Editing Frequency modulation synthesis

{{short description|Form of sound synthesis}}
{|style="float:right;width:400px;text-align:center;" <!-- class="wikitable" -->
|+ '''FM synthesis using 2 operators'''
|-
|style="line-height:1.8ex;text-align:left;"|<small>[[Image:2op FM (large font).svg|left|200px]] A 220&nbsp;Hz carrier tone ''f<sub>c</sub>'' modulated by a 440&nbsp;Hz modulating tone ''f<sub>m</sub>'', with various choices of [[frequency modulation#Modulation index|frequency modulation index]], ''β''. The time domain signals are illustrated above, and the corresponding spectra are shown below (spectrum amplitudes in [[decibel|dB]]).</small>
|-
|
; Waveforms for each ''&beta;'':
[[Image:frequencymodulationdemo-td.png]]
|-
|
; Spectra for each ''&beta;'':
[[Image:frequencymodulationdemo-fd.png]]
|}
'''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 frequency|(instantaneous) frequency]] of an oscillator is altered in accordance with the [[amplitude]] of a modulating signal.<ref>{{harvnb|Dodge|Jerse|1997|p=115}}</ref>

FM synthesis can create both harmonic and [[Inharmonicity|inharmonic]] sounds. To synthesize harmonic sounds, the modulating signal must have a [[harmonic]] relationship to the original carrier signal. As the amount of frequency modulation increases, the sound grows progressively complex. Through the use of modulators with frequencies that are non-integer multiples of the carrier signal (i.e. inharmonic), inharmonic bell-like and percussive spectra can be created.

FM synthesis using analog [[oscillator]]s may result in pitch instability.<ref>{{Cite book |last1=McGuire |first1=Sam |url=https://books.google.com/books?id=S90NEAAAQBAJ&dq=analog+fm+synthesis+unstable&pg=SA3-PA11 |title=The Art of Digital Orchestration |last2=Matějů |first2=Zbyněk |date=2020-12-28 |publisher=CRC Press |isbn=978-1-000-28699-1 |language=en}}</ref> However, FM synthesis can also be implemented digitally, which is more stable and became standard practice.

== Applications ==
=== In synthesizers ===
Digital FM synthesis (equivalent to [[phase modulation]] using the time integration of [[instantaneous frequency]]) was the basis of several musical instruments beginning as early as 1974. Yamaha built the first prototype [[digital synthesizer]] in 1974, based on FM synthesis,<ref name=yamaha2014/> before commercially releasing the Yamaha GS-1 in 1980.<ref name="roads"/> The [[Synclavier|Synclavier I]], manufactured by [[New England Digital|New England Digital Corporation]] beginning in 1978, included a digital FM synthesizer, using an FM synthesis algorithm licensed from Yamaha.<ref name=mixmag2006>{{cite journal | title	= 1978 New England Digital Synclavier | url  	= http://www.mixonline.com/news/news-products/1978-new-england-digital-synclavier/383609 | date 	= September 1, 2006 | journal	= Mix | publisher	= Penton Media}}</ref> Yamaha's groundbreaking [[Yamaha DX7]] synthesizer, released in 1983, brought FM to the forefront of synthesis in the mid-1980s.<ref name=":6">{{Cite news |title=The top 10 classic synth presets (and where you can hear them) |language=EN-GB |work=[[MusicRadar]] |url=https://www.musicradar.com/news/tech/the-top-10-classic-synth-presets-and-where-you-can-hear-them-637677 |access-date=October 19, 2018}}</ref>

=== In PCs, arcades, game consoles, and mobile phones ===
FM synthesis also became the usual setting for games and software up until the mid-nineties. Sound cards for [[IBM PC compatible]] systems like the [[AdLib]] and [[Sound Blaster]] popularized [[Yamaha Corporation|Yamaha]] chips like the [[Yamaha YM3812|OPL2]] and [[Yamaha YMF262|OPL3]]. Other computers such as the Sharp [[X68000]] and [[MSX]] ([[Yamaha CX5M|Yamaha CX5M computer unit]]) utilize the [[Yamaha YM2151|OPM]] sound chip (with later CX5M units using the [[Yamaha YM2164|OPP]] sound chip). The [[NEC]] [[PC-88]] and [[PC-98]] computers use either the [[Yamaha YM2203|OPN]] and [[OPNA]] sound chips.

