FM Synthesis Explained Simply: Ratios, Bells and Basses

FM synthesis makes sound by letting one oscillator, the modulator, wobble the pitch of another, the carrier, hundreds or thousands of times a second. At that speed you stop hearing a wobble and hear a new, richer tone. Two settings do most of the work: the ratio between the two oscillators decides which overtones appear (whole numbers sound like instruments, in-between numbers sound like bells), and the FM amount decides how many, so more amount means a brighter sound.

What is FM synthesis? Carrier and modulator

Start with vibrato. Send an LFO to a synth’s pitch at 5 Hz and the note wavers five times a second. Speed it up to 50 Hz, then 200 Hz, and the wavering stops sounding like movement and turns into tone color. John Chowning heard exactly this at Stanford in 1967. His 1973 paper in the Journal of the Audio Engineering Society showed how to predict which overtones you get. Yamaha licensed the method. Its first FM synth, the GS-1, came out in 1980, and the DX7 followed in 1983.

FM gives the two oscillators names:

• Carrier: the oscillator you hear. Its pitch is the note you play.
• Modulator: the oscillator doing the wobbling. You usually never hear it on its own, only what it does to the carrier.

Both are normally sine waves, the plainest oscillator shape, with no overtones at all. The modulation adds overtones, called sidebands, at the carrier frequency plus and minus whole multiples of the modulator frequency. Play A3 (220 Hz) with the modulator also at 220 Hz, and new partials appear at 440, 660, 880 Hz and up. That is the full harmonic series, the same ingredients as a sawtooth wave, so two plain sines give you a buzzy, saw-like tone without any filter.

Operators and algorithms

In an FM synth, each sine oscillator comes packaged with its own envelope and level control, and that package is called an operator. The DX7 had six operators per voice and 32 algorithms. An algorithm is a wiring diagram: it says which operators modulate which, and which ones are carriers you actually hear. Algorithm 32 wires all six as carriers, with no operator modulating another, which makes a drawbar-style organ out of six sines. Four-operator synths like the DX100 and TX81Z use the same idea with fewer parts. Everything below needs just two: one carrier and one modulator.

Is FM synthesis analog or digital?

The math works with any oscillator, and many analog synths have an FM input for growls and clangs. Tuned FM depends on an exact ratio between two oscillators, though. Analog oscillators drift, and when the ratio slides, the whole tone slides with it. Digital oscillators hold their tuning, so the FM instruments that made it famous were all digital. Yamaha’s version is technically phase modulation, which gives the same results mathematically, so a synth that says “PM” is doing the same job.

What is ratio in FM synthesis?

Ratio is the modulator’s frequency compared with the carrier’s, written carrier first. At 1:2 the modulator runs twice as fast as the carrier. Because it is a ratio and not a fixed pitch, the modulator follows you up the keyboard, and every note keeps the same tone color.

Whole-number ratios put the sidebands on the note’s harmonic series, so you hear a pitched instrument. In-between ratios scatter sidebands into the gaps between harmonics, and your ear hears metal.

To see why 1:1.4 clangs, put the carrier at 200 Hz and the modulator at 280 Hz. The first sidebands land at 80 and 480 Hz, the next pair at 360 and 760 Hz. A normal 200 Hz note has its harmonics at 400, 600 and 800 Hz, and none of those four sidebands lands on one of them.

For slow movement, nudge a whole-number ratio slightly off: 1:1.005, or about 9 cents of detune on the modulator. The sidebands no longer line up exactly, so neighboring partials beat against each other, about twice a second at A3 and faster on higher notes. The sound picks up a gentle, phasing shimmer. Try it on electric pianos and pads.

FM amount (modulation index) and envelopes

FM amount (also called the modulation index, or the modulator’s output level on operator synths) sets how far the modulator pushes the carrier’s pitch. At zero you hear a pure sine. As you raise it, more sidebands join, spreading out above and below the carrier. The number of sideband pairs loud enough to hear is roughly the index plus one. An index of 1 adds a soft touch of color. An index of 5 brings in around six pairs and a hard, brassy buzz.

Opening a low-pass filter brightens a sound smoothly. Raising FM amount is lumpier: individual sidebands swell, fade and come back, because each follows its own curve (Bessel functions, if you want the math). At an index of about 2.4 the carrier itself drops out completely. That uneven motion gives an FM sweep its talking, metallic quality.

