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<div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Syntonic comma in the history of music</span> </div> </a> <ul id="toc-Syntonic_comma_in_the_history_of_music-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comma_pump" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comma_pump"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Comma pump</span> </div> </a> <ul id="toc-Comma_pump-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notation</span> </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Syntonic comma</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://cs.wikipedia.org/wiki/Syntonick%C3%A9_koma" title="Syntonické koma – Czech" lang="cs" hreflang="cs" data-title="Syntonické koma" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Syntonisches_Komma" title="Syntonisches Komma – German" lang="de" hreflang="de" data-title="Syntonisches Komma" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Coma_sint%C3%B3nica" title="Coma sintónica – Spanish" lang="es" hreflang="es" data-title="Coma sintónica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Sintona_komao" title="Sintona komao – Esperanto" lang="eo" hreflang="eo" data-title="Sintona komao" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D9%85%D8%A7%DB%8C_%D9%87%D9%85%D8%B5%D8%AF%D8%A7" title="کمای همصدا – Persian" lang="fa" hreflang="fa" data-title="کمای همصدا" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Comma_syntonique" title="Comma syntonique – French" lang="fr" hreflang="fr" data-title="Comma syntonique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Comma_sintonico" title="Comma sintonico – Italian" lang="it" hreflang="it" data-title="Comma sintonico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Szintonikus_komma" title="Szintonikus komma – Hungarian" lang="hu" hreflang="hu" data-title="Szintonikus komma" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Didymisch_komma" title="Didymisch komma – Dutch" lang="nl" hreflang="nl" data-title="Didymisch komma" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B7%E3%83%B3%E3%83%88%E3%83%8B%E3%83%83%E3%82%AF%E3%82%B3%E3%83%B3%E3%83%9E" title="シントニックコンマ – Japanese" lang="ja" hreflang="ja" data-title="シントニックコンマ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1075846#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li 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For 27/26, see <a href="/wiki/Comma_(music)" title="Comma (music)">Comma (music)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:204px;max-width:204px"><div class="trow"><div class="theader">Syntonic comma (81:80) on C<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_0" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-durationhint="7" data-mwtitle="Syntonic_comma_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/8/8b/Syntonic_comma_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8b/Syntonic_comma_on_C.mid/Syntonic_comma_on_C.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8b/Syntonic_comma_on_C.mid/Syntonic_comma_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Syntonic_comma_on_C_HE_notation.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Syntonic_comma_on_C_HE_notation.png/200px-Syntonic_comma_on_C_HE_notation.png" decoding="async" width="200" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Syntonic_comma_on_C_HE_notation.png/300px-Syntonic_comma_on_C_HE_notation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Syntonic_comma_on_C_HE_notation.png/400px-Syntonic_comma_on_C_HE_notation.png 2x" data-file-width="588" data-file-height="472" /></a></span></div><div class="thumbcaption">Helmholtz-Ellis notation</div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Syntonic_comma_on_C.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Syntonic_comma_on_C.png/200px-Syntonic_comma_on_C.png" decoding="async" width="200" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Syntonic_comma_on_C.png/300px-Syntonic_comma_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Syntonic_comma_on_C.png/400px-Syntonic_comma_on_C.png 2x" data-file-width="588" data-file-height="472" /></a></span></div><div class="thumbcaption"><a href="/wiki/Ben_Johnston_notation" class="mw-redirect" title="Ben Johnston notation">Ben Johnston's notation</a></div></div></div></div></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Just_perfect_fifth_on_D.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Just_perfect_fifth_on_D.png/220px-Just_perfect_fifth_on_D.png" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Just_perfect_fifth_on_D.png/330px-Just_perfect_fifth_on_D.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Just_perfect_fifth_on_D.png/440px-Just_perfect_fifth_on_D.png 2x" data-file-width="539" data-file-height="252" /></a><figcaption>Just perfect fifth on D<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_1" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-mwtitle="Just_perfect_fifth_on_D.