For arcade systems and game consoles, OPM was used in many arcade boards from the 1980s and 1990s (including [[Sega]]'s [[Sega System 16|System 16]] and [[Capcom]]'s [[CP System]] arcade boards); OPN was also used in some arcade boards in the 1980s. [[OPNB]] was notably used in [[SNK]]'s [[Neo Geo]] arcade (MVS) and home console (AES) machines, as well as being used as the main basic sound generator in [[Taito]]'s arcade boards (with a variant of the OPNB being used in the [[Taito Z System]] board). The related [[OPN2]] was used in Sega's [[Sega Genesis|Mega Drive (Genesis)]], [[Fujitsu]]'s [[FM Towns Marty]], and some of Sega's [[List of Sega arcade system boards|arcade boards]] (e.g. Sega System C-2 and Sega System 32) as one of its sound generator chips.

FM synthesis was also used on a wide range of mobile phones in the 2000s to play ringtones and other sounds, using the [[SMAF|Yamaha SMAF]] format.

==History==

=== Don Buchla (mid-1960s) ===
[[Don Buchla]] implemented FM on his instruments in the mid-1960s, prior to Chowning's patent. His 158, 258 and 259 dual oscillator modules had a specific FM control voltage input,<ref name="buchla100">
{{cite book
 | author    = Dr. Hubert Howe<!-- (Queens College, NY) -->
 | title     = Buchla Electronic Music System: Users Manual written for CBS Musical Instruments (Buchla 100 Owner's Manual)
 | url       = https://archive.org/details/synthmanual-buchla-100-owners-manual
 | publisher = Educational Research Department, CBS Musical Instruments, Columbia Broadcasting System
 | publication-date = 1960s
 | page      = [https://archive.org/download/synthmanual-buchla-100-owners-manual/buchla100ownersmanual.pdf#page=7 7]
 | quote     = ''At this point we may consider various additional signal modifications that we may wish to make to the series of tones produced by the above example. For instance, if we would like to add frequency modulation to the tones, it is necessary to patch another audio signal into the jack connected by a line to the middle dial on the Model 158 Dual Sine-Sawtooth Oscillator. ...''
}}
</ref> and the model 208 (Music Easel) had a modulation oscillator hard-wired to allow FM as well as AM of the primary oscillator.<ref name="buchla music easel">
{{cite book
 | author    = Atten Strange
 | title     = Programming and Metaprogramming in the Electro-Organism - An Operating Directive for the Music Easel
 | url       = https://archive.org/details/synthmanual-buchla-music-easel-owners-manual
 | publisher = Buchla and Associates
 | publication-date = 1974
}}
</ref> These early applications used analog oscillators, and this capability was also followed by other modular synthesizers and portable synthesizers including [[Minimoog]] and [[ARP Odyssey]].

=== John Chowning (late-1960s–1970s) ===
[[File:Chowning.jpg|thumb|Digital frequency modulation synthesis was developed by [[John Chowning]]]]

