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Nonlinear Distortion

In §6.13, nonlinear elements were introduced in the context of general digital waveguide synthesis. In this section, we discuss specifically virtual electric guitar distortion, and mention other instances of audible nonlinearity in stringed musical instruments.

As discussed in Chapter 6, typical vibrating strings in musical acoustics are well approximated as linear, time-invariant systems, there are special cases in which nonlinear behavior is desired.

Tension Modulation

In every freely vibrating string, the fundamental frequency declines over time as the amplitude of vibration decays. This is due to tension modulation, which is often audible at the beginning of plucked-string tones, especially for low-tension strings. It happens because higher-amplitude vibrations stretch the string to a longer average length, raising the average string tension $ \Rightarrow$ faster wave propagation $ \Rightarrow$ higher fundamental frequency.

The are several methods in the literature for simulating tension modulation in a digital waveguide string model [494,233,508,512,513,495,283], as well as in membrane models [298]. The methods can be classified into two categories, local and global.

Local tension-modulation methods modulate the speed of sound locally as a function of amplitude. For example, opposite delay cells in a force-wave digital waveguide string can be summed to obtain the instantaneous vertical force across that string sample, and the length of the adjacent propagation delay can be modulated using a first-order allpass filter. In principle the string slope reduces as the local tension increases. (Recall from Chapter 6 or Appendix C that force waves are minus the string tension times slope.)

Global tension-modulation methods [495,494] essentially modulate the string delay-line length as a function of the total energy in the string.

String Length Modulation

A number of stringed musical instruments have a ``nonlinear sound'' that comes from modulating the physical termination of the string (as opposed to its acoustic length in the case of tension modulation).

The Finnish Kantele [231,513] has a different effective string-length in the vertical and horizontal vibration planes due to a loose knot attaching the string to a metal bar. There is also nonlinear feeding of the second harmonic due to a nonrigid tuning peg.

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 [263,366].

In all of these cases, the string length is physically modulated in some manner each period, at least when the amplitude is sufficiently large.

Hard Clipping

A widespread class of distortion used in electric guitars, is clipping of the guitar waveform. it is easy to add this effect to any string-simulation algorithm by passing the output signal through a nonlinear clipping function. For example, a hard clipper has the characteristic (in normalized form)

$\displaystyle f(x) = \left\{\begin{array}{ll} -1, & x\leq -1 \\ [5pt] x, & -1 \leq x \leq 1 \\ [5pt] 1, & x\geq 1 \\ \end{array} \right. \protect$ (10.4)

where $ x$ denotes the current input sample $ x(n)$, and $ f(x)$ denotes the output of the nonlinearity.

Soft Clipping

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. [489],

$\displaystyle f(x) = \left\{\begin{array}{ll} -\frac{2}{3}, & x\leq -1 \\ [5pt]... ... \leq x \leq 1 \\ [5pt] \frac{2}{3}, & x\geq 1. \\ \end{array} \right. \protect$ (10.5)

This particular soft-clipping characteristic is diagrammed in Fig.9.3. An analysis of its spectral characteristics, with some discussion of aliasing it may cause, was given in in §6.13. An input gain may be used to set the desired degree of distortion.

Figure: Soft-clipper defined by Eq.$ \,$(9.5).
\includegraphics[width=3in]{eps/cnl}

Enhancing Even Harmonics

A cubic nonlinearity, as well as any odd distortion law,10.2 generates only odd-numbered harmonics (like in a square wave). For best results, and in particular for tube distortion simulation [31,395], it has been argued that some amount of even-numbered harmonics should also be present. Breaking the odd symmetry in any way will add even-numbered harmonics to the output as well. One simple way to accomplish this is to add an offset to the input signal, obtaining

$\displaystyle y(n) = f[x(n) + c], $

where $ c$ is some small constant. (Signals $ x(n)$ in practice are typically constrained to be zero mean by one means or another.)

Another method for breaking the odd symmetry is to add some square-law nonlinearity to obtain

$\displaystyle f(x) = \alpha x^3 + \beta x^2 + \gamma x + \delta \protect$ (10.6)

where $ \beta$ controls the amount of square-law distortion in the more general third-order polynomial. The square-law is the most gentle nonlinear distortion in existence, adding only some second harmonic to a sinusoidal input signal. The constant $ \delta$ can be set to zero the mean, on average; if the input signal $ x(n)$ is zero-mean with variance is 1, then $ \delta= - \beta$ will cancel the nonzero mean induced by the squaring term $ \beta x^2$. Typically, the output of any audio effect is mixed with the original input signal to allow easy control over the amount of effect. The term $ \gamma$ can be used for this, provided the constant gains for $ x>1$ and $ x<-1$ are modified accordingly, or $ x$ is hard-clipped to the desired range at the input.

Software for Cubic Nonlinear Distortion

The function cubicnl in the Faust software distribution (file effect.lib) implements cubic nonlinearity distortion [454]. The Faust programming example cubic_distortion.dsp provides a real-time demo with adjustable parameters, including an offset for bringing in even harmonics. The free, open-source guitarix application (for Linux platforms) uses this type of distortion effect along with many other guitar effects.

In the Synthesis Tool Kit (STK) [91], the class Cubicnl.cpp implements a general cubic distortion law.

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The Extended Karplus-Strong Algorithm