4.3 Musical Distortion and Saturation Effects

4.3.1 Valve Simulation

Introduction

Valve or tube devices dominated electronic signal processing circuits during the first part of the last century and have experienced a revival in audio processing every decade since their introduction [Bar98, Ham73]. One of the most commonly used effects for electric guitars is the amplifier and especially the valve amplifier. The typical behavior of the amplifier and the connected loudspeaker cabinet have demonstrated their influence on the sound of rock music over the past decades. Besides the two most important guitars, namely the Fender Stratocaster and the Gibson Les Paul, several valve amplifiers have helped in creating exciting sounds from these classic guitars.

Valve microphones, preamplifiers and effect devices such as compressors, limiters and equalizers are also used for vocal recordings where the warm and subtle effect of valve compression is applied. A lot of vocalists prefer recording with valve condenser microphones because of their warm low end and smooth top end frequency response. Also the recording of acoustical instruments such as acoustic guitars, brass instruments and drums benefit from being processed with valve outboard devices. Valve processors also assist the mixing process for individual track enhancing and on the mix buses. The demand for valve outboard effects and classic mixing consoles used in combination with digital audio workstations has led back to entire valve technology mixing consoles. For the variety of digital audio workstations a lot of plug-in software modules for valve processing are available.

Vintage Valve Amplifiers

An introduction to valve amplifiers and their history can be found in [Fli93, Bar98], where several comments on sound characteristics are published. We will concentrate on the most important amplifier manufacturers over the years and point out some characteristic features.

• Fender: The Fender series of guitar amplifiers goes back to the year 1946 when the first devices were introduced. These were based on standard tube schematics supplied by the manufacturers of tubes. Over the years modifications of the standard design approach were integrated in response to musicians' needs and proposals. The range of Fender amplifiers is still expanding, but also reissues of the originals are very popular with musicians. The sound of Fender amplifiers is the “classic tube sound” and was made famous by blues musicians like Buddy Guy and Stevie Ray Vaughan. A detailed analysis of a Fender amplifier and a discussion on its similarity to other amplifier designs is presented in [Kue05].
• Vox: The manufacturer Vox is always associated with its most famous amplifier, the Vox AC30/4. Its sound is best characterized by guitar player Brian May in [PD93] where he states “the quality at low levels is broad and crisp and unmistakably valve like, and as the volume is turned up it slides into a pleasant, creamy compression and distortion.” There is always a ringing treble quality through all level settings of the amp. The real “soul of the amp” comes out if you play it at full volume and control your sound with the volume knob of your guitar. The heart of the sound characteristic of the Vox AC30/4 is claimed to be the use of EL84 pentodes, negative feedback and cathode biasing in Class A configuration. The four small EL84s should sound more lively than the bigger EL34s. The sound of the Vox AC30 can be found on recordings by Brian May, Status Quo, Tom Petty and Bryan Adams.
• Marshall: The Fender Bassman 5F6 was the basis for the Marshall JTM 45. The differences between both are discussed in [Doy93] and [Kue05], and are claimed to be the output transformers, speakers, input valve and feedback circuit, although the main circuit diagrams are nearly identical. The sound of Marshall is characterized by an aggressive and “crunchy” tone with brilliant harmonics, as Eric Clapton says, “I was probably playing full volume to get that sound” [Doy93]. Typical representatives of the early Marshall sound are Jimi Hendrix, Eric Clapton, Jimmy Page and Ritchie Blackmore. The JCM800 series established the second generation of Marshall amplifiers featuring the typical hardrock and heavy metal sound as played by Zakk Wylde or Slash.
• Mesa-Boogie: The “creamy” tone and the high gain of Mesa-Boogie amplifiers has its seeds in a wrongly connected test arrangement of two preamp stages. Founder Randall Smith stated in an interview [Sal02]: “Then when we plugged it in right, Lee hit a big power chord and practically blew both our bodies right through the back wall! (…) It was just HUGE sounding. And it would sustain forever. That was the beginning of cascading high-gain pre-amp architecture”. Through this assembly Mesa-Boogie amplifiers featured 50 to 80 times more gain compared to amplifier designs of that time, leading to a long sustaining tone. An ambassador for Mesa-Boogie amplifiers is Carlos Santana.

Signal Processing

The sound of a valve amplifier is based on a combination of several important factors. First of all the main processing features of valves or tubes are important [Bar98, Ham73]. Then the amplifier circuit has its influence on the sound and, last but not least, the chassis and loudspeaker combination play an important role in sound shaping. We will discuss all three factors now.