By the mid-20th century, [[frequency modulation]] (FM), a means of carrying sound, had been understood for decades and was being used to [[FM broadcasting|broadcast radio transmissions]]. FM synthesis was developed since 1967 at [[Stanford University]], California, by [[John Chowning]], {{citation needed span|through his exploration of digital synthesis and spatialization, inspired by the new possibilities of digital sound as described by [[Max Mathews]]|date=November 2024}}. His {{citation needed span|algorithm|date=March 2023|reason=next word "license" implies patent}}  was licensed to Japanese company [[Yamaha Corporation|Yamaha]] in 1973.<ref name="yamaha2014">{{cite web |ref={{sfnref|Yamaha|2014}} | title     = [Chapter 2] FM Tone Generators and the Dawn of Home Music Production | url       = http://usa.yamaha.com/products/music-production/synthesizers/synth_40th/history/chapter02/ | archive-url= https://web.archive.org/web/20170511080846/http://usa.yamaha.com/products/music-production/synthesizers/synth_40th/history/chapter02/ | archive-date=2017-05-11 | work      = Yamaha Synth 40th Anniversary - History | year      = 2014 | publisher = Yamaha Corporation}}</ref> The implementation commercialized by Yamaha (US Patent 4018121 Apr 1977<ref name="uspto">{{cite web|url=http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=4018121|publisher=patft.uspto.gov|title=U.S. Patent 4018121 Apr 1977|access-date=2017-04-30}}</ref> or U.S. Patent 4,018,121<ref name="patent">{{cite web|url=https://patents.google.com/patent/US4018121A/en=|title=Patent US4018121 - Method of synthesizing a musical sound - Google Patents |access-date=2017-04-30}}</ref>) {{citation needed span|is actually based on [[phase modulation]]|date=February 2023|reason=Although Yamaha's multiple-operator extension (FM Algorithm derived from their PAMS in the 1970s) seems possibly lacking the time integration of instantaneous frequency, clear sources are needed yet.}}, {{citation needed span|but the results end up being equivalent mathematically as both are essentially a special case of [[quadrature amplitude modulation]]|date=February 2023|reason=The omit of time integration of instantaneous frequency is not the equivalence}}.<ref>
{{cite web
 |author       = Rob Hordijk
 |title        = FM synthesis on Modular
 |url          = http://www.clavia.se/nordmodular/Modularzone/FMsynthesis.html
 |work         = Nord Modular & Micro Modular V3.03 tips & tricks
 |publisher    = Clavia DMI AB
 |access-date  = 2013-03-23
 |archive-url  = https://web.archive.org/web/20070407072439/http://www.clavia.se/nordmodular/Modularzone/FMsynthesis.html
 |archive-date = 2007-04-07
 }}</ref>

=== 1970s–1980s ===
==== Expansions by Yamaha ====
Yamaha's engineers began adapting Chowning's algorithm for use in a commercial digital synthesizer, adding improvements such as the "key scaling" method {{citation needed span|to avoid the introduction of distortion that normally occurred in analog systems during [[frequency modulation]]|date=March 2023|reason="distortion that normally occurred in analog systems" is not common. One possibility, it may be the result of Exponential FM, and not related to the Linear FM normally used on digital.}}, though it would take several years before Yamaha released their FM digital synthesizers.<ref name="holmes_257-8">{{cite book|title=Electronic and experimental music: technology, music, and culture|first=Thom|last=Holmes|edition=3rd|publisher=[[Taylor & Francis]]|year=2008|isbn=978-0-415-95781-6|chapter=Early Computer Music|pages=257–8|chapter-url=https://books.google.com/books?id=hCthQ-bec-QC&pg=PA257|access-date=2011-06-04}}</ref> In the 1970s, Yamaha were granted a number of patents, under the company's former name "Nippon Gakki Seizo Kabushiki Kaisha", evolving Chowning's work.<ref name="patent" /> Yamaha built the first prototype FM [[digital synthesizer]] in 1974.<ref name="yamaha2014" /> Yamaha eventually commercialized FM synthesis technology with the Yamaha GS-1, the first FM digital synthesizer, released in 1980.<ref name="roads">{{cite book|title=The computer music tutorial|author=Curtis Roads|publisher=[[MIT Press]]|year=1996|isbn=0-262-68082-3|page=226|url=https://books.google.com/books?id=nZ-TetwzVcIC&pg=PA226|access-date=2011-06-05}}</ref> FM synthesis was the basis of some of the early generations of [[digital synthesizer]]s, most notably those from Yamaha, as well as New England Digital Corporation under license from Yamaha.<ref name="mixmag2006"/> 

[[File:YAMAHA DX7.jpg|thumb|Yamaha DX7 FM digital synthesizer (1983)]]

Yamaha's [[Yamaha DX7|DX7]] synthesizer, released in 1983, was ubiquitous throughout the 1980s. Several other models by Yamaha provided variations and evolutions of FM synthesis during that decade.<ref name="SoS80s">{{cite web|url=http://www.soundonsound.com/sos/sep01/articles/retrofmpt2.asp|title=Sounds of the '80s Part 2: The Yamaha DX1 & Its Successors (Retro)|author=Gordon Reid|date=September 2001|work=Sound on Sound|archive-url=https://web.archive.org/web/20110917223333/http://www.soundonsound.com/sos/sep01/articles/retrofmpt2.asp|archive-date=17 September 2011|access-date=2011-06-29|df=dmy}}</ref>

Yamaha had patented its hardware implementation of FM in the 1970s,<ref name="patent"/> allowing it to nearly monopolize the market for FM technology until the mid-1990s.

