Additive synthesis

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

The timbre of an instrument is composed of multiple harmonics or "partials", in different quantities, that change over time. Additive synthesis emulates such timbres by combining numerous waveforms pitched to different harmonics, with a different amplitude envelope on each, along with inharmonic artifacts. 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 that changes over time.

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. Following this, it was established that all periodic 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:

:$s\; [n]\; =\; frac\{1\}\{2\}\; a\_0\; [n]\; +\; sum\_\{k=1\}^\{k\_\{max\; a\_k\; [n]\; cosleft(\; frac\{2\; pi\; f\_0\}\{F\_mathrm\{s\; k\; n\; ight)-b\_k\; [n]\; sinleft(\; frac\{2\; pi\; f\_0\}\{F\_mathrm\{s\; k\; n\; ight)$

or

:$s\; [n]\; =\; frac\{1\}\{2\}\; a\_0\; [n]\; +\; sum\_\{k=1\}^\{k\_\{max\; r\_k\; [n]\; cosleft(\; frac\{2\; pi\; f\_0\}\{F\_mathrm\{s\; k\; n\; +\; varphi\_k\; [n]\; ight)$

where

:$a\_k\; [n]\; =\; r\_k\; [n]\; cos\; left(\; varphi\_k\; [n]\; ight)\; quad\; b\_k\; [n]\; =\; r\_k\; [n]\; sin\; left(\; varphi\_k\; [n]\; ight)\; ,$

and "F"_s is the sampling frequency, "f"₀ is the fundamental frequency, and "k"_max < floor("F"_s/(2 "f"₀)) is the highest harmonic and below the Nyquist frequency. 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 "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 (which have non-periodic waveforms) if the individual partials are not all having a frequency that is an integer multiple of the same fundamental frequency. With time-varying and general (not necessarily harmonic) frequencies of "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:

:$s\; [n]\; =\; frac\{1\}\{2\}\; a\_0\; [n]\; +\; sum\_\{k=1\}^\{k\_\{max\; a\_k\; [n]\; cosleft(\; frac\{2\; pi\}\{F\_mathrm\{s\; sum\_\{i=1\}^\{n\}f\_k\; [i]\; ight)\; -\; b\_k\; [n]\; sinleft(\; frac\{2\; pi\}\{F\_mathrm\{s\; sum\_\{i=1\}^\{n\}f\_k\; [i]\; ight)$

or

:$s\; [n]\; =\; frac\{1\}\{2\}\; a\_0\; [n]\; +\; sum\_\{k=1\}^\{k\_\{max\; r\_k\; [n]\; cosleft(\; frac\{2\; pi\}\{F\_mathrm\{ssum\_\{i=1\}^\{n\}f\_k\; [i]\; +\; varphi\_k\; [n]\; ight)$

where

:$a\_k\; [n]\; =\; r\_k\; [n]\; cos\; left(\; varphi\_k\; [n]\; ight)\; quad\; b\_k\; [n]\; =\; r\_k\; [n]\; sin\; left(\; varphi\_k\; [n]\; ight)\; ,$.

If "f_k" ["n"] = "k f"₀, with constant "f"₀, all partials are harmonic, the synthesized waveform is quasi-periodic, and the more general equations above reduce to the simpler equations at the top. For each non-harmonic partial, the phase term &phi;_k ["n"] can be absorbed into the instantaneous frequency term, "f_k" ["n"] by the substitution:

:$f\_k\; [n]\; leftarrow\; f\_k\; [n]\; +\; frac\{F\_mathrm\{s\{2\; pi\}left(\; varphi\_k\; [n]\; -varphi\_k\; [n-1]\; ight)$

If that substitution is made, all of the &phi;_k ["n"] terms can be set to zero with no loss of generality (retaining the initial phase value at "s" [0] ) and the expressions of non-harmonic additive synthesis can be simplified (with the additional elimination of the DC term) to

:$s\; [n]\; =\; sum\_\{k=1\}^\{k\_\{max\; r\_k\; [n]\; cosleft(\; frac\{2\; pi\}\{F\_mathrm\{ssum\_\{i=1\}^\{n\}f\_k\; [i]\; +\; varphi\_k\; [0]\; ight)$ .

If this constant phase term (at time "n"=0) is expressed as

:$varphi\_k\; [0]\; =\; frac\{2\; pi\}\{F\_mathrm\{ssum\_\{i=-infty\}^\{0\}f\_k\; [i]$ ,

the general expression of additive synthesis can be further simplified:

:$s\; [n]\; =\; sum\_\{k=1\}^\{k\_\{max\; r\_k\; [n]\; cosleft(\; frac\{2\; pi\}\{F\_mathrm\{ssum\_\{i=-infty\}^\{n\}f\_k\; [i]\; ight)$

Additive synthesizers

A classic additive synthesizer was the Synclavier. Certain organ pipes, which create sinusoidal waves (mostly flute pipes) can be combined in the manner of additive synthesis. However, pipes, which generate other types of wave forms (for example square wave generating clarinet stops)are not suited to this purpose. 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 [http://www.camelaudio.com Camel Audio] Cameleon, [http://www.image-line.com/documents/morphine.html Image-Line] Morphine, the VirSyn Cube, White Noise WNAdditive, and the ConcreteFX softsynth Adder. Another instrument with this capability is the Hammond organ, which uses nine drawbars to control the harmonics. The Hammond was invented in 1935 as a substitute for the much bulkier and expensive pipe organ.

It has been shown in [http://www.musicdsp.org/files/Wavetable-101.pdf 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 (of differing fundamental frequencies) and synthesize each group separately with wavetable synthesis before mixing the results.

Additive resynthesis

As demonstrated by software such as [http://www.klingbeil.com/spear/ SPEAR] , it is possible to analyse the frequency components of a recorded sound and then resynthesize a representation of the sound using additive techniques. By calculating the frequency and amplitude weighting of discrete partials in the frequency domain (typically using a fast Fourier transform), an additive resynthesis system can construct an equally weighted sinusoid at the same frequency for each partial.

Because the sound is represented by a bank of oscillators inside the system, a user can make adjustments to the frequency and amplitude of any set of partials. The sound can be 'reshaped' - by alterations made to timbre or the overall amplitude envelope, for example. A harmonic sound could be restructured to sound inharmonic, and vice versa.

Links

* [http://users.ece.gatech.edu/lanterma/synergy/ Digital Keyboards Synergy]
* Harmonic series (music)
* Fourier series