Even vs odd harmonics

At a glance

What are even and odd harmonics?

Every pitched note contains harmonics, frequencies at whole-number multiples of its pitch, and the harmonics lesson covers where they come from. The counting starts at the note itself, the fundamental, which is the 1st harmonic. For an A at 110 Hz, the 2nd harmonic is 220 Hz, the 3rd is 330 Hz and so on up.

Even harmonics are the ones with an even number, starting with the 2nd at 220 Hz and the 4th at 440 Hz. Odd harmonics start with the 3rd at 330 Hz and carry on through the 5th at 550 Hz, the 7th and up. The note itself is number 1 and so odd by number, but when people talk about odd harmonics they mean the ones a process adds, from the 3rd up.

Watch the counting if you're reading about overtones instead. Overtones start counting above the note, so the 2nd harmonic is the 1st overtone and the 3rd harmonic is the 2nd overtone. A description of "even overtones" is talking about odd harmonics.

How even and odd harmonics sit against the note

Here are the harmonics of the A at 110 Hz that the lab's sine plays, with the note each one lands on and how far it is from the nearest piano key. A cent is a hundredth of a semitone, and a harmonic a few cents off a key sounds in tune.

Read down the even rows and a pattern shows up. Every even harmonic is exactly one octave above a lower harmonic: the 4th doubles the 2nd, the 6th doubles the 3rd, the 10th doubles the 5th. The first ones, the 2nd, 4th and 8th, are the note itself in higher octaves. The 6th and 10th are octaves of the 3rd and 5th, and a real instrument already has those, so on a bass or a voice the even harmonics mostly double what's there an octave up.

Every new note arrives on an odd harmonic. The 3rd brings in the fifth, the 5th a major third, the 7th a seventh that's 31 cents flat of any piano key, and the 11th lands almost exactly between two keys. That's the most solid reason behind the usual descriptions. Strong even harmonics leave a note sounding like itself, only bigger. Strong odd harmonics put other notes on top of it, and from the 7th up several of those notes are well out of tune with any key.

Symmetry decides which you get

A saturator is a curve. For every level that goes in, it sends a slightly different level out, and a curve that bends near the top squashes the peaks of the waveform. Whether it adds even harmonics comes down to one question: does it treat the positive and negative halves of the wave the same way?

A symmetrical curve bends the top and bottom of every cycle by the same amount, so the second half of each cycle comes out as the first half turned upside down. A wave with that symmetry can only contain odd harmonics, and the figure below shows why. A 3rd harmonic pushes the top of the note down and the bottom up, so it flattens both peaks equally and the symmetry survives. A 2nd harmonic goes through two whole cycles while the note does one, so it pushes the top and the bottom of the note the same way: one peak gets taller, the other gets shallower, and the two halves no longer match.

If you've met a little algebra, there's a neat way to see it. Squaring a sine gives a constant plus the 2nd harmonic, because sin²t = ½ − ½ cos 2t. Cubing one gives the sine back plus the 3rd harmonic, because sin³t = ¾ sin t − ¼ sin 3t, so the squared part of a curve makes even harmonics and the cubed part makes odd ones. A symmetrical curve, such as the tanh shape behind the lab's soft clipper, can be written with odd powers only, so it can't make a single even harmonic however hard you drive it.

That constant in sin² is a DC offset, which means the whole wave has been lifted off the centre line. Every lopsided curve makes some. Left in, it wastes headroom and can click wherever a clip starts or stops, so the lab runs a filter after its curves to take it out.

In the lab, the Character knob sets the lean. With it at zero, the soft clipper, hard clipper and tape curves are symmetrical and make only odd harmonics. The valve curve leans a little even at zero, which matches how a single valve stage behaves, and much further as you turn Character up.

What even and odd harmonics sound like

To hear the difference you need two settings that add the same amount of harmonics. In the lab, a soft clipper with 6 dB of drive adds the 3rd at −16 dB, the 5th at −30 and the 7th at −44, which comes to 15% of the note's level. The valve curve at the same drive with Character at 60% adds the 2nd at −16 dB, the 4th at −45 and the 3rd at only −53, about 16% in total. That's the same amount of the opposite kind.

