What are harmonics in music?

Harmonics in plain English

Pluck the open A string on a guitar and it vibrates 110 times a second, which is the pitch you hear. That's only part of what the string is doing. At the same time it also vibrates in halves, 220 times a second, and in thirds at 330, in quarters at 440, and further up in smaller and smaller sections. Each of those extra vibrations is a harmonic.

The lowest one, 110 Hz here, is the fundamental. Every harmonic above it sits at a whole-number multiple of it: two times, three times, four times and on up. You don't hear them as separate notes, because your ear fuses them into a single tone, but change their levels and the tone changes with them. A sound with a strong fundamental and quiet harmonics sounds round and soft. Turn the harmonics up, especially the higher ones, and the same note gets brighter and buzzier.

That's why harmonics belong on a page about saturation. Every saturator, distortion pedal and clipper works by bending the shape of the wave, and a bent wave contains harmonics that weren't there before. Once you can see which ones a process adds, you can predict how it will sound.

The harmonic series

The harmonics of a note, in order, are called the harmonic series. Here it is for the A at 110 Hz that the lab's sine plays, with the note each harmonic lands on. The gaps get smaller as you go up: the first step is a whole octave, the next is a fifth, then a fourth, and from the 9th harmonic up they're less than a whole tone apart.

The first six harmonics spell out an A major chord, which goes some way to explaining why octaves and fifths sound so stable together: their harmonics line up. The 5th and 7th harmonics don't land exactly on piano keys, because a piano is tuned in equal temperament and the harmonic series isn't. The 3rd harmonic is off too, but only by 2 cents, far too little to hear.

Harmonics, overtones and partials

Harmonic and overtone name the same frequencies but count them differently. Harmonics start counting at the fundamental, so the fundamental is the 1st harmonic and 220 Hz is the 2nd. Overtones start counting above it, so 220 Hz is the 1st overtone. If a manual says a valve adds "the first overtone", it means the 2nd harmonic, an octave up.

Partial is the wider word. It means any single frequency in a sound, whether or not it's a whole-number multiple of the fundamental. A plucked string's partials are almost exactly harmonic. A piano's are slightly sharp of the series, because a stiff steel string doesn't bend as freely as an ideal one, and piano tuners stretch their tuning to allow for it. A drum skin is far from harmonic: for an ideal circular membrane the first few partials sit at about 1, 1.59, 2.14 and 2.30 times the lowest one, which is why a tom gives you a pitch that's hard to sing back. Bells and cymbals are the same story taken further.

Harmonics and timbre

Timbre is the word for tone colour: whatever lets you tell a violin from a voice when both are on the same A. The balance of the harmonics is a big part of it. The other part is how that balance changes over time: a plucked or struck note is brightest in its first few milliseconds and loses its upper harmonics as it fades.

Synth waveforms show it most clearly because their harmonics follow simple rules. A sine has only the fundamental. A saw has every harmonic, each at 1/n of the fundamental's level, so the 2nd is 6 dB down, the 3rd 9.5 dB down and the 10th 20 dB down. A square has only the odd ones at those same levels. A triangle also has only odd harmonics, but they fall away twice as fast in decibels, so its 3rd is already 19 dB down and it sounds close to a sine. The oscillator waveforms lesson lets you play all four.

Drag the slider in the figure and you can watch it happen. One harmonic gives a sine. By about ten, the saw has its sharp ramp and the square has its flat top, with a little ripple at the corners that never quite goes away however many you add.

How saturation adds harmonics

A saturator is a transfer curve: for every level that goes in, it sends out a slightly different level. A straight line changes nothing. A curve that bends near the top squashes the peaks of the wave while leaving the quieter parts alone, so the shape of each cycle changes.

The changed wave still repeats 110 times a second, since the curve does the same thing to every cycle. Anything that repeats 110 times a second can only be built out of frequencies that fit a whole number of times into each cycle, which means 110, 220, 330 Hz and up. So on a single note, everything the curve adds is a harmonic of that note. Which harmonics, and how loud, depends on the shape of the curve and how hard you push into it.

In the lab, step 2 sends the full sine through a soft clipper with no drive at all. The curve rounds the peaks a tiny amount and the 3rd harmonic appears at −25 dB, with the 5th at −48 dB. Push the drive to 12 dB in step 3 and the 3rd climbs to −12 dB and the 5th to −20 dB, and the 7th, 9th and 11th now come out louder than −60 dB as well. Drive does exactly this: it puts more of the wave on the bent part of the curve.

