Additive synthesis
Encyclopedia
Additive synthesis is a technique of sound synthesis that creates music
al timbre
by explicitly adding sinusoidal overtones together.
The timbre of an instrument is composed of multiple harmonic
or inharmonic partials
(individual sine waves), of different frequencies and amplitudes, that change over time. Additive synthesis allows the emulation of a given tone or sound by giving control over the frequency and amplitude of each individual harmonic or partial. In general, each harmonic generator has its own customizable amplitude
envelope and instantaneous frequency function, creating a realistic, dynamic sound that changes over time.
. Fourier discovered that periodic function
s are formed by the summation of an infinite series of sinusoidal functions called "harmonic
s". This summation is called a 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. Additive synthesis models any periodic or quasi-periodic
tone or sound in the discrete time
domain and can be synthesized as follows:
or
where
The DC
term is normally not used in audio synthesis, so the a0[n] term can be removed. Introducing time varying coefficients rk[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 shape 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
. By replacing the k-th harmonic frequency, k f0, with a time-varying and general (not necessarily harmonic) frequency, fk[n], (the instantaneous frequency of the k-th partial at the time of sample n) the definition (removing the DC term) of the synthesized output would be:
or
where
If fk[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 φk[n] can be absorbed into the instantaneous frequency term, fk[n] by the substitution:
If that substitution is made, all of the φk[n] phase terms can be set to zero with no loss of generality (retaining the initial phase value at y[0]) and the expressions of non-harmonic additive synthesis can be simplified to
.
If this constant phase term (at time n = 0) is expressed as
the general expression of additive synthesis can be further simplified:
where
and
The Hammond organ
uses nine drawbars to mix several harmonics, which are generated by a set of tonewheel
s. It was invented in 1934 as a substitute for the much bulkier and expensive pipe organ.
One of the earliest realtime additive synthesizer may be the ANS synthesizer
developed by Evgeny Murzin
during 1937–1957. It had 720 microtonal sine waves (1/6 semitone
s × 10 octave
s) generated by five optical discs, and composers could control time evolution of these microtonal sine waves, by making scratches on black mastic
-covered glass plate user interface.
Bell Labs Digital Synthesizer (Alles Machine)
The Bell Labs Digital Synthesizer
(Alles Machine) developed in 1970s has been called the first true digital additive synthesizer, following on the earlier software experiments in Bell Labs. Based on this design, several commercial synthesizers were developed including Crumar
/DKI GDS in 1980, reduced version DK Synergy in 1982, and Atari AMY
sound chip, etc.
Others
In addition, RMI
Harmonic synthesizer, a very rare monophonic synthesizer utilizing static additive synthesis implementation, may be possibly notable as earliest commercial digital product: it had two digital harmonic oscillators with 16 harmonics, but each amplitudes of harmonics couldn't be changed in real time by envelope generators or sliders.
The Synclavier was a programmable harmonic definable FM/additive synthesizer and sampler. : one can construct a patch defining 16 partials per voice (as in the Casio FZ-1, Korg DSS-1
, and Kawai K3) and apply dynamic enveloping, and FM operator with envelope, . With Synclavier software upgrade one can specify several harmonic spectrums and crossfade between them in time.
Note that Synclavier's FM re-synthesis feature with fine grain time slice is sometimes considered as equivalent to Wavetable synthesis
, and Wavetable synthesis under some conditions is equivalent to real-time additive synthesis. (⇒ See section Wavetable synthesis)
Crumar/DKI General Development System (GDS)
DK Synergy:
The General Development System (GDS, released in 1980) and DK Synergy (released in 1982), a descendant of Alles Machine
, are user definable PM (as FM) semi algorithmic with additive capabilities, 32 digital oscillators synthesizers. This means that you could use it as a 16 partials two voice polyphony fully additive synth (with limited timbrical results) or the most usual way: complex 8 voice polyphony Yamaha FM style synthesis. One can think of it as a much more flexible algorithm, envelope (for frequency and amplitude for each oscillator) and filter equalization DX7 style synth. In fact, one can came very close to the original patches using a soft synth as FM7: you cannot do the best patches (as Wendy Carlos
's collection) on a DX7 because of the limited envelopes and operator output fixed curves, not to consider the somewhat "metallic" quality of sound that all the DXs have. In comparison, the Synergy is really warm.
