HOME

TheInfoList



OR:

Additive synthesis is a
sound synthesis A synthesizer (also synthesiser or synth) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis an ...
technique that creates
timbre In music, timbre (), also known as tone color or tone quality (from psychoacoustics), is the perceived sound of a musical note, sound or tone. Timbre distinguishes sounds according to their source, such as choir voices and musical instrument ...
by adding
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
waves together. The timbre of musical instruments can be considered in the light of Fourier theory to consist of multiple
harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
or inharmonic '' partials'' or
overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
s. Each partial is a sine wave of different
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
and
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
that swells and decays over time due to
modulation Signal modulation is the process of varying one or more properties of a periodic waveform in electronics and telecommunication for the purpose of transmitting information. The process encodes information in form of the modulation or message ...
from an
ADSR envelope In sound and music, an envelope describes how a sound changes over time. For example, a piano key, when struck and held, creates a near-immediate initial sound which gradually decreases in volume to zero. An envelope may relate to elements such ...
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 A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
.


Explanation

The sounds that are heard in everyday life are not characterized by a single
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
. Instead, they consist of a sum of pure sine frequencies, each one at a different
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
. When humans hear these frequencies simultaneously, we can recognize the sound. This is true for both "non-musical" sounds (e.g. water splashing, leaves rustling, etc.) and for "musical sounds" (e.g. a piano note, a bird's tweet, etc.). This set of parameters (frequencies, their relative amplitudes, and how the relative amplitudes change over time) are encapsulated by the ''
timbre In music, timbre (), also known as tone color or tone quality (from psychoacoustics), is the perceived sound of a musical note, sound or tone. Timbre distinguishes sounds according to their source, such as choir voices and musical instrument ...
'' of the sound.
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
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 A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
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 The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...
. For simplicity, we often say that the note is playing at that fundamental frequency (e.g. "
middle C C or Do is the first note of the C major scale, the third note of the A minor scale (the relative minor of C major), and the fourth note (G, A, B, C) of the Guidonian hand, commonly pitched around 261.63  Hz. The actual frequency has d ...
is 261.6 Hz"), even though the sound of that note consists of many other frequencies as well. The set of the remaining frequencies is called the
overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
s (or the
harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
s, if their frequencies are integer multiples of the fundamental frequency) 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 (for example, an
upright piano A piano is a keyboard instrument that produces sound when its keys are depressed, activating an action mechanism where hammers strike strings. Modern pianos have a row of 88 black and white keys, tuned to a chromatic scale in equal temper ...
vs. a
grand piano A piano is a keyboard instrument that produces sound when its keys are depressed, activating an Action (music), action mechanism where hammers strike String (music), strings. Modern pianos have a row of 88 black and white keys, tuned to a c ...
). Additive synthesis aims to exploit this property of sound in order to construct timbre from the ground up. By adding together pure frequencies (
sine wave A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic function, periodic wave whose waveform (shape) is the trigonometric function, trigonometric sine, sine function. In mechanics, as a linear motion over time, this is ''simple ...
s) 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 A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
which is a way of expressing a
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
as the sum of
sinusoidal A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is '' simple harmonic motion''; as rotation, it correspond ...
functions with
frequencies Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
equal to integer multiples of a common
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...
. These sinusoids are called
harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
s,
overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
s, 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 (one with frequency of 0 Hz). 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 : y(t) = y(t+P) for all t and for some period P . The
Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
of a periodic function is mathematically expressed as: : \begin y(t) &= \frac + \sum_^ \left a_k \cos(2 \pi k f_0 t ) - b_k \sin(2 \pi k f_0 t ) \right\\ &= \frac + \sum_^ r_k \cos\left(2 \pi k f_0 t + \phi_k \right) \\ \end where * f_0 = 1/P is the
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...
of the waveform and is equal to the reciprocal of the period, * a_k = r_k \cos(\phi_k) = 2 f_0 \int_^P y(t) \cos(2 \pi k f_0 t)\, dt, \quad k \ge 0 * b_k = r_k \sin(\phi_k) = -2 f_0 \int_^P y(t) \sin(2 \pi k f_0 t)\, dt, \quad k \ge 1 * r_k = \sqrt is the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of the kth harmonic, * \phi_k = \operatorname(b_k, a_k) is the
phase offset In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a s ...
of the kth harmonic.
atan2 In computing and mathematics, the function (mathematics), function atan2 is the 2-Argument of a function, argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi 0, \\ mu \arctan\left(\fr ...
is the four-quadrant
arctangent In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specific ...
function, Being inaudible, the DC component, a_0/2, and all components with frequencies higher than some finite limit, K f_0, are omitted in the following expressions of additive synthesis.