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/5/5a/Just_perfect_fifth_on_D.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5a/Just_perfect_fifth_on_D.mid/Just_perfect_fifth_on_D.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5a/Just_perfect_fifth_on_D.mid/Just_perfect_fifth_on_D.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span> The perfect fifth above D (A+) is a syntonic comma higher than the (A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-natural">♮</span></span></sup>) that is a <a href="/wiki/Just_major_sixth" class="mw-redirect" title="Just major sixth">just major sixth</a> above C, assuming C and D are <span style="font-size:85%;"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">9</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span></span> apart.<sup id="cite_ref-Fonville_1-0" class="reference"><a href="#cite_note-Fonville-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Major_second_on_C.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Major_second_on_C.svg/220px-Major_second_on_C.svg.png" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Major_second_on_C.svg/330px-Major_second_on_C.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Major_second_on_C.svg/440px-Major_second_on_C.svg.png 2x" data-file-width="674" data-file-height="315" /></a><figcaption>3-limit 9:8 <a href="/wiki/Major_second" title="Major second">major tone</a><span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_2" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-mwtitle="Major_tone_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/7/7b/Major_tone_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7b/Major_tone_on_C.mid/Major_tone_on_C.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7b/Major_tone_on_C.mid/Major_tone_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Minor_tone_on_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Minor_tone_on_C.png/220px-Minor_tone_on_C.png" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Minor_tone_on_C.png/330px-Minor_tone_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Minor_tone_on_C.png/440px-Minor_tone_on_C.png 2x" data-file-width="539" data-file-height="252" /></a><figcaption>5-limit 10:9 <a href="/wiki/Major_second" title="Major second">minor tone</a><span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_3" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-mwtitle="Minor_tone_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/8/89/Minor_tone_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/89/Minor_tone_on_C.mid/Minor_tone_on_C.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/89/Minor_tone_on_C.mid/Minor_tone_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <p>In <a href="/wiki/Music_theory" title="Music theory">music theory</a>, the <b>syntonic comma</b>, also known as the <b><a href="/wiki/Chromatic" class="mw-redirect" title="Chromatic">chromatic</a> diesis</b>, the <b>Didymean comma</b>, the <b><a href="/wiki/Ptolemy" title="Ptolemy">Ptolemaic</a> comma</b>, or the <b><a href="/wiki/Diatonic" class="mw-redirect" title="Diatonic">diatonic</a> comma</b><sup id="cite_ref-Johnston_2-0" class="reference"><a href="#cite_note-Johnston-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is a small <a href="/wiki/Comma_(music)" title="Comma (music)">comma</a> type <a href="/wiki/Interval_(music)" title="Interval (music)">interval</a> between two <a href="/wiki/Musical_note" title="Musical note">musical notes</a>, equal to the frequency ratio <span style="font-size:85%;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">80</span></span>⁠</span></span> (= 1.0125) (around 21.51 <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>). Two notes that differ by this interval would sound different from each other even to untrained ears,<sup id="cite_ref-BBC_3-0" class="reference"><a href="#cite_note-BBC-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a <i>Didymean comma</i> because it is the amount by which <a href="/wiki/Didymus_the_Musician" title="Didymus the Musician">Didymus</a> corrected the <a href="/wiki/Pythagorean_interval" title="Pythagorean interval">Pythagorean</a> <a href="/wiki/Major_third" title="Major third">major third</a> (<span style="font-size:85%;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">64</span></span>⁠</span></span>, around 407.82 cents)<sup id="cite_ref-Lloyd_4-0" class="reference"><a href="#cite_note-Lloyd-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> to a <a href="/wiki/Just_intonation" title="Just intonation">just</a> / <a href="/wiki/Harmonic_series" class="mw-disambig" title="Harmonic series">harmonicly</a> consonant <a href="/wiki/Major_third" title="Major third">major third</a> (<span style="font-size:85%;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span></span>, around 386.31 cents). </p><p>The word "comma" came via Latin from Greek <span