Valve basics. Valves or vacuum tubes are active electronic components used for amplifying, rectifying, switching, modulating or generating electrical signals. Prior to the invention of transistors, valves were the main active components in electronic equipment. In today's electronics, valves are replaced completely by semiconductors, except for some special applications. In the following we will discuss the reasons why these components are still very popular in hi-fi and guitar amplifiers.

Triode valves [Rat95, RCA59] consist of three electrodes, namely anode (or plate), cathode (or filament) and grid. Applying a voltage between anode and cathode, electrons are emitted by the heated cathode and flow to the positively charged anode. The grid is placed between these electrodes and can be used to modulate the rate of electron flow. A negative charge on the grid electrode affects the electron flow: the larger the charge, the smaller the current from anode to cathode. Thus, the triode can be used as voltage-controlled amplifier. The corresponding transfer function relates the anode current IA to the input grid voltage VG, as depicted in Figure 4.19. This nonlinear curve has a quadratic shape. An input signal represented by the grid voltage VG delivers an anode output current IA = f(VG) representing the output signal. The corresponding output spectrum shows a second harmonic in addition to the input frequency. This second harmonic can be lowered in amplitude when the operating point of the nonlinear curve is shifted right and the input voltage is applied to the more linear region of the quadratic curve. As a consequence of this, triodes are considered to provide a warm and soft sound coloration when used in preamplifiers.

Figure 4.19 Triode: nonlinear characteristic curve IA = f(VG) and nonlinear effect on input signal. The output spectrum consists of the fundamental input frequency and a second harmonic generated by the quadratic curve of the triode.

The dc component in the output signal can be suppressed by a subsequent highpass filter. Note also the asymmetrical soft clipping of the negative halves of the output sinusoid, which is the result of the quadratic curve of the triode. Input stages of valve amplifiers make use of these triode valves. A design parameter is the operating point which controls the amplitude of the second harmonic.

Pentode valves feature two additional electrodes, the screen and the suppressor grid. With this arrangement oscillations can be suppressed which can arise in triodes. When driving a load resistance the characteristic curve IA = f(VG) is shaped like a S-curve, as shown in Figure 4.20. Through this, the output signal is compressed for higher-input amplitudes, leading to a symmetrical soft clipping. The corresponding output spectrum shows the creation of odd order harmonics. For lower input amplitudes the static characteristic curve operates in a nearly linear region, which again shows the control of the nonlinear behavior by properly selecting the operating point.

Figure 4.20 Pentode: nonlinear characteristic curve IA = f(VG) and nonlinear effect on input signal.

The technical parameters of valves have wide variation, which leads to a wide variation of sound features, although selected valves (so-called “matched pairs”) with limited deviations of parameters are of course available. All surrounding environmental parameters like humidity and temperature have their influence as well.

Valve amplifier circuits. Valve amplifier circuits are based on the block diagram in Figure 4.21. Several measured signals from a Vox AC30 at different stages of the signal flow path are also displayed. This will give an indication of typical signal distortions in valve amplifiers. The corresponding spectra of the signals for the Vox AC30/4 measurements are shown in Figure 4.22. The distortion characteristic can be visualized by the waterfall representation of short-time FFTs for a chirp input signal in Figure 4.23. The main stages of a valve amplifier are given below:

• The input stage consists of a triode circuit providing the input matching and preamplification followed by a volume control for the next stages.
• The tone control circuitry is based on passive filter networks, typically with three controls for bass, mid and treble.
• The phase splitter stage provides symmetrical power amp feeding. This phase splitter delivers the original input for the upper power amp and a phase inverted replica of the input for the lower power amp.
• The power amp stage in push-pull configuration performs individual amplification of the original and the phase inverted replica in a class A, class B or class AB configuration (see Figure 4.24). Class A is shown in the left plot, where the output signal is valid all the time. Class B performs amplification only for one half wave, as depicted in the middle plot. The working point for class AB operation (right plot) lies in-between class A and class B, also amplifying a part of the negative half wave. Class A and class AB are the main configurations for guitar power amplifiers.
• The output transformer performs the subtraction of both waveforms delivered by the power amplifiers, which leads to a doubling of the output amplitude. Figure 4.25 shows the principle of push-pull amplification and the typical connection of the output transformer. The nonlinear behavior of transformers is beyond the scope of this discussion.

