FM synthesis explained: operators, ratios and modulation index
Ratio, index and envelope: the three settings you need to understand FM synthesis and program your own sounds, from the DX7 to current FM synths.
FM synthesis (frequency modulation) has a reputation as the hardest kind of synthesis to program. The principle fits in one sentence, though: one oscillator, the modulator, varies the pitch of a second oscillator, the carrier, so fast that the wobble turns into new harmonics. Three settings decide the result: the ratio between the two frequencies, the modulation index and the envelope that moves that index over time. Understand those three and you can get past the presets, on a DX7 or on any current FM synth.
Where FM comes from
The technique was formalised by John Chowning at Stanford University in the paper “The Synthesis of Complex Audio Spectra by Means of Frequency Modulation”, published in 1973 in the Journal of the Audio Engineering Society. Yamaha licensed it and made it mainstream in 1983 with the DX7: six operators, 32 algorithms, 16-voice polyphony. Chowning tells the story of the discovery himself in a talk published by the AES:
For a fuller mathematical treatment, Bill Schottstaedt’s introduction to FM, hosted by Stanford’s CCRMA, is still the go-to free reference.
Operators and algorithms
FM synths talk about operators rather than oscillators: an operator is a sine-wave oscillator with its own volume envelope. Depending on where it sits, it plays one of two roles:
- carrier: you hear its output directly;
- modulator: you do not hear its output; it modulates the frequency of another operator.
An algorithm is the way the operators are wired together: stacked (a modulator on a modulator on a carrier), in parallel (several carriers side by side, like organ drawbars), or a mix of both. On the DX7, the operator at the top of a stack can also modulate itself: this is feedback, which pushes the sine towards a sawtooth and then towards noise.
The ratio: harmonic or metallic
Operators are not tuned in hertz but as a ratio of the played note: at 1 an operator follows the note, at 2 it sits an octave higher. Modulation creates components called sidebands, located at the carrier frequency plus and minus multiples of the modulator frequency. The relationship between the two ratios therefore sets the tone colour:
| Carrier : modulator ratio | Resulting components | Character |
|---|---|---|
| 1:1 | every harmonic | close to a sawtooth: brass, strings |
| 1:2 | odd harmonics only | close to a square: hollow, clarinet-like |
| 1:3, 1:4… | harmonics with regular gaps | nasal, organ, vocal |
| non-integer (1:1.41, 1:3.5) | inharmonic components | bells, metal, percussion |
A worked example: a 200 Hz carrier modulated at a 1:2 ratio (modulator at 400 Hz) produces components at 200, 600, 1,000, 1,400 Hz and so on: the odd harmonics of 200 Hz. Sidebands that fall below zero fold back into the spectrum and, here, land on those same frequencies.
The index: how bright it gets
The modulation index measures how hard the modulation is: the frequency deviation forced on the carrier, divided by the modulator frequency. On a synth it is simply the modulator’s output level. At zero you hear a pure sine; as it rises, more sidebands appear and the sound gets brighter. The useful bandwidth roughly follows Carson’s rule: about twice the sum of the frequency deviation and the modulator frequency.
The index does the job that filter cutoff does in subtractive synthesis, with one big difference: brightness does not change smoothly, because each harmonic rises and falls at its own pace as the index goes up. That is what makes FM tones sound alive, and what makes FM less intuitive.
The modulator envelope: making the sound move
The secret of classic FM sounds is the envelope applied to the modulator. Because the modulator level sets the index, an ADSR envelope (attack, decay, sustain, release, covered in our ADSR envelope guide) on that level changes the timbre over time:
- a fast attack and short decay on the modulator gives a bright transient that fades out: electric piano, marimba, plucky bass;
- a slightly slower attack makes the sound open up like breath in a brass instrument, the example Chowning described back in 1973;
- a high sustain keeps a rich, steady tone, useful for pads and gritty basses.
Routing velocity to the modulator level as well makes brightness follow how hard you play: that is what made DX7 electric pianos so expressive.
Three starting recipes
| Sound | Carrier : modulator ratio | Index | Modulator envelope |
|---|---|---|---|
| Electric piano | 1:1, plus a light modulator at 14 for the “tine” | medium | zero attack, short decay, low sustain |
| FM bass | 1:1 or 1:2 | high | medium decay, medium sustain |
| Bell | 1:3.5 | medium to high | long decay, no sustain |
Treat these values as starting points: the same ratio sounds different depending on the algorithm and how many operators are stacked.
FM or phase modulation?
Most so-called “FM” synths, the DX7 included, actually perform phase modulation: the modulator shifts the carrier’s phase instead of its frequency. To the ear the result is equivalent, but pitch stays stable even under heavy modulation, which makes tuning easier. The controls, ratio, index and envelopes, work in exactly the same way.
Where to practise today
FM now turns up on very affordable instruments. The M-VAVE FM1 and its open firmware puts a six-operator architecture in a small box, the Twisted Electrons FLEXFM combines FM with sampling and analogue filters, and the free software synth Micronix by Push and Groove includes a six-operator FM engine among its models. To compare FM with other digital techniques, see our explainer on wavetable synthesis and the FM and additive section.