Waveforms

Repeating a pattern gives you two things to control independently. How often the pattern repeats sets the pitch: repeat it 440 times a second and you hear the A above middle C, whatever the pattern looks like. The shape of the pattern sets the tone colour of the sound. The four shapes below are the ones almost every oscillator offers, and played at the same frequency they give the same note with four clearly different tones.

Those four shapes are built the same way. Any repeating pattern can be made by adding sine waves together: one at the repeat rate itself, which is called the fundamental, then quieter ones at two times that rate, three times, four times and onward. Each waveform has its own set of levels for those sines, and the curve on screen is their sum. A sine wave is the simplest case, with the fundamental alone and nothing stacked above it. For the rest, the sound depends on two things: which of the higher sines are present, and how quickly they get quieter going up.

What each waveform contains

The spectrum underneath plots the same sound against frequency rather than against time. Each bar is one harmonic: a sine wave at a whole-number multiple of the note you are playing, and the height of the bar is how loud that sine is in the mix. The two rows underneath give the harmonic number and the frequency it falls on, so at 220 Hz the fourth harmonic sits at 880. Move the frequency slider and the whole row of frequencies moves with it, because every harmonic is tied to the fundamental.

The sawtooth has a bar at every multiple, each one falling off as one over its number, so the second harmonic is half the height of the first, the third a third, and so on. Because every harmonic is present, the sawtooth sounds the brightest of the four. It is also why subtractive synthesis usually starts from a sawtooth, since a filter can only reduce harmonics that the oscillator produces. The square has bars only on the odd multiples, and the missing even harmonics give it a hollow tone. The low register of a clarinet sounds similar, because a cylindrical tube closed at one end also favours the odd harmonics, although the reason in the tube is acoustic rather than the symmetry of a waveform.

The triangle also contains only odd harmonics, but they fall off as one over the square of the number rather than one over the number, so the third harmonic is a ninth of the fundamental rather than a third. Its upper harmonics are quiet enough that a triangle sounds close to a sine. The sine shows a single bar, because a sine wave contains only its fundamental and has no harmonics above it.

Adding harmonics one at a time is additive synthesis, and it shows directly that a waveform is the sum of its harmonics. By about sixteen harmonics the drawing is recognisably a sawtooth, although every part of it was built from smooth sine waves. Switch to odd harmonics only and the same stack becomes a square wave. The harmonic series guide explains where those whole-number multiples come from and how they line up with musical intervals.

A subtractive synthesiser does not build sounds this way, because it would need a separate oscillator for every harmonic. It starts from a sawtooth, which already contains them all, and reduces the upper ones with a filter. The chapter on harmonics and overtones looks more closely at the harmonics themselves.