Additive synthesis: Definition from Answers.com

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

The timbre of an instrument is composed of multiple harmonics (sine waves), at 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 harmonic. Often, each harmonic generator has its own customizable volume envelope, creating a realistic, dynamic sound that changes over time.

1 Theory
2 Additive synthesizers
3 Additive resynthesis
4 See also
5 External links

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:

$s[n] = \frac{1}{2} a_0[n] + \sum_{k=1}^{k_{\max}} a_k[n] \cos\left( \frac{2 \pi f_0}{F_\mathrm{s}} k n \right)-b_k[n] \sin\left( \frac{2 \pi f_0}{F_\mathrm{s}} k n \right)$

$s[n] = \frac{1}{2} a_0[n] + \sum_{k=1}^{k_{\max}} r_k[n] \cos\left( \frac{2 \pi f_0}{F_\mathrm{s}} k n + \varphi_k[n] \right)$

where

$a_k[n] = r_k[n] \cos \left( \varphi_k[n] \right) \quad b_k[n] = r_k[n] \sin \left( \varphi_k[n] \right) \,$

and F_s is the sampling frequency, f₀ is the fundamental frequency, and k_max < floor(F_s/(2 f₀)) 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 (which have non-periodic waveforms) if the individual harmonics do not all have a frequency that is an integer multiple of the 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:

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

$s[n] = \frac{1}{2} a_0[n] + \sum_{k=1}^{k_{\max}} r_k[n] \cos\left( \frac{2 \pi}{F_\mathrm{s}}\sum_{i=1}^{n}f_k[i] + \varphi_k[n] \right)$

where

$a_k[n] = r_k[n] \cos \left( \varphi_k[n] \right) \quad b_k[n] = r_k[n] \sin \left( \varphi_k[n] \right) \,$ .

If f_k[n] = k f₀, with constant f₀, 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 φ_k[n] can be absorbed into the instantaneous frequency term, f_k[n] by the substitution:

$f_k[n] \leftarrow f_k[n] + \frac{F_\mathrm{s}}{2 \pi}\left( \varphi_k[n]-\varphi_k[n-1] \right)$

If that substitution is made, all of the φ_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

$s[n] = \sum_{k=1}^{k_{\max}} r_k[n] \cos\left( \frac{2 \pi}{F_\mathrm{s}}\sum_{i=1}^{n}f_k[i] + \varphi_k[0] \right)$ .

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

$\varphi_k[0] = \frac{2 \pi}{F_\mathrm{s}}\sum_{i=-\infty}^{0}f_k[i]$ ,

the general expression of additive synthesis can be further simplified:

$s[n] = \sum_{k=1}^{k_{\max}} r_k[n] \cos\left( \frac{2 \pi}{F_\mathrm{s}}\sum_{i=-\infty}^{n}f_k[i] \right)$

Additive synthesizers

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 Camel Audio Alchemy, Image-Line Morphine, the VirSyn Cube, White Noise Audio Soft 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 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 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.

External links

Digital Keyboards Synergy

v • d • e Sound synthesis types

Frequency modulation synthesis \| Phase distortion synthesis \| Scanned synthesis \| Subtractive synthesis \| Additive synthesis \| HCM synthesis \| Distortion synthesis \| Linear Arithmetic synthesis

Sample-based synthesis	Wavetable synthesis \| Granular synthesis \| Vector synthesis

Physical modelling synthesis	Digital waveguide synthesis \| Karplus-Strong string synthesis \| Formant synthesis

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Additive synthesis

Contents

Theory

Additive synthesizers

Additive resynthesis

See also

External links