The Karplus-Strong Algorithm

The simulation diagram for the ideal string with the simplest frequency-dependent loss filter is shown in Fig. 4.12. Readers of the computer music literature will recognize this as the structure of the Karplus-Strong algorithm [241,213,502].

The Karplus-Strong algorithm, per se, is obtained when the delay-line initial conditions used to ``pluck'' the string consist of random numbers, or ``white noise.'' We know the initial shape of the string is obtained by adding the upper and lower delay lines of Fig. 4.10, i.e., . It is shown in §H.7.4 that the initial velocity distribution along the string is determined by the difference between the upper and lower delay lines. Thus, in the Karplus-Strong algorithm, the string is ``plucked'' by a random initial displacement and initial velocity distribution. This is a very energetic excitation, and usually in practice the white noise is lowpass filtered; the lowpass cut-off frequency gives an effective dynamic level control since natural stringed instruments are typically brighter at louder dynamic levels [437,213].

Karplus-Strong sound examples are available on the Web.

Appendix D contains a section (§D.1) on implementing the Karplus-Strong algorithm in the Faust programming language.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.