Accoustic theory, developed originally by Helmholz in the 19th century, holds that all periodic (repeating) waveforms, of whatever shape, are made up of a mix of sine waves of various frequencies. For most musical instruments and many natural sounds, the frequencies of these sine waves obey the harmonic series. A harmonic series consists of a sine wave at a particular frequency (referred to as the fundamental), and a series of higher tones which are referred to as harmonics. Each harmonic is a positive integer multiple of the frequency of the fundamental; the second harmonic is twice the fundamental’s frequency, the third harmonic is 3x the fundamental, and so on. Usually, the frequency of the fundamental determines the pitch of the note that the listener perceives, and the strength of the various harmonics determine the timbre and character of the note. (When a guitarist plays a “harmonic” at the 12th fret, the guitarist is effectively demonstrating this theory; the finger touched lightly to the string mutes the fundamental and some of the odd-numbered harmonics, changing the character of the normal string sound.) Understanding of the harmonic series, and its effect on the ear, is essential to determining why a particular sound has the tonal characteristics that it has. Some well-known waveform types can be precisely described mathematically in terms of harmonic series; for example, a square wave is made up of a fundamental and all odd-multiple harmonics mixed in a certain proportion.
Most synthesis techniques involved altering the harmonics of a waveform in some form. Subtractive synthesis, the method used by most analog synthesizers, starts with a certain waveform (such as a square wave) and uses filters to increase or decrease the strength of certain harmonics, altering the timbre. Additive synthesis takes the opposite approach; it starts with a sine-wave fundemental and then adds other sine waves at the harmonic frequencies, at various strengths, to build up timbre.