Nonlinear Distortion

While most musical vibrating strings are well approximated as linear, time-invariant systems, there are special cases in which nonlinear behavior is desired. This section summarizes a few examples.

One class of nonlinearity is purposely created in the design of some acoustic stringed instruments. For example, the Finnish Kantele exhibits nonlinear characteristics [234]. Perhaps a better known example is the Indian sitar, in which a curved ``jawari'' (functioning as a nonlinear bridge) serves to shorten the string gradually as it displaces toward the bridge. The Indian tambura also employs a thread perpendicular to the strings a short distance from the bridge, which serves to shorten the string whenever string displacement toward the bridge exceeds a certain distance. Finally, the slap bass playing technique for bass guitars involves hitting the string hard enough to cause it to beat against the neck during vibration [375]. In all of these cases, the string length is physically modulated in some manner each period, at least when the amplitude is sufficiently large.

Another widespread class of distortion, used in electric guitars, is clipping of the guitar waveform. While this more properly belongs in the category of ``virtual analog'' synthesis techniques [59,491,492,278,69], it is easy to add this effect to any string-simulation algorithm by passing the output signal through a nonlinear clipping function [502]. For example, a hard clipper has the characteristic (in normalized form)

where denotes the current input sample , and denotes the output of the nonlinearity.

A soft clipper is similar to a hard clipper, but with the corners smoothed. A common choice of soft-clipper is the cubic nonlinearity, e.g. [502],

This particular soft-clipping characteristic is diagrammed in Fig.D.12. An analysis of its spectral characteristics, with some discussion of aliasing it may cause, appears in Appendix T. An input gain may be used to set the desired degree of distortion.

A cubic nonlinearity, as well as any odd distortion law,^5.19 generates only odd-numbered harmonics (like in a square wave). For best results, and in particular for tube distortion simulation [