What is FM synthesis?

FM synthesis in plain English

Start with vibrato. A singer holds a note and lets the pitch drift a little above and below it, five or six times a second. On a synth you get the same thing by pointing a slow LFO at an oscillator's pitch, and you hear it as the note wobbling.

Now speed the LFO up. Past somewhere around 20 Hz the wobble gets too fast to follow as movement, and the movement turns into a new tone colour, because the note has picked up extra frequencies that weren't there before. That is frequency modulation, and FM synthesis is the craft of controlling those extra frequencies.

The oscillator you hear is the carrier. The one doing the pushing is the modulator, and you never hear it directly, only its effect on the carrier. The new frequencies are called sidebands, and they appear in pairs either side of the carrier, spaced exactly one modulator frequency apart.

Classic FM synths such as the DX7 use plain sine waves for both, and so does the lab. A sine has no harmonics of its own, so every partial you see in the lab above, apart from the first one, was made by the modulation.

Reading the lab

The lab is the smallest possible FM synth: one modulator, one carrier and an envelope on each. Press Play riff and it loops a two-bar riff in A minor at 120 BPM. The screens all show a single A at 220 Hz rendered through the same voice with your settings, so the drawings are the sound you're hearing.

The big screen is a spectrogram of that note. Time runs left to right over 1.6 seconds, frequency runs up from 0 to 8 kHz, and brighter means more energy. Frequency goes up in even steps here, unlike most spectrum displays, because FM sidebands are evenly spaced in hertz and this way you can see the gaps. The dashed amber line is Carson's rule, a quick estimate of where the sidebands run out, which is explained further down.

Underneath, the scope shows two cycles of the wave with the modulator drawn over it in teal. Where the modulator is high, the carrier runs faster and its cycles bunch up. The partials screen measures every frequency in the first 50 ms of the note against the level of the whole note, so a plain sine reads 0 dB, and the bar at the carrier's frequency is drawn in white so you can watch it lose level to the sidebands. Hover over any bar to see which sideband it is.

Mod off sets the index to 0 for new notes, which leaves the carrier on its own. The Whole-number ratios switch makes the Ratio knob step through 0.5, 1, 2, 3 and up, the way the coarse ratio does on a DX7. The Right now panel describes what your settings are doing, with numbers measured from the rendered note.

Carrier and modulator

Take an A at 220 Hz as the carrier. Set the modulator to 440 Hz and connect it so that it pushes the carrier's frequency 660 Hz up and 660 Hz down. That 660 Hz is the deviation. The carrier's frequency now sweeps between 880 Hz and −440 Hz and back, 440 times a second. A negative frequency sounds strange, but it only means the wave runs backwards for a moment, and the maths and the lab both handle it.

You can't hear that sweep as a sweep, because it repeats 440 times a second. What you hear is the result: a note at 220 Hz with sidebands at 220 Hz plus and minus 440, 880, 1,320 Hz and so on. Those below 0 Hz come back up the other side, so −220 Hz is heard at 220 Hz and −660 Hz at 660 Hz. All of those frequencies are odd multiples of 220, so the result still sounds like an A.

On a hardware FM synth the carrier and modulator are both called operators. An operator is an oscillator, usually a sine, with its own envelope and output level, and whether it's a carrier or a modulator depends only on where its output goes.

The modulation index

The modulation index is the deviation divided by the modulator frequency. In the example above that's 660 ÷ 440, an index of 1.5. It's the single biggest control over how bright an FM sound is, and on a DX7 it's set by the modulator's output level.

At an index of 0 the carrier is a plain sine, and raising it a little brings in the first pair of sidebands. Keep going and more pairs join, each one further from the note than the last. A handy rule of thumb is that you get one strong pair of sidebands for every step of the index, plus one more. It's called Carson's rule, after the radio engineer John Carson, and it says the sound spreads across roughly 2 × (deviation + modulator frequency).

Here are the numbers for the lab's 220 Hz note with the modulator also at 220 Hz. At an index of 1.5 the deviation is 330 Hz, so Carson's rule gives a spread of 2 × (330 + 220) = 1,100 Hz, centred on the carrier: sidebands out to about 770 Hz. At an index of 5 the deviation is 1,100 Hz and the spread is 2 × (1,100 + 220) = 2,640 Hz, which takes the top sideband out to about 1.5 kHz. That's the amber line in the lab.

The sidebands don't simply grow as the index goes up. Each one rises, peaks and falls back, then rises again, following curves called Bessel functions. You don't need the maths to use FM, but one result is useful: at an index of about 2.4 the carrier itself drops almost to nothing, and the note is made entirely of sidebands. Set the ratio to 3 : 1 and the index to 2.4 in the lab, and the white bar on the partials screen sinks out of sight. At 1 : 1 and 2 : 1 it doesn't, because sidebands reflected up from below 0 Hz land on the carrier's frequency and fill the gap.

