additive synthesis

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 (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 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 (e.g. "middle C is 261.6 Hz"),{{Cite web|url=http://www.liutaiomottola.com/formulae/freqtab.htm|title=Table of Musical Notes and Their Frequencies and Wavelengths|last=Mottola|first=Liutaio|date=31 May 2017}} even though the sound of that note consists of many other frequencies as well. The set of the remaining frequencies is called the overtones (or the harmonics, if their frequencies are integer multiples of the fundamental frequency) of the sound.{{Cite web|url=http://www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics|title=Fundamental Frequency and Harmonics}} 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 vs. a grand 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.

{{See also|Fourier series|Fourier analysis}}

File:Additive synthesis.svg

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 (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 of a periodic function is mathematically expressed as:

: $\backslash begin\{align\}$

y(t) &= \frac{a_0}{2} + \sum_{k=1}^{\infty} \left[ a_k \cos(2 \pi k f_0 t ) - b_k \sin(2 \pi k f_0 t ) \right] \\

&= \frac{a_0}{2} + \sum_{k=1}^{\infty} r_k \cos\left(2 \pi k f_0 t + \phi_k \right) \\

\end{align}

where

• $f\_0\; =\; 1/P$ is the fundamental frequency of the waveform and is equal to the reciprocal of the period,
• $a\_k\; =\; r\_k\; \backslash cos(\backslash phi\_k)\; =\; 2\; f\_0\; \backslash int\_\{0\}^P\; y(t)\; \backslash cos(2\; \backslash pi\; k\; f\_0\; t)\backslash ,\; dt,\; \backslash quad\; k\; \backslash ge\; 0$
• $b\_k\; =\; r\_k\; \backslash sin(\backslash phi\_k)\; =\; -2\; f\_0\; \backslash int\_\{0\}^P\; y(t)\; \backslash sin(2\; \backslash pi\; k\; f\_0\; t)\backslash ,\; dt,\; \backslash quad\; k\; \backslash ge\; 1$
• $r\_k\; =\; \backslash sqrt\{a\_k^2\; +\; b\_k^2\}$ is the amplitude of the $k$th harmonic,
• $\backslash phi\_k\; =\; \backslash operatorname\{atan2\}(b\_k,\; a\_k)$ is the phase offset of the $k$th harmonic. atan2 is the four-quadrant arctangent 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.

The simplest harmonic additive synthesis can be mathematically expressed as:

{{NumBlk|:|$y(t)\; =\; \backslash sum\_\{k=1\}^\{K\}\; r\_k\; \backslash cos\backslash left(2\; \backslash pi\; k\; f\_0\; t\; +\; \backslash phi\_k\; \backslash right),$|{{EquationRef|1}}}}

where $y(t)$ is the synthesis output, $r\_k$, $k\; f\_0$, and $\backslash phi\_k$ are the amplitude, frequency, and the phase offset, respectively, of the $k$th harmonic partial of a total of $K$ harmonic partials, and $f\_0$ is the fundamental frequency of the waveform and the frequency of the musical note.

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

{{NumBlk|:|$y(t)\; =\; \backslash sum\_\{k=1\}^\{K\}\; r\_k(t)\; \backslash cos\backslash left(2\; \backslash pi\; k\; f\_0\; t\; +\; \backslash phi\_k\; \backslash right)$.|{{EquationRef|2}}}}

Each envelope $r\_k(t)\backslash ,$ should vary slowly relative to the frequency spacing between adjacent sinusoids. The bandwidth of $r\_k(t)$ should be significantly less than $f\_0$.

Additive synthesis can also produce inharmonic sounds (which are aperiodic waveforms) in which the individual overtones need not have frequencies that are integer multiples of some common fundamental frequency.

