Making sounds with analogue electronics – Part 2: Subtractive synthesis
[Part 1 briefly reviews the differences between analogue and digital synthesis, and discusses "one of the major innovations in the development of the synthesizer" – voltage control.]
3.3 Subtractive synthesis
Subtractive synthesis is often mistakenly regarded as the only method of analogue sound synthesis. Although there are other 
methods of synthesis, the majority of commercial analogue synthesizers use subtractive synthesis. Because it is often presented with a user interface consisting of a large number of knobs and switches, it can be intimidating to the beginner.
Because there is often a one-to-one relationship between the available controls and the knobs and switches, it is well suited to educational purposes. It can also be used to illustrate a number of important principles and models that are used in acoustics and sound theory.
3.3.1 Theory: source and modifier
Subtractive synthesis is based around the idea that real instruments can be broken down into three major parts: a source of sound, a modifier (which processes the output of the source) and some controllers (which act as the interface between the performer and the instrument). This is most obviously apparent in many wind instruments, where the individual parts can be examined in isolation (Figure 3.3.1).
For example, a clarinet, where a vibrating reed is coupled to a tube, can be taken apart and the two parts can be investigated independently. On its own, the reed produces a harsh, strident tone, whilst the body of the instrument is merely a tube that can be shown to have a series of acoustic resonances related to its length, the diameter of the longitudinal hole and other physical characteristic; in other words, it behaves like a series of resonant filters. Put together, the reed produces a sound which is then modified by the resonances of the body of the instrument to produce the final characteristic sound of the clarinet.
Although this model is a powerful metaphor for helping to understand how some musical instruments work, it is by no means a complete or unique answer. Attempting to apply the same concept to an instrument such as a guitar is more difficult, since the source of the sound appears to be the plucked string, and the body of the guitar must therefore be the modifier of the sound produced by the string.
Unfortunately, in a guitar, the source and the modifier are much more closely coupled, and it is much harder to split them into separate parts. For example, the string cannot be played in isolation in quite the same way as the reed of a clarinet can, and all of the resonances of the guitar body cannot be determined without the strings being present and under tension.
FIGURE 3.3.1 The performer uses the instrument controllers to alter the source and modifier parameters.
Despite this, the idea of modifying the output of a sound source is easy to grasp and it can be used to produce a wide range of synthetic and imitative timbres. In fact, the underlying idea of source and modifier is a common theme in most types of sound synthesis.
3.3.2 Subtractive synthesis
Subtractive synthesis uses a subset of this generalized idea of source and modifier, where the source produces a sound that contains all the required harmonic content for the final sound, whilst the modifier is used to filter out any unwanted harmonics and shape the sound’s volume envelope. The filter thus ‘subtracts’ the harmonics that are not required; hence the name of the synthesis method (Figure 3.3.2).
3.3.3 Sources
The sound sources used in analogue subtractive synthesizers tend to be based on mathematics. There are two basic types: waveforms and random. The waveforms are typically named after simple waveshapes: sawtooth, square, pulse, sine and triangle are the most common. The shapes are the ones which are easy to describe mathematically and also to produce electronically. Random waveshapes produce noise, which contains a constantly changing mixture of all frequencies.
Oscillators are related to one of the component parts of analogue synthesizers: function generators. A function generator produces an output waveform, and this can be of arbitrary shape and can be continuous or triggered. An oscillator that is intended to be used in a basic analogue subtractive synthesizer normally produces just a few continuous waveshapes, and the frequency needs to be controlled by a voltage.
The waveshapes in analogue synthesizers are only approximations to the mathematical shapes and the differences give part of the appeal of analogue sounds.
FIGURE 3.3.2 The source produces a constant raw waveform. The filter changes the harmonic structure, whilst the envelope shapes the sound.
It should also be noted that, in general, sources produce continuous outputs. You need to use a modifier in order to alter the timbre or apply an envelope to the sound.
VCOs
The VCOs provide voltage control of the frequency or pitch of their output. Some VCOs also provide voltage control inputs for modulation (usually FM) and for varying the shape of the output waveforms (usually the pulse width of the rectangular waveshape, although some VCOs allow the shape of other waveforms to be altered as well).
Many VCOs have an additional input for another VCO audio signal, to which the VCO can be synchronized. Hard synchronization forces the VCO to reset its output to keep in sync with the incoming signal, which means that the VCO can only operate at the same or multiple frequencies of the input frequency. This produces a characteristic harsh sound. Other ‘softer’ synchronization schemes can be used to produce timbral changes in the output rather than locking of the VCO frequency.