Harmonic form

The simplest harmonic additive synthesis can be mathematically expressed as: where y(t) is the synthesis output, r_k, k f_0, and \phi_k are the amplitude, frequency, and the phase offset, respectively, of the kth harmonic partial of a total of K harmonic partials, and f_0 is the
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...
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, r_k(t), in which case the synthesis output is Each
envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card. Traditional envelopes are made from sheets of paper cut to one o ...
r_k(t)\, should vary slowly relative to the frequency spacing between adjacent sinusoids. The
bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
of r_k(t) should be significantly less than f_0.


Inharmonic form

Additive synthesis can also produce
inharmonic In music, inharmonicity is the degree to which the frequencies of overtones (also known as partials or partial tones) depart from whole multiples of the fundamental frequency ( harmonic series). Acoustically, a note perceived to have a sin ...
sounds (which are
aperiodic A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the tr ...
waveforms) in which the individual overtones need not have frequencies that are integer multiples of some common fundamental frequency.
online reprint
While many conventional musical instruments have harmonic partials (e.g. an
oboe The oboe ( ) is a type of double-reed woodwind instrument. Oboes are usually made of wood, but may also be made of synthetic materials, such as plastic, resin, or hybrid composites. The most common type of oboe, the soprano oboe pitched in C, ...
), some have inharmonic partials (e.g. bells). Inharmonic additive synthesis can be described as : y(t) = \sum_^ r_k(t) \cos\left(2 \pi f_k t + \phi_k \right), where f_k is the constant frequency of kth partial.


Time-dependent frequencies

In the general case, the
instantaneous frequency Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''compl ...
of a sinusoid is the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
(with respect to time) of the argument of the sine or cosine function. If this frequency is represented in
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in ter ...
, rather than in
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
form, then this derivative is divided by 2 \pi. 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, f_k(t), 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 Subtractive synthesis is a method of sound synthesis in which Harmonic_series_(music)#Partial.2C_harmonic.2C_fundamental.2C_inharmonicity.2C_and_overtone, overtones of an audio signal are attenuated by a audio filter, filter to alter the timbre of ...
, nonlinear synthesis, and physical modeling. In this broad sense,
pipe organ The pipe organ is a musical instrument that produces sound by driving pressurised air (called ''wind'') through the organ pipes selected from a Musical keyboard, keyboard. Because each pipe produces a single tone and pitch, the pipes are provide ...
s, 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. (See section '' Discrete-time equations'' for the underlying discrete-time theory)


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 Wavetable synthesis is a sound synthesis technique used to create quasi-periodic waveforms often used in the production of musical tones or notes. Development Wavetable synthesis was invented by Max Mathews in 1958 as part of MUSIC II. MU ...
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 (having different fundamental frequencies) and synthesize each group separately with ''wavetable synthesis'' before mixing the results.


Inverse FFT synthesis

An inverse
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
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 A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...
.


