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Subtractive Synthesis

Basic Subtractive SynthesisOscillators and WaveformsFiltersEnvelopesModulationAdvanced WaveformsApplying Principles to Other Synthesizers

Terms & Abbreviations Used
  Osc        - Oscillator.
  Kybd       - Keyboard
  LPF        - Low Pass Filter.
  HPF        - High Pass Filter.
  Amp        - Amplifier (not Ampere; the measure of electrical current).
  Env        - Envelope.
  Mod        - Modulation.
  LFO        - Low Frequency Oscillator: usually used for modulation.
  Kybd.Trk.  - Keyboard Tracking: aka KF (Keyboard Follow) or KS (Keyboard Scaling).
  Vel.       - Velocity : information on how hard an instrument note is struck.
  VCO or DCO - Voltage Controlled Oscillator or Digitally Controlled Oscillator respectively.
  VCF or DCF - Voltage Controlled Filter or Digitally Controlled Filter respectively.
  VCA or DCA - Voltage Controlled Amplifier or Digitally Controlled Amplifier respectively.
  ADSR       - Attack, Decay, Sustain & Release: the typical components of an envelope

Basic Subtractive Synthesis

Subtractive synthesis is often referred to as analogue synthesis because most analogue synthesizers (ie non-digital) use this method of generating sounds. In it's most basic form, subtractive synthesis is a very simple process as follows:

OSCILLATOR ---------> FILTER ---------> AMPLIFIER
In essence, this is subtractive synthesis. You start with a sound, subtract out the unwanted bits and control its loudness over time.

Oscillators and Waveforms

OSCILLATORS - An Oscillator generates a sound. It actually generates a waveform or a shape. The oscillator does this continuously. The rate at which it generates each cycle of the waveform is what we hear as pitch. Pitch is measured in Hertz (Hz) [where one Hertz is one cycle per second].

There are many types of oscillators but the main "Oscillator" usually refers to a "Keyboard-Controlled Oscillator". On a synthesizer, the Keyboard-Controlled Oscillator may be labled as VCO, DCO, or sometimes as Waveform.

All this means is that when you play a note on the keyboard, the oscillator will generate the waveform (or shape) at that intended pitch continuously (for as long as you hold down that note on the keyboard). If you pressed "A4" (the note "A" just above middle "C"), the waveform will be generated at a rate of 440Hz (ie 440 cycles a second). In short, the keyboard controls the pitch of the oscillator (hence, keyboard controlled oscillator) .

A Waveform (or Wave) is a shape which the oscillator generates. The shape determines the "timbre" or quality, characteristic or brightness of a sound. While pitch tells us which note is being played, "timbre" tells us which instrument is being played. While pitch is a basic frequency which identifys the note, "timbre" is made up of many other frequencies or overtones which gives the instrument its overall character and identity.

WAVE CONCEPTS - The basic set of waveforms covered by this document are simple mathematical shapes which are found on analogue synthesizers. The first waveform to look at is the "Sine" wave. Although it is not always available on analogue synthesizers, it is the starting point to understanding waveforms and their "harmonic content" (ie timbre).

WAVEFORMS - Let's look at the harmonic spectrum of the basic waveforms available on subtractive synthesizers. The graph shows Waveforms (left) and Harmonic Spectrum (right). Top to bottom: Tri, Square, Saw, Pulse.

 Amplitude of Waveform Harmonics
 Harmonic       F     2F     3F     4F     5F     6F     7F     8F     9F 
 Triangle       1      -    -1/9     -     1/25    -    -1/49    -     1/81 
 Square         1      -     1/3     -     1/5     -     1/7     -     1/9 
 Saw            1     1/2    1/3    1/4    1/5    1/6    1/7    1/8    1/9 
Note - Some amplitudes have alternating negative values which means inverted phase. To the ear, it really doesn't make any difference.

Note - The fundamental is usually labled "F". The first harmonic is actually at 2F and the second harmonic is at 3F etc. We usually refer to Odd-Numbered harmonics as being the series F, 3F, 5F, 7F etc; and refer to Even-Numbered harmonics as being the series 2F, 4F, 6F, 8F etc.

Filters

The Filter section of a synthesizer may be labled as VCF (Voltage Controlled Filter) or DCF (Digitally Controlled Filter). A Filter is used to remove frequencies from the waveform so as to alter the timbre.

LOW PASS FILTERS - The main filter in subtractive synthesis is the Low-Pass Filter (LPF). Low-Pass Filters allows the lower frequencies to pass through unaffected and filters out (or blocks out) the higher frequencies. This is controlled by a parameter labled as "Cut-Off Frequency" (or "Cut-Off" or "Frequency").

