A pianist plays the same note on two instruments and nobody confuses them. A synthesiser player turns one knob and a sound goes from a flute to a buzz saw. Behind that knob is a choice that was made decades ago by someone building the machine: describe a sound as a sum of pure tones, or as a bright sound that is filtered, or as one wave bending another, or as a recording chopped into specks, or as a simulation of the plucked string itself.
Each choice gives a family of sounds that is easy to make and a family that is almost impossible. Producers and sound designers keep this map in their heads. When a patch is not working they do not turn random knobs; they ask which method would make this sound naturally, and switch. The tour below takes six methods, with one thing to play for each.
The sound is produced in your browser from oscillators and arithmetic. No recordings are used, and every interactive shows its picture and numbers with the sound off.
- Who takes it
- The Music Production specialization of the Music major, in the studio years; it also appears as a core or elective course in music-technology, electronic-music and game-audio programmes. Most courses teach subtractive synthesis first, then FM and sampling, then one or two newer methods, with a patch-building project.
- What it is for
- Designing a sound on purpose: knowing which method suits it, which controls matter, and which defects (harshness, aliasing, repetition) each method brings.
- Comes after
- Digital Audio & the DAW (36/62) and Acoustics, Tuning & Timbre (35/68).
- Leads to
- No later subject lists it as a prerequisite. It feeds Beatmaking, Sampling & Arrangement (36/69), where sampled and synthesised drums are the raw material, and Sound for Film, TV & Games (36/67), where creature voices, engines and interface sounds are designed, not recorded.
- What this piece gives you
- You will not be able to recreate a famous patch by ear after this. You will know six ways of building a sound, what each is good and bad at, and what is happening inside each one. Easier first read: Music Technology & Production (35/73), which introduces the filter, the envelope and a two-oscillator FM voice. This piece goes further and does not repeat them.
Additive: every sound is a stack of sine waves
The oldest idea is also the most general. In 1822 Joseph Fourier showed that any repeating wave can be written as a sum of sine waves whose frequencies are whole multiples of the lowest one, the fundamental. Each sine is a partial (or harmonic), and what makes a clarinet different from a violin is mostly the list of strengths of those partials, their spectrum. Building a sound by adding sine waves is additive synthesis. Thaddeus Cahill's Telharmonium (1900, commercial 1906) did this with giant tonewheels and alternators, and the Hammond organ (commercial from 1935) does it with the nine drawbars you pull out to set how much of each harmonic is added.
The classic waves are just recipes. A sawtooth has every harmonic with strength 1/n. A square wave has only the odd ones, again 1/n. A triangle has the odd ones with strength 1/n². A sine is only the first.
Try it. Pick a shape and add partials one at a time, or tap the bars to draw your own spectrum.
Tap or drag on the bars to set the strength of a partial.
Sound isn't available in this reader; the bars and the wave still work.
The wave is computed by summing the sines at 340 points. Organ drawbars are a fixed example of eight partials (strengths 1.0, 0.6, 0.9, 0.5, 0.2, 0.4, 0.1, 0.3, invented for illustration). The sound is the same sum, made by the browser's periodic-wave oscillator.
With a square wave, add more and more partials and the wave squares up, but a little spike remains at each edge. This is the Gibbs phenomenon: the sum overshoots the jump by about 9% of its height, however many partials you add. With the first 3 partials of a square wave the peak is 1.20; with 15, 1.18; with infinitely many, still 1.18 (the jump runs from −1 to +1). Additive synthesis is exact but expensive, because a rich sound needs dozens of partials, each with its own moving strength. That is why it is the idea underneath everything and rarely the method used on its own.
Subtractive and modulation: carve it, then let something else turn the knobs
Subtractive synthesis turns additive on its head. Start with a bright oscillator, already rich in harmonics, and remove what you do not want with a filter. It is how the Moog and nearly every analogue synthesiser worked, and Music Technology & Production (35/73) shows the oscillator, filter and envelope. The step that distinguishes a sound designer from a user of presets is the next one: modulation.