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 (STFT)-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 (SMS), 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 The Synclavier is an early digital synthesizer, polyphonic digital sampling system, and music workstation manufactured by New England Digital Corporation of Norwich, Vermont. It was produced in various forms from the late 1970s into the ea ...
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 is a singing Speech synthesis, voice synthesizer software product. Its signal processing part was developed through a joint research project between Yamaha Corporation and the Music Technology Group at Pompeu Fabra University, Barcelona. The s ...
have been implemented on the basis of additive analysis/resynthesis: its spectral voice model called Excitation plus Resonances (EpR) model
PDF
is extended based on Spectral Modeling Synthesis (SMS), and its
diphone In phonetics, a diphone is an adjacent pair of phones in an utterance. For example, in aɪfəʊn the diphones are a ɪ �f ə �ʊ �n The term is usually used to refer to a recording of the transition between two phones. In the following ...
concatenative synthesis is processed using ''spectral peak processing'' (SPP) technique similar to modified phase-locked vocoder (an improved phase vocoder for formant processing). Using these techniques, spectral components (''
formant In speech science and phonetics, a formant is the broad spectral maximum that results from an acoustic resonance of the human vocal tract. In acoustics, a formant is usually defined as a broad peak, or local maximum, in the spectrum. For harmo ...
s'') consisting of purely harmonic partials can be appropriately transformed into desired form for sound modeling, and sequence of short samples (''diphones'' or ''
phoneme A phoneme () is any set of similar Phone (phonetics), speech sounds that are perceptually regarded by the speakers of a language as a single basic sound—a smallest possible Phonetics, phonetic unit—that helps distinguish one word fr ...
s'') constituting desired phrase, can be smoothly connected by interpolating matched partials and formant peaks, respectively, in the inserted transition region between different samples. (See also Dynamic timbres)


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 Linguistics is the scientific study of language. The areas of linguistic analysis are syntax (rules governing the structure of sentences), semantics (meaning), Morphology (linguistics), morphology (structure of words), phonetics (speech sounds ...
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 In speech science and phonetics, a formant is the broad spectral maximum that results from an acoustic resonance of the human vocal tract. In acoustics, a formant is usually defined as a broad peak, or local maximum, in the spectrum. For harmo ...
frequencies and amplitudes derived by
linear predictive coding Linear predictive coding (LPC) is a method used mostly in audio signal processing and speech processing for representing the spectral envelope of a digital signal of speech in compressed form, using the information of a linear predictive model ...
were synthesized additively as pure tone whistles. This method is called sinewave synthesis. Also the composite sinusoidal modeling (CSM) used on a singing
speech synthesis Speech synthesis is the artificial production of human speech. A computer system used for this purpose is called a speech synthesizer, and can be implemented in software or hardware products. A text-to-speech (TTS) system converts normal langua ...
feature on the
Yamaha CX5M Yamaha CX5M is an MSX-system compatible computer that expands upon the normal features expected from these systems with a built-in eight-voice FM synthesizer module, introduced in 1984 by Yamaha Corporation. This FM synth itself has stereo audi ...
(1984), 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 Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
. This principle was also utilized on a
physical modeling synthesis Physical modelling synthesis refers to sound synthesis methods in which the waveform of the sound to be generated is computed using a mathematical model, a set of equations and algorithms to simulate a physical source of sound, usually a musical i ...
method, called modal synthesis.  (See als
companion page


