Additive synthesis example

A bell-like sound generated by additive synthesis of 21 inharmonic partials

---

Problems playing this file? Seemedia help.

Additive synthesis is asound synthesis technique that createstimbre by addingsine waves together.^[1][2]

The timbre of musical instruments can be considered in the light ofFourier theory to consist of multipleharmonic or inharmonicpartials orovertones. Each partial is a sine wave of differentfrequency andamplitude that swells and decays over time due tomodulation from anADSR envelope orlow frequency oscillator.

Additive synthesis most directly generates sound by adding the output of multiple sine wave generators. Alternative implementations may use pre-computedwavetables or the inversefast Fourier transform.

The sounds that are heard in everyday life are not characterized by a singlefrequency. Instead, they consist of a sum of pure sine frequencies, each one at a differentamplitude. 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 thetimbre 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 theFourier series of the original sound signal.

In the case of a musical note, the lowest frequency of its timbre is designated as the sound'sfundamental frequency. For simplicity, we often say that the note is playing at that fundamental frequency (e.g. "middle C is 261.6 Hz"),^[3] even though the sound of that note consists of many other frequencies as well. The set of the remaining frequencies is called theovertones (or theharmonics, if their frequencies are integer multiples of the fundamental frequency) of the sound.^[4] 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, anupright piano vs. agrand piano).

Additive synthesis aims to exploit this property of sound in order to construct timbre from the ground up. By adding together pure frequencies (sine waves) of varying frequencies and amplitudes, we can precisely define the timbre of the sound that we want to create.

Harmonic additive synthesis is closely related to the concept of aFourier series which is a way of expressing aperiodic function as the sum ofsinusoidal functions withfrequencies equal to integer multiples of a commonfundamental frequency. These sinusoids are calledharmonics,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 aDC component (one with frequency of 0Hz).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 beperiodic if

${\displaystyle y(t)=y(t+P)}$

for all${\displaystyle t}$ and for some period${\displaystyle P}$.

TheFourier series of a periodic function is mathematically expressed as:

where

• ${\displaystyle f_0=1/P}$ is thefundamental frequency of the waveform and is equal to the reciprocal of the period,
• ${\displaystyle a_k=r_k\cos ({\varphi }_k)=2f_0{\int }_0^{P}y(t)\cos (2\pi k{f}_{0}t)dt,k\ge 0}$
• ${\displaystyle b_k=r_k\sin ({\varphi }_k)=-2f_0{\int }_0^{P}y(t)\sin (2\pi k{f}_{0}t)dt,k\ge 1}$
• ${\displaystyle r_k=\sqrt{a_k^2+b_k^2}}$ is theamplitude of the${\displaystyle k}$th harmonic,
• ${\displaystyle {\varphi }_k=\mathrm{atan2}(b_k,a_k)}$ is thephase offset of the${\displaystyle k}$th harmonic.atan2 is the four-quadrantarctangent function,

Being inaudible, theDC component,${\displaystyle a_0/2}$, and all components with frequencies higher than some finite limit,${\displaystyle K{f}_0}$, are omitted in the following expressions of additive synthesis.

The simplest harmonic additive synthesis can be mathematically expressed as:

${\displaystyle y(t)=\sum _{k=1}^K{r}_k\cos \left(2\pi k{f}_{0}t+{\varphi }_k\right),}$

1

where${\displaystyle y(t)}$ is the synthesis output,${\displaystyle r_k}$,${\displaystyle k{f}_0}$, and${\displaystyle {\varphi }_k}$ are the amplitude, frequency, and the phase offset, respectively, of the${\displaystyle k}$th harmonic partial of a total of${\displaystyle K}$ harmonic partials, and${\displaystyle f_0}$ is thefundamental frequency of the waveform and thefrequency of the musical note.

Example of harmonic additive synthesis in which each harmonic has a time-dependent amplitude. The fundamental frequency is 440 Hz. Problems listening to this file? SeeMedia help

More generally, the amplitude of each harmonic can be prescribed as a function of time,${\displaystyle r_k(t)}$, in which case the synthesis output is

${\displaystyle y(t)=\sum _{k=1}^K{r}_k(t)\cos \left(2\pi k{f}_{0}t+{\varphi }_k\right)}$.

2

Eachenvelope${\displaystyle r_k(t)}$ should vary slowly relative to the frequency spacing between adjacent sinusoids. Thebandwidth of${\displaystyle r_k(t)}$ should be significantly less than${\displaystyle f_0}$.

Additive synthesis can also produceinharmonic sounds (which areaperiodic waveforms) in which the individual overtones need not have frequencies that are integer multiples of some common fundamental frequency.^[5][6] While many conventional musical instruments have harmonic partials (e.g. anoboe), some have inharmonic partials (e.g.bells). Inharmonic additive synthesis can be described as

${\displaystyle y(t)=\sum _{k=1}^K{r}_k(t)\cos \left(2\pi f_{k}t+{\varphi }_k\right),}$

where${\displaystyle f_k}$ is the constant frequency of${\displaystyle k}$th partial.

