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<title>Frequency modulation synthesis</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Frequency modulation synthesis</span></span>

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<caption><b>FM synthesis using 2 operators</b>

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<td style="line-height:1.8ex;text-align:left;"><small> A 220 Hz carrier tone <i>f<sub>c</sub></i> modulated by a 440 Hz modulating tone <i>f<sub>m</sub></i>, with various choices of <a href="Frequency_modulation#Modulation_index" title="Frequency modulation">frequency modulation index</a>, <i>β</i>. The time domain signals are illustrated above, and the corresponding spectra are shown below (spectrum amplitudes in <a href="Decibel" title="Decibel">dB</a>).</small>

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<dl><dt>Waveforms for each <i>β</i></dt>

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<dl><dt>Spectra for each <i>β</i></dt>

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<p><b>Frequency modulation synthesis</b> (or <b>FM synthesis</b>) is a form of <a href="Synthesizer#Sound_synthesis" title="Synthesizer">sound synthesis</a> whereby the frequency of a <a href="Waveform" title="Waveform">waveform</a> is changed by <a href="Frequency_modulation" title="Frequency modulation">modulating its frequency</a> with a modulator. The <a href="Instantaneous_frequency" class="mw-redirect" title="Instantaneous frequency">(instantaneous) frequency</a> of an oscillator is altered in accordance with the <a href="Amplitude" title="Amplitude">amplitude</a> of a modulating signal.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>FM synthesis can create both harmonic and <a href="Inharmonicity" title="Inharmonicity">inharmonic</a> sounds. To synthesize harmonic sounds, the modulating signal must have a <a href="Harmonic" title="Harmonic">harmonic</a> 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.

</p><p>FM synthesis using analog <a href="Oscillator" class="mw-redirect" title="Oscillator">oscillators</a> may result in pitch instability.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> However, FM synthesis can also be implemented digitally, which is more stable and became standard practice.

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="In_synthesizers">In synthesizers</h3></div>

<p>Digital FM synthesis (equivalent to <a href="Phase_modulation" title="Phase modulation">phase modulation</a> using the time integration of <a href="Instantaneous_frequency" class="mw-redirect" title="Instantaneous frequency">instantaneous frequency</a>) was the basis of several musical instruments beginning as early as 1974. Yamaha built the first prototype <a href="Digital_synthesizer" title="Digital synthesizer">digital synthesizer</a> in 1974, based on FM synthesis,<sup id="cite_ref-yamaha2014_3-0" class="reference"><a href="#cite_note-yamaha2014-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> before commercially releasing the Yamaha GS-1 in 1980.<sup id="cite_ref-roads_4-0" class="reference"><a href="#cite_note-roads-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The <a href="Synclavier" title="Synclavier">Synclavier I</a>, manufactured by <a href="New_England_Digital" title="New England Digital">New England Digital Corporation</a> beginning in 1978, included a digital FM synthesizer, using an FM synthesis algorithm licensed from Yamaha.<sup id="cite_ref-mixmag2006_5-0" class="reference"><a href="#cite_note-mixmag2006-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Yamaha's groundbreaking <a href="Yamaha_DX7" title="Yamaha DX7">Yamaha DX7</a> synthesizer, released in 1983, brought FM to the forefront of synthesis in the mid-1980s.<sup id="cite_ref-:6_6-0" class="reference"><a href="#cite_note-:6-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="In_PCs,_arcades,_game_consoles,_and_mobile_phones">In PCs, arcades, game consoles, and mobile phones</h3></div>

<p>FM synthesis also became the usual setting for games and software up until the mid-nineties. Sound cards for <a href="IBM_PC_compatible" title="IBM PC compatible">IBM PC compatible</a> systems like the <a href="AdLib" class="mw-redirect" title="AdLib">AdLib</a> and <a href="Sound_Blaster" title="Sound Blaster">Sound Blaster</a> popularized <a href="Yamaha_Corporation" title="Yamaha Corporation">Yamaha</a> chips like the <a href="Yamaha_YM3812" class="mw-redirect" title="Yamaha YM3812">OPL2</a> and <a href="Yamaha_YMF262" class="mw-redirect" title="Yamaha YMF262">OPL3</a>. Other computers such as the Sharp <a href="X68000" title="X68000">X68000</a> and <a href="MSX" title="MSX">MSX</a> (<a href="Yamaha_CX5M" title="Yamaha CX5M">Yamaha CX5M computer unit</a>) utilize the <a href="Yamaha_YM2151" title="Yamaha YM2151">OPM</a> sound chip (with later CX5M units using the <a href="Yamaha_YM2164" title="Yamaha YM2164">OPP</a> sound chip). The <a href="NEC" title="NEC">NEC</a> <a href="PC-88" class="mw-redirect" title="PC-88">PC-88</a> and <a href="PC-98" title="PC-98">PC-98</a> computers use either the <a href="Yamaha_YM2203" title="Yamaha YM2203">OPN</a> and <a href="OPNA" class="mw-redirect" title="OPNA">OPNA</a> sound chips.