That uneven movement is why sweeping the index sounds different from opening a filter. A filter lets more harmonics through smoothly, from the bottom up. Turning up the index makes partials bulge and dip as it goes, so an index sweep has a watery, shifting quality of its own.

FM moves energy out of the carrier and into the sidebands without adding any. The level can still shift by a few decibels with whole-number ratios, because some reflected sidebands land on the same frequency as others and either add up or partly cancel, and the lab's coach measures that change against a plain sine. A high index also sounds louder than its level suggests, because it moves energy up from 220 Hz towards the range from about 2 to 5 kHz, where ears are most sensitive.

The ratio

The ratio is the modulator's frequency as a multiple of the carrier's, and it decides which frequencies the sidebands land on. This page writes it modulator first: 2 : 1 is a modulator at 440 Hz. On most FM synths the modulator follows the keyboard, so the ratio stays the same whichever note you play.

With a whole-number ratio, every sideband, including the ones that reflect back up from below 0 Hz, lands on a harmonic of the note. The ratio then decides which harmonics are there.

The 2 : 1 row is a good example of the arithmetic. With the modulator at 440 Hz the sidebands sit at 220 ± 440, ± 880, ± 1,320 Hz, which gives 660, 1,100 and 1,540 Hz above the carrier and −220, −660 and −1,100 Hz below 0. Reflected back up, the negative ones land on 220, 660 and 1,100 Hz. Every partial is an odd multiple of 220, which is why the sound turns hollow. The lesson on even and odd harmonics covers why odd harmonics on their own sound that way.

At 1 : 1 there's a quirk. The first lower sideband is the carrier minus the modulator, 220 − 220 = 0 Hz. A partial at 0 Hz is a constant offset you can't hear, and how big it is depends on where the two oscillators start in their cycles. In this lab it's small enough that the scope barely shows it.

Move away from whole numbers and the sidebands stop lining up. At 1.41 : 1 the modulator sits at about 310 Hz, so the partials going up fall at 220 Hz, 530 Hz, 840 Hz and 1.15 kHz, and the reflected ones come back in the gaps, starting at 90 Hz and climbing in the same 310 Hz steps. No single fundamental fits all of those, so the note loses its pitch and you hear a clang. The word for that is inharmonic, and it's the sound FM is best known for.

A ratio that's only a fraction away from a whole number does something gentler. At 1.003 : 1 the sidebands land a hair off the harmonics, so the partials drift slowly in and out of step with each other and the tone shimmers. It's a common trick for making a static FM patch move.

Putting an envelope on the index

A static index gives a static tone. Most acoustic sounds change as they go: a struck bell, a plucked string or a piano hammer starts bright and loses its upper partials as it fades. FM copies that with an envelope on the modulator's level, so the index starts high and falls. It's the FM version of a filter envelope, and the ADSR envelopes lesson covers how envelope stages work in general.

John Chowning's 1973 paper on FM synthesis made bell tones this way: an in-between ratio, a high index at the strike, and the index and the level dying away together. In the lab, the Mod decay knob sets how long the index takes to fall to nothing, and the Amp decay knob sets how long the note rings. Watch the big screen as you change them: the top edge of each note follows the amber line down.

Short mod decays give plucks. With a 7 : 1 ratio and a mod decay of 150 ms, the high sidebands only exist for the first moment of each note, which gives a glassy ping on a tone that's close to a sine. FM electric piano sounds build their attack from the same idea.

Basses work the other way round from bells. Keep the ratio at 1 : 1 so the note stays in tune, start the index at 3 to 5, let it fall over 100 to 300 ms and hold the amp level. Each note bites and then rounds off, much like a quick filter envelope on a subtractive bass.

Operators, algorithms and feedback

The lab has two operators. The Yamaha DX7 has six, and it offers 32 ways of connecting them, called algorithms. An algorithm decides which operators are carriers going straight to the output, and which are modulators feeding other operators.

Most algorithms mix two arrangements, stacks and carriers side by side. In a stack, one modulator drives another modulator that drives a carrier, and the spectrum gets much denser than a single pair can make, because the carrier is being pushed by a signal that already has sidebands of its own. Carriers side by side are simply mixed together, so an algorithm with several carriers can layer a bright strike over a softer body in one patch, each with its own envelope.

Many FM synths also let one operator modulate itself, called feedback. A little feedback brightens a sine; turn it up and the wave leans over towards a sawtooth, and at the top it breaks up into noise. Every one of the DX7's 32 algorithms includes one feedback loop.