A modulation source is any signal that moves a control. An envelope is a source that fires once per note: attack, decay, sustain, release. A low-frequency oscillator (LFO) is one that repeats slowly, from about 0.1 to 20 cycles a second, too slow to be heard as pitch. Route an LFO to pitch and you get vibrato; to loudness, tremolo; to the filter cutoff, the "wobble" at the heart of a dubstep bass. Route the envelope to the filter and you get the pluck: bright at the start, dark as it decays. A modulation matrix is the table that says which source goes to which destination and by how much.
Modulation matrix
Sound isn't available in this reader; the chart and numbers still work.
The chart shows the cutoff (100 Hz to 12 kHz, log scale) over 2 seconds for a note held for 1 second. The cutoff knob maps 0 to 100 onto 80 Hz to 5 kHz on a log scale; the filter is the browser's second-order low-pass; the LFO is a sine. Notes are one at a time (a new key releases the old one). Hold a key; release to hear the release stage.
Set the envelope to open the filter by two octaves, then bring up the LFO to the cutoff: the chart shows the cutoff bobbing around the envelope curve, and the ear hears a bass that pulses. Move LFO to pitch to 30 cents and a slow LFO and the note leans in and out of tune, which is vibrato. Pushing 100 cents at 6 Hz sounds like a siren. Every classic sound is a short list of these routings.
Digital oscillators and aliasing: the harmonics that fold back
A computer cannot make a perfect sawtooth. It makes numbers at a fixed rate, say 44,100 a second, and a sawtooth has harmonics all the way to infinity. Any harmonic above half the sample rate, the Nyquist frequency (22,050 Hz here), cannot be represented. It does not vanish. It aliases: it folds back down as a different, wrong frequency, which is not a harmonic of the note. Take a harmonic at frequency h; at sample rate fs it appears at the distance from h to the nearest multiple of fs.
Aliased components are quiet (the 23rd harmonic of a saw is 27 dB below the fundamental) but they are inharmonic: they do not belong to any note, so the ear hears them as a metallic fizz. And they move the wrong way. As you play higher, the saw's harmonics climb and their aliases fall, crossing the real ones: the "sweeping" sound of a cheap digital synth. Compute it for a note of your choice.
Sound isn't available in this reader; the chart and numbers still work.
Harmonics 1 to 2,000 of a sawtooth with strength 1/n, folded at the chosen sample rate. The sound is a 2-second sweep from 200 to 3,000 Hz at your device's sample rate: the first button writes a plain ramp one sample at a time; the second uses the browser's band-limited sawtooth. Listen for the fizz that falls as the note rises.
The cures all trim or smooth the jump before it is sampled: band-limited oscillators built from stored tables that stop below Nyquist, corrections at each jump (the polyBLEP family of methods), or simply running everything at two or four times the sample rate and filtering down. Raising the sample rate helps but does not cure it, as the 96 kHz button above shows: the lowest audible alias just comes from a much higher, much quieter harmonic.
FM: two oscillators and one number
Frequency modulation is the method that made digital synthesis a commercial success. John Chowning, at Stanford, found in 1967 that if one oscillator (the modulator) varies the frequency of another (the carrier) at an audible rate, the result is not vibrato but a new, rich spectrum, built from only two sines and one control. Stanford patented the method; after American organ makers declined, Yamaha took the licence and had exclusive rights from 1975. The Yamaha DX7 (1983) used six sine-wave "operators" in 32 arrangements ("algorithms") and 16 voices, and was the first synthesiser to sell more than 100,000 units.
The spectrum has a line at the carrier and pairs of lines either side, spaced by the modulator frequency. How strong each pair is depends on the modulation index I (the depth of the modulation divided by the modulator frequency) through Bessel functions Jn(I). Index zero is a pure sine; raising it pushes energy outward into more lines, so the sound brightens without a filter in sight. Music Technology & Production (35/73) shows this pair. The extra step is to read the ratio.
Sound isn't available in this reader; the chart and numbers still work.
Bessel functions of the first kind are computed by their power series; the squared strengths add to 1, which the readout confirms. Axis 0 to 6 kHz; lines beyond it are counted but not drawn. Lines that would sit at negative frequency are folded to positive, adding to what is there.
With the ratio at 1 : 1 every line is a multiple of the carrier: a clear, brassy note whose brightness follows the index. At 2 : 1 the lines fall on odd multiples of the carrier only, a hollow sound like a clarinet's. At 1.41 : 1, close to the square root of 2, which is not a fraction at all, the lines fall on no common series and you hear a bell. That is how FM makes bells, tines and electric pianos so cheaply. The trick that made the DX7 famous is the third control: let the index fall as the note decays, and the note starts bright and mellows, like a struck piece of metal.