A typical VCO has controls for the coarse (semitones) and fine (cents) tuning of its pitch, some sort of waveform selector (usually one of sine, triangle, square, sawtooth and pulse), a pulse width control for the shape of the pulse waveform and an output level control (Figure 3.3.3). Sometimes multiple simultaneous output waveforms are available, and some VCOs also provide ‘sub-octave’ outputs that are one or two octaves lower in pitch. A CV for the pulse width allows the shape of the pulse waveform (and sometimes other waveforms as well) to be altered. This is called pulse width modulation (PWM) or shape modulation.
Harmonic content of waveforms
The ordering of waveforms on some early analogue synthesizers was not random. The waveforms are deliberately arranged so that the harmonic content increases as the rotary control is twisted.
One example: the Minimoog waveforms are arranged in the order of increasing harmonic content.
FIGURE 3.3.3 A block diagram of a typical VCO.
Arguably the simplest waveshape is the sine wave (Figure 3.3.4). This is a smooth, rounded waveform based on the mathematical sine function. A sine wave contains just one ‘harmonic’, the first or fundamental. This makes it somewhat unsuitable for subtractive synthesis since it has no harmonics to be filtered.
FIGURE 3.3.4 A sine waveform and harmonic spectrum and the same diagrams with actual frequencies shown.
A triangle waveshape has two linear slopes (Figure 3.3.5). It has small amounts of odd-numbered harmonics, which give it enough harmonic content for a filter to work on.
FIGURE 3.3.5 A triangle waveform and spectrum.
A square wave contains only odd harmonics (Figure 3.3.6). It has a distinctive ‘hollow’ sound and a very synthetic feel.
FIGURE 3.3.6 A square waveform and spectrum, with a typical clarinet spectrum for comparison.
A sawtooth wave contains both odd and even harmonics (Figure 3.3.7). It sounds bright, although many pulse waves can actually have more harmonic content. ‘Super-sawtooth’ waveshapes replace the linear slope with exponential slopes, as well as gapped sawtooths: these can contain greater levels of the upper harmonics than the basic sawtooth.
FIGURE 3.3.7 A sawtooth waveform and spectrum, with the spectrum also shown on a vertical decibel scale.
Depending on the ratio between the two parts (known as the mark–space ratio, shape, duty cycle or symmetry), pulse waveforms (Figure 3.3.8) can contain both odd and even harmonics, although not all of the harmonics are always present. The overall harmonic content of pulse waves increases as the pulse width narrows, although if a pulse gets too narrow, it can completely disappear (the depth of PWM needs to be carefully adjusted to prevent this).
FIGURE 3.3.8 A pulse wave and spectrum. The relative levels of the harmonics depend on the width of the pulse.
A special case of a pulse waveshape is the 50:50 equal ratio square wave, where the even harmonics are not present. Pulse width modulated pulse waveforms are known as PWM waveforms and their harmonic content changes as the width of the pulse varies. PWM waveforms are normally controlled with LFO or an envelope, so that the pulse width changes with time. The audible effect when a PWM waveform is cyclically changed by an LFO is similar to two oscillators beating together.
It is possible to adjust the pulse width to give a square by ear: listening to the fundamental, the pulse width is adjusted until the note one octave up fades away. This note is the second harmonic and is thus not present in a square waveform. See also Figure 3.3.8.
All of the waveshapes and harmonic contents shown previously are idealized. In the real world the edges are not as sharp, the shapes are not so linear and the spectra are not as mathematically precise. Figure 3.3.9 shows a more realistic spectrum with dotted lines. This is a result of the filtering process used in producing the spectrum display and does not mean that there are extra frequencies present.
FIGURE 3.3.9 Analogue waveshaping allows the conversion of one waveform shape into others. In this example the sawtooth is the source waveform, although others are possible.
Although the waveshapes are based on mathematical functions, this does not always mean that they are all produced directly from mathematical formulas expressed in analogue electronics. For example, the ‘sine’ wave output on many VCOs is produced by shaping a triangle wave through a non-linear amplifier which rounds off the top of the triangle so that it looks like a true sine wave (Figure 3.3.6). The resulting waveform resembles a sine wave, although it will have some additional harmonics – but for the purposes of subtractive synthesis, it is perfectly adequate. Section 3.4 on additive synthesis shows what real-world waveforms look like when they are constructed from simpler waveforms, rather than the perfect cases shown earlier.
