string is resonating, half of the wave in the string will be time-compressed at it reflects off the end of the string, because the string is shortening, and the other half of the wave will be expanded, because the string is lengthening. In this case, the string's length changes smoothly, so that the compressed part of the wave appears to have been pitch-shifted upwards, and the expanded part appears to have been pitch-lowered (indeed, a degenerate time-domain pitch shifter can be visualized as a delay whose length is being modulated by a sawtooth wave). Thus we get a simultaneous upward and downward shift of the spectrum of the sound that exists in the string. When A! is not too close to zero, the effect will soon reverse itself, because the modulating oscillator becomes out of 110 I C M C P ROCE E D I N G S 1995

Page  111 ï~~phase from the string. Thus sections of the wave that had been shifted up, get shifted down, and vice versa. If A f is too small, waveform will become too distorted before the effect reverses, and will succumb to various numerical effects, which kill it ("blowing up") Half-Cycle Effects: Quite a few algorithms treat different parts of a cycle differently, such as a string with a passive nonlinear filter (Van Duyne et al 94), where the PNF's action is dependent on the state itself, so can, when placed in a string loop, appear to give two different string lengths for different parts of the signal, which produces a sound quite similar to the WeirdString effect. A simple variation of the patch modulates the string length by a function of the string state, rather than simply by the string fundamental, which produces a slightly different class of sounds and behaviors ("it sounds sort of like a speed boat on the open sea..."). Regeneration: The signal-controlled one-pole filter in the loop has its gain set such that during half of the cycle, the filter has a large gain around 4Lrather than an attenuation (acenter determines how much of the cycle is gain and how much is attenuation). This causes the loop to become unstable under certain circumstances. If the standard Karplus-Strong loop filter is used for the FIR filter in the loop (a two-point average), the instability is never seen, because the two-point average has a zero at 4, which quells the gain of the one-pole filter, but if the feedback FIR filter is changed, even if only a little bit, so that the zero moves off 4, the instability manifests itself. The previously-mentioned pitch-shifting effect of the modulation causes the high frequencies generated by the instability to be pushed down in frequency into the rest of the spectrum, thus regenerating the effect over the whole frequency range. Regeneration allows the sound to continue for quite a long time, and adds new dimensions to the effect. In the shown patch, the "intensity" of the instability is proportional to the modulation index, so the amount of regeneration is connected to the magnitude of the whole effect. If the modulation index gets too high, the regeneration can end up overpowering everything else, and degenerating the sound into noise. Additional: A stereo version of this patch has been constructed by making a nearly identical copy of the string and placing it on the other channel. Because of the complexity of the action of this system, most differences between the two strings would cause their sounds to quickly become uncorrelated, killing any stereo image, thus only the tiniest of changes is made to the copy: the modulating oscillator is temporarily detuned to cause the modulators to become out of phase between the two channels, otherwise the channels are identical and are excited and controlled identically. The repetition period of this system can be even further lengthened by adding an LFO tuned to some very low frequency (such as -u Hz) to the modulation index. This has produced patterns whose general sound 'shape' changes on the scale of minutes. References [Jaffe and Smith, 1983] David A. Jaffe, Julius 0. Smith III. Extensions of the Karplus-Strong PluckedString Algorithm, Computer Music Journal 7(2): 56-69, 1983. [Karplus and Strong, 1983] Kevin Karplus, Alex Strong. Digital Synthesis of Plucked-String and Drum Timbres, Computer Music Journal 7(2): 43-55, 1983. [Van Duyne et. al, 1994] Scott A. Van Duyne, John R. Pierce, and Julius 0. Smith III. Traveling Wave Implementation of A Lossless Mode-Coupling filter and the Wave Digital Hammer, ICMC-94, Aarhus, Denmark. 94 [Van Duyne and Smith, 1992] Scott A. Van Duyne and Julius 0. Smith III. Implementation of a Variable Pick- Up Point on a Waveguide String Model with FM/AM Applications, ICMC-92, San Jose. 92 This material is based upon work supported under a National Science Foundation Graduate Fellowship. I C M C P ROC E E D I N G S 199511 111