Waveforms and Fourier Synthesis

The waveform of the oscillator affects its harmonic content and thereby its timbre. The three most common waveforms are sawtooth, pulse wave and triangle.

Looking at the shape of a waveform tells very little about how it sounds. Instead, there's a better way to show it, with a frequency spectrum. Let's introduce some quick theory: Mathematically, all waveforms can be considered as built from a number of harmonics, added together. Each of the harmonics consists of a sine wave, the purest and simplest waveform there is (a sine wave has no harmonics at all). In other words, if you add a number of sine waves together, each one with its own pitch (frequency) and volume (amplitude), then you can build any waveform you like.

The lowest harmonic is called the fundamental. The fundamental determines the basic pitch of the sound. If the fundamental has a frequency of 440Hz, we will perceive the entire sound as having a pitch of 440Hz.

Other harmonics are then added to the fundamental, called overtones. Normally the first overtone appears at a frequency twice the fundamental (in our example 880 Hz). The next harmonic appears at a frequency three times the fundamental (in our example 1320Hz) and so on. In a spectral display of a waveform you can see the frequency (pitch) of each harmonic and its amplitude (level). This is done by drawing each harmonic as a line raising up from a horizontal scale. Each line's position on this scale indicates the harmonic's frequency. The line furthest to the left is the fundamental, the next is the first harmonic etc. To make life easier, one usually doesn't label the horizontal scale with frequency in Hz, but rather with the number of the harmonic. The height of each line represents the amplitude of each harmonic.

If you understand the principle, you also understand that if the harmonics with high numbers have a high amplitude, the sound will be perceived as bright.

Sawtooth

The Sawtooth wave has a simple spectrum. All harmonics are present in the wave, in proportional values. As you can see, the high harmonics have a fairly high amplitude, which makes this waveform sound bright.

Triangle

The triangle wave does not have very strong harmonics. Furthermore they only appear at odd harmonic numbers. The first fact makes the tone pure, a bit like a flute, and the second fact gives the sound a slightly "hollow" character.

Pulse Wave

The pulse wave is slightly more complicated, because it is not one waveform, it is many different ones. A pulse wave is a waveform that during one period jumps once between full positive amplitude and full negative and then back.The thing that can be varied is where within the period you jump from maximum to minimum amplitude. This is the pulse width. A pulse width of 50% causes a special case of the pulse wave, called a square wave, and this has one peculiarity, it only contains odd number harmonics, which gives it a "hollow" quality. On many synthesizers the pulse width can be adjusted, to set the timbre of the pulse wave. The more narrow the pulse width, the "thinner" the sound will be. You can also have the pulse width vary continuously, for example from an LFO or envelope. This is referred to as pulse width modulation (PWM). Modulating pulse widths from an LFO creates a rich, chorus-like effect often used in "string" sounds.

Inharmonic Spectra

Above we have only discussed spectra where the overtones appear at perfect harmonics. While this is true for the basic waveforms discussed above, it is definitely not true for all sounds. If you for example use tFM or ring modulation with two oscillators set to an "unusual" interval (not octaves or fifths, for example), you will get a spectrum where the overtones appear at frequencies somewhere between the perfect harmonics. This results in an inharmonic sound, which often sounds "metallic".

Sync

On some instruments, two oscillators can be synchronized. If you for example synchronize Oscillator 2 to Oscillator 1, Oscillator 2 will start over with a new period of the waveform, each time Oscillator 1 does so. If Oscillator 2 then has a higher frequency than 1, it will get a complex waveform that depends both on its own pitch and on that of the other oscillator.

When sync is applied, the basic pitch of Oscillator 2 is locked to that of Oscillator 1. If you change the pitch of Oscillator 1 you will affect the basic pitch of both oscillators. Furthermore, when you vary the pitch of the synchronized oscillator (Oscillator 2), this will be perceived as a change in timbre, rather than in pitch. If you go even further and let the pitch of the synchronized oscillator vary continuously, for example from an LFO or envelope, you will change the harmonic content of the sound in an interesting and very characteristic way.

This site has a java applet which lets you see the frequencies in different waveforms, and play around with other harmonics to create new ones.

Basic synthesis tutorial for the Nord Modular.

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