</p><p>For arcade systems and game consoles, OPM was used in many arcade boards from the 1980s and 1990s (including <a href="Sega" title="Sega">Sega</a>'s <a href="Sega_System_16" class="mw-redirect" title="Sega System 16">System 16</a> and <a href="Capcom" title="Capcom">Capcom</a>'s <a href="CP_System" title="CP System">CP System</a> arcade boards); OPN was also used in some arcade boards in the 1980s. <a href="OPNB" class="mw-redirect" title="OPNB">OPNB</a> was notably used in <a href="SNK" title="SNK">SNK</a>'s <a href="Neo_Geo" title="Neo Geo">Neo Geo</a> arcade (MVS) and home console (AES) machines, as well as being used as the main basic sound generator in <a href="Taito" title="Taito">Taito</a>'s arcade boards (with a variant of the OPNB being used in the <a href="Taito_Z_System" class="mw-redirect" title="Taito Z System">Taito Z System</a> board). The related <a href="OPN2" class="mw-redirect" title="OPN2">OPN2</a> was used in Sega's <a href="Sega_Genesis" title="Sega Genesis">Mega Drive (Genesis)</a>, <a href="Fujitsu" title="Fujitsu">Fujitsu</a>'s <a href="FM_Towns_Marty" title="FM Towns Marty">FM Towns Marty</a>, and some of Sega's <a href="List_of_Sega_arcade_system_boards" title="List of Sega arcade system boards">arcade boards</a> (e.g. Sega System C-2 and Sega System 32) as one of its sound generator chips.

</p><p>FM synthesis was also used on a wide range of mobile phones in the 2000s to play ringtones and other sounds, using the <a href="SMAF" class="mw-redirect" title="SMAF">Yamaha SMAF</a> format.

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Don_Buchla_(mid-1960s)">Don Buchla (mid-1960s)</h3></div>

<p><a href="Don_Buchla" title="Don Buchla">Don Buchla</a> 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,<sup id="cite_ref-buchla100_7-0" class="reference"><a href="#cite_note-buchla100-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and the model 208 (Music Easel) had a modulation oscillator hard-wired to allow FM as well as AM of the primary oscillator.<sup id="cite_ref-buchla_music_easel_8-0" class="reference"><a href="#cite_note-buchla_music_easel-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> These early applications used analog oscillators, and this capability was also followed by other modular synthesizers and portable synthesizers including <a href="Minimoog" title="Minimoog">Minimoog</a> and <a href="ARP_Odyssey" title="ARP Odyssey">ARP Odyssey</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="John_Chowning_(late-1960s–1970s)">John Chowning (late-1960s–1970s)</h3></div>

<p>By the mid-20th century, <a href="Frequency_modulation" title="Frequency modulation">frequency modulation</a> (FM), a means of carrying sound, had been understood for decades and was being used to <a href="FM_broadcasting" title="FM broadcasting">broadcast radio transmissions</a>. FM synthesis was developed since 1967 at <a href="Stanford_University" title="Stanford University">Stanford University</a>, California, by <a href="John_Chowning" title="John Chowning">John Chowning</a>, <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">through his exploration of digital synthesis and spatialization, inspired by the new possibilities of digital sound as described by <a href="Max_Mathews" title="Max Mathews">Max Mathews</a></span>. His <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">algorithm</span> was licensed to Japanese company <a href="Yamaha_Corporation" title="Yamaha Corporation">Yamaha</a> in 1973.<sup id="cite_ref-yamaha2014_3-1" class="reference"><a href="#cite_note-yamaha2014-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The implementation commercialized by Yamaha (US Patent 4018121 Apr 1977<sup id="cite_ref-uspto_9-0" class="reference"><a href="#cite_note-uspto-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> or U.S. Patent 4,018,121<sup id="cite_ref-patent_10-0" class="reference"><a href="#cite_note-patent-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>) <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">is actually based on <a href="Phase_modulation" title="Phase modulation">phase modulation</a></span>, <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">but the results end up being equivalent mathematically as both are essentially a special case of <a href="Quadrature_amplitude_modulation" title="Quadrature amplitude modulation">quadrature amplitude modulation</a></span>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading4"><h4 id="Expansions_by_Yamaha">Expansions by Yamaha</h4></div>