Odd and even harmonics

With its Character knob at zero, the soft clipper only adds odd harmonics: the 3rd, 5th, 7th and so on, with an empty slot at every even number. That happens with any curve that treats the top and bottom of the wave the same way. Lean the curve so it bends one side harder than the other and even harmonics appear. On the valve curve in step 4, the 2nd harmonic comes in at −12 dB while the 3rd falls to −30 dB.

The 2nd harmonic is an octave above the note and the 4th is two octaves up, so even harmonics reinforce the note you're already hearing, and they tend to sound full. Odd ones bring in the fifth and the slightly flat major third, and on their own they sound hollower and edgier. The even vs odd harmonics comparison goes into how to hear the difference.

From a sine to a square

A hard clipper cuts the peaks flat. Drive a sine far enough into it and almost every cycle becomes a flat top and a flat bottom with a fast jump in between, which is a square wave. The harmonics follow: at 24 dB of drive the lab measures the 3rd at −10 dB, the 5th at −14, the 7th at −17, the 9th at −20 and the 11th at −22. An ideal square wave has them at −9.5, −14, −16.9, −19.1 and −20.8 dB. That's why heavy distortion on a bass or a guitar sounds buzzy and reedy: you've turned each note into something close to a square.

Reading the lab

The lab at the top of the page is the saturation lab, set up for this lesson with a sine wave loaded first. The screen to watch is the one labelled Harmonics on a 110 Hz sine. The lab runs a pure 110 Hz sine through your current settings and measures what comes out, so the first bar is the note and every bar after it is a harmonic, in dB against the note. Odd harmonics are pink and even ones are amber. The grey bar at the end is everything that isn't a harmonic, which on these curves is mostly aliasing.

The harmonics screen always measures the sine, even when you switch the sound to bass or chords, so you can compare settings on a fixed test tone. For the sound you're actually hearing, use the spectrum beside it: the pale shape is the original and the coloured line is the output, so wherever the line rises above the shape, the curve has added something.

Each of the five cards along the top is a different curve, with a drawing of what it does to one cycle of a sine. The panel on the right describes your current settings in numbers the lab has just measured. Leave Auto gain on: it keeps the output at the same loudness as the original, which you need here, because harmonics on their own make things sound louder. You can also upload your own sound or record one, and it plays through the lab in place of ours.

What each curve adds

These figures come from the lab, with a 110 Hz sine at 0.9 of full scale going in. The shape of the curve decides far more than the amount of drive does.

The wavefolder is the odd one out. Its 3rd harmonic comes out 11 dB louder than the note itself, so the result sounds less like a coloured version of the note and more like a new, ringing tone. That's why wavefolding lives in synths more than in mixing.

Total harmonic distortion (THD)

Total harmonic distortion is a single number for how much a device adds. You play a sine in, add up the level of every harmonic that comes out, and give it as a percentage of the fundamental. 1% THD means the harmonics together are 40 dB below the note, and 0.1% means 60 dB below. Amplifier and interface specs quote it because for those, less is better.

On a saturator the harmonics are what you're paying for, so a high figure is expected. Turn the drive knob in the lab and the panel on the right tells you the total: tape at 6 dB of drive adds about 9% on the sine, the soft clipper at 12 dB adds about 28%, and a hard clipper at 24 dB adds about 45%. Two curves with the same THD can still sound very different, because the figure says how much and nothing about which harmonics. Spec sheets also often list THD+N, which counts noise along with the harmonics.

Harmonics on bass and small speakers

Phone and laptop speakers are too small to move much air at low frequencies, so a sub-bass note at 55 Hz can all but disappear on them. Your brain has a way around it. Play 110 Hz and 165 Hz together, the 2nd and 3rd harmonics of 55 Hz, and you'll hear a pitch of 55 Hz even if nothing at 55 Hz is playing. This is called the missing fundamental, and it's why a bass line with some saturation on it can be followed on a phone while a pure sine sub vanishes.

Step 6 puts the valve curve on the sine. On the spectrum, the original is a single peak at 110 Hz and the output has a row of new ones above it, and on laptop speakers the saturated tone is far easier to hear at the same loudness. Plug-ins such as Waves MaxxBass are built around this effect: they generate harmonics of the bass and let you blend them in, so you can turn the real low end down without the bass line going missing.

Distorting chords: intermodulation

Everything above holds for one note at a time. Put two notes through a curve together and it adds their harmonics plus new frequencies at the sums and differences between them, which is called intermodulation. Those extra frequencies are often not harmonics of either note.