==== Related development by Casio ====
[[Casio]] developed a related form of synthesis called [[phase distortion synthesis]], used in its [[Casio CZ synthesizers|CZ range of synthesizers]]. It had a similar (but slightly differently derived) sound quality to the DX series.

=== 1990s ===
==== Free use of FM after the patent expiration ====
With the expiration of the Stanford University FM patent in 1995, digital FM synthesis can now be implemented freely by other manufacturers. The FM synthesis patent brought Stanford $20&nbsp;million before it expired, making it (in 1994) "the second most lucrative licensing agreement in Stanford's history".<ref>Stanford University News Service (06/07/94), [http://news.stanford.edu/pr/94/940607Arc4222.html Music synthesis approaches sound quality of real instruments]</ref>

Today, FM is mostly found in software-based synths such as [[Native Instruments]]' FM8 or [[Image-Line]]'s [[Sytrus]] plug-ins, but it has also been incorporated into the synthesis repertoire of some modern digital synthesizers, usually coexisting as an option alongside other methods of synthesis such as [[subtractive synthesis|subtractive]], [[sample-based synthesis]], [[additive synthesis]], and other techniques. The degree of complexity of the FM in such hardware synths may vary from simple 2-operator FM, to the highly flexible 6-operator engines of the [[Korg Kronos]] and [[Alesis Fusion]], to creation of FM in extensively modular engines such as those in the latest synthesisers by [[Kurzweil Music Systems]].{{Citation needed|date=July 2019}}

==== Later use of FM and other technologies: Realtime Convolution & Modulation (AFM + Sample) and Formant Shaping Synthesis ====
The [[Yamaha SY99]]<ref name=YamahaSY99>{{cite web |title=Yamaha SY99 spec |url=https://jp.yamaha.com/products/music_production/synthesizers/sy99/specs.html |language=Japanese |website=[[Yamaha Corporation]]}}</ref> and [[Yamaha FS1R|FS1R]]<ref name=SOS1998>{{cite magazine |first1=Debbie |last1=Poyser |first2=Derek |last2=Johnson |date=1998 |title=Yamaha FS1R - FM Synthesis / Formant-shaping Tone Generator |url=https://www.soundonsound.com/reviews/yamaha-fs1r |magazine=[[Sound on Sound]] |issue=December 1998}}</ref> synthesizers marketed their highly powerful FM abilities as counterparts to [[sample-based synthesis]] and [[formant synthesis]] respectively. New hardware synths specifically marketed for their FM capabilities disappeared from the market after the release of FS1R in 1999, {{Citation needed span|however, well-developed FM synthesis options are a feature of [[Nord Lead]] synths manufactured by Clavia, the [[Alesis Fusion]] range, the [[Korg Oasys]] and [[Korg Kronos|Kronos]] and the Modor NF-1. Various other synthesizers offer limited FM abilities to supplement their main engines.|date=July 2019}}

The FS1R had 16 operators, 8 standard FM operators and 8 additional operators that used a noise source rather than an oscillator as its sound source. By adding in tuneable noise sources the FS1R could model the sounds produced in the human voice and in a wind instrument, along with making percussion instrument sounds. The FS1R also contained an additional wave form called the Formant wave form. Formants can be used to model resonating body instrument sounds like the cello, violin, acoustic guitar, bassoon, English horn, or human voice. Formants can even be found in the harmonic spectrum of several brass instruments.<ref name="zollinger2016" />

=== 2000s–present ===
==== Additional improvements: Variable Phase Modulation, FM-X Synthesis, Altered FM, etc. ====
{{expand section|date=February 2023}}
In 2016, [[Korg]] released the Korg Volca FM, a, 3-voice, 6 operators FM iteration of the Korg [[Volca]] series of compact, affordable desktop modules.<ref>[https://www.korg.com/us/products/dj/volca_fm/ Volca FM product page]</ref> Korg has also released the [https://www.korg.com/us/products/synthesizers/opsix/ opsix] (2020) and opsix SE (2023), integrating 6 operators FM synthesis with subtractive, analogue modeling, additive, semi-modular and Waveshaping.

