Subtractive synthesis is easy to teach because it matches physical intuition. Start with a bright, buzzy wave, remove the parts you do not want with a filter, shape the result with an envelope. Sculpture.
FM synthesis does not work like that, and the usual explanation — “one oscillator modulates another’s frequency” — is true and nearly useless, because it does not tell you what will come out. Here is the version that lets you actually program it.
Operators, carriers and modulators
An operator is an oscillator with its own envelope. That is the whole definition. In classic FM instruments the operator produces a sine wave — and a sine wave has exactly one harmonic, which is the point. It has nothing in it to filter.
Operators do one of two jobs depending on how they are wired:
- A carrier is an operator you hear.
- A modulator is an operator you do not hear. Its output is added to another operator’s frequency.
The arrangement of who modulates whom is called an algorithm. A two-operator stack — one modulator feeding one carrier — is where everybody should start, and most of what you need to understand is visible there.
Control one: the ratio
The frequency ratio between modulator and carrier determines where the new harmonics land.
When a modulator at frequency M modulates a carrier at frequency C, the output contains the carrier plus a series of sidebands at C ± M, C ± 2M, C ± 3M and so on, spreading outward in both directions.
This is the single fact that makes FM predictable:
- Integer ratios (1:1, 2:1, 3:1, 1:2) put the sidebands exactly on harmonics of the carrier. Harmonic spectrum, definite pitch, musical.
- Non-integer ratios (1:1.41, 1:3.7) put sidebands between the harmonics. Inharmonic spectrum — bells, metal, gongs, mallets.
So the famous FM bell is not a trick. It is what a non-integer ratio sounds like, because that is what struck metal sounds like: a spectrum whose partials are not whole-number multiples of a fundamental.
| Ratio (mod:carrier) | Character |
|---|---|
| 1:1 | Sawtooth-ish, full harmonic series, good for basses and brass |
| 2:1 | Odd-harmonic emphasis, hollow, clarinet-like |
| 3:1, 4:1 | Brighter, more nasal; formant-like peaks |
| 1:2 | Sidebands below the carrier — thickening, sub-harmonic weight |
| 7:1, 11:1 | Sparse, clangorous |
| 1:1.41, 1:2.76 | Inharmonic — bells, chimes, tubular metal |
Start with integers. Make a 1:1 bass, a 2:1 reed and a 3:1 brass before touching a decimal.
Control two: the index
The modulation index — how much the modulator is driving the carrier — determines how many sidebands are audible and how strong they are.
Low index: almost pure carrier, a near-sine tone. Raise it and sidebands appear, and keep appearing, spreading further out. Very high index: dense, noisy, often ugly.
The crucial consequence is this: in FM, brightness is the modulation amount. There is no filter. A modulator envelope that starts high and decays is what produces the bright attack and mellow tail of a plucked or struck sound — and that is the whole mechanism behind the FM electric piano.
This is the thing to internalise. In subtractive synthesis, you shape timbre over time by enveloping a filter cutoff. In FM, you shape it by enveloping the modulator’s level. The modulator’s envelope is your filter envelope.
Three sounds from one two-operator patch
Keep the same algorithm — one modulator into one carrier — and change only three things.
Electric piano. Ratio 1:1. Carrier envelope: fast attack, long decay. Modulator envelope: fast attack, fast decay, moderate level. The bright knock disappears quickly and leaves a soft sine-ish tone. That decaying modulator is the hammer.
Bass. Ratio 1:1 or 2:1. Carrier: fast attack, sustained. Modulator: low to moderate level, slight decay. Keep the index modest — too much and the low fundamental disappears into sidebands, which is the most common FM bass mistake.
Bell. Ratio 1:1.41 (or anything non-integer). Both envelopes: instant attack, long exponential decay, no sustain. Moderate-to-high modulator level. The inharmonic sidebands plus a long decay is functionally the definition of a bell.
Same two operators, same wiring. Only the ratio and the envelopes moved.
Reading an algorithm diagram
Once you move past two operators, the algorithm chart becomes the map. The conventions are consistent across instruments:
- Operators stacked vertically are a modulation chain: the top one modulates the one below it.
- Operators side by side at the bottom are parallel carriers — their outputs are summed.
- A loop arrow on an operator is feedback: it modulates itself, which generates a progressively noisier, sawtooth-to-noise spectrum as it increases.
Two useful readings: more carriers in parallel means additive-style layering, so that is how you build a sound with two distinct registers. A longer modulation chain means the spectrum gets complicated fast and becomes hard to control, so a three-deep stack is already an experimental patch rather than a workhorse.
The three mistakes everyone makes
Turning everything up. FM’s dynamic range of timbre is enormous, and the top half of the modulator range is mostly unusable noise. Most good FM sounds live at surprisingly low modulation indices.
Forgetting that modulators have envelopes too. A static modulator level gives you a static timbre, which is the thing that makes beginner FM patches sound synthetic and dead. Almost every FM sound that feels alive has a modulator envelope doing the work.
Treating ratios as a continuous knob to explore. Detuning a ratio slightly off an integer does not get you “a bit more interesting” — it collapses the harmonic relationship, and the pitch goes ambiguous. Integer for pitched, deliberately non-integer for inharmonic. The territory in between is narrow and mostly sounds broken.
Where to practise
Any FM instrument will do, and the free options are good: Dexed (a DX7 emulation, so patches and algorithms transfer directly to decades of existing presets), Vital and Surge XT both have FM routing, and VCV Rack has several FM operators you can patch explicitly, which is the best way to see the signal flow rather than infer it from a diagram.
If you have hardware with a 4-operator or 6-operator engine, find the two-operator algorithm — usually the last one in the list — and stay there until the ratio and index behaviour is boring. Everything else in FM is that same relationship, repeated.
Sources
- FM Synthesis: How does it work? — Benn Jordan
- FM Algorithms: How to read them, how to choose one — Floyd Steinberg
- Dexed — free DX7 emulation
- Surge XT