{{Cite book

| last1 = Smith III | first1 = Julius O.

| last2 = Serra | first2 = Xavier

| year = 2005

| chapter = Additive Synthesis

| chapter-url = https://ccrma.stanford.edu/~jos/parshl/Additive_Synthesis.html

| title = PARSHL: An Analysis/Synthesis Program for Non-Harmonic Sounds Based on a Sinusoidal Representation

| url = https://ccrma.stanford.edu/~jos/parshl/

| series = Proceedings of the International Computer Music Conference (ICMC-87, Tokyo), Computer Music Association, 1987

| publisher = CCRMA, Department of Music, Stanford University

| access-date = 11 January 2015

}} ([https://ccrma.stanford.edu/STANM/stanms/stanm43/stanm43.pdf online reprint])

{{Cite book

| last = Smith III | first = Julius O.

| year = 2011

| chapter = Additive Synthesis (Early Sinusoidal Modeling)

| chapter-url = https://ccrma.stanford.edu/~jos/sasp/Additive_Synthesis_Early_Sinusoidal.html

| title = Spectral Audio Signal Processing

| url = https://ccrma.stanford.edu/~jos/sasp/

| publisher = CCRMA, Department of Music, Stanford University

| isbn = 978-0-9745607-3-1

| access-date = 9 January 2012

}} While many conventional musical instruments have harmonic partials (e.g. an oboe), some have inharmonic partials (e.g. bells). Inharmonic additive synthesis can be described as

: $y(t)\; =\; \backslash sum\_\{k=1\}^\{K\}\; r\_k(t)\; \backslash cos\backslash left(2\; \backslash pi\; f\_k\; t\; +\; \backslash phi\_k\; \backslash right),$

where $f\_k$ is the constant frequency of $k$th partial.

In the general case, the instantaneous frequency of a sinusoid is the derivative (with respect to time) 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 $2\; \backslash pi$. 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

{{NumBlk|:|$y(t)\; =\; \backslash sum\_\{k=1\}^\{K\}\; r\_k(t)\; \backslash cos\backslash left(2\; \backslash pi\; \backslash int\_0^t\; f\_k(u)\backslash \; du\; +\; \backslash phi\_k\; \backslash right).$|{{EquationRef|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.

{{cite book

| last = Roads | first = Curtis

| author-link = Curtis Roads

| year = 1995

| title = The Computer Music Tutorial

| url=https://archive.org/details/computermusictut00road

| url-access=limited

| publisher = MIT Press

| isbn = 978-0-262-68082-0

| page = [https://archive.org/details/computermusictut00road/page/n152 134]

}}

{{cite book

| last = Moore | first = F. Richard

| year = 1995

| title = Foundations of Computer Music

| publisher = Prentice Hall

| isbn = 978-0-262-68082-0

| page = 16

}}

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.

{{cite book

| last = Roads | first = Curtis

| author-link = Curtis Roads

| year = 1995

| title = The Computer Music Tutorial

| url=https://archive.org/details/computermusictut00road

| url-access=limited

| publisher = MIT Press

| isbn = 978-0-262-68082-0

| pages = [https://archive.org/details/computermusictut00road/page/n168 150]–153

}}

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

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

{{Main|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.

{{cite web

|author = Robert Bristow-Johnson

|date = November 1996

|title = Wavetable Synthesis 101, A Fundamental Perspective

|url = http://www.musicdsp.org/files/Wavetable-101.pdf

|access-date = 21 May 2005

|archive-url = https://web.archive.org/web/20130615202748/http://musicdsp.org/files/Wavetable-101.pdf

|archive-date = 15 June 2013

|df = dmy-all

}}

{{cite journal

| author = Andrew Horner

| date = November 1995

| title = Wavetable Matching Synthesis of Dynamic Instruments with Genetic Algorithms

| journal = Journal of the Audio Engineering Society

| volume = 43 | issue = 11 | pages = 916–931

| url = http://www.aes.org/e-lib/browse.cfm?elib=7923

}} 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

{{cite web

| author = Julius O. Smith III

| title = Group Additive Synthesis

| url = https://ccrma.stanford.edu/~jos/sasp/Group_Additive_Synthesis.html

| publisher = CCRMA, Stanford University

| access-date = 12 May 2011

| archive-url = https://web.archive.org/web/20110606200135/https://ccrma.stanford.edu/~jos/sasp/Group_Additive_Synthesis.html