class="texhtml">κόμμα</span>, from earlier <span class="texhtml">*κοπ-μα</span> = "a thing cut off", or "a hair", as in "off by just a hair". </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Relationships">Relationships</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syntonic_comma&action=edit&section=1" title="Edit section: Relationships"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The prime factors of the just interval <span style="font-size:85%;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">80</span></span>⁠</span></span> known as the syntonic comma can be separated out and reconstituted into various sequences of two or more intervals that arrive at the comma, such as <span class="nowrap"><span class="texhtml"><span style="font-size:85%;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">1</span></span>⁠</span> × <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">80</span></span>⁠</span> </span></span>  </span> or (fully expanded and sorted by prime) <span class="nowrap"><span class="texhtml"><span style="font-size:85%;"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3 × 3 × 3 × 3</span><span class="sr-only">/</span><span class="den">  2 × 2 × 2 × 2 × 5  </span></span>⁠</span> </span></span> .</span> All sequences of notes that produce that fraction are mathematically valid, but some of the more musical sequences people use to remember and explain the comma's composition, occurrence, and usage are listed below: </p> <ul><li>The ratio of the two kinds of <a href="/wiki/Major_second" title="Major second">major second</a> which occur in <a href="/wiki/5-limit_tuning" class="mw-redirect" title="5-limit tuning">5-limit tuning</a>: the greater <a href="/wiki/Whole_tone" class="mw-redirect" title="Whole tone">tone</a> (9:8, about 203.91 cents, or <span class="nowrap">C <span style="font-size:85%;">↗</span> D</span> in <a href="/wiki/Just_intonation" title="Just intonation">just</a> C major) and lesser tone (10:9, about 182.40 cents, or <span class="nowrap">D <span style="font-size:85%;">↗</span> E</span>). Namely, <span class="nowrap"> 9:8 ÷ 10:9 = 81:80 ,</span> or equivalently, sharpening by a comma promotes a lesser <a href="/wiki/Major_second" title="Major second">major second</a> to a greater second <span class="nowrap"> <span class="texhtml"><span style="font-size:85%;"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">10</span><span class="sr-only">/</span><span class="den">9</span></span>⁠</span> × <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">80</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">9</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span> </span></span> .</span><sup id="cite_ref-Lloyd_4-1" class="reference"><a href="#cite_note-Lloyd-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>The difference in <a href="/wiki/Interval_(music)#size" title="Interval (music)">size</a> between a Pythagorean <a href="/wiki/Ditone" title="Ditone">ditone</a> (<a href="/wiki/Interval_ratio" title="Interval ratio">frequency ratio</a> 81:64, or about 407.82 <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>) and a just major third (5:4, or about 386.31 cents). Namely, <span class="nowrap"><span style="font-size:85%;"><span class="texhtml"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">64</span></span>⁠</span> ÷ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">80</span></span>⁠</span> </span></span> .</span></li> <li>The difference between four <a href="/wiki/Just_intonation" title="Just intonation">justly</a> tuned <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifths</a>, and two <a href="/wiki/Octave" title="Octave">octaves</a> plus a justly tuned <a href="/wiki/Major_third" title="Major third">major third</a>. A just perfect fifth has a size of 3:2 (about 701.96 cents), and four of them are equal to 81:16 (about 2807.82 cents). A just major third has a size of <a href="/wiki/Sesquiquartum" class="mw-redirect" title="Sesquiquartum">5:4</a> (about 386.31 cents), and one of them plus two octaves (4:1 or exactly 2400 cents) is equal to 5:1 (about 2786.31 cents). The difference between these is the syntonic comma. Namely, <span class="nowrap"> 81:16 ÷ 5:1 = 81:80 .</span></li> <li>The difference between one octave plus a justly tuned <a href="/wiki/Minor_third" title="Minor third">minor third</a> (12:5, about 1515.64 cents), and three justly tuned <a href="/wiki/Perfect_fourth" title="Perfect fourth">perfect fourths</a> (64:27, about 1494.13 cents). Namely, 12:5 ÷ 64:27 = 81:80.</li> <li>The difference between a <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean</a> <a href="/wiki/Major_sixth" title="Major sixth">major sixth</a> (27:16, about 905.87 cents) and a <a href="/wiki/5-limit_tuning#The_justest_ratios" class="mw-redirect" title="5-limit tuning">justly tuned</a> or "pure" <a href="/wiki/Major_sixth" title="Major sixth">major sixth</a> (5:3, about 884.36 cents). Namely, 27:16 ÷ 5:3 = 81:80.