<p>Yamaha's engineers began adapting Chowning's algorithm for use in a commercial digital synthesizer, adding improvements such as the "key scaling" method <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">to avoid the introduction of distortion that normally occurred in analog systems during <a href="Frequency_modulation" title="Frequency modulation">frequency modulation</a></span>, though it would take several years before Yamaha released their FM digital synthesizers.<sup id="cite_ref-holmes_257-8_12-0" class="reference"><a href="#cite_note-holmes_257-8-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-patent_10-1" class="reference"><a href="#cite_note-patent-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Yamaha built the first prototype FM <a href="Digital_synthesizer" title="Digital synthesizer">digital synthesizer</a> in 1974.<sup id="cite_ref-yamaha2014_3-2" class="reference"><a href="#cite_note-yamaha2014-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Yamaha eventually commercialized FM synthesis technology with the Yamaha GS-1, the first FM digital synthesizer, released in 1980.<sup id="cite_ref-roads_4-1" class="reference"><a href="#cite_note-roads-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> FM synthesis was the basis of some of the early generations of <a href="Digital_synthesizer" title="Digital synthesizer">digital synthesizers</a>, most notably those from Yamaha, as well as New England Digital Corporation under license from Yamaha.<sup id="cite_ref-mixmag2006_5-1" class="reference"><a href="#cite_note-mixmag2006-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

<p>Yamaha's <a href="Yamaha_DX7" title="Yamaha DX7">DX7</a> synthesizer, released in 1983, was ubiquitous throughout the 1980s. Several other models by Yamaha provided variations and evolutions of FM synthesis during that decade.<sup id="cite_ref-SoS80s_13-0" class="reference"><a href="#cite_note-SoS80s-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p><p>Yamaha had patented its hardware implementation of FM in the 1970s,<sup id="cite_ref-patent_10-2" class="reference"><a href="#cite_note-patent-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> allowing it to nearly monopolize the market for FM technology until the mid-1990s.

</p>

<div class="mw-heading mw-heading4"><h4 id="Related_development_by_Casio">Related development by Casio</h4></div>

<p><a href="Casio" title="Casio">Casio</a> developed a related form of synthesis called <a href="Phase_distortion_synthesis" title="Phase distortion synthesis">phase distortion synthesis</a>, used in its <a href="Casio_CZ_synthesizers" title="Casio CZ synthesizers">CZ range of synthesizers</a>. It had a similar (but slightly differently derived) sound quality to the DX series.

</p>

<div class="mw-heading mw-heading4"><h4 id="Free_use_of_FM_after_the_patent_expiration">Free use of FM after the patent expiration</h4></div>

<p>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 million before it expired, making it (in 1994) "the second most lucrative licensing agreement in Stanford's history".<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>

</p><p>Today, FM is mostly found in software-based synths such as <a href="Native_Instruments" title="Native Instruments">Native Instruments</a>' FM8 or <a href="Image-Line" title="Image-Line">Image-Line</a>'s <a href="Sytrus" title="Sytrus">Sytrus</a> 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 <a href="Subtractive_synthesis" title="Subtractive synthesis">subtractive</a>, <a href="Sample-based_synthesis" title="Sample-based synthesis">sample-based synthesis</a>, <a href="Additive_synthesis" title="Additive synthesis">additive synthesis</a>, 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 <a href="Korg_Kronos" title="Korg Kronos">Korg Kronos</a> and <a href="Alesis_Fusion" title="Alesis Fusion">Alesis Fusion</a>, to creation of FM in extensively modular engines such as those in the latest synthesisers by <a href="Kurzweil_Music_Systems" title="Kurzweil Music Systems">Kurzweil Music Systems</a>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Later_use_of_FM_and_other_technologies:_Realtime_Convolution_&_Modulation_(AFM_+_Sample)_and_Formant_Shaping_Synthesis">Later use of FM and other technologies: Realtime Convolution & Modulation (AFM + Sample) and Formant Shaping Synthesis</h4></div>

<p>The <a href="Yamaha_SY99" title="Yamaha SY99">Yamaha SY99</a><sup id="cite_ref-YamahaSY99_15-0" class="reference"><a href="#cite_note-YamahaSY99-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and <a href="Yamaha_FS1R" title="Yamaha FS1R">FS1R</a><sup id="cite_ref-SOS1998_16-0" class="reference"><a href="#cite_note-SOS1998-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> synthesizers marketed their highly powerful FM abilities as counterparts to <a href="Sample-based_synthesis" title="Sample-based synthesis">sample-based synthesis</a> and <a href="Formant_synthesis" class="mw-redirect" title="Formant synthesis">formant synthesis</a> respectively. New hardware synths specifically marketed for their FM capabilities disappeared from the market after the release of FS1R in 1999, <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">however, well-developed FM synthesis options are a feature of <a href="Nord_Lead" title="Nord Lead">Nord Lead</a> synths manufactured by Clavia, the <a href="Alesis_Fusion" title="Alesis Fusion">Alesis Fusion</a> range, the <a href="Korg_Oasys" class="mw-redirect" title="Korg Oasys">Korg Oasys</a> and <a href="Korg_Kronos" title="Korg Kronos">Kronos</a> and the Modor NF-1. Various other synthesizers offer limited FM abilities to supplement their main engines.</span>