FM's weakness is its controls. Two numbers shape the whole spectrum, and moving either by a little can change the sound entirely. Programmers praised its range and cursed the editing: the DX7's tiny screen and 155 parameters are the reason many players kept to presets.
Wavetable and granular: scan it, scatter it
Two methods start from stored sound and differ in how finely they slice it. Wavetable synthesis stores a short list of single-cycle waves and plays one on repeat at the pitch you want; the sound changes by moving between tables. Wolfgang Palm's PPG Wave (published in 1979) swept a position through a table of waves, and the method returned in the Waldorf synthesisers and in many software instruments. Because a morph between two waves is a morph between two spectra, a slow sweep produces timbres that no analogue filter makes: a sine that gets a vowel that gets a saw. (The word "wavetable" was also used on 1990s sound cards for something else, sample playback.)
Sound isn't available in this reader; the chart and numbers still work.
Four invented 16-harmonic tables: sine, hollow (odd harmonics 1/n²), vowel-like (peaks at harmonics 3 and 9) and saw (1/n). The shown wave and bars are a linear blend of the two nearest tables; the spectral centroid is the strength-weighted mean harmonic number.
Granular synthesis slices much finer, into grains of about 1 to 100 milliseconds, each with a smooth fade in and out so it does not click. The idea comes from the physicist Dennis Gabor's 1947 suggestion that sound is made of small elementary pieces; Iannis Xenakis built music from them on tape in the late 1950s; Curtis Roads made the first computer implementation in 1974; Barry Truax made it work in real time in the 1980s. The controls are the shape of a cloud: where in the source the grains start (the position), how long they are, how many per second (the density), how much the start point is scattered, and how much each is transposed.
Hold the position fixed and the cloud repeats the same slice, turning a moment into a drone: that is time-stretching to infinity. Scatter the position and one note melts into the next. Raise the density with short grains and it is a continuous tone; lower it and you hear individual specks.
The source is a made-up 3-second sound that steps through four notes, so the position decides which note the grains sample.
Sound isn't available in this reader; the chart and numbers still work.
Overlap = density × grain length: the average number of grains sounding at once. The page keeps it at 8 or below, lowering the density if needed. Grains are shaped with a raised-cosine window; transposition plays a grain faster or slower, as sampling does.
Physical modelling: simulate the thing that makes the sound
The last method starts neither from spectra nor from recordings but from physics. A plucked string is a pulse travelling along it, reflecting at each end, losing a little energy each time. Physical modelling simulates that directly. The simplest and most famous example was found by Alexander Strong and analysed by Kevin Karplus, published in 1983: fill a delay line with a burst of noise, then read it round in a loop, averaging each sample with its neighbour as it goes. The length of the loop sets the pitch, because the pulse takes that long to come round. The averaging is a gentle low-pass that eats the highest frequencies a little more on every trip, and that is exactly how a real string loses its brightness before its fundamental. Julius O. Smith later showed this is a special case of digital waveguide modelling, which also covers tubes, membranes and bowed strings.
The catch shows up when you try it. N must be a whole number of samples, so only certain pitches are available, and at high pitches they are far apart: at 44.1 kHz a note near 1,760 Hz comes out 30 cents flat. Real implementations use a fractional delay to tune precisely. Pluck it.
Sound isn't available in this reader; the chart and numbers still work.
The string is simulated sample by sample at 44.1 kHz in the page (at your device's rate when playing), with g = 0.996 per trip and a random starting burst, so each pluck differs slightly. The curves are windowed spectra of 1,024 samples, 0 to 5 kHz in decibels.
Damping at 100% rings with a soft, guitar-like decay: the red curve at 0.5 s has lost the high end while the fundamental survives. At 0% there is no smoothing, so the string keeps all its brightness and sounds like a stretched wire or a harsh buzz that goes on far longer than any real one. The method's range is the real strength: change the length while it rings and you get a slide, change the burst and you get a different pluck or a drum, add a second loop and you get a bowed or blown tone. Its limit is that it only makes sounds that physical things make, and each new instrument needs a model of its own.