Figure 4.21 Main stages of a valve amplifier. Upper left plot shows signal after pre-amplifier, lower left plot shows signal after phase splitter, upper right plot shows signal after power amplifier and lower right plot shows signal after output transformer.

Figure 4.22 Vox AC30/4 spectra at different stages: (a) input stage, (b) output phase splitter, (c) output power amp and (d) output of transformer.

Figure 4.23 Short-time FFTs (waterfall representation) of Vox AC30 with a chirp input signal. The amplifier is operating with full volume setting.

Figure 4.24 Power amplifier operation with idealized transfer characteristic (left class A, middle class B, right class AB operation).

Figure 4.25 Power amplifier stage and output transformer. Left: principle and waveforms for class AB push-pull power amplifier. Right: simplified circuit with two pentodes, output transformer and loudspeaker.

Speaker and cabinet. Guitar amplifiers are built either as a combo, where amplifier chassis and one or more loudspeakers are combined in the same enclosure or as a stack with separated amplifier head and loudspeaker cabinet. The traditional guitar cabinet is closed-back and houses four 12′′ speakers (4×12), but different combinations with loudspeakers in the range from 8′′ to 15′′ in closed or open enclosures are also available. The frequency responses of common guitar cabinets show an uneven bandpass characteristic with many resonances at mid frequencies. Simulations can be done by impulse response measurements of the loudspeaker and cabinet combination. The nonlinear behavior of a loudspeaker cabinet was analyzed and modeled in [YBK08].

As well as the discussed topics, the influence of the power supply with valve rectifier [zL97, pp. 51–54] is claimed to be of importance. A soft reduction of the power supply voltage occurs when in a high-power operation short transients need a high current. This power supply effect leads to a soft clipping of the audio signal.

A proposal for tube simulation by using asymmetrical clipping [Ben97] is given by

4.13

The underlying design parameters for the simulation of tube distortion are based on the mathematical model [Ben97] where no distortion should occur when the input level is low (the derivative of f(x) has to be f′(0) ≈ 1 and f(0) = 0). The static characteristic curve should perform clipping and limiting of large negative input values and be approximately linear for positive values. The result of Equation (4.13) is shown in Figure 4.26.

Figure 4.26 Static characteristic curve of asymmetric soft clipping for tube simulation Q = − 0.2 and dist = 8.

The following M-file 4.4 performs Equation (4.13) from [Ben97]. To remove the dc component and to shape higher harmonics, additional lowpass and highpass filtering of the output signal is performed.

Short-time FFTs (waterfall representation) of this algorithm applied to a 1 kHz sinusoid are shown in Figure 4.27. The waterfall representation shows strong even-order harmonics and also odd-order harmonics.

Figure 4.27 Short-time FFTs (waterfall representation) of asymmetrical soft clipping.

Digital Amp Modeling

New guitar amplifier designs with digital preamplifiers are based on digital modeling technology, featuring the simulation of classic valve amplifiers. Available are combos, amplifier heads or separated preamplifiers, mostly providing a wide variety of amplifier models in combination with additional effects. Besides these guitar amplifiers, software models are very popular which can be used as plug-ins together with a recording software. The principles of established modeling techniques will be introduced in Section 12.3.

Musical Applications

Musical applications of valve amplifiers can be found on nearly every recording featuring guitar tracks. Ambassadors of innovative guitar players from the blues to the early rock period are B. B. King, Albert King and Chuck Berry, who mainly used valve amplifiers for their warm and soft sound. Representatives of the classic rock period are Jimi Hendrix [m-Hen67a, m-Hen67b, m-Hen68], Eric Clapton [m-Cla67], Jimmy Page [m-Pag69], Ritchie Blackmore [m-Bla70], Jeff Beck [m-Bec89] and Carlos Santana [m-San99]. All make extensive use of valve amplification and special guitar effect units. There are also players from the new classic period like Eddie van Halen, Steve Ray Vaughan and Steve Morse up to the new guitar heroes such as Steve Lukather, Joe Satriani, Gary Moore, Steve Vai and Paul Gilbert, who are using effect devices together with valve amplifiers.