History

Harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
was discovered by
Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre, Burgundy and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analys ...
, who published an extensive treatise of his research in the context of
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
in 1822. The theory found an early application in prediction of tides. Around 1876, William Thomson (later ennobled as
Lord Kelvin William Thomson, 1st Baron Kelvin (26 June 182417 December 1907), was a British mathematician, Mathematical physics, mathematical physicist and engineer. Born in Belfast, he was the Professor of Natural Philosophy (Glasgow), professor of Natur ...
) constructed a mechanical tide predictor. It consisted of a ''harmonic analyzer'' and a ''harmonic synthesizer'', as they were called already in the 19th century. The analysis of tide measurements was done using James Thomson's '' integrating machine''. The resulting
Fourier coefficient A Fourier series () is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the fun ...
s were input into the synthesizer, which then used a system of cords and pulleys to generate and sum harmonic sinusoidal partials for prediction of future tides. In 1910, a similar machine was built for the analysis of periodic waveforms of sound. The synthesizer drew a graph of the combination waveform, which was used chiefly for visual validation of the analysis.
Georg Ohm Georg Simon Ohm (; ; 16 March 1789 – 6 July 1854) was a German mathematician and physicist. As a school teacher, Ohm began his research with the new electrochemical cell, invented by Italian scientist Alessandro Volta. Using equipment of his o ...
applied Fourier's theory to sound in 1843. The line of work was greatly advanced by
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (; ; 31 August 1821 – 8 September 1894; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...
, who published his eight years worth of research in 1863. Helmholtz believed that the psychological perception of tone color is subject to learning, while hearing in the sensory sense is purely physiological. He supported the idea that perception of sound derives from signals from nerve cells of the basilar membrane and that the elastic appendages of these cells are sympathetically vibrated by pure sinusoidal tones of appropriate frequencies. Helmholtz agreed with the finding of
Ernst Chladni Ernst Florens Friedrich Chladni (, , ; 30 November 1756 – 3 April 1827) was a German physicist and musician. His most important work, for which he is sometimes labeled the father of acoustics, included research on vibrating plates and th ...
from 1787 that certain sound sources have inharmonic vibration modes. In Helmholtz's time, electronic amplification was unavailable. For synthesis of tones with harmonic partials, Helmholtz built an electrically excited array of
tuning fork A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs ( ''tines'') formed from a U-shaped bar of elastic metal (usually steel). It resonates at a specific constant pitch when set vibrating by striking it ag ...
s and acoustic resonance chambers that allowed adjustment of the amplitudes of the partials. Built at least as early as in 1862, these were in turn refined by Rudolph Koenig, who demonstrated his own setup in 1872. For harmonic synthesis, Koenig also built a large apparatus based on his ''wave siren''. It was pneumatic and utilized cut-out
tonewheel A tonewheel or tone wheel is a simple electromechanical apparatus used for generating electric musical notes in electromechanical electronic organ, organ instruments such as the Hammond Organ, Hammond organ and in telephony to generate audible ...
s, and was criticized for low purity of its partial tones. Also tibia pipes of
pipe organ The pipe organ is a musical instrument that produces sound by driving pressurised air (called ''wind'') through the organ pipes selected from a Musical keyboard, keyboard. Because each pipe produces a single tone and pitch, the pipes are provide ...
s have nearly sinusoidal waveforms and can be combined in the manner of additive synthesis. In 1938, with significant new supporting evidence, it was reported on the pages of
Popular Science Monthly Popular science (also called pop-science or popsci) is an interpretation of science intended for a general audience. While science journalism focuses on recent scientific developments, popular science is more broad ranging. It may be written ...
that the human vocal cords function like a fire siren to produce a harmonic-rich tone, which is then filtered by the vocal tract to produce different vowel tones. By the time, the additive Hammond organ was already on market. Most early electronic organ makers thought it too expensive to manufacture the plurality of oscillators required by additive organs, and began instead to build subtractive ones. In a 1940
Institute of Radio Engineers The Institute of Radio Engineers (IRE) was a professional organization which existed from 1912 until December 31, 1962. On January 1, 1963, it merged with the American Institute of Electrical Engineers (AIEE) to form the Institute of Electrical ...
meeting, the head field engineer of Hammond elaborated on the company's new ''Novachord'' as having a ''"subtractive system"'' in contrast to the original Hammond organ in which ''"the final tones were built up by combining sound waves"''. Alan Douglas used the qualifiers ''additive'' and ''subtractive'' to describe different types of electronic organs in a 1948 paper presented to the Royal Musical Association. The contemporary wording ''additive synthesis'' and ''subtractive synthesis'' can be found in his 1957 book ''The electrical production of music'', in which he categorically lists three methods of forming of musical tone-colours, in sections titled ''Additive synthesis'', ''Subtractive synthesis'', and ''Other forms of combinations''. A typical modern additive synthesizer produces its output as an
electrical Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
,
analog signal An analog signal (American English) or analogue signal (British and Commonwealth English) is any continuous-time signal representing some other quantity, i.e., ''analogous'' to another quantity. For example, in an analog audio signal, the ins ...
, or as
digital audio Digital audio is a representation of sound recorded in, or converted into, digital signal (signal processing), digital form. In digital audio, the sound wave of the audio signal is typically encoded as numerical sampling (signal processing), ...
, such as in the case of
software synthesizers A software synthesizer or softsynth is a computer program that generates digital audio, usually for music. Computer software that can create sounds or music is not new, but advances in processing speed now allow softsynths to accomplish the same t ...
, which became popular around year 2000.