</p><p>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.<sup id="cite_ref-zollinger2016_17-0" class="reference"><a href="#cite_note-zollinger2016-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="2000s–present">2000s–present</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Additional_improvements:_Variable_Phase_Modulation,_FM-X_Synthesis,_Altered_FM,_etc.">Additional improvements: Variable Phase Modulation, FM-X Synthesis, Altered FM, etc.</h4></div>

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<p>In 2016, <a href="Korg" title="Korg">Korg</a> released the Korg Volca FM, a, 3-voice, 6 operators FM iteration of the Korg <a href="Volca" class="mw-redirect" title="Volca">Volca</a> series of compact, affordable desktop modules.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Korg has also released the <a rel="nofollow" class="external text" href="https://www.korg.com/us/products/synthesizers/opsix/">opsix</a> (2020) and opsix SE (2023), integrating 6 operators FM synthesis with subtractive, analogue modeling, additive, semi-modular and Waveshaping.

</p><p>Yamaha released the <a href="List_of_Yamaha_products" class="mw-redirect" title="List of Yamaha products">Montage</a> 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.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> It was then followed by the more affordable Yamaha <a href="List_of_Yamaha_products" class="mw-redirect" title="List of Yamaha products">MODX</a> in 2018, with 64-voice, 8 operators FM-X architecture in addition to a 128-voice sample-based engine.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The MODX+ released in 2022 increased the number of voices of the FM-X engine to 128, the same as with the Montage.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

</p><p>Elektron launched the <a href="Elektron_(company)#Music_Hardware" title="Elektron (company)">Digitone</a> in 2018, which is an 8-voice, 4 operators FM synth featuring Elektron's renowned sequence engine.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>

</p><p>FM-X synthesis was first introduced with the <a href="List_of_Yamaha_products" class="mw-redirect" title="List of Yamaha products">Yamaha Montage</a> 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.<sup id="cite_ref-zollinger2016_17-1" class="reference"><a href="#cite_note-zollinger2016-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p><p>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.

</p>

<div class="mw-heading mw-heading2"><h2 id="Spectral_analysis">Spectral analysis</h2></div>

<p>There are multiple variations of FM synthesis, including:

</p>

<ul><li>Various operator arrangements (known as "FM Algorithms" in Yamaha terminology)

<ul><li>2 operators</li>

<li>Serial FM (multiple stages)</li>

<li>Parallel FM (multiple modulators, multiple-carriers),</li>

<li>Various waveform of operators

<ul><li>Sinusoidal waveform</li>

<li>Other waveforms</li></ul></li>

<li>Additional modulation

<ul><li>Linear FM</li>

<li>Exponential FM (preceded by the <a href="Anti-logarithm" class="mw-redirect" title="Anti-logarithm">anti-logarithm</a> conversion for CV/oct. interface of analog synthesizers)</li>

<li><a href="Oscillator_sync" title="Oscillator sync">Oscillator sync</a> with FM</li></ul></li></ul>

<p><i>etc</i>.

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

</p>

<div class="mw-heading mw-heading3"><h3 id="2_operators">2 operators</h3></div>

<p>The spectrum generated by FM synthesis with one modulator is expressed as follows:<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>

</p><p>For modulation signal <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(t)=B\,\sin(\omega _{m}t)\,}">

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</math></span><img src="./877edb2026cd934a42cb1d4746f2f30d2ed7c0df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.338ex; height:2.843ex;" alt="{\displaystyle m(t)=B\,\sin(\omega _{m}t)\,}" loading="lazy"></span>, the carrier signal is:<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

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</math></span><img src="./ee7642f6eacb573f45e814032a3af664b7507245.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:52.76ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}" loading="lazy"></span></dd></dl>

<p>If we were to ignore the constant phase terms on the carrier <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{c}=B/\omega _{m}\,}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{c}=B/\omega _{m}\,}</annotation>

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</math></span><img src="./140cdf2d2db5bcc428d76ec612a8f764555eb6a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.862ex; height:2.843ex;" alt="{\displaystyle \phi _{c}=B/\omega _{m}\,}" loading="lazy"></span> and the modulator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{m}=-\pi /2\,}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{m}=-\pi /2\,}</annotation>