Seiko Digital Sound System
In 1984, Seiko
introduced its "Digital Sound System" line of keyboard instruments, which utilized a 16-operator additive synthesis engine. Although the keyboards (the DS-101, DS-202, and DS-250) were not directly programmable, a separate programming device was available that allowed the user to create new presets.
Kurzweil K150
The Kurtzweil K150 (released in 1986) is an additive engine that trades off quantity of oscillators vs. polyphony and where one can program each partial individually with envelopes. Full programming (known as Fourier Synthesis option) is only possible using an old Apple II
computer, and cannot be done from the front panel.
Kawai K5
The Kawai K5 (released in 1987) does the same but is a simplification, being able to control only 4 groups of harmonics and not each one: practice shows that individual control is desirable up to the 16th partial... The much more modern Kawai K5000
is the classic additive synth, but combined with samples: it's considered quite powerful but clumsy to work compared with software synthesis.
series of synthesizers in the 1990s and, more recently, software synthesizer
s such as discoDSP Vertigo, Camel Audio Alchemy, Image-Line Morphine, the VirSyn Cube, White Noise Audio Soft WNAdditive, and the ConcreteFX softsynth Adder.
can be as general as additive synthesis but requires less real-time computation. In this case, 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.
for non periodic waves), an additive re-synthesis 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.
models
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
al timbre
Timbre
In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...
by explicitly adding sinusoidal overtones together.
The timbre of an instrument is composed of multiple harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
or inharmonic partials
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
(individual sine waves), of different frequencies and amplitudes, that change over time. Additive synthesis allows the emulation of a given tone or sound by giving control over the frequency and amplitude of each individual harmonic or partial. In general, each harmonic generator has its own customizable amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
envelope and instantaneous frequency function, 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 FourierJoseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...
. Fourier discovered that periodic function
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
s are formed by the summation of an infinite series of sinusoidal functions called "harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
s". This summation is called a Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
. 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. Additive synthesis models any periodic or quasi-periodic
Almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov,...
tone or sound in the discrete time
Discrete time
Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...
domain and can be synthesized as follows:
or
where
- y[n] is the output sample at discrete time n,
- ak[n] = rk[n] cos(φk[n])
- bk[n] = rk[n] sin(φk[n])
- rk[n] is the amplitudeAmplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
envelope of the k-th harmonic at discrete time n, - φk[n] is the instantaneous phaseInstantaneous phaseThe notions of Instantaneous Phase and Instantaneous Frequency are important concepts in Signal Processing that occur in the context of the representation and analysis of time-varying signals....
function of the k-th harmonic at discrete time n, - Fs is the sampling frequency,
- f0 is the fundamental frequencyFundamental frequencyThe fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
of the waveform or the note frequencyPiano key frequenciesThis is a virtual keyboard showing the absolute frequencies in hertz of the notes on a modern piano in twelve-tone equal temperament, with the 49th key, the fifth A , tuned to 440 Hz...
, - k f0 is the frequency of the k-th harmonic,
- K is the number of harmonics; K < floor(Fs/(2 f0)),
- K f0 is the frequency of the highest harmonic and is below the Nyquist frequencyNyquist frequencyThe Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...
, Fs/2.
The DC
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...
term is normally not used in audio synthesis, so the a0[n] term can be removed. Introducing time varying coefficients rk[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 shape over the longer term).