3.3.4 Modifiers
There are two major modifiers for audio signals in analogue synthesizers: filters and amplifiers. Filtering is used to change the harmonic content or timbre of the sound, whilst amplification is used to change the volume or ‘shape’ of the sound. Both types of modifiers are typically controlled by EGs, which produce complex CVs that change with time.
Effects such as reverb and chorus are not normally included as ‘modifiers’ in analogue synthesizers, although there are some notable exceptions: For instance, the EMS (Electronic Music Studios) VCS-3 has a built-in spring-line reverb unit.
Filters
A filter is an amplifier whose gain changes with frequency. It is usually the convention to have filters whose maximum gain is one, and so it is more correct to say that for a filter, the attenuation changes with frequency. A VCF is one where one or more parameters can be altered using a CV. Filters are powerful modifiers of timbre, because they can change the relative proportions of harmonics in a sound.
Filters come in many different forms. One classification method is based on the shape of the attenuation curve. If a sine wave test signal is passed through a filter, then the output represents the attenuation of the filter at that frequency; this is called the frequency response of the filter. An alternative method injects a noise signal into the filter and then monitors the output spectrum, but the sine wave method is easier to carry out. The major types of frequency response curve are
- low-pass
- band-pass
- high-pass
- notch.
Low-pass
A low-pass filter has more attenuation as the frequency increases. The point at which the attenuation is 3 dB is called the cut-off frequency, since this is the frequency at which the attenuation first becomes apparent. It is also the point at which half of the power in the audio signal has been lost and so it is sometimes called the half-power point. Below the cut-off frequency, a low-pass filter has no effect on the audio signal and it is said to have a fl at response (the attenuation does not change with frequency). Above the cut-off frequency, the attenuation increases at a rate which is called a slope. The slope of the attenuation varies with the design of the filter.
In general, analogue synthesizer filters have two or four poles, whilst digital filters can have up to eight or more.
Simple filters with one resistor and capacitor (RC) will have slopes of 6 dB/octave, which means that for each doubling of frequency, the attenuation increases by 6 dB. Each pair of RC elements is called a pole and the slope increases as the number of poles increases. A two-pole filter will have an attenuation of 12 dB/octave, whilst a four-pole filter will have 24 dB/octave. Audibly, a four-pole filter has a more ‘synthetic’ tone and makes much larger changes to the timbre of the sound as the cut-off frequency is changed. A two-pole filter is usually associated with a more ‘natural’ sound and more subtle changes to the timbre (Figure 3.3.10).
FIGURE 3.3.10 Filter responses are normally shown on a log frequency scale since a dB/octave cut-off slope then appears as a straight line. But harmonics are based on linear frequency scales and on these graphs the filter appears as a curve. Low-pass filtering a sawtooth waveform with the cut-off frequency set to four different values:
(i) At 100 Hz, the filter cut-off frequency is the same as the fundamental frequency of the sawtooth waveform. The second harmonic is 30 dB below the fundamental and so the ear will hear an impure sine wave at 100 Hz.
(ii) At 300 Hz, the first three harmonics are in the pass-band of the filter and the output will sound considerably brighter.
(iii) At 500 Hz, the first five harmonics are in the filter pass-band, and so the output will sound like a slightly dull sawtooth waveform.
(iv) At 1 kHz, the first ten harmonics are all in the pass-band of the filter and the output will sound like a sawtooth waveform.
Low-pass VCFs usually have the cut-off frequency as the main controlled parameter. A sweep of cut-off frequency from high to low frequencies makes any audio signal progressively ‘darker’, with the lower frequencies emphasized and less high frequencies present. A filter sweeping from high frequency to low frequency of cut-off is often referred to as changing from ‘open’ to ‘closed’. When the cut-off frequency is set to maximum, and the filter is ‘open’, then all frequencies can pass through the filter.
As the cut-off frequency of a low-pass filter is raised from zero, the first frequency that is heard is usually the fundamental. As the frequency rises, each of the successive harmonics (if any) of the sound will be heard. The audible effect of this is an initial sine wave (the fundamental), followed by a gradual increase in the ‘brightness’ of the sound as any additional frequencies are allowed through the filter. If the cut-off frequency of a low-pass filter is set to allow just the fundamental to pass through the filter, then the resulting sine wave will be identical for any input signal waveform. It is only when the cut-off frequency is increased and additional harmonics are heard, the differences between the different waveforms will become apparent. For example, a sawtooth will have a second harmonic, whilst a square wave will not.