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</math></span><img src="./8f34ea719306cba4b65afb04d41300f244df6781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.011ex; height:2.843ex;" alt="{\displaystyle \phi _{m}=-\pi /2\,}" loading="lazy"></span>, finally we would get the following expression, as seen on <a href="#CITEREFChowning1973">Chowning 1973</a> and <a href="#CITEREFRoads1996">Roads 1996</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nZ-TetwzVcIC&&pg=PA232">232</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

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</math></span><img src="./2245df5dbe5a6c9f04835df2d4e89f07728a81e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:92.552ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{c}\,,\,\omega _{m}\,}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{c}\,,\,\omega _{m}\,}</annotation>

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</math></span><img src="./49137a668444f7aed72e95bb3d8619e69c087fe5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.706ex; height:2.009ex;" alt="{\displaystyle \omega _{c}\,,\,\omega _{m}\,}" loading="lazy"></span> are <a href="Angular_frequency" title="Angular frequency">angular frequencies</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\omega =2\pi f\,}">

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</math></span><img src="./75df73e4c9db6ce9e8cad78e79fbda1ed371f531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.092ex; height:2.509ex;" alt="{\displaystyle \,\omega =2\pi f\,}" loading="lazy"></span>) of carrier and modulator, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =B/\omega _{m}\,}">

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</math></span><img src="./2c9cc23de50d64e4c08a6baf7d1574f17c7c1f45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.865ex; height:2.843ex;" alt="{\displaystyle \beta =B/\omega _{m}\,}" loading="lazy"></span> is <a href="Frequency_modulation#Modulation_index" title="Frequency modulation">frequency modulation index</a>, and <a href="Amplitude" title="Amplitude">amplitudes</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{n}(\beta )\,}">

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</math></span><img src="./dbf6efb055ed7299405aa976490105e93b822836.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.037ex; height:2.843ex;" alt="{\displaystyle J_{n}(\beta )\,}" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\,}">

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</math></span><img src="./205e33e6845813cc72ca346b896a7945f90ca373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle n\,}" loading="lazy"></span>-th <a href="Bessel_function#Bessel_functions_of_the_first_kind_:_Jα" title="Bessel function">Bessel function of first kind</a>, respectively.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Additive_synthesis" title="Additive synthesis">Additive synthesis</a></li>

<li><a href="Chiptune" title="Chiptune">Chiptune</a></li>

<li><a href="Digital_synthesizer" title="Digital synthesizer">Digital synthesizer</a></li>

<li><a href="Electronic_music" title="Electronic music">Electronic music</a></li>

<li><a href="Sound_card" title="Sound card">Sound card</a></li>

<li><a href="Sound_chip" title="Sound chip">Sound chip</a></li>

<li><a href="Video_game_music" title="Video game music">Video game music</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Footnotes">Footnotes</h3></div>

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<li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Note that modulation signal <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(t)}">

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</math></span><img src="./ea26578dd72bf4dbc3fa391c9feb11eed495699b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:2.843ex;" alt="{\displaystyle m(t)}" loading="lazy"></span> as <a href="Instantaneous_frequency" class="mw-redirect" title="Instantaneous frequency">instantaneous frequency</a> is converted to the <a href="Phase_(waves)" title="Phase (waves)">phase</a> of carrier signal <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FM(t)}">

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</math></span><img src="./47c85e8057155b46040b87ed062ea4d3038b30f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.832ex; height:2.843ex;" alt="{\displaystyle FM(t)}" loading="lazy"></span>, by time integral between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,t]}">

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</li>

<li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">The above expression is transformed using <a href="List_of_trigonometric_identities#Angle_sum_and_difference_identities" title="List of trigonometric identities">trigonometric addition formulas</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(x\pm y)&=\sin x\cos y\pm \cos x\sin y\end{aligned}}}</annotation>

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</math></span><img src="./9b4088a1437c06844604aba3dc25087b334a805e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.907ex; height:2.843ex;" alt="{\displaystyle {\begin{aligned}\sin(x\pm y)&=\sin x\cos y\pm \cos x\sin y\end{aligned}}}" loading="lazy"></span></dd></dl>

and a lemma of Bessel function

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

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</math></span><img src="./ee2f3ba5bd350d90d227ba636b8445c7dc40e3c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:43.867ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\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{aligned}}}" loading="lazy"></span></dd>

<dd>(<b>Source</b>: <a href="#CITEREFKreh2012">Kreh 2012</a>)</dd></dl>

as following:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\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{aligned}}}</annotation>