Plenty of software works the same way. Dexed is a free model of the DX7 that reads its patch files, Ableton's Operator has four operators, Native Instruments FM8 has six, and Logic's EFM1 is a two-operator design like this lab. Wavetable synths such as Serum and Vital can also use one oscillator to frequency modulate another.

FM and vibrato

Vibrato is FM with a slow modulator. A 5 Hz modulator with a deviation of 6.6 Hz, an index of 1.3, swings a 220 Hz note about half a semitone each way. The sidebands are there, but they're only 5 Hz apart, far too close for your ears to separate, so you hear the pitch move instead.

Raise the modulator and the sidebands spread out. Somewhere above 20 Hz they're far enough apart to be heard as tone, and by the time the modulator is in the same range as the note, as it is everywhere in this lab, there's no wobble left at all. The vibrato lesson covers the slow end of that range.

FM, ring modulation and subtractive synthesis

Ring modulation also makes sum and difference frequencies, but it only makes one pair for each frequency going in. FM makes a whole row of pairs, and the index decides how far along the row the energy reaches. So turning up a ring modulator makes its one pair louder without adding new ones, while turning up the index on an FM patch keeps adding partials further out.

Subtractive synthesis works in the opposite direction. It starts with a wave that already has every harmonic, such as a saw, and a filter takes them away. FM starts with a sine and adds partials. Subtractive is easier to steer by ear, while FM reaches inharmonic sounds like bells and metal that a filter can't make. Subtractive vs FM vs wavetable puts the three side by side.

A high index on a high note can push sidebands past the highest frequency the sample rate can hold, and they fold back down as aliasing, out of tune with the note. The lab's coach warns you when Carson's rule passes that limit. The lesson on aliasing and oversampling explains why it happens.

Chowning, Yamaha and the DX7

John Chowning, a composer at Stanford University, found in 1967 that vibrato pushed into the audio range made rich, controllable tones from just two oscillators. He published the method in the Journal of the Audio Engineering Society in 1973, and Stanford licensed it to Yamaha.

Yamaha released the DX7 in 1983, with six operators, 32 algorithms and 16-note polyphony, at a US list price of $1,995, a small fraction of what digital instruments such as the Fairlight CMI and the Synclavier cost. Yamaha's FM chips then spread into home computers and games consoles. The YM3812, a two-operator chip, was the sound of AdLib and Sound Blaster cards on PCs, and the four-operator YM2612 made most of the music on the Sega Mega Drive.

How to program an FM sound, step by step

This order works on a two-operator synth like the lab and carries straight over to bigger ones, one carrier and modulator pair at a time.

1. Start with the index at 0You're hearing the carrier alone. Shape its amp envelope first, so you know whether you're making a pluck, a pad or a bass.
2. Choose the ratio by character1 : 1 for full and bright, 2 : 1 for hollow, 3 : 1 for nasal, an in-between value for bells and metal. Stick to whole numbers until you want a clang.
3. Bring the index up slowlyStop when the brightness is right. Indexes from about 1 to 4 cover most sounds that still read as notes; past 6 or so, most ratios turn harsh.
4. Give the index an envelopeLet it fall over the first part of the note. On a DX7 that's the envelope on the modulator operator, and in the lab it's Mod decay.
5. Play it across the keyboardA fixed index spreads sidebands wider in hertz on high notes, so they can get thin or harsh at the top. The DX7 has keyboard level scaling on every operator, so a modulator can be turned down as you play higher.
6. Compare with the modulator offPress Mod off. If the sine on its own sounds better in the track, the patch needs less index or a different ratio.

Starting settings for common FM sounds

These are places to start in this lab, with the 220 Hz A in mind. A six-operator synth can build richer versions of every one of them.

Common FM mistakes

Too much index

A high index sounds exciting on its own and harsh in a mix, because it fills the 2 to 5 kHz range where vocals and snares live. Set the index while the rest of the track is playing, and pull it back until the sound sits.

Wandering off the whole numbers by accident

On a synth with a continuous ratio knob it's easy to land on 2.08 : 1 and wonder why the patch sounds out of tune. Use the coarse ratio, or a snap like the lab's Whole-number ratios switch, and only move off it when you want the clang.

Leaving the index static

An FM patch with no envelope on the modulator sounds like an organ stop, the same from the first moment to the last. Even a slow, small fall in the index makes a held note feel more alive.

Judging the patch on one note

The same index sounds darker on low notes and brighter on high ones, and very high notes can alias. Play the patch across the range you'll use it in before you decide it's finished.

Train your ears

The quiz under the lab covers the screens and plays the riff with hidden settings. Some rounds ask whether the ratio is a whole number, which you hear as a clear pitch or a clang; the others ask whether the index is low or high. Every round is level matched, so loudness can't give the answer away. Eight rounds take about three minutes, and your best score is saved in your browser.

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