Additive synthesis


Additive synthesis is a sound synthesis technique that creates timbre by adding sine waves together.
The timbre of musical instruments can be considered in the light of Fourier theory to consist of multiple harmonic or inharmonic partials or overtones. Each partial is a sine wave of different frequency and amplitude that swells and decays over time due to modulation from an ADSR envelope or low frequency oscillator.
Additive synthesis most directly generates sound by adding the output of multiple sine wave generators. Alternative implementations may use pre-computed wavetables or the inverse fast Fourier transform.

Explanation

The sounds that are heard in everyday life are not characterized by a single frequency. Instead, they consist of a sum of pure sine frequencies, each one at a different amplitude. When humans hear these frequencies simultaneously, we can recognize the sound. This is true for both "non-musical" sounds and for "musical sounds". This set of parameters are encapsulated by the timbre of the sound. Fourier analysis is the technique that is used to determine these exact timbre parameters from an overall sound signal; conversely, the resulting set of frequencies and amplitudes is called the Fourier series of the original sound signal.
In the case of a musical note, the lowest frequency of its timbre is designated as the sound's fundamental frequency. For simplicity, we often say that the note is playing at that fundamental frequency, even though the sound of that note consists of many other frequencies as well. The set of the remaining frequencies is called the overtones of the sound. In other words, the fundamental frequency alone is responsible for the pitch of the note, while the overtones define the timbre of the sound. The overtones of a piano playing middle C will be quite different from the overtones of a violin playing the same note; that's what allows us to differentiate the sounds of the two instruments. There are even subtle differences in timbre between different versions of the same instrument.
Additive synthesis aims to exploit this property of sound in order to construct timbre from the ground up. By adding together pure frequencies of varying frequencies and amplitudes, we can precisely define the timbre of the sound that we want to create.

Definitions

Harmonic additive synthesis is closely related to the concept of a Fourier series which is a way of expressing a periodic function as the sum of sinusoidal functions with frequencies equal to integer multiples of a common fundamental frequency. These sinusoids are called harmonics, overtones, or generally, partials. In general, a Fourier series contains an infinite number of sinusoidal components, with no upper limit to the frequency of the sinusoidal functions and includes a DC component. Frequencies outside of the human audible range can be omitted in additive synthesis. As a result, only a finite number of sinusoidal terms with frequencies that lie within the audible range are modeled in additive synthesis.
A waveform or function is said to be periodic if
for all and for some period.
The Fourier series of a periodic function is mathematically expressed as:
where
  • is the fundamental frequency of the waveform and is equal to the reciprocal of the period,
  • is the amplitude of the th harmonic,
  • is the phase offset of the th harmonic. atan2 is the four-quadrant arctangent function,
Being inaudible, the DC component,, and all components with frequencies higher than some finite limit,, are omitted in the following expressions of additive synthesis.

Harmonic form

The simplest harmonic additive synthesis can be mathematically expressed as:
where is the synthesis output,,, and are the amplitude, frequency, and the phase offset, respectively, of the th harmonic partial of a total of harmonic partials, and is the fundamental frequency of the waveform and the frequency of the musical note.

Time-dependent amplitudes

More generally, the amplitude of each harmonic can be prescribed as a function of time,, in which case the synthesis output is
Each envelope should vary slowly relative to the frequency spacing between adjacent sinusoids. The bandwidth of should be significantly less than.

Inharmonic form

Additive synthesis can also produce inharmonic sounds in which the individual overtones need not have frequencies that are integer multiples of some common fundamental frequency. While many conventional musical instruments have harmonic partials, some have inharmonic partials. Inharmonic additive synthesis can be described as
where is the constant frequency of th partial.
Example of inharmonic additive synthesis in which both the amplitude and frequency of each partial are time-dependent.
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Time-dependent frequencies

In the general case, the instantaneous frequency of a sinusoid is the derivative of the argument of the sine or cosine function. If this frequency is represented in hertz, rather than in angular frequency form, then this derivative is divided by. This is the case whether the partial is harmonic or inharmonic and whether its frequency is constant or time-varying.
In the most general form, the frequency of each non-harmonic partial is a non-negative function of time,, yielding