</semantics>

</math></span><img src="./75361ca6bdbbe660138b6b060475a5284a9d7b9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.005ex; width:98.376ex; height:31.176ex;" alt="{\displaystyle {\begin{aligned}&\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{aligned}}}" loading="lazy"></span></dd></dl>

</span></li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFDodgeJerse1997">Dodge & Jerse 1997</a>, p. 115</span>

</li>

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</style><cite id="CITEREFMcGuireMatějů2020" class="citation book cs1">McGuire, Sam; Matějů, Zbyněk (2020-12-28). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=S90NEAAAQBAJ&dq=analog+fm+synthesis+unstable&pg=SA3-PA11"><i>The Art of Digital Orchestration</i></a>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-000-28699-1</bdi>.</cite></span>

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<li id="cite_note-yamaha2014-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-yamaha2014_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-yamaha2014_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-yamaha2014_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFYamaha2014" class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170511080846/http://usa.yamaha.com/products/music-production/synthesizers/synth_40th/history/chapter02/">"[Chapter 2] FM Tone Generators and the Dawn of Home Music Production"</a>. <i>Yamaha Synth 40th Anniversary - History</i>. Yamaha Corporation. 2014. Archived from <a rel="nofollow" class="external text" href="http://usa.yamaha.com/products/music-production/synthesizers/synth_40th/history/chapter02/">the original</a> on 2017-05-11.</cite></span>

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<li id="cite_note-roads-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-roads_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-roads_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFCurtis_Roads1996" class="citation book cs1">Curtis Roads (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nZ-TetwzVcIC&pg=PA226"><i>The computer music tutorial</i></a>. <a href="MIT_Press" title="MIT Press">MIT Press</a>. p. 226. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-68082-3</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">2011-06-05</span></span>.</cite></span>

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<li id="cite_note-mixmag2006-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-mixmag2006_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mixmag2006_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="http://www.mixonline.com/news/news-products/1978-new-england-digital-synclavier/383609">"1978 New England Digital Synclavier"</a>. <i>Mix</i>. Penton Media. September 1, 2006.</cite></span>

</li>

<li id="cite_note-:6-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-:6_6-0">^</a></b></span> <span class="reference-text"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.musicradar.com/news/tech/the-top-10-classic-synth-presets-and-where-you-can-hear-them-637677">"The top 10 classic synth presets (and where you can hear them)"</a>. <i><a href="MusicRadar" class="mw-redirect" title="MusicRadar">MusicRadar</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">October 19,</span> 2018</span>.</cite></span>

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<cite id="CITEREFDr._Hubert_Howe1960s" class="citation book cs1">Dr. Hubert Howe (1960s). <a rel="nofollow" class="external text" href="https://archive.org/details/synthmanual-buchla-100-owners-manual"><i>Buchla Electronic Music System: Users Manual written for CBS Musical Instruments (Buchla 100 Owner's Manual)</i></a>. Educational Research Department, CBS Musical Instruments, Columbia Broadcasting System. p. <a rel="nofollow" class="external text" href="https://archive.org/download/synthmanual-buchla-100-owners-manual/buchla100ownersmanual.pdf#page=7">7</a>. <q><i>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. ...</i></q></cite></span>

</li>

<cite id="CITEREFAtten_Strange1974" class="citation book cs1">Atten Strange (1974). <a rel="nofollow" class="external text" href="https://archive.org/details/synthmanual-buchla-music-easel-owners-manual"><i>Programming and Metaprogramming in the Electro-Organism - An Operating Directive for the Music Easel</i></a>. Buchla and Associates.</cite></span>

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<li id="cite_note-uspto-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-uspto_9-0">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=4018121">"U.S. Patent 4018121 Apr 1977"</a>. patft.uspto.gov<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-04-30</span></span>.</cite></span>

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<li id="cite_note-patent-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-patent_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-patent_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-patent_10-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://patents.google.com/patent/US4018121A/en=">"Patent US4018121 - Method of synthesizing a musical sound - Google Patents"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2017-04-30</span></span>.</cite></span>

</li>

<cite id="CITEREFRob_Hordijk" class="citation web cs1">Rob Hordijk. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070407072439/http://www.clavia.se/nordmodular/Modularzone/FMsynthesis.html">"FM synthesis on Modular"</a>. <i>Nord Modular & Micro Modular V3.03 tips & tricks</i>. Clavia DMI AB. Archived from <a rel="nofollow" class="external text" href="http://www.clavia.se/nordmodular/Modularzone/FMsynthesis.html">the original</a> on 2007-04-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-03-23</span></span>.</cite></span>