4.3.2 Overdrive, Distortion and Fuzz

Introduction

As pointed out in the section on valve simulation, the distorted electric guitar is a central part of rock music. In addition to the guitar amplifier as a major sound effect device, several stomp boxes (foot-operated pedals) have been used by guitar players for the creation of their typical guitar sound. Guitar heroes like Jimi Hendrix have made use of several small analog effect devices to achieve their unmistakable sound. Most of these effect devices have been used to create higher harmonics for the guitar sound in a faster way and at a much lower sound level compared to valve amplifiers. In this context terms like overdrive, distortion and fuzz are used. Several definitions of these terms for musical applications especially in the guitar player world are available [Kee00]. For our discussion we will define overdrive as a first state where a nearly linear audio effect device at low input levels is driven by higher input levels into the nonlinear region of its characteristic curve. The operating region is in the linear region as well as in the nonlinear region, with a smooth transition. The main sound characteristic is of course from the nonlinear part. Overdrive has a warm and smooth sound. The second state is termed distortion, where the effects device mainly operates in the nonlinear region of the characteristic curve and reaches the upper input level, where the output level is fixed to a maximum level. Distortion covers a wide tonal area starting beyond tube warmth to buzz saw effects. All metal and grunge sounds fall into this category. The operating status of fuzz is represented by a completely nonlinear behavior of the effect device with a sound characterized by the guitar player terms “harder” and “harsher” than distortion. The fuzz effect is generally used on single-note lead lines.

Signal Processing

Overdrive. For overdrive simulations a soft clipping of the input values has to be performed. One possible approach for a soft saturation nonlinearity [Sch80] is given by

4.14

The static input to output relation is shown in Figure 4.28. Up to the threshold of 1/3 the input is multiplied by two and the characteristic curve is in its linear region. Between input values of 1/3 up to 2/3, the characteristic curve produces a soft compression described by the middle term of Equation (4.14). Above input values of 2/3 the output value is set to one. The corresponding M-file 4.5 for overdrive with symmetrical soft clipping is shown next.

Figure 4.28 Static characteristic curve of symmetrical soft clipping (right part shows logarithmic output value versus input value).

Figure 4.29 shows the waveforms of a simulation with the above-described characteristic curve and a decaying sinusoid of 1 kHz. In the upper left plot the first part of the output signal is shown with high signal levels, which corresponds to the saturated part of the characteristic curve. The tops and bottoms of the sinusoid run with a soft curve towards the saturated maximum values. The upper right plot shows the output signal where the maximum values are in the soft clipping region of the characteristic curve. Both the negative and the positive top of the sinusoid are rounded in their shape. The lower waterfall representation shows the entire decay of the sinusoid down to −12 dB. Notice the odd order harmonics produced by this nonlinear symmetrical characteristic curve, which appear in the nonlinear region of the characteristic curve and disappear as soon as the lower threshold of the soft compression is reached. The prominent harmonics are the third and the fifth harmonic. The slow increase or decrease of higher harmonics is the major property of symmetrical soft clipping. As soon as simple hard clipping without a soft compression is performed, higher harmonics appear with significantly higher levels (see Figure 4.30). The discussion of overdrive and distortion has so far only considered the creation of harmonics for a single sinusoid as the input signal. Since a single guitar tone itself consists of the fundamental frequency plus all odd- and even-order harmonics, it is always the sum of sinusoids that is processed by the nonlinearity. The nonlinearity also produces sum and difference frequencies.

Figure 4.29 Short-time FFTs (waterfall representation) of symmetrical soft clipping for a decaying sinusoid of 1 kHz.

Figure 4.30 Short-time FFTs (waterfall representation) of symmetrical hard clipping for a decaying sinusoid of 1 kHz.

Distortion.

A nonlinearity suitable for the simulation of distortion [Ben97] is given by

4.15

The M-file 4.6 for performing Equation (4.15) is shown next.

The static characteristic curve is illustrated in Figure 4.31 and short-time FFTs of a decaying 1 kHz sinusoid are shown in Figure 4.32.

Figure 4.31 Static characteristic curve of exponential distortion.

Figure 4.32 Short-time FFTs (waterfall representation) of exponential distortion.

Fuzz.

We have already discussed the behavior of triode valves in the previous section which produce an asymmetrical overdrive. One famous representative of asymmetrical clipping is the Fuzz Face [Kee98a], which was used by Jimi Hendrix. The basic analog circuit is shown in Figure 4.33 and consists only of a few components with two transistors in a feedback arrangement.

Figure 4.33 Analog circuit of Fuzz Face.

The output signals for various input levels are presented in Figure 4.34 in conjunction with the corresponding spectra for a 1 kHz sinusoid. The upper plots down to the lower plots show an increasing input level. For low-level input signals the typical second harmonic of a triode valve can be noticed, although the time signal shows no distortion components. With increasing input level the second harmonic and all even-order harmonics as well as odd-order harmonics appear. The asymmetrical clipping produces enhanced even-order harmonics, as shown in the third row of Figure 4.34. Notice that only the top of the positive maximum values are clipped. As soon as the input level further increases, the negative part of the waveform is clipped. The negative clipping level is lower than the positive clipping value and so asymmetrical clipping is performed.