<sup id="cite_ref-Lloyd_4-2" class="reference"><a href="#cite_note-Lloyd-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <p>On a <a href="/wiki/Piano" title="Piano">piano</a> keyboard (typically tuned with <a href="/wiki/Equal_temperament#Twelve-tone_equal_temperament" title="Equal temperament">12-tone equal temperament</a>) a stack of four fifths (700 × 4 = 2800 cents) is exactly equal to two octaves (1200 × 2 = 2400 cents) plus a major third (400 cents). In other words, starting from a C, both combinations of intervals will end up at E. Using <a href="/wiki/Just_intonation" title="Just intonation">justly tuned</a> octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes. The ratio between their frequencies, as explained above, is a syntonic comma (81:80). <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a> uses justly tuned fifths (3:2) as well, but uses the relatively complex ratio of 81:64 for major thirds. <a href="/wiki/Quarter-comma_meantone" title="Quarter-comma meantone">Quarter-comma meantone</a> uses justly tuned major thirds (5:4), but flattens each of the fifths by a quarter of a syntonic comma, relative to their just size (3:2). Other systems use different compromises. This is one of the reasons why <a href="/wiki/Equal_temperament#Twelve-tone_equal_temperament" title="Equal temperament">12-tone equal temperament</a> is currently the preferred system for tuning most musical instruments<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (July 2022)">clarification needed</span></a></i>]</sup>. </p><p>Mathematically, by <a href="/wiki/St%C3%B8rmer%27s_theorem" title="Størmer's theorem">Størmer's theorem</a>, 81:80 is the closest <a href="/wiki/Superparticular_number" class="mw-redirect" title="Superparticular number">superparticular ratio</a> possible with <a href="/wiki/Regular_number" title="Regular number">regular numbers</a> as numerator and denominator. A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5:4, and a regular number is one whose <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a> are limited to 2, 3, and 5. Thus, although smaller intervals can be described within 5-limit tunings, they cannot be described as superparticular ratios. </p> <div class="mw-heading mw-heading2"><h2 id="Syntonic_comma_in_the_history_of_music">Syntonic comma in the history of music</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syntonic_comma&action=edit&section=2" title="Edit section: Syntonic comma in the history of music"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:304px;max-width:304px"><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Syntonic_comma_major_third_Cuisenaire_rods_just.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Syntonic_comma_major_third_Cuisenaire_rods_just.png/300px-Syntonic_comma_major_third_Cuisenaire_rods_just.png" decoding="async" width="300" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Syntonic_comma_major_third_Cuisenaire_rods_just.png/450px-Syntonic_comma_major_third_Cuisenaire_rods_just.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Syntonic_comma_major_third_Cuisenaire_rods_just.png/600px-Syntonic_comma_major_third_Cuisenaire_rods_just.png 2x" data-file-width="1350" data-file-height="99" /></a></span></div><div class="thumbcaption">Syntonic comma (the mismatch at the top)</div></div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Syntonic_comma_major_third_Cuisenaire_rods_ET.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Syntonic_comma_major_third_Cuisenaire_rods_ET.png/300px-Syntonic_comma_major_third_Cuisenaire_rods_ET.png" decoding="async" width="300" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Syntonic_comma_major_third_Cuisenaire_rods_ET.png/450px-Syntonic_comma_major_third_Cuisenaire_rods_ET.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Syntonic_comma_major_third_Cuisenaire_rods_ET.png/600px-Syntonic_comma_major_third_Cuisenaire_rods_ET.png 2x" data-file-width="1349" data-file-height="100" /></a></span></div><div class="thumbcaption">is tempered out in 12TET (bottom)</div></div></div></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:304px;max-width:304px"><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Syntonic_comma_major_and_minor_tone_Cuisenaire_rods_just.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Syntonic_comma_major_and_minor_tone_Cuisenaire_rods_just.png/300px-Syntonic_comma_major_and_minor_tone_Cuisenaire_rods_just.png" decoding="async" width="300" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Syntonic_comma_major_and_minor_tone_Cuisenaire_rods_just.png/450px-Syntonic_comma_major_and_minor_tone_Cuisenaire_rods_just.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/43/Syntonic_comma_major_and_minor_tone_Cuisenaire_rods_just.png 2x" data-file-width="578" data-file-height="99" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:302px;max-width:302px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Septimal_and_syntonic_comma_whole_tones_Cuisenaire_rods_ET.