Additive synthesis can also create non-harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
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
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
. By replacing the k-th harmonic frequency, k f0, with a time-varying and general (not necessarily harmonic) frequency, fk[n], (the instantaneous frequency of the k-th partial at the time of sample n) the definition (removing the DC term) of the synthesized output would be:
or
where
-
.
If fk[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 φk[n] can be absorbed into the instantaneous frequency term, fk[n] by the substitution:
If that substitution is made, all of the φk[n] phase terms can be set to zero with no loss of generality (retaining the initial phase value at y[0]) and the expressions of non-harmonic additive synthesis can be simplified to
.If this constant phase term (at time n = 0) is expressed as
the general expression of additive synthesis can be further simplified:
where
and
Implementations
Pipe organs
Certain organ pipes, which create nearly sinusoidal/triangle waves (mostly flute pipes), can be combined in the manner of additive synthesis: it's the classical organ plenum practice. However, pipes which generate other types of wave forms (for example square wave generating clarinet stops) are not suited to this purpose: they are used more as solo stops.Electronic organs
The Hammond organ
Hammond organ
The Hammond organ is an electric organ invented by Laurens Hammond in 1934 and manufactured by the Hammond Organ Company. While the Hammond organ was originally sold to churches as a lower-cost alternative to the wind-driven pipe organ, in the 1960s and 1970s it became a standard keyboard...
uses nine drawbars to mix several harmonics, which are generated by a set of tonewheel
Tonewheel
A tonewheel is a simple electromechanical apparatus for generating electronic musical notes. The tonewheel assembly consists of a synchronous AC motor and an associated gearbox that drives a series of rotating disks...
s. It was invented in 1934 as a substitute for the much bulkier and expensive pipe organ.
Early experimental models
ANS synthesizerOne of the earliest realtime additive synthesizer may be the ANS synthesizer
ANS synthesizer
The ANS synthesizer is a photoelectronic musical instrument created by Russian engineer Evgeny Murzin from 1937 to 1957. The technological basis of his invention was the method of graphical sound recording used in cinematography , which made it possible to obtain a visible image of a sound wave, as...
developed by Evgeny Murzin
Evgeny Murzin
Yevgeny Murzin was a Russian audio engineer and inventor of the ANS synthesizer.-Murzin's synthesizer:In 1938, invented a design for composers based on synthesizing complex musical sounds from a limited number of pure tones; this proposed system was to perform music without musicians or musical...
during 1937–1957. It had 720 microtonal sine waves (1/6 semitone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
s × 10 octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
s) generated by five optical discs, and composers could control time evolution of these microtonal sine waves, by making scratches on black mastic
Mastic
Pistacia lentiscus is a dioecious evergreen shrub or small tree of the Pistacio genus growing up to tall which is cultivated for its aromatic resin, mainly on the Greek island of Chios.- Description :...
-covered glass plate user interface.
Bell Labs Digital Synthesizer (Alles Machine)
The Bell Labs Digital Synthesizer
Bell Labs Digital Synthesizer
The Bell Labs Digital Synthesizer, better known as the Alles Machine or Alice, was an experimental additive synthesizer designed by Hal Alles at Bell Labs during the 1970s. The Alles Machine used 72 computer controlled oscillators whose output was mixed to produce a number of discrete "voices" for...
(Alles Machine) developed in 1970s has been called the first true digital additive synthesizer, following on the earlier software experiments in Bell Labs. Based on this design, several commercial synthesizers were developed including Crumar
Crumar
Crumar is an Italian company which manufactured synthesizers and keyboards in the 60's, 70s and 80s. Its name is taken from that of its founder, Mario Crucianelli...
/DKI GDS in 1980, reduced version DK Synergy in 1982, and Atari AMY
Atari AMY
Atari's AMY was a 64-oscillator additive synthesizer implemented as a single-IC sound chip. For a time, AMY was slated to be included in newer versions of the Atari 8-bit family, but this did not occur before development of that line was discontinued...
sound chip, etc.