| archive-date = 6 June 2011 | url-status= live}}

{{cite journal

| author = P. Kleczkowski

| title = Group additive synthesis

| journal = Computer Music Journal

| volume = 13 | issue = 1 | pages = 12–20

| year = 1989

| doi=10.2307/3679851

| jstor = 3679851

}}

{{cite book

| author = B. Eaglestone and S. Oates

| chapter = Analytical tools for group additive synthesis

| title = Proceedings of the 1990 International Computer Music Conference, Glasgow

| publisher = Computer Music Association

| year = 1990

| chapter-url = http://quod.lib.umich.edu/cgi/p/pod/dod-idx?c=icmc;idno=bbp2372.1990.015

}} 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.

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.

{{cite journal

| last1 = Rodet | first1 = X.

| last2 = Depalle | first2 = P.

| year = 1992

| title = Spectral Envelopes and Inverse FFT Synthesis

| citeseerx = 10.1.1.43.4818

| journal = Proceedings of the 93rd Audio Engineering Society Convention

}}

[[Image:Sinusoidal Analysis & Synthesis (McAulay-Quatieri 1988).svg|thumb|350px|Sinusoidal analysis/synthesis system for Sinusoidal Modeling (based on {{harvnb|McAulay|Quatieri|1988|p=161}}){{cite journal

|last1 = McAulay

|first1 = R. J.

|last2 = Quatieri

|first2 = T. F.

|author-link2 = Thomas F. Quatieri

|year = 1988

|title = Speech Processing Based on a Sinusoidal Model

|url = http://www.ll.mit.edu/publications/journal/pdf/vol01_no2/1.2.3.speechprocessing.pdf

|journal = The Lincoln Laboratory Journal

|volume = 1

|issue = 2

|pages = 153–167

|access-date = 9 December 2013

|archive-url = https://web.archive.org/web/20120521071601/http://www.ll.mit.edu/publications/journal/pdf/vol01_no2/1.2.3.speechprocessing.pdf

|archive-date = 21 May 2012

|df = dmy-all

}}]]

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.

{{cite journal

| last1 = McAulay | first1 = R. J.

| last2 = Quatieri| first2 = T. F.

| date = Aug 1986

| title = Speech analysis/synthesis based on a sinusoidal representation

| journal = IEEE Transactions on Acoustics, Speech, and Signal Processing

| volume = 34 | issue = 4 | pages = 744–754

| doi = 10.1109/TASSP.1986.1164910 }}

{{cite web

| title = McAulay-Quatieri Method

| url = http://www.clear.rice.edu/elec301/Projects02/lorisFor/mqmethod2.html

}}

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.

{{cite thesis

| degree = PhD

| last = Serra | first = Xavier

| date = 1989

| title = A System for Sound Analysis/Transformation/Synthesis based on a Deterministic plus Stochastic Decomposition

| url = http://mtg.upf.edu/node/304

| publisher = Stanford University

| access-date = 13 January 2012

}}

Additive analysis/resynthesis has been employed in a number of techniques including Sinusoidal Modelling,

{{cite web

| last1 = Smith III | first1 = Julius O.

| last2 = Serra | first2 = Xavier

| title = PARSHL: An Analysis/Synthesis Program for Non-Harmonic Sounds Based on a Sinusoidal Representation

| url = https://ccrma.stanford.edu/~jos/parshl/Additive_Synthesis.html

| access-date = 9 January 2012

}} Spectral Modelling Synthesis (SMS), and the Reassigned Bandwidth-Enhanced Additive Sound Model.

