math lessons - Additive synthesis

Additive synthesis is a technique of audio synthesis which creates musical timbre.

Since different instruments' timbre is composed of varying amounts of harmonics that change over time, with respect to a base tone, additive synthesis emulates this behavior similarly by creating a different amplitude envelope on each harmonic, as well as adding non-harmonic artifacts aiming to result in a realistic timbre recreation.

Usually this involves a bank of oscillators tuned to multiples of the base frequency. Often, each oscillator has its own customizable volume envelope, creating a realistic, dynamic sound.

The concept behind additive synthesis may be recalled to discoveries by the French mathematician Jean Baptiste Joseph Fourier. Fourier discovered that discontinuous functions are formed by the summation of an infinite series. Following this, it was established that all signals, when represented as a mathematical function, can be composed as a sum of sine functions ( sin(x) ) of various frequencies. More rigorously, any periodic sound in the discrete time domain can be synthesized as follows:

$s(n) = \frac{1}{2} a_0 + \sum_{k=1}^{k_{\max}} a_k\cos\left(kn 2{\pi}\frac{F_{1}}{F_s}\right)-b_k\sin\left(kn 2{\pi}\frac{F_{1}}{F_s}\right)$

$s(n) = \frac{1}{2} a_0 + \sum_{k=1}^{k_{\max}} r_k\cos\left(kn 2{\pi}\frac{F_{1}}{F_s}+\varphi_k\right)$

where

$a_k = r_k\cos(\varphi_k) \,$ , $b_k = r_k\sin(\varphi_k) \,$

and $F_s \,$ is the sampling frequency and $k_{\max}<\operatorname{floor}(F_s/2F_1) \,$ is the highest harmonic and below the Nyquist frequency. The DC term is generally undesirable in audio synthesis, so the a₀ term can be removed. Introducing time varying coefficients r_k(n) allows for the dynamic use of envelopes to modulate oscillators creating a "quasi-periodic" waveform (one that is periodic over the short term but changes its waveform over the longer term). Additive synthesis can also create non-harmonic sounds if the individual partials are not all having a frequency that is an integer multiple of the same fundamental frequency.

A classic additive synthesizer was the Synclavier. The pipe organ may also qualify as an additive synthesizer because its pipes generate sine waves when blown and are added each other to generate tones. More contemporary popular implementations of additive synthesis include the Kawai K5000 series of synthesizers in the 1990s and, more recently, software synthesizers such as the Camel Audio Cameleon and the VirSyn Cube.

It has been shown in Wavetable Synthesis 101, A Fundamental Perspective, that wavetable synthesis is equivalent to additive synthesis in the case that all partials or overtones are harmonic (that is all overtones are at frequencies that are an integer multiple of a fundamental frequency of the tone as shown in the equation above). Not all musical sounds have harmonic partials, (e.g. bells) but many do. In these cases, an efficient implementation of additive synthesis can be accomplished with wavetable synthesis. Group additive synthesis is a method to group partials into harmonic groups and synthesize each group separately with wavetable synthesis before mixing the results.

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