The Harmonic Series

Two terms first

Equal temperament is the tuning almost every keyboard, guitar and synth uses. It cuts the octave into twelve equal steps, and that evenness is what lets one instrument play in any key without being retuned. It is usually written 12-TET, for twelve-tone equal temperament, and it is the grid this page keeps comparing things to.

A cent is a hundredth of one of those steps, so an octave is 1200 cents and a semitone is 100. It is the unit for saying how far apart two pitches are, and it works at any register, where a gap counted in hertz does not. On a sustained tone most people begin to hear a difference somewhere around 5 to 10 cents.

Both matter here for one reason: the harmonic series does not line up with equal temperament, and cents are how the gap gets measured.

One sine at a time

Every row is one sine wave. Turn them on from the top down and the tone thickens without its pitch changing. The fundamental stays where it is and the harmonics above it change its colour.

Switch harmonic 1 off and leave the rest running. The pitch does not move. Your ear reconstructs the fundamental from the spacing of what is left, which is how a small speaker with no low end still sounds like it is playing a bass note. With the levels set to 1/n, sixteen sine waves add up to something close to a sawtooth, which is where a sawtooth oscillator's buzz comes from. Overtone is this table as a module, with eight harmonics and a switch on each.

Why a string does this

A string fixed at both ends can only hold a wave that has a node at each end. One half wavelength fits, and so do two, three, four and onward, but nothing in between. Each of those modes vibrates at its own frequency, and because each one fits a whole number of half wavelengths into the same length, those frequencies are whole-number multiples of the lowest. The series is a consequence of the geometry.

The dots are nodes, where the string stays still. The second mode has one in the middle, which is why touching a string at its midpoint and plucking it gives you the octave.

What one string actually does

Those six are not six strings. They are six of the things one string does at once, and the real motion is their sum. Pluck a string and you do not see a sawtooth travelling along it. You see a kink: the triangle you pulled into it, splitting in two and running back and forth between the ends.

So the answer to whether the string looks like a sawtooth is no, and the reason is worth following. What reaches your ear is not the shape of the string. It is the force the string drags the bridge with, and that force follows the slope of the string where it meets the bridge, not its displacement.

Taking the slope is what turns one into the other. The modes of a plucked string fall off as 1/n², which is a gentle enough roll-off to give you that rounded kink. Slope multiplies each mode by n, which leaves 1/n, and 1/n is the sawtooth you stacked up at the top of this page. The kink is on the string; the sawtooth is at the bridge. Drag the pluck toward the middle and watch harmonics drop out of the lower trace: a mode with a node where you plucked never gets started, which is why plucking a guitar over the 12th fret sounds hollow and plucking by the bridge sounds thin and bright.

A bowed string is the tidier case. The bow drags the string and lets it snap back once per cycle, so the string takes up two straight segments with a kink circling between them, and the force at the bridge is very nearly a true sawtooth. That is why bowed string models start from one.

Where they land on a keyboard

Equal temperament cuts the octave into twelve equal steps of 100 cents each. The harmonic series does not. Below, the vertical lines are the twelve-tone grid over four octaves and the marks are the harmonics. The octaves agree exactly. The fifths are about 2 cents out. The 7th, 11th and 13th are nowhere near a key on the keyboard.

Red marks are more than 25 cents from the nearest tempered note, amber more than 10.

Written out, the series is the shape every brass player knows, because a bugle plays it and nothing else. Notice that the steps start wide and close up as they go: an octave, then a fifth, then a fourth, and by the top of the staff the harmonics are a tone apart.

Harmonic numbers run along the foot. An arrow marks every harmonic more than 10 cents from the note it is written as, with the distance in cents: the 7th, 11th, 13th and 14th are the ones a keyboard cannot play. Writing them as ordinary notes is already a rounding.

True harmonics against equal temperament

Switch the tuning control in the tool above to hear this, or use the two buttons here. Both play harmonics 1 to 16. The first plays them where physics puts them. The second rounds each one to the nearest note on a keyboard.

Deviation from the nearest equal-tempered note, in cents. The track spans a quarter tone either way.

The tempered version sounds restless, and the waveform shows the same thing. True harmonics are whole-number multiples, so the sum repeats exactly at the fundamental's period and the trace on the display above holds still. Tempered ones are not, so the sum never quite repeats and the trace keeps crawling.

Equal temperament is a compromise, and a useful one. It lets a single fixed-pitch instrument play in every key, and the price is that every interval except the octave is slightly off. The harmonic series is not a tuning system at all. It is what a vibrating string does whether anyone tunes it or not.

Where this model stops

A real string is stiff, and stiffness makes its higher modes vibrate slightly faster than a whole-number multiple would. The effect is called inharmonicity, and it is why piano tuners stretch the tuning: the octaves of a real piano are tuned slightly wide to match the piano's own sharp harmonics rather than the arithmetic. It is strongest on short thick strings, which is why the bottom of a small upright is the worst case.

Struck solids depart much further. A bell, a drum head or a metal bar has modes set by its shape rather than by a length of string, and those modes are not whole-number multiples at all. That is why their components are usually called partials rather than harmonics, and why a bell can sound like it has no single pitch. The Kit drum model is built on exactly that: a struck membrane solved mode by mode, with ratios like 1, 1.59, 2.14, 2.30 rather than 1, 2, 3, 4.