CUT-OFF - The Cut-Off Frequency determines the position in the harmonic spectrum where the filter will begin to filter (or block) the higher harmonics (ie frequencies). If the Cut-Off is set to a high position, then the timbre will be bright. If the Cut-Off is set to a low position, then the timbre will be dull.

What does it sound like? Try this! Slowly say the words "Wow, wow, wow". This is what an LPF sounds like as the Cut-Off position moves up the harmonic spectrum and back. The first sound was "wuuu" which is fairly pure and dull. The next sound is "aaaa" which is bright and full of harmonics.

FILTER RESPONSE - The way in which filters block out frequencies beyond the Cut-Off position is not abrupt but is instead a gradual rate (which is more natural). So, the further beyond the Cut-Off position, the more the subsequent frequencies are filtered. This occurs up to a point where all further frequencies are completely filtered out.

In Low-Pass Filters, the gradual rate of filtration (beyond the Cut-Off) is usually available as 4-pole or 2-pole filters (which are 24dB/octave and 12dB/octave respectively). So a 4-pole filter has steeper rate of filtration and will filter out more as compared to a 2-pole filter.

The graphs show Filter Response with Amplitude (vertical axis) and Frequency (horizontal axis). The horizontal dotted lines indicate the "unity gain" level, where the timbre is unaffected. Below this dotted line, the harmonics are filtered (cut); and above the line, the harmonics are boosted. Where the graph is at zero (horizontal axis), the harmonics are completely filtered out.
Top graph = Low-Pass Filter with 4-pole and 2-pole response (4-pole is the steeper sloped).
Bottom graph = LPF with Resonance and shows the filter response as Resonance is increased.

RESONANCE - Resonance is another parameter available in a Low-Pass Filter. Resonance is where the frequencies at the Cut-Off position are boosted (made louder). When a little resonance is introduced, the frequencies around the Cut-Off are boosted a little. With a lot of resonance, the frequencies around the Cut-Off are boosted a lot while the lower frequencies will start to diminish a little. When too much resonance is applied, the frequencies around the Cut-Off are so pronounced that the filter will itself start to oscillate and "howl" (like feedback, but in a tuned way).

HIGH-PASS FILTERS - Another common (but less used) filter in subtractive synthesis is the High-Pass Filter (or HPF). A HPF does the opposite of a LPF. A High-Pass Filter allows the higher frequencies to pass through unaffected and filters out (blocks-out) the lower frequencies.

Top graphs show the filter responses. Left to right: Low-Pass Filter; Resonance; and, High-Pass Filter. Cut-Off is indicated by vertical dotted lines.
Middle graphs - harmonic spectrums of the original waveform.
Bottom graphs - harmonic spectrums of the resultant waveforms (affected by the filters).

Envelopes

AMPLIFIER ENVELOPE - The amplifier section on a synthesizer may be labled as VCA (Voltage Controlled Amplifier) or DCA (Digitally Controlled Amplifier). An Amplifier uses an Envelope to control the overall loudness (or amplitude) of the sound over time. Envelopes are activated by pressing the keys on a keyboard.

ENVELOPE PARAMETERS - Nowadays, there are many variations of envelopes available. However, the most common envelope has 4 parameters: (1) Attack time, (2) Decay time, (3) Sustain level, and (4) Release time. This is usually called ADSR Envelope.

Apart from controlling the Amplifier, envelopes can also be used to control other sections of a synthesizer. But for now, let us just look at controlling the Amplifier.

Basically, an ASDR envelope only has 3 loudness (amplitude) levels; zero (off), full (the maximum) and the Sustain level.
- Attack time it the time taken from zero to full.
- Decay time is the time taken from full to Sustain level.
- Release time is the time taken from Sustain level to zero.

Let us look at what happens to an ADSR envelope when a key is pressed on the keyboard {Note - Before any key of pressed, the envelope is at zero and so there is no sound}.

ASDR EXAMPLES - For amplifier envelope, the graphs show Amplitude (vertical axis) and Time (horizontal axis).
Top - shows key-pressed and key-released information (also called "Gate", equivalent to ADSR = 0%, NA, 100%, 0%).
Second ADSR = 0%, 80%, 50%, 60%. Perhaps for Piano.
Third ADSR = 10%, 20%, 75%, 40%. Perhaps for Strings.
Bottom ADSR = 20%, 40%, 25%, 20%. Perhaps for Brass.

OTHER ENVELOPE TYPES - Apart from the ADSR envelope, there are many other variations but they are usually based on the same principles.
One variant with 2 decays is the ADBDSR envelope; Attack, Decay-1, Break-Point, Decay-2, Sustain, Release [where Decay-1 is from full to the Break-Point level; and Decay-2 is from Break-Point level to Sustain level].
Another variant would be something like Rate-1, Level-1, Rate-2, Level-2, Rate-3, Sus, Rate-4 where the envelope is very flexible (Rate is the inverse of Time). Rate = High (a fast rate) is equivalent to Time = Low (a short time).