</li>

<li id="cite_note-holmes_257-8-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-holmes_257-8_12-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFHolmes2008" class="citation book cs1">Holmes, Thom (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hCthQ-bec-QC&pg=PA257">"Early Computer Music"</a>. <i>Electronic and experimental music: technology, music, and culture</i> (3rd ed.). <a href="Taylor_%26_Francis" title="Taylor & Francis">Taylor & Francis</a>. pp. <span class="nowrap">257–</span>8. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-415-95781-6</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">2011-06-04</span></span>.</cite></span>

</li>

<li id="cite_note-SoS80s-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-SoS80s_13-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGordon_Reid2001" class="citation web cs1">Gordon Reid (September 2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110917223333/http://www.soundonsound.com/sos/sep01/articles/retrofmpt2.asp">"Sounds of the '80s Part 2: The Yamaha DX1 & Its Successors (Retro)"</a>. <i>Sound on Sound</i>. Archived from <a rel="nofollow" class="external text" href="http://www.soundonsound.com/sos/sep01/articles/retrofmpt2.asp">the original</a> on 17 September 2011<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-06-29</span></span>.</cite></span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Stanford University News Service (06/07/94), <a rel="nofollow" class="external text" href="http://news.stanford.edu/pr/94/940607Arc4222.html">Music synthesis approaches sound quality of real instruments</a></span>

</li>

<li id="cite_note-YamahaSY99-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-YamahaSY99_15-0">^</a></b></span> <span class="reference-text"><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://jp.yamaha.com/products/music_production/synthesizers/sy99/specs.html">"Yamaha SY99 spec"</a>. <i><a href="Yamaha_Corporation" title="Yamaha Corporation">Yamaha Corporation</a></i> (in Japanese).</cite></span>

</li>

<li id="cite_note-SOS1998-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-SOS1998_16-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFPoyserJohnson1998" class="citation magazine cs1">Poyser, Debbie; Johnson, Derek (1998). <a rel="nofollow" class="external text" href="https://www.soundonsound.com/reviews/yamaha-fs1r">"Yamaha FS1R - FM Synthesis / Formant-shaping Tone Generator"</a>. <i><a href="Sound_on_Sound" title="Sound on Sound">Sound on Sound</a></i>. No. December 1998.</cite></span>

</li>

<li id="cite_note-zollinger2016-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-zollinger2016_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-zollinger2016_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFZollinger2017" class="citation web cs1">Zollinger, W. Thor (Dec 2017). <a rel="nofollow" class="external text" href="http://javelinart.com/FM_Synthesis_of_Real_Instruments.pdf">"FM_Synthesis_of_Real_Instruments"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170925230705/http://javelinart.com/FM_Synthesis_of_Real_Instruments.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2017-09-25.</cite></span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.korg.com/us/products/dj/volca_fm/">Volca FM product page</a></span>

</li>

<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://usa.yamaha.com/products/music_production/synthesizers/montage/features.html#product-tabs">Yamaha Montage Product Features Page</a></span>

</li>

<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://usa.yamaha.com/products/music_production/synthesizers/modx/features.html#product-tabs">Yamaha MODX Product Features Page</a></span>

</li>

<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://usa.yamaha.com/products/music_production/synthesizers/modxplus/features.html#product-tabs">MODX8+, MODX7+, and MODX6+ Features</a></span>

</li>

<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://usa.yamaha.com/products/music_production/synthesizers/montagem/index.html#d2153572">MONTAGE M Synthesizer</a></span>

</li>

<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.elektron.se/products/digitone/">Digitone product page</a></span>

</li>

<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a href="#CITEREFChowning1973">Chowning 1973</a>, pp. 1–2</span>

</li>

<cite id="CITEREFDoering" class="citation web cs1">Doering, Ed. <a rel="nofollow" class="external text" href="http://cnx.org/content/m15482/latest/">"Frequency Modulation Mathematics"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-11</span></span>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3></div>

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</style><div class="div-col">

<ul><li><cite id="CITEREFChowning1973" class="citation journal cs1">Chowning, J. (1973). <a rel="nofollow" class="external text" href="https://web.eecs.umich.edu/~fessler/course/100/misc/chowning-73-tso.pdf">"The Synthesis of Complex Audio Spectra by Means of Frequency Modulation"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of the Audio Engineering Society</i>. <b>21</b> (7).</cite></li>

<li><cite id="CITEREFChowningBristow1986" class="citation book cs1">Chowning, John; Bristow, David (1986). <i>FM Theory & Applications - By Musicians For Musicians</i>. Tokyo: Yamaha. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>4-636-17482-8</bdi>.</cite></li>

<li><cite id="CITEREFDodgeJerse1997" class="citation book cs1">Dodge, Charles; Jerse, Thomas A. (1997). <i>Computer Music: Synthesis, Composition and Performance</i>. New York: Schirmer Books. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-02-864682-7</bdi>.</cite></li>