{{cite thesis

| degree = PhD

| last = Fitz | first = Kelly

| date = 1999

| title = The Reassigned Bandwidth-Enhanced Method of Additive Synthesis

| publisher = Dept. of Electrical and Computer Engineering, University of Illinois Urbana-Champaign

| citeseerx = 10.1.1.10.1130

}} Software that implements additive analysis/resynthesis includes: SPEAR,[http://www.klingbeil.com/spear/ SPEAR Sinusoidal Partial Editing Analysis and Resynthesis for Mac OS X, MacOS 9 and Windows] LEMUR, LORIS,{{Cite web |url=http://www.hakenaudio.com/Loris/ |title=Loris Software for Sound Modeling, Morphing, and Manipulation |access-date=13 January 2012 |archive-url=https://web.archive.org/web/20120730195624/http://www.hakenaudio.com/Loris/ |archive-date=30 July 2012 |df=dmy-all }} SMSTools,[http://mtg.upf.edu/technologies/sms SMSTools application for Windows] ARSS.[http://arss.sourceforge.net/ ARSS: The Analysis & Resynthesis Sound Spectrograph]

{{multiple image |direction=vertical |align=right |width=165

| header = Additive re-synthesis using timbre-frame concatenation:

| image1 = Wavesequence.svg

| caption1 = Concatenation with crossfades (on Synclavier)

| image2 = Vocaloid's phonemes crossfading - en.jpg

| caption2 = Concatenation with spectral envelope interpolation (on Vocaloid)

}}

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 (EpR) model

{{cite journal

| last1 = Bonada | first1 = J.

| last2 = Celma | first2 = O.

| last3 = Loscos | first3 = A.

| last4 = Ortola | first4 = J.|first5=X. |last5=Serra |first6=Y. |last6=Yoshioka |first7=H. |last7=Kayama |first8=Y. |last8=Hisaminato |first9=H. |last9=Kenmochi

| year = 2001

| title = Singing voice synthesis combining Excitation plus Resonance and Sinusoidal plus Residual Models

| periodical = Proc. Of ICMC

| citeseerx = 10.1.1.18.6258

}} ([http://mtg.upf.edu/files/publications/icmc2001-celma.pdf PDF])

{{cite thesis

| degree = PhD

| last = Loscos | first = A.

| year = 2007

| title = Spectral processing of the singing voice

| location = Barcelona, Spain

| publisher = Pompeu Fabra University

| hdl= 10803/7542

}} ([http://www.tdx.cat/bitstream/handle/10803/7542/talm.pdf?sequence=1 PDF]).

See "Excitation plus resonances voice model" (p. 51)

is extended based on Spectral Modeling Synthesis (SMS),

and its diphone concatenative synthesis is processed using

spectral peak processing (SPP){{harvnb|Loscos|2007|p=44}}, "Spectral peak processing" technique similar to modified phase-locked vocoder{{harvnb|Loscos|2007|p=44}}, "Phase locked vocoder" (an improved phase vocoder for formant processing).

{{cite journal

| last1 = Bonada | first1 = Jordi

| last2 = Loscos | first2 = Alex

| year = 2003

| title = Sample-based singing voice synthesizer by spectral concatenation: 6. Concatenating Samples

| url = http://mtg.upf.edu/node/322

| periodical = Proc. of SMAC 03

| pages = 439–442

}} 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 or phonemes) 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)

{{main|Synthesizer|Electronic musical instrument|Software synthesizer}}

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

{{main|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.

{{cite journal

| last = Remez | first = R.E.

|author2= Rubin, P.E.|author3= Pisoni, D.B.|author4= Carrell, T.D.

| s2cid = 13039853 | year = 1981

| title = Speech perception without traditional speech cues

| journal = Science

| volume = 212 | issue = 4497 | pages = 947–950 | doi=10.1126/science.7233191 | pmid=7233191

| bibcode = 1981Sci...212..947R }}{{cite journal | last = Rubin | first = P.E. | year = 1980 | title = Sinewave Synthesis Instruction Manual (VAX) | url = http://www.haskins.yale.edu/featured/sws/SWSmanual.pdf | journal = Internal Memorandum | publisher = Haskins Laboratories, New Haven, CT | access-date = 27 December 2011 | archive-date = 29 August 2021 | archive-url = https://web.archive.org/web/20210829131739/http://www.haskins.yale.edu/featured/sws/SWSmanual.pdf | url-status = dead }} Also the composite sinusoidal modeling (CSM)

