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Electric Guitars

For electric guitars, which are mainly about the vibrating string and subsequent effects processing, we pick up where §6.7 left off, and work our way up to practical complexity step by step. We begin with practical enhancements for the waveguide string model that are useful for all kinds of guitar models.

Length Three FIR Loop Filter

The simplest nondegenerate example of the loop filters of §6.8 is the three-tap FIR case ( $ N_{\hat g}=3$). The symmetry constraint leaves two degrees of freedom in the frequency response:10.1

$\displaystyle {\hat G}(e^{j\omega T}) = {\hat g}(0) + 2{\hat g}(1) \cos(\omega T)

If the dc gain is normalized to unity, then $ {\hat g}(0)+2{\hat g}(1)=1$, and there is only one remaining degree of freedom which can be interpreted as a damping control. As damping is increased, the duration of free vibration is reduced at all nonzero frequencies, and the decay of higher frequencies is accelerated relative to lower frequencies, provided

$\displaystyle {\hat g}(0) \ge 2{\hat g}(1) > 0.

In this coefficient range, the string-loop amplitude response can be described as a ``raised cosine'' having a unit-magnitude peak at dc, and minimum gains $ {\hat g}(0)-2{\hat g}(1)\ge 0$ at plus and minus half the sampling rate ( $ \omega T=\pm\pi$).

Length $ 3$ FIR Loop Filter Controlled by ``Brightness'' and ``Sustain''

Another convenient parametrization of the second-order symmetric FIR case is when the dc normalization is relaxed so that two degrees of freedom are retained. It is then convenient to control them as brightness $ B$ and sustain $ S$ according to the formulas
$\displaystyle g_0$ $\displaystyle =$ $\displaystyle \exp( - 6.91 P / S)$ (10.1)
$\displaystyle {\hat g}(0)$ $\displaystyle =$ $\displaystyle g_0 (1 + B)/2$ (10.2)
$\displaystyle {\hat g}(1)$ $\displaystyle =$ $\displaystyle g_0 (1 - B)/4
\protect$ (10.3)

where $ P$ is the period in seconds (total loop delay), $ S$ is the desired sustain time in seconds, and $ B$ is the brightness parameter in the interval $ [0,1]$. The sustain parameter $ S$ is defined here as the time to decay by $ -60$ dB (or $ \approx 6.91$ time-constants) when brightness $ B$ is maximum ($ B=1$) in which case the loop gain is $ g_0$ at all frequencies, or $ {\hat G}(e^{j\omega T}) = g_0$. As the brightness is lowered, the dc gain remains fixed at $ g_0$ while higher frequencies decay faster. At the minimum brightness, the gain at half the sampling rate reaches zero, and the loop-filter amplitude-response assumes the form

$\displaystyle {\hat G}(e^{j\omega T}) = g_0\frac{1 + \cos(\omega T)}{2} = g_0 \cos^2\left(\frac{\omega T}{2}\right).

A Faust function implementing this FIR filter as the damping filter in the Extended Karplus Strong (EKS) algorithm is described in [454].

One-Zero Loop Filter

If we relax the constraint that $ N_{\hat g}$ be odd, then the simplest case becomes the one-zero digital filter:

$\displaystyle {\hat G}(z) = {\hat g}(0) + {\hat g}(1) z^{-1}

When $ {\hat g}(0)={\hat g}(1)$, the filter is linear phase, and its phase delay and group delay are equal to $ 1/2$ sample [362]. In practice, the half-sample delay must be compensated elsewhere in the filtered delay loop, such as in the delay-line interpolation filter [207]. Normalizing the dc gain to unity removes the last degree of freedom so that $ {\hat g}(0) = {\hat g}(1) = 1/2$, and $ {\hat G}(e^{j\omega T}) = \cos\left({\omega T/ 2}\right),\,\left\vert\omega\right\vert\leq \pi f_s$. See [454] for related discussion from a software implementation perspective.

The Karplus-Strong Algorithm

The simulation diagram for the ideal string with the simplest frequency-dependent loss filter is shown in Fig. 9.1. Readers of the computer music literature will recognize this as the structure of the Karplus-Strong algorithm [236,207,489].
Figure 9.1: Rigidly terminated string with the simplest frequency-dependent loss filter. All $ N$ loss factors (possibly including losses due to yielding terminations) have been consolidated at a single point and replaced by a one-zero filter approximation.
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. 6.11, i.e., $ y(t_n,x_m) = y^{+}(n-m) +
y^{-}(n+m)$. It is shown in §C.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 [428,207]. Karplus-Strong sound examples are available on the Web. An implementation of the Karplus-Strong algorithm in the Faust programming language is described (and provided) in [454].