Because amount controls brightness, an envelope on it does the job a filter envelope does on an analog-style synth. The DX7 had no filter at all, so its sounds got brighter or darker through the modulators’ output levels, moved over time by their envelopes. On any FM synth, think in two envelopes:

• The carrier’s ADSR envelope shapes the volume.
• The modulator’s envelope shapes the brightness.

Struck and plucked instruments get darker as they fade, so FM versions of them give the modulator a shorter decay than the carrier. Some starting points:

If your synth can send velocity to the modulator’s level, do it. Harder notes come out brighter, the way a real piano or a struck bell behaves.

Many FM synths also have a feedback control, which routes an operator’s output back into its own input. Turned up, it bends a sine toward a sawtooth, so one operator can give you a saw-like tone. A feedback loop that runs through two operators can break the tone down into noise, which is useful for hi-hats and the rattle of a snare.

What is FM synthesis used for? Bells, basses and electric pianos

FM is at its best on sounds whose brightness changes fast within one note: the hard strike and soft ring of things that are hit or plucked.

• Electric pianos. The DX7’s E. PIANO 1 preset became famous on 1980s power ballads. Build one from a 1:1 pair with the electric piano envelopes above, then, if you have a third operator, add a second modulator at a high ratio (try 1:14) with a decay of 20 to 50 ms for the metallic knock of the tine. More in classic electric piano sounds.
• Bells and mallets. Use 1:3.5 for a tubular bell or 1:1.4 for a church bell, with an instant attack, a long decay, zero sustain, and FM that fades well before the volume does. The full recipe is in how to make an FM bell sound.
• Basses. A 1:1 or 1:0.5 pair with a high amount that decays in 100 to 200 ms gives a punchy, slap-like attack over a round low end. Kenny Loggins’ “Danger Zone” uses the DX7’s BASS 1 preset.
• Brass and organ. Brass is a 1:1 pair whose amount rises and falls with the volume. For organ, you can skip the modulation and mix carriers at ratios 0.5, 1, 2 and 4, the same pitches as a Hammond’s 16’, 8’, 4’ and 2’ drawbars, which is how algorithm 32 is wired.
• Game music. Sega’s Mega Drive (Genesis in North America) used Yamaha’s OPN2 FM chip, and the AdLib and Sound Blaster PC sound cards used its OPL2 and OPL3 chips. Much of the music in Mega Drive and DOS-era PC games is FM.

FM is weaker at the slow, resonant filter sweep of an acid bassline or a warm pad that opens up over several bars, which is subtractive territory. For the side-by-side, read FM vs subtractive, wavetable and additive synthesis.

New FM gear keeps coming. The Korg Volca FM (2016), the Elektron Digitone (2018) and the Korg opsix (2020) are all FM instruments, and the free, open-source Dexed plugin is closely modeled on the DX7.

Is FM synthesis hard? A five-minute first patch

FM got its reputation from the DX7, where six operators, each with its own envelope, had to be edited through menus on a small LCD. Most owners stuck to the factory presets. With two operators the whole thing shrinks to four controls (ratio, amount and one envelope per operator), and you can hear each one work.

On iPhone, Gelato Synth keeps FM to a few big controls: a Ratio slider on the Osc B page that runs from x0.5 to x8, and an FM page with Amount and Envelope to FM, which gives a bright attack that settles into a purer tail. A live oscilloscope shows the wave you hear, so you can watch the wave change shape as you raise Amount.

These steps work on any FM synth with ratio and level controls:

1. Start from an init patch. Two sine operators, one carrier and one modulator, both at ratio 1, with the amount at zero. Play A3 and you hear a plain sine, like a tuning fork.
2. Raise the amount slowly while you hold the note. The tone moves from a soft flute toward a brassy buzz. Stop about a third of the way up.
3. Change the modulator’s ratio to 2. Now it is hollow and woody, like a clarinet. Try 3 for a nasal reed, then 0.5 for extra weight an octave down.
4. Set the ratio to 3.5. Each note now rings like metal. Then go back to 1.
5. Give the modulator an envelope. Attack 0, decay 400 ms, sustain 10%. Set the carrier to attack 1 ms, decay 3 s, sustain 0. Play a few notes around middle C (C4) and you have a bright electric piano that softens as it fades. Save it.