A power chord avoids most of the trouble. Play A at 110 Hz with the E a fifth above at about 165 Hz and the difference is 55 Hz and the sum 275 Hz, and both are harmonics of 55 Hz, an A an octave down. Everything new fits one harmonic series, so the chord stays solid under heavy drive. Add the major third and the new frequencies land on notes outside the chord: A at 110 Hz and C♯ at 138.6 Hz give a sum of 248.6 Hz, close to a B, and a difference of 28.6 Hz, a rumble far below the bass. With three or four notes those products pile up, which is why a full chord through heavy distortion turns into mush. Step 7 runs the chord loop through a hard clipper so you can hear it.

When harmonics fold back: aliasing

A hard clipper's harmonics don't stop at the twelve bars on the screen. They carry on up, and the 200th harmonic of 110 Hz is 22 kHz. At a 44.1 kHz sample rate nothing above 22.05 kHz can be stored, so harmonics past that point fold back down as aliasing, at frequencies that have nothing to do with the note. That's the grey bar in the lab, which reads −57 dB at the step 5 settings.

Aliasing sounds like a faint, out-of-tune whistle under the note, and it's worst on high notes and hard curves, because their harmonics reach the top sooner. Most saturation plug-ins offer oversampling to deal with it, and the aliasing and oversampling lesson shows what it does.

Harmonics and EQ

An EQ can only turn up what's already there. Put a 3 kHz boost on a pure sine sub and you won't hear any change, because the sine has nothing at 3 kHz to boost. Saturation is how you get something there in the first place, and then EQ decides how much of it you keep.

So saturate a dull bass first and shape the new harmonics with EQ afterwards. A low-pass filter at 3 to 5 kHz takes off the fizz, and a bell around 700 Hz to 1 kHz can bring out the part that helps the bass cut through. The lab's Tone knob is a low-pass filter after the curve for the same reason, and the EQ lesson covers the filters themselves.

How to add harmonics to a sound

This is the usual way to make a bass audible on small speakers without changing how it sounds on big ones. It works on kicks, 808s and dull synths too.

1. Make a copy for the harmonicsDuplicate the bass track, or send it to a return track. The original stays clean and carries the low end.
2. Saturate the copyPut a tape or valve saturator on it and drive it until the harmonics are obvious when you solo it. Too much is fine at this stage.
3. Filter out the low end of the copyA high-pass filter around 100 to 150 Hz leaves only the harmonics, so the copy can't add more sub or fight the original's phase down there.
4. Trim the topA low-pass around 3 to 5 kHz takes off the fizz, which is usually the part that makes saturated bass sound cheap.
5. Blend it in on small speakersBring the copy up under the original while listening on a phone or laptop until you can follow the bass line, then check that it still sounds right on headphones.
6. Compare at matched loudnessMute the copy and turn the original up to match. If you can't hear what the copy added, it's too quiet or doing nothing useful.

A saturator with a mix knob does a simpler version of this in one plug-in. Step 8 runs heavy valve drive at 35% mix, and the 2nd harmonic drops from −12 dB to −23 dB under the clean sine. The parallel saturation lesson has more on blending.

Common mistakes with harmonics

Judging by loudness

Harmonics in the upper mids make a sound seem louder, and louder seems better for the first few seconds. Leave Auto gain on in the lab, and in your DAW turn a saturator's output down until bypassing it doesn't change the level.

Saturating the sub and the chords together

Drive a full mix or a pad into a curve and intermodulation fills the spaces between notes with frequencies that don't belong to the key. Saturate the bass and the chords on separate channels, or keep the drive low on anything with several notes at once.

Reaching for EQ to add brightness that isn't there

If a boost at 5 kHz does nothing but raise the hiss, the sound has no harmonics up there to bring out. Add a little saturation first and then boost.

Stacking small amounts everywhere

A saturator on every channel, then on the bus, then on the master, each adding a little, can add up to a harsh, tiring mix. Every stage adds harmonics to the harmonics the last one made. Bypass them all at matched loudness now and then and check that the mix really did get better.

Training your ears to hear harmonics

The quiz under the lab plays the current sound through the soft clipper, the valve or the hard clipper, at the same loudness as the original, with the screens covered. Start with the chords, where the differences are easiest to hear. The valve's strong 2nd harmonic makes notes sound fuller and closer, while the hard clipper's long row of odd harmonics sounds like buzz on the front of every note. Your best score is kept in your browser.

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