FM synthesis explained: ratios, index and algorithms without the maths anxiety

Most producers learn synthesis the subtractive way: start with a buzzy waveform and carve it down with a filter. FM works the other way round. You start with plain sine waves and grow harmonics by letting one wave push the pitch of another up and down, hundreds or thousands of times a second. The results are the glassy electric pianos, struck bells, metallic basses and clangy percussion that a filter alone struggles to make.

If you have not built a patch from scratch before, read our synth sound design basics first; it covers oscillators, envelopes and filters, which FM still uses. This guide stays on FM.

Where it came from

FM as a music tool starts with John Chowning at Stanford. His 1973 paper in the Journal of the Audio Engineering Society showed that modulating a sine wave's frequency at audio rate produces rich, controllable spectra from very little computation. Stanford patented the method and licensed it to Yamaha, whose DX7 made FM the sound of mid-1980s pop. That history matters for one practical reason: the DX7's vocabulary (operators, ratios, algorithms) is still the vocabulary of almost every FM synth you will open today.

The three ideas you actually need

Operators. An operator is a sine oscillator with its own envelope. A carrier is an operator you hear. A modulator is one you don't hear directly; it bends the pitch of the carrier.

Ratio. Each operator's frequency is usually set as a multiple of the note you play. A carrier at 1 and a modulator at 1 gives a tone with every harmonic, close to a sawtooth. Modulator at 2 gives only odd harmonics relative to the carrier, nearer a square. Whole-number ratios give harmonic, pitched tones. Non-whole ratios such as 1.41 or 3.5 give inharmonic partials: bells, gongs, metal.

Index (or modulator level). How hard the modulator pushes. At zero, you hear a pure sine. As you raise it, sidebands appear at the carrier frequency plus and minus multiples of the modulator frequency, and the sound gets brighter and wider. Chowning's key insight was that if the modulator has an envelope, the brightness changes over time, which is exactly what real struck and plucked instruments do. In FM, the modulator's envelope does the job a filter envelope does in subtractive synthesis.

Algorithm is simply the wiring diagram: which operators modulate which, and which ones are heard. The DX7 manual lists 32 algorithms for its six operators. Ableton's Operator uses four oscillators and 11 algorithms, which is plenty to learn on.

Step by step: three patches in any four-operator FM synth

Use Operator, the free open-source DX7 emulation Dexed, or any FM synth that shows ratios. Start each patch from an init preset.

1. An electric piano

1. Choose an algorithm with two separate stacks: modulator into carrier, twice.
2. Stack one: carrier ratio 1, modulator ratio 1. Give the modulator a fast attack and a decay of about a second, so the tone starts bright and mellows.
3. Stack two: carrier ratio 1, modulator ratio 14 at a low level with a very short decay. That brief high sideband is the "tine" click.
4. Set both carriers' envelopes to a medium decay with some sustain. Map velocity to modulator level so harder playing sounds brighter.

2. A bell

1. One modulator into one carrier.
2. Carrier ratio 1, modulator ratio 3.5 (or try 1.41). The non-whole ratio is what makes it a bell rather than an organ.
3. Long carrier decay, several seconds, no sustain. Modulator decay a little shorter, so the clang fades into a purer hum.
4. Play it an octave higher than you think; bells sit well above the mix.

3. A punchy bass

1. One modulator into one carrier, carrier ratio 1, modulator ratio 1 (or 0.5 for a sub-octave growl).
2. Give the modulator a short, snappy decay: that is your "filter pluck".
3. Add a little feedback on the modulator (the operator modulating itself) for grit. Feedback pushes a sine towards a sawtooth-like tone.
4. Keep the carrier mono and check it against your kick; our 808 and sub bass guide covers making low end translate.

Common mistakes

• Turning up the index with no envelope. You get a static, harsh buzz. Put the movement on the modulator.
• Random ratios everywhere. Change one ratio at a time and listen. Most good patches use a small set: 0.5, 1, 2, 3, 14, and one deliberately odd value.
• Too much at the top of the keyboard. High notes with high index can fold sidebands into harsh, unrelated tones. Use key scaling (the DX7 calls it keyboard level scaling) to lower modulator levels as you play higher.
• Expecting a filter to fix it. Many FM synths have one now, but if a patch sounds wrong, the fix is usually a ratio or an envelope.

What we think

FM's reputation as the "hard" synth is mostly a hangover from the DX7's tiny screen and membrane buttons, where you edited one number at a time with no picture of what was going on. On a modern plug-in with visible algorithms and envelopes, it is no harder than subtractive, just different, and the payoff is sounds that are yours rather than the same filtered sawtooth everyone else starts from.

For artists, the bigger point is control. Presets and AI sound generators can hand you a finished tone in seconds, but FM rewards knowing why a sound behaves as it does: a ratio is a decision about harmony, an envelope is a decision about gesture. Learn three or four ratios and the modulator envelope properly, and you will hear FM everywhere, in 1980s ballads, in modern hyperpop percussion and in game music, and you will be able to make your own version instead of hunting for a preset. We would rather an artist spent an evening with one sine wave and one modulator than a month scrolling sample packs. Start with the bell; it is the fastest route to the "oh, that's how it works" moment.