What a graduate just sees
- Every pitched sound is a stack of partials; a sound's character is the list of their strengths, and how it changes.
- Subtractive means carving a rich wave; the skill is in modulation, since a filter that never moves sounds dead.
- An LFO is a slow oscillator wired to a control: pitch gives vibrato, loudness tremolo, cutoff wobble.
- A digital oscillator has to stop at Nyquist; harmonics beyond it fold back as inharmonic fizz that moves the wrong way.
- FM builds a spectrum from two sines; whole-number ratios are pitched, odd ratios are bells, and the index is a brightness knob.
- Wavetables change timbre by moving between stored cycles; granular clouds turn a moment into a texture.
- Physical models need a model of the thing, and then do what the thing does for free, such as a string losing its highs first.
- Pick the method that makes the sound naturally; fighting a method for a sound it cannot make is how patches go wrong.
Where it shows up next
- Beatmaking, Sampling & Arrangement (36/69): the drum machines in it are synthesis (a kick is a sine that falls in pitch) or sampling (a recorded hit).
- Sound for Film, TV & Games (36/67): creature voices, vehicles, magic and interface beeps are designed from oscillators and processing, and game engines synthesise engines and wind.
- Acoustics, Tuning & Timbre (35/68): partials, formants and inharmonicity are why each method sounds as it does.
- Signals & Systems (34/37): aliasing, filters and convolution are the same mathematics, written without the music.
- In the world: the preset browser of every DAW, modular and Eurorack systems, hardware and software synthesisers from every continent, and the phone's notification sound.
If you want to go deeper
- 📘 Textbook. Julius O. Smith III, Physical Audio Signal Processing (free online at CCRMA, Stanford), the standard treatment of waveguides and string models; and Miller Puckette, The Theory and Technique of Electronic Music (World Scientific, 2007), a clear course on oscillators, filters, FM and granular methods, also free from his site.
- 📘 Textbook. Curtis Roads, Microsound (MIT Press, 2001), the definitive book on granular synthesis; and Andy Farnell, Designing Sound (MIT Press, 2010), which builds sound effects from first principles.
- ▶️ Gentler. Syntorial, a paid interactive synthesis tutorial, or the free Learning Synths site from Ableton at learningsynths.ableton.com, which runs in a browser and teaches subtractive synthesis by playing.
- 🌐 Free course. Julius O. Smith III, Introduction to Digital Filters with Audio Applications, for the filter mathematics behind every module; and MIT OpenCourseWare 21M.380, Music and Technology.
Making a sound is choosing a description of it: partials, a filtered wave, a bending wave, stored cycles, scattered grains or a simulated object. Each description makes some sounds easy and others impossible, and every digital one must respect the sample rate. A good sound designer reaches for the method first and the knob second.
Try this: pick three sounds you hear this week (a ringtone, a game's coin, a keyboard patch in a song) and for each guess the method. Pitched with a changing brightness? Filter or FM. Bell-like? FM. Natural but impossibly smooth? Probably a model or a sample.
Sources. Fourier 1822, Telharmonium (1900, commercial 1906), Hammond organ (commercial 1935): encyclopaedia summary of additive synthesis. Gibbs overshoot (about 9% of the jump, peak 1.179 for ±1): computed in the page and standard. Chowning (FM found 1967, published 1973; Stanford patent; Yamaha exclusive rights 1975) and the Yamaha DX7 (1983; six operators, 32 algorithms, 16 voices; first synthesiser to sell over 100,000): encyclopaedia summary and Chowning's 1973 Journal of the Audio Engineering Society paper. Karplus and Strong, "Digital Synthesis of Plucked-String and Drum Timbres," Computer Music Journal 7(2), 1983; Jaffe and Smith in the same issue. Granular synthesis: Gabor 1947, Xenakis (Analogique A-B, 1959), Roads 1974, Truax in the 1980s: encyclopaedia summary. PPG Wave and Palm's published method (1979): encyclopaedia summary. Nyquist folding, Bessel sidebands, the DX7's 155 parameters and the polyBLEP family: standard texts.
- 📘 Puckette, The Theory and Technique of Electronic Music
- 📘 Roads, Microsound
- 🌐 Julius O. Smith III, Introduction to Digital Filters