Others
In addition, RMI
Rocky Mount Instruments
Rocky Mount Instruments or RMI was a subsidiary of the Allen Organ Company, established in about 1966. It was based in Rocky Mount, North Carolina...
Harmonic synthesizer, a very rare monophonic synthesizer utilizing static additive synthesis implementation, may be possibly notable as earliest commercial digital product: it had two digital harmonic oscillators with 16 harmonics, but each amplitudes of harmonics couldn't be changed in real time by envelope generators or sliders.
Early products
New England Digital SynclavierThe Synclavier was a programmable harmonic definable FM/additive synthesizer and sampler. : one can construct a patch defining 16 partials per voice (as in the Casio FZ-1, Korg DSS-1
Korg DSS-1
The Korg DSS-1 was a 12-bit polyphonic sampling synthesizer released in September 1986. It came out at a time when many of the popular synthesizer companies were beginning to get into sampling, an area of sound design that had previously been left to a handful of fledgling companies such as...
, and Kawai K3) and apply dynamic enveloping, and FM operator with envelope, . With Synclavier software upgrade one can specify several harmonic spectrums and crossfade between them in time.
Note that Synclavier's FM re-synthesis feature with fine grain time slice is sometimes considered as equivalent to Wavetable synthesis
Wavetable synthesis
Wavetable synthesis is used in certain digital music synthesizers to implement a restricted form of real-time additive synthesis. The technique was first developed by Wolfgang Palm of PPG in the late 1970s and published in 1979, and has since been used as the primary synthesis method in...
, and Wavetable synthesis under some conditions is equivalent to real-time additive synthesis. (⇒ See section Wavetable synthesis)
Crumar/DKI General Development System (GDS)
DK Synergy:
The General Development System (GDS, released in 1980) and DK Synergy (released in 1982), a descendant of Alles Machine
Bell Labs Digital Synthesizer
The Bell Labs Digital Synthesizer, better known as the Alles Machine or Alice, was an experimental additive synthesizer designed by Hal Alles at Bell Labs during the 1970s. The Alles Machine used 72 computer controlled oscillators whose output was mixed to produce a number of discrete "voices" for...
, are user definable PM (as FM) semi algorithmic with additive capabilities, 32 digital oscillators synthesizers. This means that you could use it as a 16 partials two voice polyphony fully additive synth (with limited timbrical results) or the most usual way: complex 8 voice polyphony Yamaha FM style synthesis. One can think of it as a much more flexible algorithm, envelope (for frequency and amplitude for each oscillator) and filter equalization DX7 style synth. In fact, one can came very close to the original patches using a soft synth as FM7: you cannot do the best patches (as Wendy Carlos
Wendy Carlos
Wendy Carlos is an American composer and electronic musician. Carlos first came to notice in the late 1960s with recordings made on the Moog synthesizer, then a relatively new and unknown instrument; most notable were LPs of synthesized Bach and the soundtrack for Stanley Kubrick's film A...
's collection) on a DX7 because of the limited envelopes and operator output fixed curves, not to consider the somewhat "metallic" quality of sound that all the DXs have. In comparison, the Synergy is really warm.
Seiko Digital Sound System
In 1984, Seiko
Seiko
, more commonly known simply as Seiko , is a Japanese watch company.-History and ongoing developments:The company was founded in 1881, when Kintarō Hattori opened a watch and jewelry shop called in the Ginza area of Tokyo, Japan. Eleven years later, in 1892, he began to produce clocks under the...
introduced its "Digital Sound System" line of keyboard instruments, which utilized a 16-operator additive synthesis engine. Although the keyboards (the DS-101, DS-202, and DS-250) were not directly programmable, a separate programming device was available that allowed the user to create new presets.