<li><cite id="CITEREFKreh2012" class="citation cs2">Kreh, Martin (2012), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171118214412/http://www.math.psu.edu/papikian/Kreh.pdf">"Bessel Functions"</a> <span class="cs1-format">(PDF)</span>, <i>The Pennsylvania State University</i>, pp. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171118214412/http://www.math.psu.edu/papikian/Kreh.pdf#page=7">5</a>–6, archived from <a rel="nofollow" class="external text" href="http://www.math.psu.edu/papikian/Kreh.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-11-18<span class="reference-accessdate">, retrieved <span class="nowrap">2014-08-22</span></span></cite></li>

<li><cite id="CITEREFRoads1996" class="citation book cs1">Roads, Curtis (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nZ-TetwzVcIC"><i>The Computer Music Tutorial</i></a>. MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-262-68082-0</bdi>.</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://ccrma.stanford.edu/software/snd/snd/fm.html">An Introduction To FM</a>, by Bill Schottstaedt</li>

<li><a rel="nofollow" class="external text" href="https://www.sfu.ca/~truax/fmtut.html">FM tutorial</a></li>

<li><a rel="nofollow" class="external text" href="http://www.soundonsound.com/sos/apr00/articles/synthsecrets.htm">Synth Secrets, Part 12: An Introduction To Frequency Modulation</a>, by Gordon Reid</li>

<li><a rel="nofollow" class="external text" href="http://www.soundonsound.com/sos/may00/articles/synth.htm">Synth Secrets, Part 13: More On Frequency Modulation</a>, by Gordon Reid</li>

<li><a rel="nofollow" class="external text" href="http://www.soundonsound.com/sos/1997_articles/sep97/synthschool3.html">Paul Wiffens Synth School: Part 3</a></li>

<li><a rel="nofollow" class="external text" href="http://yala.freeservers.com/2fmsynth.htm#2Mod">F.M. Synthesis including complex operator analysis</a> <a rel="nofollow" class="external text" href="https://sites.google.com/view/yala-music1/fm-synthesis">mirror site of F.M. Synthesis, 2019</a></li></ul>

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</style><div id="Sound_synthesis_types519" style="font-size:114%;margin:0 4em"><a href="Synthesizer" title="Synthesizer">Sound synthesis</a> types</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul>

<li><a href="Linear_arithmetic_synthesis" title="Linear arithmetic synthesis">Linear arithmetic</a></li>

<li><a href="Phase_distortion_synthesis" title="Phase distortion synthesis">Phase distortion</a></li>

<li><a href="Scanned_synthesis" title="Scanned synthesis">Scanned</a></li>

<li><a href="Subtractive_synthesis" title="Subtractive synthesis">Subtractive</a></li>

<li><a href="Additive_synthesis" title="Additive synthesis">Additive</a></li>

<li><a href="Distortion_synthesis" title="Distortion synthesis">Distortion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Sample-based_synthesis" title="Sample-based synthesis">Sample-based</a> or <a href="Sampler_(musical_instrument)" title="Sampler (musical instrument)">Sampler</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Wavetable_synthesis" title="Wavetable synthesis">Wavetable</a></li>

<li><a href="Granular_synthesis" title="Granular synthesis">Granular</a></li>

<li><a href="Vector_synthesis" title="Vector synthesis">Vector</a></li>

<li><a href="Concatenative_synthesis" title="Concatenative synthesis">Concatenative</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Physical_modelling_synthesis" title="Physical modelling synthesis">Physical modelling</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banded_waveguide_synthesis" title="Banded waveguide synthesis">Banded waveguide</a></li>

<li><a href="Digital_waveguide_synthesis" title="Digital waveguide synthesis">Digital waveguide</a></li>

<li><a href="Direct_digital_synthesizer" class="mw-redirect" title="Direct digital synthesizer">Direct digital</a></li>

<li><a href="Formant_synthesis" class="mw-redirect" title="Formant synthesis">Formant</a></li>

<li><a href="Karplus%E2%80%93Strong_string_synthesis" title="Karplus–Strong string synthesis">Karplus–Strong string</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Analog_synthesizer" title="Analog synthesizer">Analog synthesizer</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Graphical_sound" title="Graphical sound">Graphical sound</a></li>

<li><a href="Modular_synthesizer" title="Modular synthesizer">Modular</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Digital_synthesizer" title="Digital synthesizer">Digital synthesizer</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analog_modeling_synthesizer" title="Analog modeling synthesizer">Analog modeling</a></li>

<li><a href="Scanned_synthesis" title="Scanned synthesis">Scanned synthesis</a></li>

<li><a href="Software_synthesizer" title="Software synthesizer">Software synthesizer</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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