Additive synthesis is a technique of audio synthesis which creates musical timbre.
The timbre of an instrument is composed of multiple harmonic or nonharmonic partials (individual sine waves), of different frequencies and amplitudes, that change over time. Additive synthesis allows the emulation of sounds by giving control over the frequency and amplitude of each individual harmonic or partial. Often, each harmonic generator has its own customizable volume envelope, creating a realistic, dynamic sound that changes over time.
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Theory
The concept behind additive synthesis is directly related to work done by the French mathematician Joseph Fourier. Fourier discovered that periodic functions are formed by the summation of an infinite series, called Fourier series. Following this, it was established that all periodic signals, when represented as a mathematical function, can be composed as a sum of sinusoidal functions ( sin(x), cos(x) ) of various frequencies. More rigorously, any periodic sound in the discrete time domain can be synthesized as follows:
or
where
The DC term is generally undesirable in audio synthesis, so the a_{0} term can be removed. Introducing time varying coefficients r_{k}[n] allows for the dynamic use of envelopes to modulate oscillators creating a "quasiperiodic" waveform (one that is periodic over the short term but changes its waveform shape over the longer term).
Additive synthesis can also create nonharmonic sounds (which have nonperiodic waveforms) if the individual harmonics do not all have a frequency that is an integer multiple of the fundamental frequency. By replacing the k^{th} harmonic frequency, k f_{0}, with timevarying and general (not necessarily harmonic) frequencies, f_{k}[n], (the instantaneous frequency of the k^{th} partial at the time of sample n) the definition of the synthesized output would be, (also eliminating the DC term):
or
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