{{citation

| last1 = Sagayama | first1 = S. | author-link1 = :ja:嵯峨山茂樹

| last2 = Itakura | first2 = F. | author-link2 = Fumitada Itakura

| year = 1979

|script-title=ja:複合正弦波による音声合成

| trans-title = Speech Synthesis by Composite Sinusoidal Wave

| periodical = Speech Committee of Acoustical Society of Japan

| id = S79-39 | publication-date = October 1979

}}

{{cite conference

| last1 = Sagayama | first1 = S.

| last2 = Itakura | first2 = F.

| date = October 1979

|script-chapter=ja:複合正弦波による簡易な音声合成法

|trans-chapter=Simple Speech Synthesis method by Composite Sinusoidal Wave

|title=Proceedings of Acoustical Society of Japan, Autumn Meeting

| volume = 3-2-3 | pages = 557–558

}}

used on a singing speech synthesis feature on the Yamaha CX5M (1984), is known to use a similar approach which was independently developed during 1966–1979.

{{cite book

| last1 = Sagayama | first1 = S.

| title = ICASSP '86. IEEE International Conference on Acoustics, Speech, and Signal Processing

| last2 = Itakura | first2 = F.

| year = 1986

| author-link2 = Fumitada Itakura

| chapter = Duality theory of composite sinusoidal modeling and linear prediction

| publisher = Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP '86.

| volume = 11 | pages = 1261–1264 | publication-date = April 1986

| doi = 10.1109/ICASSP.1986.1168815

| s2cid = 122814777

}}

{{cite journal

| last = Itakura | first = F.

| author-link = Fumitada Itakura

| year = 2004

| title = Linear Statistical Modeling of Speech and its Applications -- Over 36-year history of LPC --

| url = http://www.icacommission.org/Proceedings/ICA2004Kyoto/pdf/We3.D.pdf

| periodical = Proceedings of the 18th International Congress on Acoustics (ICA 2004), We3.D, Kyoto, Japan, Apr. 2004.

| volume = 3 | pages = III–2077–2082 | publication-date = April 2004

| quote = 6. Composite Sinusoidal Modeling(CSM) In 1975, Itakura proposed the line spectrum representation (LSR) concept and its algorithm to obtain a set of parameters for new speech spectrum representation. Independently from this, Sagayama developed a composite sinusoidal modeling (CSM) concept which is equivalent to LSR but give a quite different formulation, solving algorithm and synthesis scheme. Sagayama clarified the duality of LPC and CSM and provided the unified view covering LPC, PARCOR, LSR, LSP and CSM, CSM is not only a new concept of speech spectrum analysis but also a key idea to understand the linear prediction from a unified point of view. ...

}}

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.

{{cite book |last = Adrien

|first = Jean-Marie

|chapter = The missing link: modal synthesis

|chapter-url = http://dl.acm.org/citation.cfm?id=131158

|editor= Giovanni de Poli |editor2=Aldo Piccialli |editor3=Curtis Roads |editor3-link=Curtis Roads |title = Representations of Musical Signals

|url = https://archive.org/details/representationso0000unse_k7n4/page/269

|publisher = MIT Press

|location = Cambridge, MA

|date = 1991

|isbn = 978-0-262-04113-3

|pages = [https://archive.org/details/representationso0000unse_k7n4/page/269 269–298]

|url-access = registration

}}

{{cite journal

| last1 = Morrison | first1 = Joseph Derek (IRCAM)

| last2 = Adrien | first2 = Jean-Marie

| title = MOSAIC: A Framework for Modal Synthesis

| journal = Computer Music Journal

| volume = 17 | issue = 1 | publication-date = 1993

| pages = 45–56 | doi=10.2307/3680569

| jstor = 3680569

| year = 1993

}}

{{citation

| last = Bilbao | first = Stefan

| chapter = Modal Synthesis

| chapter-url= https://ccrma.stanford.edu/~bilbao/booktop/node14.html

| title = Numerical Sound Synthesis: Finite Difference Schemes and Simulation in Musical Acoustics