Yamaha released the [[List of Yamaha products#Synthesizers|Montage]] in 2016, which combines a 128-voice sample-based engine with a 128-voice FM engine. This iteration of FM is called FM-X, and features 8 operators; each operator has a choice of several basic wave forms, but each wave form has several parameters to adjust its spectrum.<ref>[https://usa.yamaha.com/products/music_production/synthesizers/montage/features.html#product-tabs Yamaha Montage Product Features Page]</ref> It was then followed by the more affordable Yamaha [[List of Yamaha products#Synthesizers|MODX]] in 2018, with 64-voice, 8 operators FM-X architecture in addition to a 128-voice sample-based engine.<ref>[https://usa.yamaha.com/products/music_production/synthesizers/modx/features.html#product-tabs Yamaha MODX Product Features Page]</ref> The MODX+ released in 2022 increased the number of voices of the FM-X engine to 128, the same as with the Montage.<ref>[https://usa.yamaha.com/products/music_production/synthesizers/modxplus/features.html#product-tabs MODX8+, MODX7+, and MODX6+ Features]</ref> The Montage was succeeded by the Montage M in 2023, which uses the same 128-voice, 8 operators FM-X engine alongside a 128-voice sample-based engine and a newly-introduced 16-voice 3 oscillator analog-based engine known as AN-X.<ref>[https://usa.yamaha.com/products/music_production/synthesizers/montagem/index.html#d2153572 MONTAGE M Synthesizer]</ref>

Elektron launched the [[Elektron (company)#Music Hardware|Digitone]] in 2018, which is an 8-voice, 4 operators FM synth featuring Elektron's renowned sequence engine.<ref>[https://www.elektron.se/products/digitone/ Digitone product page]</ref>

FM-X synthesis was first introduced with the [[List of Yamaha products#Synthesizers|Yamaha Montage]] synthesizers in 2016. FM-X uses 8 operators. Each FM-X operator has a set of multi-spectral wave forms to choose from, which means each FM-X operator can be equivalent to a stack of 3 or 4 DX7 FM operators. The list of selectable wave forms includes sine waves, the All1 and All2 wave forms, the Odd1 and Odd2 wave forms, and the Res1 and Res2 wave forms. The sine wave selection works the same as the DX7 wave forms. The All1 and All2 wave forms are a saw-tooth wave form. The Odd1 and Odd2 wave forms are pulse or square waves. These two types of wave forms can be used to model the basic harmonic peaks in the bottom of the harmonic spectrum of most instruments.  The Res1 and Res2 wave forms move the spectral peak to a specific harmonic and can be used to model either triangular or rounded groups of harmonics further up in the spectrum of an instrument. Combining an All1 or Odd1 wave form with multiple Res1 (or Res2) wave forms (and adjusting their amplitudes) can model the harmonic spectrum of an instrument or sound.<ref name="zollinger2016">{{Cite web|url=http://javelinart.com/FM_Synthesis_of_Real_Instruments.pdf |archive-url=https://web.archive.org/web/20170925230705/http://javelinart.com/FM_Synthesis_of_Real_Instruments.pdf |archive-date=2017-09-25 |url-status=live|title=FM_Synthesis_of_Real_Instruments|last=Zollinger|first=W. Thor|date=Dec 2017}}</ref>  {{citation needed|date=August 2017}}

Combining sets of 8 FM operators with multi-spectral wave forms was first introduced in the FS1R, released in 1999 by Yamaha. It was able to achieve similar results to that of FM-X using 8 noise operators.

== Spectral analysis ==
[[File:DemoFMfreq.wav|thumb|2-operator demonstration: if the frequency of the modulator is lower than that of the carrier, the output note will be that of the modulator.]]
There are multiple variations of FM synthesis, including:
*Various operator arrangements (known as "FM Algorithms" in Yamaha terminology)
**2 operators
**Serial FM (multiple stages)
**Parallel FM (multiple modulators, multiple-carriers),
**Mix of them
*Various waveform of operators
**Sinusoidal waveform
**Other waveforms
*Additional modulation
**Linear FM
**Exponential FM (preceded by the [[anti-logarithm]] conversion for CV/oct. interface of analog synthesizers)
**[[Oscillator sync]] with FM
''etc''.