On the bass, the even version sounds thicker and a little closer, and it takes an A/B to notice that anything was added at all. The odd version has more edge on the front of each note and a slightly hollow, reedy quality. Switch between steps 3 and 4 while it plays. Auto gain keeps both at the same loudness, so the tone is all that changes.

Acoustic instruments give you a reference for each. A clarinet in its low register is the textbook odd-heavy sound: its tube is closed at the reed end, which makes the even harmonics weak, and the result is the tone most people would call hollow. A square wave sounds hollow for the same reason, since it has only odd harmonics. A saw wave has every harmonic, both kinds at the same 1/n levels, and sounds bright and full by comparison. You can play all of them in the oscillator waveforms lesson.

Order and level matter more than odd or even

"Even is warm, odd is harsh" is half right. The 3rd harmonic is odd, and in small amounts it sounds thick and smooth. Tape adds mostly 3rd harmonic and still sounds warm at gentle levels. Harshness comes from high harmonics, the 7th, 9th, 11th and beyond, and those sound rough whichever row they sit in, because many of them are out of tune with the note and they crowd into the range where your ears are most sensitive. On a vocal note at 440 Hz, the 7th harmonic is at about 3 kHz.

The lab shows this two ways. In step 5, leaning a soft clipper at 12 dB of drive adds the 2nd, 4th and 6th at −22, −26 and −30 dB while the 3rd stays at −12, so asymmetry adds even harmonics on top of the odd ones instead of swapping one kind for the other. In steps 7 and 8, a hard clipper on the chords makes a long odd row, with the 3rd at −10 dB and the 9th still at −32. Adding even harmonics with the Character knob makes the chords thicker and leaves them just as buzzy, because the high odd harmonics haven't moved.

So pick a curve by how high its harmonics go first, and by odd or even second. A curve with a sharp corner, like a hard clipper, sends harmonics much further up the spectrum than a gentle one like tape or a valve.

Where even and odd harmonics come from in real gear

Valves

In a single-ended valve stage, one valve amplifies the whole wave, and its curve is lopsided: it runs out of room sooner on one side than the other. At moderate levels that adds mostly 2nd harmonic, which is one reason valves get called warm. Small single-ended guitar amps like the Fender Champ work this way. Most bigger guitar amps use a push-pull power stage instead, where two valves each handle one half of the wave. Their 2nd harmonics cancel out, so the output leans odd. The tube saturation lesson goes further into both.

Drive any valve stage hard and it clips on both sides, which lets the odd harmonics catch up. Step 6 shows it on the lab's valve curve: at 24 dB of drive the 3rd harmonic, at −11 dB, is louder than the 2nd at −18.

Tape and transformers

Magnetic tape saturates the same way in both directions, so it adds mostly odd harmonics, led by the 3rd. Tape's maximum output level is usually quoted as the level where the 3rd harmonic reaches 3%. Transformers behave similarly: their cores saturate symmetrically, adding mostly 3rd harmonic, and they saturate more easily at low frequencies than high ones, so a driven transformer thickens the bass first. The tape saturation lesson covers what tape does to the top end.

Clippers and pedals

A clipper that cuts the top and bottom at the same level is symmetrical and makes only odd harmonics. That includes a plain digital clipper and overdrive pedals with a matched pair of clipping diodes, such as the Ibanez Tube Screamer. Some pedal designs put two diodes on one side and one on the other so the clipping is lopsided on purpose, which adds even harmonics. In a plug-in, look for a control called asymmetry, bias or character; the lab in soft clipping vs hard clipping vs limiting has an Asymmetry knob that does it.

Even and odd harmonics on bass and small speakers

Phone and laptop speakers play very little below 150 to 200 Hz, so a 55 Hz bass note mostly vanishes on them. Harmonics bring it back, because your brain works out the pitch from the spacing between them. Hear 110, 165 and 220 Hz together and you'll hear a pitch of 55 Hz, even though nothing at 55 Hz is playing. This is called the missing fundamental.

