Harmonics and overtones

The waveforms chapter built a sawtooth by adding sine waves one at a time. The sections below cover how the levels of the harmonics set the brightness of a sound, why their phases change the waveform but hardly change what you hear, and what happens when the partials move away from whole-number multiples.

The lowest partial is the fundamental, and it usually sets the pitch you hear. The second harmonic is at twice its frequency, the third at three times, and so on. Partial is the general word for any of these components, and overtone means any partial above the fundamental, counted from one, so the second harmonic is the first overtone. The harmonic series guide lists the first sixteen harmonics as musical intervals and lets you play them.

Brightness

The brightness of a sound depends mostly on how quickly the levels of its harmonics fall as the harmonic number rises. In a sawtooth the level of harmonic n is 1/n, so the tenth harmonic is a tenth as strong as the fundamental. In a triangle wave it is 1/n² and only the odd harmonics are present, so by the ninth harmonic the level is already below 1/80, which is why a triangle sounds close to a sine. The Brightness slider moves between those curves and beyond them in both directions. At 1/n with all harmonics the levels are those of a sawtooth. At 1/n² with odd harmonics only they are those of a triangle wave, although the waveform drawn is a different shape, because in a triangle wave every other harmonic is inverted. It sounds almost the same, for the reason given under Phase below.

A filter changes brightness in a similar way. A lowpass filter reduces the harmonics above its cutoff, which makes the levels fall faster towards the top of the spectrum. Subtractive synthesis relies on this: it starts from an oscillator with many strong harmonics and uses a filter to reduce the upper ones.

Phase

Every harmonic has a level and a phase, which is the point in its cycle where it starts. The waveform is the sum of all the harmonics, so moving their phases relative to each other changes the waveform's shape a great deal. A sawtooth with shuffled phases no longer looks like a sawtooth at all. It sounds almost the same, because for a steady tone the ear responds mainly to the level of each harmonic and only slightly to how the harmonics line up. The physicist Georg Ohm described this in 1843, and it is known as Ohm's acoustic law. The differences that remain are easiest to hear on low notes with many harmonics, as a change in buzziness rather than in brightness. For a steady tone, the spectrum is therefore often a more useful picture of a sound than the waveform: a sawtooth with shuffled phases has the same spectrum as an ordinary sawtooth and sounds almost the same, although the two waveforms look very different on the plot.

Inharmonic partials

Strings and air columns vibrate with partials at, or very close to, whole-number multiples of their lowest frequency, which is why strings, wind instruments and voices have a clear pitch. Bells, metal bars, cymbals and drum heads vibrate in more complicated patterns, and their partials fall at frequencies that are not whole-number multiples. These are called inharmonic partials. Because they do not line up as a harmonic series, they do not fuse into a single pitch as clearly, and the sound has the metallic quality those instruments share.

Piano strings are close to harmonic but not exactly. A real string is slightly stiff, and the stiffness raises each partial a little above its harmonic position, by more for higher partials. Piano tuners stretch the tuning of the top and bottom octaves to match. In a synthesiser, inharmonic partials come from FM at ratios that are not whole numbers, from ring modulation, and from physical models of bars and membranes.