Modulation

To "modulate" is to exert control or affect or influence or shape. Modulation is used to breathe more life into a sound and make it more expressive. Modulation has 2 components; the Modulator and the Carrier. As an example, if you were humming a note and you used your hand to vibrate your throat, then your hand would be the Modulator and your humming would be the Carrier.

ENVELOPE - We have already seen how the amplifier is triggered by the pressing of the keys on a keyboard but is shaped by an ADSR envelope. You could say that the amplitude is being modulated by the ASDR envelope. But an envelope can also be used to modulate other parameters.

LOW FREQUENCY OSCILLATOR - A Low Frequency Oscillator (LFO) is a very common Modulator available on most synthesizers. Basically, an LFO is an oscillator which works at a set slow rate (perhaps between 0.1Hz to 10 Hz). Typically, an LFO will have the following parameters: Wave (usu Triangle, Square, Saw, Sine), Rate (the speed of the oscillation) and perhaps Delay (a time delay before the oscillator starts). The normal selection would be to use either Tri or Sine waves because of their regular rises and falls without any sudden jumps (Square and Saw waves have sharp jumps). KEYBOARD TRACKING - The keyboard not only controls the pitch of the oscillators but can itself be used as Modulator. Keyboard Tracking (KT) is where the position of the notes being played are used to influence other parameters. KT is sometimes labled as Keyboard Follow or Keyboard Scaling. Keyboard Tracking can best be viewed as either off or a positive amount or an negative amount. VELOCITY - Nowadays, many synthesizers respond to Velocity. As standard, the keyboard transmits information on which note is being played. If Velocity is available, then the keyboard will also transmit information on how hard the note was struck. Velocity information (sometimes labled as Velocity Sensitivity) can be used in many ways.

Advanced Waveforms

MIXED WAVEFORMS - Many synthesizers allow two or more oscillators to be mixed together. The resultant waveform is simply the sum of the waveforms (ie both added together). The resultant harmonic series would also be the sum of the harmonics.
Some synthesizers only have on/off switches for its various selectable waveforms. Switching on two or more waveforms will produce a new waveform based on the sum of the parts. Below is the harmonic series and amplitudes for a Square wave and Saw wave summed together:-
 HARMONIC  F    2F    3F    4F     5F     6F     7F     8F     9F
 Square  100%    -    33.3%  -     20%     -     14.3%   -     11.1%
 Saw     100%   50%   33.3% 25%    20%    16.7%  14.3%  12.5%  11.1%
 RESULT  100%   25%   33.3% 12.5%  20%     8.3%  14.3%   6.25% 11.1%
I have used "average results" to better illustrate the "relative" strengths of the resultant waveform.

Some synthesizers offer level-controls (amplitude) for each different waveform. The ability to control the "mix" of waveform levels means you can have greater control over the resultant harmonics. Below is the harmonic series for Square and Saw wave mixed together at 100% and 75% respectively:-

 HARMONIC  F    2F    3F     4F     5F     6F     7F     8F     9F
 Square  100%    -    33.3%  -      20%     -     14.3%   -     11.1%
 Saw      75%  37.5%  25%    18.8%  15%   12.5%   10.7%  9.38%   8.33%
 RESULT  100%  21.4%  33.3%  10.7%  20%    7.14%  14.3%  5.36%  11.1%
TWO DETUNED OSCILLATORS - Where two detuned oscillators are mixed together, the result is still the sum of the waveforms. But this time, the oscillators are not synchronised (ie not perfectly in tune).

The diagram shows the resultant waveforms. The top and middle waves are the two individual detuned waves, while the bottom wave is the added result. The left shows 2 detuned Square waves added together and the right shows 2 detuned Tri-waves added together.
The resultant waveform is quite dynamic. It changes over time, due to the interference of the detuning. This evolution of waveform over time gives a sense of movement to the timbre because of the "phase" differences. Sometimes the two waves are in-phase (ie both going up at the same time) and sometimes they are out-of-phase (ie one going up and one going down). Mostly, it's something in between.
Note that, when they are in-phase, the resultant amplitude is strong (ie it is loud) and, when it is out-of-phase, the resultant amplitude is weak (ie it is quiet). This will cause a bit of "tremolo" or beating of amplitude (The greater the detuning, the faster the rate or speed of the tremolo). The tremolo effect can be reduced by making one oscillator louder than the other.