| publisher = John Wiley and Sons | location = Chichester, UK

| date = October 2009

| isbn = 978-0-470-51046-9

| quote = A different approach, with a long history of use in physical modeling sound synthesis, is based on a frequency-domain, or modal description of vibration of objects of potentially complex geometry. Modal synthesis [1,148], as it is called, is appealing, in that the complex dynamic behaviour of a vibrating object may be decomposed into contributions from a set of modes (the spatial forms of which are eigenfunctions of the particular problem at hand, and are dependent on boundary conditions), each of which oscillates at a single complex frequency. ...

}}  (See also [http://www2.ph.ed.ac.uk/~sbilbao/nsstop.html companion page])

{{cite journal

| last1 = Doel | first1 = Kees van den

| last2 = Pai | first2 = Dinesh K.

| title = Modal Synthesis For Vibrating Object

| url = http://www.cs.ubc.ca/~kvdoel/publications/modalpaper.pdf

| editor-last = Greenebaum | editor-first = K.

| journal = Audio Anecdotes

| publisher = AK Peter | location = Natick, MA

| date = 2003

| quote = When a solid object is struck, scraped, or engages in other external interactions, the forces at the contact point causes deformations to propagate through the body, causing its outer surfaces to vibrate and emit sound waves. ... A good physically motivated synthesis model for objects like this is modal synthesis ... where a vibrating object is modeled by a bank of damped harmonic oscillators which are excited by an external stimulus.

}}

{{multiple image |direction=horizontal

|header = Lord Kelvin's Tide-predicting machine

|caption1 = Harmonic synthesizer

|image1 = DSCN1739-thomson-tide-machine.jpg |width1=109

|caption2 = Harmonic analyzer

|image2 = Harmonic analyser.jpg |width2=224

}}

Harmonic analysis was discovered by Joseph Fourier,{{Cite book |last=Prestini |first=Elena |url=https://books.google.com/books?id=fye--TBu4T0C |title=The Evolution of Applied Harmonic Analysis: Models of the Real World |publisher=Birkhäuser Boston |others=trans. |year=2004 |isbn=978-0-8176-4125-2 |location=New York, USA |pages=114–115 |access-date=6 February 2012 |orig-date=Rev. ed of: Applicazioni dell'analisi armonica. Milan: Ulrico Hoepli, 1996}} who published an extensive treatise of his research in the context of heat transfer in 1822.{{Cite book |last=Fourier |first=Jean Baptiste Joseph |author-link=Joseph Fourier |url=https://archive.org/details/thorieanalytiqu00fourgoog |title=Théorie analytique de la chaleur |publisher=Chez Firmin Didot, père et fils |year=1822 |isbn=9782876470460 |location=Paris, France |language=fr |trans-title=The Analytical Theory of Heat}} The theory found an early application in prediction of tides. Around 1876, William Thomson (later ennobled as Lord Kelvin) constructed a mechanical tide predictor. It consisted of a harmonic analyzer and a harmonic synthesizer, as they were called already in the 19th century.{{Cite journal |year=1875 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |publisher=Taylor & Francis |volume=49 |page=490}}{{failed verification|date=July 2024}}{{Cite journal |last=Thomson |first=Sir W. |year=1878 |title=Harmonic analyzer |url=https://zenodo.org/record/1432065 |journal=Proceedings of the Royal Society of London |publisher=Taylor and Francis |volume=27 |issue=185–189 |pages=371–373 |doi=10.1098/rspl.1878.0062 |jstor=113690 |doi-access=free}} The analysis of tide measurements was done using James Thomson's integrating machine. The resulting Fourier 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. The synthesizer drew a graph of the combination waveform, which was used chiefly for visual validation of the analysis.