Figure 4.34 Signals and corresponding spectra of Fuzz Face.

Short-time Fourier transforms (in waterfall representation) for an increasing 1 kHz sinusoid, together with two waveforms are shown in Figure 4.35.

Figure 4.35 Short-time FFTs (waterfall representation) of Fuzz Face for an increasing 1 kHz sinusoid. The upper plots show segments of samples from the complete analyzed signal.

Musical Applications

There are a lot of commercial stomp effects for guitarists on the market. Some of the most interesting distortion devices for guitars are the Fuzz Face which performs asymmetrical clipping towards symmetrical soft clipping and the Tube Screamer [Kee98b], which performs symmetrical soft clipping. The Fuzz Face was used by Jimi Hendrix and the Tube Screamer by Stevie Ray Vaughan. They both offer classical distortion and are well known because of their famous users. It is difficult to explain the sound of a distortion unit without listening personally to it. The technical specifications for the sound of distortion are missing, so the only way to choose a distortion effect is by a comparative listening test. Investigations about the sound coloration caused by typical distortion or overdrive effects can be found in [MM05, DHMZ09].

4.3.3 Harmonic and Subharmonic Generation

Introduction

Harmonic and subharmonic generation are performed by simple analog or digital effect devices, which should produce an octave above and/or an octave below a single note. Advanced techniques to achieve pitch shifting of instrument sounds will be introduced in Chapter 6. Here, we will focus on simple techniques, which lead to the generation of harmonics and subharmonics.

Signal Processing

The signal-processing algorithms for harmonic and subharmonic generation are based on simple mathematical operations like absolute value computation and counting of zero crossings, as shown in Figure 4.36 when an input sinusoid has to be processed (first row shows time signal and corresponding spectrum).

Figure 4.36 Signals and corresponding spectra of half-wave rectification, full-wave rectification and octave division.

The second row of Figure 4.36 demonstrates half-wave rectification, where positive values are kept and negative values are set to zero. This operation leads to the generation of even-order harmonics. Full-wave rectification, where the absolute value is taken from the input sequence, leads to even-order harmonics, as shown in the third row of Figure 4.36. Notice the absence of the fundamental frequency. If a zero crossing counter is applied to the half-wave or the full-wave rectified signal, a predefined number of positive wave parts can be set to zero to achieve the signal in the last row of Figure 4.36. This signal has a fundamental frequency which is one octave lower than the input frequency in the first row of the figure, but also shows harmonics of this new fundamental frequency. If appropriate lowpass filtering is applied to such a signal, only the fundamental frequency can be obtained, which is then added to the original input signal.

Musical Applications

Harmonic and subharmonic generation is mostly used on single-note lead lines, where an additional harmonic or subharmonic frequency helps to enhance the octave effect. Harmonic generators can be found in stomp boxes for guitar or bass guitar and appear under the name octaver. Subharmonic generation is often used for solo and bass instruments to give them an extra bass boost or simply a fuzz bass character.

4.3.4 Tape Saturation

Introduction and Musical Application

The special sound characteristic of analog tape recordings has been acknowledged by a variety of producers and musicians in the field of rock music. They prefer doing multi track recordings with analog tape-based machines and use the special physics of magnetic tape recording as an analog effects processor for sound design. One reason for their preference for analog recording is the fact that magnetic tape goes into distortion gradually [Ear76, pp. 216-218] and produces those kinds of harmonics which help special sound effects on drums, guitars and vocals.

Signal Processing

Tape saturation can be simulated by the already introduced techniques for valve simulation. An input-level-derived weighting curve is used for generating a gain factor, which is used to compress the input signal. A variety of measurements of tape recordings can help in the design of such processing devices. An example of the input/output behavior is shown in Figure 4.37 and a short-time FFT of a sinusoid input signal in Figure 4.38 illustrates a tape saturation algorithm. For low-level inputs the transfer characteristic is linear without any distortions. A smooth soft compression simulates the gradually increasing distortion of magnetic tape.

Figure 4.37 Tape saturation: input and output signal (left) and characteristic curve.

Figure 4.38 Tape saturation: short-time FFTs (waterfall representation) for decaying sinusoid of 1 kHz.