Broader definitions

Additive synthesis more broadly may mean sound synthesis techniques that sum simple elements to create more complex timbres, even when the elements are not sine waves. For example, F. Richard Moore listed additive synthesis as one of the "four basic categories" of sound synthesis alongside subtractive synthesis, nonlinear synthesis, and physical modeling. In this broad sense, pipe organs, which also have pipes producing non-sinusoidal waveforms, can be considered as a variant form of additive synthesizers. Summation of principal components and Walsh functions have also been classified as additive synthesis.

Implementation methods

Modern-day implementations of additive synthesis are mainly digital.

Oscillator bank synthesis

Additive synthesis can be implemented using a bank of sinusoidal oscillators, one for each partial.

Wavetable synthesis

In the case of harmonic, quasi-periodic musical tones, wavetable synthesis can be as general as time-varying additive synthesis, but requires less computation during synthesis. As a result, an efficient implementation of time-varying additive synthesis of harmonic tones can be accomplished by use of wavetable synthesis.

Group additive 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.

Inverse FFT synthesis

An inverse fast Fourier transform can be used to efficiently synthesize frequencies that evenly divide the transform period or "frame". By careful consideration of the DFT frequency-domain representation it is also possible to efficiently synthesize sinusoids of arbitrary frequencies using a series of overlapping frames and the inverse fast Fourier transform.

Additive analysis/resynthesis

It is possible to analyze the frequency components of a recorded sound giving a "sum of sinusoids" representation. This representation can be re-synthesized using additive synthesis. One method of decomposing a sound into time varying sinusoidal partials is short-time Fourier transform -based McAulay-Quatieri Analysis.
By modifying the sum of sinusoids representation, timbral alterations can be made prior to resynthesis. For example, a harmonic sound could be restructured to sound inharmonic, and vice versa. Sound hybridisation or "morphing" has been implemented by additive resynthesis.
Additive analysis/resynthesis has been employed in a number of techniques including Sinusoidal Modelling, Spectral Modelling Synthesis, and the Reassigned Bandwidth-Enhanced Additive Sound Model. Software that implements additive analysis/resynthesis includes: SPEAR, LEMUR, LORIS, SMSTools, ARSS.

Products

New England Digital Synclavier had a resynthesis feature where samples could be analyzed and converted into "timbre frames" which were part of its additive synthesis engine. Technos Acxel, launched in 1987, utilized the additive analysis/resynthesis model, in an FFT implementation.
Also a vocal synthesizer, Vocaloid have been implemented on the basis of additive analysis/resynthesis: its spectral voice model called Excitation plus Resonances model is extended based on Spectral Modeling Synthesis,
and its diphone concatenative synthesis is processed using
spectral peak processing technique similar to modified phase-locked vocoder. Using these techniques, spectral components consisting of purely harmonic partials can be appropriately transformed into desired form for sound modeling, and sequence of short samples constituting desired phrase, can be smoothly connected by interpolating matched partials and formant peaks, respectively, in the inserted transition region between different samples.

Applications

Musical instruments

Additive synthesis is used in electronic musical instruments. It is the principal sound generation technique used by Eminent organs.

Speech synthesis

In linguistics research, harmonic additive synthesis was used in the 1950s to play back modified and synthetic speech spectrograms.
Later, in the early 1980s, listening tests were carried out on synthetic speech stripped of acoustic cues to assess their significance. Time-varying formant frequencies and amplitudes derived by linear predictive coding were synthesized additively as pure tone whistles. This method is called sinewave synthesis. Also the composite sinusoidal modeling used on a singing speech synthesis feature on the Yamaha CX5M, is known to use a similar approach which was independently developed during 1966-1979. These methods are characterized by extraction and recomposition of a set of significant spectral peaks corresponding to the several resonance modes occurring in the oral cavity and nasal cavity, in a viewpoint of acoustics. This principle was also utilized on a physical modeling synthesis method, called modal synthesis.