High-pass
A high-pass filter has the opposite filtering action to a low-pass filter: it attenuates all frequencies that are below the cut-off frequency. As with the low-pass VCF, the primary parameter that is voltage controlled is the cut-off frequency. High-pass filters remove harmonics from a signal waveform, but as the frequency is raised from zero, it is the fundamental which is removed first. As additional harmonics are removed, the timbre becomes ‘thinner’ and brighter, with less low-frequency content and more high-frequency content, and the perceived pitch of the sound may change because the fundamental is missing.
Some subtractive synthesizers have a high-pass (not voltage-controlled) filter connected either before or after the low-pass VCF in the signal path. This allows limited additional control over the low frequencies that are passed by the low-pass filter. It is usually used to remove or change the level of the fundamental, which is useful for imitating the timbre of instruments where the fundamental is not the largest frequency component.
Band-pass
A band-pass filter only allows a set range of frequencies to pass through it unchanged – all other frequencies are attenuated. The range of frequencies that are passed is called the bandwidth, or more usually, the pass-band, of the filter. Band-pass VCFs usually have control over the cut-off frequency and the bandwidth.
Band-pass (and notch) filters are the equivalent of the resonances that happen in the real world. A wine-glass can be stimulated to oscillate at its resonant frequency by running a wet finger around the rim.
A band-pass filter can be thought of as a combination of a high-pass and a low-pass filters, connected in series, one after the other in the signal path. By using the same CV to the cut-off frequency inputs of two VCFs (one high-pass and the other low-pass), the cut-off frequencies will ‘track’ each other and the effective bandwidth of the band-pass filter will stay constant as the cut-off frequencies are changed. The width of the band-pass filter’s pass-band can be controlled by adding an extra CV offset to one of the filters. If the cut-off frequency of the low-pass filter is set below that of the high-pass filter, then the pass-band does not exist, and no frequencies will pass through the filter (Figure 3.3.11).
FIGURE 3.3.11 A band-pass filter only passes frequencies in a specific range. This is normally the two points at which the filter attenuates by 3 dB. It can be thought of as a low-pass and a high-pass filter connected in series (one after the other). In the example shown, the lower cut-off frequency is about 0.6f (for the high-pass filter), whilst the upper cut-off frequency is about 1.6f (for the low-pass filter). The bandwidth of the filter is the difference between these two cut-off frequencies. Small differences are referred to as ‘narrow’, whilst large differences are known as ‘wide’.
Band-pass filters are often described in terms of the shape of their pass-band response. Narrow pass-bands are referred to as ‘narrow’ or ‘sharp’, and they produce marked changes in the frequency content of an audio signal. Wider passbands have less effect on the timbre, since they merely emphasize a range of frequencies. The middle frequency of the pass-band is called the center frequency.
Very narrow band-pass filters can be used to examine a waveform and determine its frequency content. By sweeping through the frequency range, each harmonic frequency will be heard as a sine wave when the center frequency of the band-pass filter is the same as the frequency of the harmonic (Figure 3.3.12).
FIGURE 3.3.12 If a narrow band-pass filter is used to process a sound that has a rich harmonic content, then the harmonics which are in the pass-band of the filter will be emphasized, whilst the remainder will be attenuated. This produces a characteristic resonant sound. If the band-pass filter is moved up and down the frequency axis, then a characteristic ‘wah-wah’ sound will be heard – this is sometimes used on electric guitar sounds.
Notch Filters, Resonance and Oscillation
A notch filter is the opposite of a band-pass filter. Instead of passing a band of frequencies, it attenuates just those frequencies and allows all others to pass through unaffected. Notch filters are used to remove or attenuate specific ranges of frequencies and narrow ‘notches’ can be used to remove single harmonic frequencies from a sound. Notch VCFs usually provide control over both the cut-off and the bandwidth (or ‘stop-band’) of the filter (Figure 3.3.13).
FIGURE 3.3.13 A notch filter is the opposite of a band-pass filter, which it attenuates a band of frequencies. It can also be formed from a series combination of a low- and a high-pass filters, provided that the low-pass cut-off frequency is lower than the high-pass cut-off frequency. If not, then no notch will be present.