Kurzweil K150
The Kurtzweil K150 (released in 1986) is an additive engine that trades off quantity of oscillators vs. polyphony and where one can program each partial individually with envelopes. Full programming (known as Fourier Synthesis option) is only possible using an old Apple II
Apple II
The Apple II is an 8-bit home computer, one of the first highly successful mass-produced microcomputer products, designed primarily by Steve Wozniak, manufactured by Apple Computer and introduced in 1977...
computer, and cannot be done from the front panel.
Kawai K5
The Kawai K5 (released in 1987) does the same but is a simplification, being able to control only 4 groups of harmonics and not each one: practice shows that individual control is desirable up to the 16th partial... The much more modern Kawai K5000
Kawai K5000
The Kawai K5000 is a series of digital synthesizers / music workstation manufactured by Kawai Musical Instruments of Japan.-History:The first of the series was the K5000W keyboard workstation introduced in 1996. Soon afterwards the K5000S was introduced, followed its rackmounted sibling the K5000R...
is the classic additive synth, but combined with samples: it's considered quite powerful but clumsy to work compared with software synthesis.
Contemporary models
More contemporary popular implementations of additive synthesis include the Kawai K5000Kawai K5000
The Kawai K5000 is a series of digital synthesizers / music workstation manufactured by Kawai Musical Instruments of Japan.-History:The first of the series was the K5000W keyboard workstation introduced in 1996. Soon afterwards the K5000S was introduced, followed its rackmounted sibling the K5000R...
series of synthesizers in the 1990s and, more recently, software synthesizer
Software synthesizer
A software synthesizer, also known as a softsynth is a computer program or plug-in for digital audio generation. Computer software which can create sounds or music is not new, but advances in processing speed are allowing softsynths to accomplish the same tasks that previously required dedicated...
s such as discoDSP Vertigo, Camel Audio Alchemy, Image-Line Morphine, the VirSyn Cube, White Noise Audio Soft WNAdditive, and the ConcreteFX softsynth Adder.
Wavetable synthesis
As shown by Bristow-Johnson, in the case of quasi-periodic musical tones, wavetable synthesisWavetable synthesis
Wavetable synthesis is used in certain digital music synthesizers to implement a restricted form of real-time additive synthesis. The technique was first developed by Wolfgang Palm of PPG in the late 1970s and published in 1979, and has since been used as the primary synthesis method in...
can be as general as additive synthesis but requires less real-time computation. In this case, an efficient implementation of additive synthesis can be accomplished with wavetable synthesis
Wavetable synthesis
Wavetable synthesis is used in certain digital music synthesizers to implement a restricted form of real-time additive synthesis. The technique was first developed by Wolfgang Palm of PPG in the late 1970s and published in 1979, and has since been used as the primary synthesis method in...
. 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 analyze the frequency components of a recorded sound and then re-synthesize a representation of the sound using additive techniques. By calculating the frequency and amplitude weighting of discrete partials in the frequency domain (typically used Short-time Fourier transformShort-time Fourier transform
The short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....
for non periodic waves), an additive re-synthesis 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.
See also
- Subtractive synthesisSubtractive synthesisSubtractive synthesis is a method of sound synthesis in which partials of an audio signal are attenuated by a filter to alter the timbre of the sound...
- Harmonic series (music)Harmonic series (music)Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
- Fourier seriesFourier seriesIn mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
models
- Hammond organ (1934)
- ANS synthesizerANS synthesizerThe ANS synthesizer is a photoelectronic musical instrument created by Russian engineer Evgeny Murzin from 1937 to 1957. The technological basis of his invention was the method of graphical sound recording used in cinematography , which made it possible to obtain a visible image of a sound wave, as...
(1937-1957) - Bell Labs Digital SynthesizerBell Labs Digital SynthesizerThe Bell Labs Digital Synthesizer, better known as the Alles Machine or Alice, was an experimental additive synthesizer designed by Hal Alles at Bell Labs during the 1970s. The Alles Machine used 72 computer controlled oscillators whose output was mixed to produce a number of discrete "voices" for...
(mid 1970s)