Example of inharmonic additive synthesis in which both the amplitude and frequency of each partial are time-dependent. Problems listening to this file? SeeMedia help

In the general case, theinstantaneous frequency of a sinusoid is thederivative (with respect to time) of the argument of the sine or cosine function. If this frequency is represented inhertz, rather than inangular frequency form, then this derivative is divided by${\displaystyle 2\pi }$. 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,${\displaystyle f_k(t)}$, yielding

${\displaystyle y(t)=\sum _{k=1}^K{r}_k(t)\cos \left(2\pi {\int }_0^t{f}_k(u)du+{\varphi }_k\right).}$

3

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.^[7][8] For example, F. Richard Moore listed additive synthesis as one of the "four basic categories" of sound synthesis alongsidesubtractive synthesis, nonlinear synthesis, andphysical modeling.^[8] 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 ofprincipal components andWalsh functions have also been classified as additive synthesis.^[9]

Modern-day implementations of additive synthesis are mainly digital. (See sectionDiscrete-time equations for the underlying discrete-time theory)

Additive synthesis can be implemented using a bank of sinusoidal oscillators, one for each partial.^[1]

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.^[10][11] As a result, an efficient implementation of time-varying additive synthesis of harmonic tones can be accomplished by use ofwavetable synthesis.

Group additive synthesis^[12][13][14] is a method to group partials into harmonic groups (having different fundamental frequencies) and synthesize each group separately withwavetable synthesis before mixing the results.

An inversefast Fourier transform can be used to efficiently synthesize frequencies that evenly divide the transform period or "frame". By careful consideration of theDFT frequency-domain representation it is also possible to efficiently synthesize sinusoids of arbitrary frequencies using a series of overlapping frames and the inversefast Fourier transform.^[15]

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 isshort-time Fourier transform (STFT)-based McAulay-Quatieri Analysis.^[17][18]

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.^[19]

Additive analysis/resynthesis has been employed in a number of techniques including Sinusoidal Modelling,^[20]Spectral Modelling Synthesis (SMS),^[19] and the Reassigned Bandwidth-Enhanced Additive Sound Model.^[21] Software that implements additive analysis/resynthesis includes: SPEAR,^[22] LEMUR, LORIS,^[23] SMSTools,^[24] ARSS.^[25]

Additive re-synthesis using timbre-frame concatenation:

Concatenation with crossfades (on Synclavier)

Concatenation with spectral envelope interpolation (on Vocaloid)

New England DigitalSynclavier 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 anFFT implementation.

Also a vocal synthesizer,Vocaloid have been implemented on the basis of additive analysis/resynthesis: its spectral voice model calledExcitation plus Resonances (EpR) model^[26][27] is extended based on Spectral Modeling Synthesis (SMS),and itsdiphoneconcatenative synthesis is processed usingspectral peak processing (SPP)^[28] technique similar to modifiedphase-locked vocoder^[29] (an improvedphase vocoder for formant processing).^[30] Using these techniques, spectral components (formants) consisting of purely harmonic partials can be appropriately transformed into desired form for sound modeling, and sequence of short samples (diphones orphonemes) constituting desired phrase, can be smoothly connected by interpolating matched partials and formant peaks, respectively, in the inserted transition region between different samples. (See alsoDynamic timbres)

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

Inlinguistics research, harmonic additive synthesis was used in the 1950s to play back modified and synthetic speech spectrograms.^[31]

Later, in the early 1980s, listening tests were carried out on synthetic speech stripped of acoustic cues to assess their significance. Time-varyingformant frequencies and amplitudes derived bylinear predictive coding were synthesized additively as pure tone whistles. This method is calledsinewave synthesis.^[32][33] Also the composite sinusoidal modeling (CSM)^[34][35] used on a singingspeech synthesis feature on theYamaha CX5M (1984), is known to use a similar approach which was independently developed during 1966–1979.^[36][37] 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 ofacoustics. This principle was also utilized on aphysical modeling synthesis method, calledmodal synthesis.^{[38][39][40][41]}

Lord Kelvin'sTide-predicting machine

Harmonic synthesizer

Harmonic analyzer

Harmonic analysis was discovered byJoseph Fourier,^[42] who published an extensive treatise of his research in the context ofheat transfer in 1822.^[43] The theory found an early application inprediction of tides. Around 1876,^[44] William Thomson (later ennobled asLord Kelvin) constructed a mechanicaltide predictor. It consisted of aharmonic analyzer and aharmonic synthesizer, as they were called already in the 19th century.^[45][46] The analysis of tide measurements was done usingJames Thomson'sintegrating machine. The resultingFourier coefficients 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.^[47] The synthesizer drew a graph of the combination waveform, which was used chiefly for visual validation of the analysis.^[47]