{{multiple image |direction=horizontal

|caption1 = Helmholtz resonator

|image1 = Helmholtz_Resonator.png |width1=100

|caption2 = Tone-generator utilizing it

|image2 = Helmholtz resonator 2.jpg |width2=148

}}

Georg Ohm applied Fourier's theory to sound in 1843. The line of work was greatly advanced by Hermann von Helmholtz, who published his eight years worth of research in 1863.{{Cite book |last=Helmholtz, von |first=Hermann |url=http://vlp.mpiwg-berlin.mpg.de/library/data/lit3483/index_html?pn=1&ws=1.5 |title=Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik |publisher=Leopold Voss |year=1863 |edition=1st |location=Leipzig |pages=v |language=de |trans-title=On the sensations of tone as a physiological basis for the theory of music}} Helmholtz believed that the psychological perception of tone color is subject to learning, while hearing in the sensory sense is purely physiological.{{Cite book |last=Christensen |first=Thomas Street |url=https://books.google.com/books?id=ioa9uW2t7AQC |title=The Cambridge History of Western Music |publisher=Cambridge University Press |year=2002 |isbn=978-0-521-62371-1 |location=Cambridge, United Kingdom |pages=251, 258}} 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.{{Cite book |last=Cahan |first=David |url=https://books.google.com/books?id=lfdJNRgzKyUC |title=Hermann von Helmholtz and the foundations of nineteenth-century science |publisher=University of California Press |year=1993 |isbn=978-0-520-08334-9 |editor-last=Cahan |editor-first=David |location=Berkeley and Los Angeles, USA |pages=110–114, 285–286}} Helmholtz agreed with the finding of Ernst Chladni from 1787 that certain sound sources have inharmonic vibration modes.

{{multiple image |direction=horizontal

|header = Rudolph Koenig's sound analyzer and synthesizer

|caption1 = sound synthesizer

|image1 = Synthesizer after Helmholtz by Koenig 1865.jpg |width1=231

|caption2 = sound analyzer

|image2 = Koenig - klankanalysator purchased in 1996.jpg |width2=102

}}

In Helmholtz's time, electronic amplification was unavailable. For synthesis of tones with harmonic partials, Helmholtz built an electrically excited array of tuning forks 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.{{Cite book |last=von Helmholtz |first=Hermann |url=https://archive.org/details/onsensationston00helmgoog |title=On the sensations of tone as a physiological basis for the theory of music |publisher=Longmans, Green, and co. |year=1875 |location=London, United Kingdom |pages=xii, 175–179}} For harmonic synthesis, Koenig also built a large apparatus based on his wave siren. It was pneumatic and utilized cut-out tonewheels, and was criticized for low purity of its partial tones. Also tibia pipes of pipe organs have nearly sinusoidal waveforms and can be combined in the manner of additive synthesis.{{Cite book |last=Miller |first=Dayton Clarence |author-link=Dayton Miller |url=https://archive.org/details/scienceofmusical028670mbp |title=The Science of Musical Sounds |publisher=The Macmillan Company |year=1926 |location=New York |pages=[https://archive.org/details/scienceofmusical028670mbp/page/n127 110], 244–248 |orig-date=1916}}