Scaling
If the keyboard pitch voltage is connected to the cut-off frequency CV input of a VCF, then the cut-off frequency can be made to track the pitch being played on the keyboard. This means that any note played on the keyboard is subjected to the same relative filtering, since the cut-off frequency will follow the pitch being played. This is called pitch tracking or keyboard scaling (Figure 3.3.14).
FIGURE 3.3.14 Filter scaling, tracking or following is the term used to describe changing the filter cut-off so that it follows changes in the pitch of a sound. This allows the spectrum of the sound produced to stay the same. In the example shown, the filter peak tracks the changes in the pitch of the sound when two notes two octaves apart are played – the peak coincides with the fundamental frequency in each case. With no filter scaling then the note with a fundamental of 4f two octaves up would be strongly attenuated if the filter cut-off frequency did not change from the peak at a frequency of f.
Resonance
Low-pass and high-pass filters can have different response curves depending on a parameter called resonance or Q (short for ‘quality’, but rarely referred to as such). Resonance is a peaking or accentuation of the frequency response of the filter at a specific frequency. For band-pass filters, the Q figure is given by the formula:
Q = Center frequency/Bandwidth (or pass-band)
This formula is often also used for the resonance in the low-pass and high-pass filters used in synthesizers. For these low-pass and high-pass filters, the resonance is usually at the cut-off frequency and it forms a ‘peak’ in the frequency response (Figure 3.3.15).
FIGURE 3.3.15 Resonance changes the shape of a lowpass filter response most markedly at the cut-off frequency. The result is a smooth and continuous transition from a low-pass to something like a narrow band-pass filter.
In many VCFs, internal feedback is used to produce resonance. By taking some of the output signal and adding it back into the input of the filter, the response of the filter can be emphasized at the cut-off frequency. This also means that the resonance of the filter can be made voltage controllable by varying the amount of feedback with a VCA. See Section 3.3.5 for more on VCAs and see Section 3.6 for more information on the implementation of filters.
Most subtractive synthesizers implement only low-pass and band-pass filtering, where the band-pass is often produced by increasing the Q of the lowpass filter so that it is a ‘peaky’ low-pass rather than a true band-pass filter. This phenomenon of a peak of gain in an otherwise low-pass (or high-pass) response is called ‘corner peaking’. Some models of analogue synthesizer also have an additional simple high-pass filter, whilst notch filters or band-rejects are very uncommon.
There are two types of filters: constant-Q and constant bandwidth. Constant-Q filters do not change their Q as the frequency of the filter is changed. This means that they are good for applications where the filter is used to produce a sense of pitch from an unpitched source such as noise. Since the Q is constant, the bandwidth varies with the filter frequency and so sounds ‘musical’.
Constant-bandwidth filters have the same bandwidth regardless of the filter frequency. This means that a relatively narrow bandwidth of 100 Hz for a filter frequency of 4 kHz, is very wide for a 400-Hz frequency: the Q of a constant-bandwidth filter changes with the filter frequency. Most analogue synthesizer filters are constant-Q.
The effect of changing the cut-off frequency of a highly resonant low-pass filter in ‘real time’, with a source sound rich in harmonics, is quite distinctive and can be approximated by singing ‘eee-yah-oh-ooh’ as a continuous sweep of vowel sounds.
Filter oscillation
If the resonance of a peaky low-pass or a band-pass VCF is increased to the point at which the filter plus its feedback has a cumulative gain of more than one at the cut-off frequency, then it will break into self-oscillation. In fact, this is one method of producing an oscillator – you put a circuit with a narrow band-pass frequency response into the feedback loop of an amplifier or operational amplifier (op-amp) (Figure 3.3.16). The oscillation produces a sine wave, sometimes much purer than the ‘sine’ waves produced by the VCOs!
FIGURE 3.3.16 If a filter with a strong resonant peak in its response is connected around an amplifier, then the circuit will tend to oscillate at the frequency with the highest gain – at the peak of the filter response. This can be easily demonstrated (perhaps too easily) with a microphone and a PA system.
Coming up in Part 3: Envelopes.
Printed with permission from Focal Press, a division of Elsevier. Copyright 2009. "Sound Synthesis and Sampling" by Martin Russ. For more information about this title and other similar books, please visit www.elsevierdirect.com.
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