Helmholtz resonator

Tone-generator utilizing it

Georg Ohm applied Fourier's theory to sound in 1843. The line of work was greatly advanced byHermann von Helmholtz, who published his eight years worth of research in 1863.^[48] Helmholtz believed that the psychological perception of tone color is subject to learning, while hearing in the sensory sense is purely physiological.^[49] 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.^[47] Helmholtz agreed with the finding ofErnst Chladni from 1787 that certain sound sources have inharmonic vibration modes.^[49]

Rudolph Koenig's sound analyzer and synthesizer

sound synthesizer

sound analyzer

In Helmholtz's time,electronic amplification was unavailable. For synthesis of tones with harmonic partials, Helmholtz built an electricallyexcited array oftuning forks and acousticresonance chambers that allowed adjustment of the amplitudes of the partials.^[50] Built at least as early as in 1862,^[50] these were in turn refined byRudolph Koenig, who demonstrated his own setup in 1872.^[50] For harmonic synthesis, Koenig also built a large apparatus based on hiswave siren. It was pneumatic and utilized cut-outtonewheels, and was criticized for low purity of its partial tones.^[44] Alsotibia pipes ofpipe organs have nearly sinusoidal waveforms and can be combined in the manner of additive synthesis.^[44]

In 1938, with significant new supporting evidence,^[51] it was reported on the pages ofPopular Science Monthly 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.^[52] 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 buildsubtractive ones.^[53] In a 1940Institute of Radio Engineers meeting, the head field engineer of Hammond elaborated on the company's newNovachord as having a"subtractive system" in contrast to the original Hammond organ in which"the final tones were built up by combining sound waves".^[54] Alan Douglas used the qualifiersadditive andsubtractive to describe different types of electronic organs in a 1948 paper presented to theRoyal Musical Association.^[55] The contemporary wordingadditive synthesis andsubtractive synthesis can be found in his 1957 bookThe electrical production of music, in which he categorically lists three methods of forming of musical tone-colours, in sections titledAdditive synthesis,Subtractive synthesis, andOther forms of combinations.^[56]

A typical modern additive synthesizer produces its output as anelectrical,analog signal, or asdigital audio, such as in the case ofsoftware synthesizers, which became popular around year 2000.^[57]

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

In digital implementations of additive synthesis,discrete-time equations are used in place of the continuous-time synthesis equations. A notational convention for discrete-time signals uses brackets i.e.${\displaystyle y[n]}$ and the argument${\displaystyle n}$ can only be integer values. If the continuous-time synthesis output${\displaystyle y(t)}$ is expected to be sufficientlybandlimited; below half thesampling rate or${\displaystyle f_{\mathrm{s}}/2}$, it suffices to directly sample the continuous-time expression to get the discrete synthesis equation. The continuous synthesis output can later bereconstructed from the samples using adigital-to-analog converter. The sampling period is${\displaystyle T=1/f_{\mathrm{s}}}$.

Beginning with (3),

${\displaystyle y(t)=\sum _{k=1}^K{r}_k(t)\cos \left(2\pi {\int }_0^t{f}_k(u)du+{\varphi }_k\right)}$

and sampling at discrete times${\displaystyle t=nT=n/f_{\mathrm{s}}}$ results in

where

${\displaystyle r_k[n]=r_k(nT)}$ is the discrete-time varying amplitude envelope

${\displaystyle f_k[n]=\frac{1}{T}{\int }_{(n-1)T}^{nT}{f}_k(t)dt}$ is the discrete-timebackward difference instantaneous frequency.

This is equivalent to

${\displaystyle y[n]=\sum _{k=1}^K{r}_k[n]\cos \left({\theta }_k[n]\right)}$

where

for all${\displaystyle n>0}$^[15]

and

${\displaystyle {\theta }_k[0]={\varphi }_k.}$

• Frequency modulation synthesis
• Subtractive synthesis
• Speech synthesis
• Harmonic series (music)

• Digital Keyboards Synergy

• Sound synthesis types

• Pages using the Phonos extension
• CS1 interwiki-linked names
• CS1 uses Japanese-language script (ja)
• CS1: long volume value
• CS1 errors: ISBN date
• CS1 French-language sources (fr)
• CS1 errors: missing title
• All articles with failed verification
• Articles with failed verification from July 2024
• CS1 German-language sources (de)
• CS1: unfit URL
• CS1 Italian-language sources (it)
• Articles with short description
• Short description matches Wikidata
• Use dmy dates from January 2020
• Articles with hAudio microformats
• Webarchive template wayback links