LFO ~MOD~ PULSE WIDTH - When a LFO is used to modulate the Pulse Width, then the Pulse Width will change over time, which means the harmonic amplitudes vary too. This is usually called Pulse Width Modulation or PWM (although not necessarily modulated with a LFO).

As the LFO varies, so will the Pulse Width. The top left graph shows the original Square wave, the middle left graph shows the modulating Tri-wave LFO, and the bottom left graph shows the result. PWM looks and sounds a little bit like two detuned oscillators and is an excellent way to create some movement in a timbre.

SYNC - Sync is where the cycle of one oscillator (the synchroniser) is used to synchronise or "reset" the oscillator of another (the slave). The top right hand graph shows a Saw wave before sync, the middle right graph shows the synchroniser (in this case a Square wave at a lower frequency), and the bottom right graph shows the effect of synchronisation. As the Square wave completes each cycle, the Saw wave is "reset" to start again. This results in a new static wave whose frequency is that of the synchroniser but whose shape is a portion of the original. Note that the choice of wave for the synchroniser is irrelevant and only it's frequency matters.

CROSS-MODULATION - Cross modulation is where the Pitch of one oscillator is modulated by another oscillator; and both are keyboard controlled oscillators. It is a form of pitch modulation except that the modulating oscillator is "keyboard tuned" (determined by the key pressed on the keyboard). This is also called FM (Frequency Modulation) and is very good for producing bell-like sounds.

RING MODULATION - Ring Modulators are circults which output the "Sum and Difference" of its inputs. If the inputs are two Sine waves at 440Hz and 420Hz, the output would be 860Hz (sum) and 20Hz (difference). Depending on the harmonic content of the input, the output can be very complex and it is generally used for gong-like sounds.

CROSS FILTER MODULATION - A rare feature on synthesizers, Cross Filter Modulation is where the Filter Cut-Off is modulated by a keyboard controlled oscillator. It is a form of Filter Modulation except that the modulating oscillator is "keyboard tuned".

Applying Principles to Other Synthesizers

With sampling and digitised waveforms, there are now more alternatives to the simple mathematical waveforms. Let us look at how the subtractive synthesizer process applies to these:

OSCILLATOR ---------> FILTER ---------> AMPLIFIER

If sampled or digitised waveforms are used, the only real difference is the waveform used by the oscillators. The process is unchanged. The difference in approach would be knowing the harmonic content of the waveforms. And the best way to learn this is to experiment with the waveforms and let your ears be the judge. Things really haven't changed that much.

DIGITISED WAVEFORMS - A digitised waveform (sometimes referred to as Pulse Code Modulated [PCM] waveform) is a waveform stored in a memory chip. It may originate from a sample (ie real sound) or it could be created. Typically, the waveforms stored would be single cycles only. Put tons of these digitised waveforms on a memory chip and you have the makings of a wave-table synthesizer.
The quality of the digitised waveform will depend on the Resolution and Rate (as well as the inherent quality of the sound itself). Compact Disk quality uses 16-bit resolution and 44.1kHz sample rate.
Resolution of 16-bits means that the sound "amplitude" is captured and stored in levels from 0~65,535 (2^16). On the waveform graphs, this would be the vertical axis. The horizontal axis (time) is captured by the sample rate. A sample rate of 44.1kHz means that 44,100 of the amplitude "levels" (0~65,535) are taken per second.

PHASE DISTORTION - This is a synthesis method popularised by Casio's CZ-101 and other CZ-synths. Although it is not a subtractive synth, it can actually be treated as one. A CZ-synth basically has 3 envelopes for DCO (pitch), DCW (distortion), and, DCA (amplifier). Phase Distortion relies on "reading" the digitised waveform in a non-linear way. Normally, when you play a digitised waveform, the processor "reads" the data at the intended rate (eg 44.1kHz). A Phase Distorted "reading" would be where the part of the data is read at one rate and the remainder at another rate (eg first half at 22kHz and the second half at 66.2kHz [average = 44.1kHz]). This distorted data-reading in effect changes the waveform. In theory, you get a brand new waveform. In practice, it sounds a lot like a LPF at work.

OTHER SYNTHESIS ENGINES - In a lot of cases, the principles of subtractive synthesis will still apply. Many new synthesizers will still include a Filter of some sort because it is a simple and easy method of controlling timbre. However, this article has not been about filters only. When encountering a new synthesis engine, the principles of Waveforms, Envelopes and Modulation will still apply. With new synths, the challenge will be learning the routing and modulation processes available. But this is true of the older analogue synths too. Learning to control a synth has much to do with familiarisation of the synth layout and how it works. There are no short-cuts. In the end, it's still about knowledge, experimentation and imagination.


For a familiarisation of a selection of synthesizers, see Synthesizer Layouts.

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