In 1938, with significant new supporting evidence,{{Cite book |last=Russell |first=George Oscar |author-link=George Oscar Russell |url=https://archive.org/details/yearbookcarne35193536carn |title=Year book - Carnegie Institution of Washington (1936) |publisher=Carnegie Institution of Washington |year=1936 |series=Carnegie Institution of Washington: Year Book |volume=35 |location=Washington |pages=[https://archive.org/details/yearbookcarne35193536carn/page/359 359]–363}} it was reported on the pages of Popular 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.{{Cite journal |last=Lodge |first=John E. |date=April 1938 |editor-last=Brown |editor-first=Raymond J. |title=Odd Laboratory Tests Show Us How We Speak: Using X Rays, Fast Movie Cameras, and Cathode-Ray Tubes, Scientists Are Learning New Facts About the Human Voice and Developing Teaching Methods To Make Us Better Talkers |url=https://books.google.com/books?id=wigDAAAAMBAJ&pg=PA32 |journal=Popular Science Monthly |location=New York, USA |publisher=Popular Science Publishing |volume=132 |issue=4 |pages=32–33}} 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.{{Cite journal |last=Comerford |first=P. |year=1993 |title=Simulating an Organ with Additive Synthesis |journal=Computer Music Journal |volume=17 |issue=2 |pages=55–65 |doi=10.2307/3680869 |jstor=3680869}} In a 1940 Institute of Radio Engineers 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".{{Cite journal |year=1940 |title=Institute News and Radio Notes |journal=Proceedings of the IRE |volume=28 |issue=10 |pages=487–494 |doi=10.1109/JRPROC.1940.228904}} 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.{{Cite journal |last=Douglas |first=A. |year=1948 |title=Electrotonic Music |journal=Proceedings of the Royal Musical Association |volume=75 |pages=1–12 |doi=10.1093/jrma/75.1.1}} 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.{{Cite book |last=Douglas |first=Alan Lockhart Monteith |url=https://archive.org/details/electricalproduc00doug |title=The Electrical Production of Music |publisher=Macdonald |year=1957 |location=London, UK |pages=[https://archive.org/details/electricalproduc00doug/page/140 140], 142 |url-access=limited}}

A typical modern additive synthesizer produces its output as an electrical, analog signal, or as digital audio, such as in the case of software synthesizers, which became popular around year 2000.{{Cite book |last=Pejrolo |first=Andrea |title=Acoustic and MIDI orchestration for the contemporary composer |last2=DeRosa |first2=Rich |publisher=Elsevier |year=2007 |location=Oxford, UK |pages=53–54}}

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. $y[n]\backslash ,$ and the argument $n\backslash ,$ can only be integer values. If the continuous-time synthesis output $y(t)\backslash ,$ is expected to be sufficiently bandlimited; below half the sampling rate or $f\_\backslash mathrm\{s\}/2\backslash ,$, 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. The sampling period is $T=1/f\_\backslash mathrm\{s\}\backslash ,$.

Beginning with ({{EquationNote|3}}),

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

and sampling at discrete times $t\; =\; nT\; =\; n/f\_\backslash mathrm\{s\}\; \backslash ,$ results in

:$\backslash begin\{align\}$

y[n] & = y(nT) = \sum_{k=1}^{K} r_k(nT) \cos\left(2 \pi \int_0^{nT} f_k(u)\ du + \phi_k \right) \\

& = \sum_{k=1}^{K} r_k(nT) \cos\left(2 \pi \sum_{i=1}^{n} \int_{(i-1)T}^{iT} f_k(u)\ du + \phi_k \right) \\

& = \sum_{k=1}^{K} r_k(nT) \cos\left(2 \pi \sum_{i=1}^{n} (T f_k[i]) + \phi_k \right) \\

& = \sum_{k=1}^{K} r_k[n] \cos\left(\frac{2 \pi}{f_\mathrm{s}} \sum_{i=1}^{n} f_k[i] + \phi_k \right) \\

\end{align}

where

: $r\_k[n]\; =\; r\_k(nT)\; \backslash ,$ is the discrete-time varying amplitude envelope

: $f\_k[n]\; =\; \backslash frac\{1\}\{T\}\; \backslash int\_\{(n-1)T\}^\{nT\}\; f\_k(t)\backslash \; dt\; \backslash ,$ is the discrete-time backward difference instantaneous frequency.

This is equivalent to

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

where

:$\backslash begin\{align\}$

\theta_k[n] &= \frac{2 \pi}{f_\mathrm{s}} \sum_{i=1}^{n} f_k[i] + \phi_k \\

&= \theta_k[n-1] + \frac{2 \pi}{f_\mathrm{s}} f_k[n] \\

\end{align} for all $n>0\backslash ,$

and

: $\backslash theta\_k[0]\; =\; \backslash phi\_k.\; \backslash ,$

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

{{Reflist}}

• [http://users.ece.gatech.edu/lanterma/synergy/ Digital Keyboards Synergy]

{{Sound synthesis types}}

{{DEFAULTSORT:Additive Synthesis}}

Category:Sound synthesis types