This is where the kind of harmonic makes a difference. The even harmonics of a 55 Hz note sit at 110, 220 and 330 Hz, which is also the harmonic series of 110 Hz, so on a speaker that can't play the note itself you tend to hear them an octave up. Add the 3rd at 165 Hz and the set only fits 55 Hz. For bass that has to read on small speakers, a mix of 2nd and 3rd works better than either kind alone. The valve curve at 12 dB of drive with Character at 30% gives the 2nd at −16 dB and the 3rd at −24.

Even and odd harmonics on chords

With one note, everything a curve adds is a harmonic of that note. With two or more it also adds new frequencies made from combinations of them, which is called intermodulation. The squared part of a curve makes sums and differences: an A at 220 Hz and a C♯ at 277.2 Hz produce 57.2 Hz, a rumble below both notes, and 497.2 Hz. The cubed part makes, among others, frequencies at twice one note minus the other, here 162.8 Hz and 334.4 Hz, which land 20 to 25 cents off the E in two different octaves.

A symmetrical curve spares you the sums and differences and still makes the second set, right next to the notes you played, which is why heavy distortion and full chords fight whichever curve you use. Distort the bass and the chords on separate channels and keep the drive low on anything with several notes at once. The harmonics lesson explains how power chords get away with it.

How to add the harmonics you want

This works in any DAW with any saturation plug-in. It starts with finding out what the plug-in really does, since mode names like "tube" and "warm" don't tell you which harmonics you'll get.

1. Test it on a sinePut a sine test tone between 100 and 200 Hz through the saturator, with an analyser after it such as Ableton's Spectrum or Voxengo's free SPAN. A peak at twice the tone's frequency is the 2nd harmonic, and one at three times is the 3rd.
2. Find the control that changes the balanceSwitch between its modes and turn any asymmetry, bias or character control while you watch. Note which settings make the 2nd louder than the 3rd.
3. Set the drive on the test toneFor colour you barely notice, stop when the loudest new harmonic is somewhere between −30 and −20 dB. Much past −15 dB, most parts start to sound distorted.
4. Move it onto your trackSwap the test tone for the real part and match the output level to the bypassed sound before you judge it.
5. Trim the top if it fizzesA low-pass filter after the saturator, somewhere between 5 and 10 kHz, takes off the highest harmonics, which are usually the harsh part.
6. Blend it in if you want more without the edgeA mix control, or the parallel saturation method on a send, keeps the clean note on top and tucks the harmonics underneath.

Common mistakes

Treating "even is good, odd is bad" as a rule

Odd harmonics are what make a distorted guitar or a driven snare cut through a mix. Choose by the job, with even harmonics for weight and odd ones for bite, and remember that either kind sounds mild in small amounts.

Comparing at different loudness

Set two saturators to different output levels and the louder one wins. Leave Auto gain on in the lab, and in your DAW trim each plug-in's output until bypassing it changes the tone and leaves the level alone.

Expecting a valve plug-in to stay even at any drive

Push a valve curve hard and it clips both sides, so the 3rd harmonic overtakes the 2nd. For the even character, keep the drive moderate and use the mix control if you want more of it.

Forgetting polarity before a lopsided curve

An asymmetric curve treats the top and bottom of the wave differently, so flipping the polarity of the sound going into it changes which side gets squashed. Voices and many acoustic instruments have lopsided waveforms of their own, so once you've dialled in some lean, try the polarity switch before the saturator and keep whichever way sounds better.

Training your ears to tell them apart

The quiz under the lab plays the current sound through either the soft clipper or the valve, at the same loudness as the original, with the screens covered. The soft clipper adds only odd harmonics and the valve leads with the 2nd, so what you're being asked is which kind you can hear. Start with the bass or the chords. The even version sounds bigger and closer, as if the note had more body, and the odd version has a hollow edge on the front of each note. Your best score is kept in your browser.

Questions people ask

Keep going