As the basic of these variations, we analyze the spectrum of 2 operators (linear FM synthesis using two sinusoidal operators) on the following.

=== 2 operators ===
The spectrum generated by FM synthesis with one modulator is expressed as follows:<ref>{{harvnb|Chowning|1973|pp=1–2}}</ref><ref>
{{cite web
 | last      = Doering | first = Ed
 | title     = Frequency Modulation Mathematics
 | url       = http://cnx.org/content/m15482/latest/
 | access-date =  2013-04-11
 }}</ref>

For modulation signal <math>m(t) = B\,\sin(\omega_m t)\,</math>, the carrier signal is:<ref group="note">Note that modulation signal <math>m(t)</math> as [[instantaneous frequency]] is converted to the [[phase (waves)|phase]] of carrier signal <math>FM(t)</math>, by time integral between <math>[0, t]</math>.</ref>
:<math>\begin{align}
FM(t)	& \ =\ A\,\sin\left(\,\int_0^t \left(\omega_c + B\,\sin(\omega_m\,\tau)\right)d\tau\right) \\
	& \ =\ A\,\sin\left(\omega_c\,t - \frac{B}{\omega_m}\left(\cos(\omega_m\,t) - 1\right)\right) \\
	& \ =\ A\,\sin\left(\omega_c\,t + \frac{B}{\omega_m}\left(\sin(\omega_m\,t - \pi/2) + 1\right)\right) \\
\end{align}</math>

If we were to ignore the constant phase terms on the carrier <math>\phi_c = B/\omega_m\,</math> and the modulator <math>\phi_m = - \pi/2\,</math>, finally we would get the following expression, as seen on {{harvnb|Chowning|1973}} and {{harvnb|Roads|1996|p=[https://books.google.com/books?id=nZ-TetwzVcIC&&pg=PA232 232]}}:
:<math>\begin{align}
FM(t)	& \ \approx\ A\,\sin\left(\omega_c\,t + \beta\,\sin(\omega_m\,t)\right) \\
	& \ =\ A\left( J_0(\beta) \sin(\omega_c\,t)
	     + \sum_{n=1}^{\infty} J_n(\beta)\left[\,\sin((\omega_c+n\,\omega_m)\,t)\ +\ (-1)^{n}\sin((\omega_c-n\,\omega_m)\,t)\,\right] \right) \\
	& \ =\ A\sum_{n=-\infty}^{\infty} J_n(\beta)\,\sin((\omega_c+n\,\omega_m)\,t)
\end{align}</math>
where <math>\omega_c\,,\,\omega_m\,</math> are [[angular frequency|angular frequencies]] (<math>\,\omega = 2\pi f\,</math>) of carrier and modulator, <math>\beta = B / \omega_m\,</math> is [[frequency modulation#Modulation index|frequency modulation index]], and [[amplitude]]s <math>J_n(\beta)\,</math> is <math>n\,</math>-th [[Bessel function#Bessel functions of the first kind : Jα|Bessel function of first kind]], respectively.<ref group="note">The above expression is transformed using [[List of trigonometric identities#Angle sum and difference identities|trigonometric addition formulas]]
: <math>\begin{align}
	\sin(x \pm y) &= \sin x \cos y \pm \cos x \sin y
    \end{align}</math>
and a lemma of Bessel function

: <math>\begin{align}
	\cos(\beta\sin \theta) & = J_0(\beta) + 2\sum_{n=1}^{\infty}J_{2n}(\beta)\cos(2n\theta) \\
	\sin(\beta\sin \theta) & = 2\sum_{n=0}^{\infty}J_{2n+1}(\beta)\sin((2n+1)\theta)
    \end{align}</math>
: ('''Source''': {{harvnb|Kreh|2012}})
as following:

: <math>\begin{align}
	& \sin\left(\theta_c + \beta\,\sin(\theta_m)\right) \\
	& \ =\ \sin(\theta_c)\cos(\beta\sin(\theta_m)) + \cos(\theta_c)\sin(\beta\sin(\theta_m)) \\
	& \ =\ \sin(\theta_c)\left[J_0(\beta) + 2\sum_{n=1}^{\infty}J_{2n}(\beta)\cos(2n \theta_m)\right] 
	     + \cos(\theta_c)\left[2\sum_{n=0}^{\infty}J_{2n+1}(\beta)\sin((2n+1)\theta_m)\right] \\
	& \ =\ J_0(\beta) \sin(\theta_c)
        +  J_1(\beta) 2\cos(\theta_c)\sin(\theta_m)
    	+  J_2(\beta) 2\sin(\theta_c)\cos(2\theta_m)
    	+  J_3(\beta) 2\cos(\theta_c)\sin(3\theta_m)
        +  ... \\
	& \ =\ J_0(\beta) \sin(\theta_c)
	     + \sum_{n=1}^{\infty} J_n(\beta)\left[\,\sin(\theta_c + n\theta_m)\ +\ (-1)^{n}\sin(\theta_c - n\theta_m)\,\right] \\
	& \ =\ \sum_{n=-\infty}^{\infty} J_n(\beta)\,\sin(\theta_c + n\theta_m)\qquad(\because\ J_{-n}(x) = (-1)^n J_{n}(x))
    \end{align}</math></ref>

==See also==
*[[Additive synthesis]]
*[[Chiptune]]
*[[Digital synthesizer]]
*[[Electronic music]]
*[[Sound card]]
*[[Sound chip]]
*[[Video game music]]

==References==

=== Footnotes ===
{{reflist|group=note}}

===Citations===
{{reflist}}

===Bibliography===
{{div col}}
* {{cite journal 
 | last      = Chowning | first = J.
 | year      = 1973
 | title     = The Synthesis of Complex Audio Spectra by Means of Frequency Modulation
 | url       = https://web.eecs.umich.edu/~fessler/course/100/misc/chowning-73-tso.pdf
 | journal   = Journal of the Audio Engineering Society
 | volume    = 21 | issue = 7
 }}
* {{cite book 
 | last1     = Chowning | first1 = John
 | last2     = Bristow  | first2 = David
 | year      = 1986
 | title     = FM Theory & Applications - By Musicians For Musicians
 | publisher = Yamaha   | location  = Tokyo
 | isbn      = 4-636-17482-8
 }}
* {{cite book 
 | last1     = Dodge    | first1 = Charles
 | last2     = Jerse    | first2 = Thomas A.
 | year      = 1997
 | title     = Computer Music: Synthesis, Composition and Performance
 | publisher = Schirmer Books | location = New York
 | isbn      = 0-02-864682-7
}}
*{{citation|last=Kreh|first=Martin|title=Bessel Functions |url=http://www.math.psu.edu/papikian/Kreh.pdf |pages=[https://web.archive.org/web/20171118214412/http://www.math.psu.edu/papikian/Kreh.pdf#page=7 5]–6 |date=2012|archive-url=https://web.archive.org/web/20171118214412/http://www.math.psu.edu/papikian/Kreh.pdf |archive-date=2017-11-18 |access-date=2014-08-22 |website=The Pennsylvania State University}}
* {{cite book 
 | last      = Roads | first = Curtis
 | year      = 1996
 | title     = The Computer Music Tutorial
 | url       = https://books.google.com/books?id=nZ-TetwzVcIC
 | publisher = MIT Press
 | isbn      = 978-0-262-68082-0
 }}
{{div col end}}

== External links ==
* [http://ccrma.stanford.edu/software/snd/snd/fm.html An Introduction To FM], by Bill Schottstaedt
* [https://www.sfu.ca/~truax/fmtut.html FM tutorial]
* [http://www.soundonsound.com/sos/apr00/articles/synthsecrets.htm Synth Secrets, Part 12: An Introduction To Frequency Modulation], by Gordon Reid
* [http://www.soundonsound.com/sos/may00/articles/synth.htm Synth Secrets, Part 13: More On Frequency Modulation], by Gordon Reid
* [http://www.soundonsound.com/sos/1997_articles/sep97/synthschool3.html Paul Wiffens Synth School: Part 3]
* [http://yala.freeservers.com/2fmsynth.htm#2Mod F.M. Synthesis including complex operator analysis] [http://sites.google.com/view/yala-music1/fm-synthesis mirror site of F.M. Synthesis, 2019]

{{Sound synthesis types}}

[[Category:Sound synthesis types]]