Timeline

The following is a timeline of historically and technologically notable analog and digital synthesizers and devices implementing additive synthesis.


Discrete-time equations

In digital implementations of additive synthesis,
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
equations are used in place of the continuous-time synthesis equations. A notational convention for discrete-time signals uses brackets i.e. y , and the argument n\, can only be integer values. If the continuous-time synthesis output y(t)\, is expected to be sufficiently
bandlimited Bandlimiting is the process of reducing a signal’s energy outside a specific frequency range, keeping only the desired part of the signal’s spectrum. This technique is crucial in signal processing and communications to ensure signals stay cl ...
; below half the
sampling rate In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or s ...
or f_\mathrm/2\,, it suffices to directly sample the continuous-time expression to get the discrete synthesis equation. The continuous synthesis output can later be reconstructed from the samples using a
digital-to-analog converter In electronics, a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function. DACs are commonly used in musi ...
. The sampling period is T=1/f_\mathrm\,. Beginning with (), : y(t) = \sum_^ r_k(t) \cos\left(2 \pi \int_0^t f_k(u)\ du + \phi_k \right) and sampling at discrete times t = nT = n/f_\mathrm \, results in : \begin y & = y(nT) = \sum_^ r_k(nT) \cos\left(2 \pi \int_0^ f_k(u)\ du + \phi_k \right) \\ & = \sum_^ r_k(nT) \cos\left(2 \pi \sum_^ \int_^ f_k(u)\ du + \phi_k \right) \\ & = \sum_^ r_k(nT) \cos\left(2 \pi \sum_^ (T f_k + \phi_k \right) \\ & = \sum_^ r_k \cos\left(\frac \sum_^ f_k + \phi_k \right) \\ \end where : r_k = r_k(nT) \, is the discrete-time varying amplitude envelope : f_k = \frac \int_^ f_k(t)\ dt \, is the discrete-time
backward difference A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly d ...
instantaneous frequency. This is equivalent to : y = \sum_^ r_k \cos\left( \theta_k \right) where : \begin \theta_k &= \frac \sum_^ f_k + \phi_k \\ &= \theta_k -1+ \frac f_k \\ \end for all n>0\, and : \theta_k = \phi_k. \,


See also

*
Frequency modulation synthesis Frequency modulation synthesis (or FM synthesis) is a form of Synthesizer#Sound synthesis, sound synthesis whereby the frequency of a waveform is changed by Frequency modulation, modulating its frequency with a modulator. The instantaneous frequen ...
*
Subtractive synthesis Subtractive synthesis is a method of sound synthesis in which Harmonic_series_(music)#Partial.2C_harmonic.2C_fundamental.2C_inharmonicity.2C_and_overtone, overtones of an audio signal are attenuated by a audio filter, filter to alter the timbre of ...
*
Speech synthesis Speech synthesis is the artificial production of human speech. A computer system used for this purpose is called a speech synthesizer, and can be implemented in software or hardware products. A text-to-speech (TTS) system converts normal langua ...
*
Harmonic series (music) The harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a ''fundamental frequency''. Definite pitch, Pitched musical instruments are often based on an Acoust ...


References


External links


Digital Keyboards Synergy
{{DEFAULTSORT:Additive Synthesis Sound synthesis types