The Extended Karplus-Strong Algorithm

Figure 9.2 shows a block diagram of the Extended Karplus-Strong (EKS) algorithm [207].
Figure 9.2: Extended Karplus-Strong (EKS) algorithm.
The EKS adds the following features to the KS algorithm:
H_p(z) &=& \frac{1-p}{1 - p\,z^{-1}}\eqsp \mbox{pick-direction...
...1-R_L}{1 - R_L\,z^{-1}}\eqsp \mbox{dynamic-level lowpass filter}
N &=& \mbox{pitch period ($2\times$\ string length) in samples...
...e^{j\omega T})\right\vert &\le& 1 \mbox{ required for stability}
Note that while $ \eta\in[0,1)$ can be used in the tuning allpass, it is better to offset it to $ [\epsilon,1+\epsilon)$ to avoid delays close to zero in the tuning allpass. (A zero delay is obtained by a pole-zero cancellation on the unit circle.) First-order allpass interpolation of delay lines was discussed in §4.1.2. A history of the Karplus-Strong algorithm and its extensions is given in §A.8. EKS sound examples are also available on the Web. Techniques for designing the string-damping filter $ H_d(z)$ and/or the string-stiffness allpass filter $ H_s(z)$ are summarized below in §6.11. An implementation of the Extended Karplus-Strong (EKS) algorithm in the Faust programming language is described (and provided) in [454].

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).

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.

Amplifier Feedback

A more extreme effect used with distorted electric guitars is amplifier feedback. In this case, the amplified guitar waveforms couple back to the strings with some gain and delay, as depicted schematically in Fig.9.4 [489].
Figure 9.4: Simulation of a basic distorted electric guitar with amplifier feedback.
The Amplifier Feedback Delay in the figure can be adjusted to emphasize certain partial overtones over others. If the loop gain, controllable via the Amplifier Feedback Gain, is greater than 1 at any frequency, a sustained ``feedback howl'' will be produced. Note that in commercial devices, the Pre-distortion gain and Post-distortion gain are frequency-dependent, i.e., they are implemented as pre- and post-equalizers (typically only a few bands, such as three). Another simple choice is an integrator $ g/(1-rz^{-1})$ for the pre-distortion gain, and a differentiator $ (1-rz^{-1})$ for the post-distortion gain. Faust software implementing electric-guitar amplifier feedback may be found in [454].

Cabinet Filtering

The distortion output signal is often further filtered by an amplifier cabinet filter, representing the enclosure, driver responses, etc. It is straightforward to measure the impulse response (or frequency response) of a speaker cabinet and fit it with a recursive digital filter design function such as invfreqz in matlab8.6.4). Individual speakers generally have one major resonance--the ``cone resonance''--which is determined by the mass, compliance, and damping of the driver subassembly. The enclosing cabinet introduces many more resonances in the audio range, generally tuned to even out the overall response as much as possible. Faust software implementing an idealized speaker filter may be found in [454].

Duty-Cycle Modulation

For class A tube amplifier simulation, there should be level-dependent duty-cycle modulation as a function:10.3
  • 50% at low amplitude levels (no duty-cycle modulation)
  • 55-65% duty cycle at high levels
A simple method for implementing duty-cycle modulation is to offset the input to nonlinearity Eq.$ \,$(9.5) by a constant, such as input RMS level times some scale factor.

Vacuum Tube Modeling

The elementary nonlinearities discussed above are aimed at approximating the sound of amplifier distortion, such as often used by guitar players. Generally speaking, the distortion characteristics of vacuum-tube amplifiers are considered superior to those of solid-state amplifiers. A real-time numerically solved nonlinear differential algebraic model for class A single-ended guitar power amplification is presented in [84]. Real-time simulation of a tube guitar-amp (triode preamp, inverter, and push-pull amplifier) based on a precomputed approximate solution of the nonlinear differential equations is presented in [293]. Other references in this area include [338,228,339].
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Acoustic Guitars
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Virtual Analog Example: Phasing