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Septimal_and_syntonic_comma_whole_tones_Cuisenaire_rods_ET.png/300px-Septimal_and_syntonic_comma_whole_tones_Cuisenaire_rods_ET.png" decoding="async" width="300" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Septimal_and_syntonic_comma_whole_tones_Cuisenaire_rods_ET.png/450px-Septimal_and_syntonic_comma_whole_tones_Cuisenaire_rods_ET.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e5/Septimal_and_syntonic_comma_whole_tones_Cuisenaire_rods_ET.png 2x" data-file-width="578" data-file-height="99" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Syntonic comma, such as between the 9/8 (203.91 approximate cents) and 10/9 (182.40 approximate cents) major and minor tones (top), is tempered out in 12TET, leaving one 200 cent tone (bottom).</div></div></div></div> <p>The syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds. In Pythagorean tuning, the only highly consonant intervals were the <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> and its inversion, the <a href="/wiki/Perfect_fourth" title="Perfect fourth">perfect fourth</a>. The Pythagorean <a href="/wiki/Major_third" title="Major third">major third</a> (81:64) and <a href="/wiki/Minor_third" title="Minor third">minor third</a> (32:27) were <a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">dissonant</a>, and this prevented musicians from using <a href="/wiki/Triad_(music)" title="Triad (music)">triads</a> and <a href="/wiki/Chord_(music)" title="Chord (music)">chords</a>, forcing them for centuries to write music with relatively simple <a href="/wiki/Texture_(music)" title="Texture (music)">texture</a>. </p><p>The syntonic tempering dates to <a href="/wiki/Didymus_the_Musician" title="Didymus the Musician">Didymus the Musician</a>, whose tuning of the <a href="/wiki/Diatonic_genus" class="mw-redirect" title="Diatonic genus">diatonic genus</a> of the <a href="/wiki/Tetrachord" title="Tetrachord">tetrachord</a> replaced one 9:8 interval with a 10:9 interval (<a href="/wiki/Lesser_tone" class="mw-redirect" title="Lesser tone">lesser tone</a>), obtaining a just major third (5:4) and semitone (16:15). This was later revised by Ptolemy (swapping the two tones) in his "syntonic diatonic" scale (συντονόν διατονικός, <i>syntonón diatonikós</i>, from συντονός + διάτονος). The term <i>syntonón</i> was based on <a href="/wiki/Aristoxenus" title="Aristoxenus">Aristoxenus</a>, and may be translated as "tense" (conventionally "intense"), referring to tightened strings (hence sharper), in contrast to μαλακόν (<i>malakón</i>, from μαλακός), translated as "relaxed" (conventional "soft"), referring to looser strings (hence flatter or "softer"). </p><p>This was rediscovered in the late <a href="/wiki/Middle_Ages" title="Middle Ages">Middle Ages</a>, where musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made <a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">consonant</a>. For instance, if the frequency of E is decreased by a syntonic comma (81:80), C-E (a major third), and E-G (a minor third) become just. Namely, C-E is narrowed to a <a href="/wiki/Just_intonation" title="Just intonation">justly intonated</a> ratio of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {81 \over 64}\cdot {80 \over 81}={{1\cdot 5} \over {4\cdot 1}}={5 \over 4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mn>64</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>80</mn> <mn>81</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>⋅</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {81 \over 64}\cdot {80 \over 81}={{1\cdot 5} \over {4\cdot 1}}={5 \over 4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8d00ddfcf8e576c711bab104d383ed717f018c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.037ex; height:5.343ex;" alt="{\displaystyle {81 \over 64}\cdot {80 \over 81}={{1\cdot 5} \over {4\cdot 1}}={5 \over 4}}"></span></dd></dl> <p>and at the same time E-G is widened to the just ratio of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {32 \over 27}\cdot {81 \over 80}={{2\cdot 3} \over {1\cdot 5}}={6 \over 5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>27</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mn>80</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>⋅</mo> <mn>5</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {32 \over 27}\cdot {81 \over 80}={{2\cdot 3} \over {1\cdot 5}}={6 \over 5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/930fb679616fb5cf50e345da24a7da62fe07bac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.037ex; height:5.343ex;" alt="{\displaystyle {32 \over 27}\cdot {81 \over 80}={{2\cdot 3} \over {1\cdot 5}}={6 \over 5}}"></span></dd></dl> <p>The drawback is that the fifths A-E and E-B, by flattening E, become almost as dissonant as the Pythagorean <a href="/wiki/Wolf_interval" title="Wolf interval">wolf fifth</a>. But the fifth C-G stays consonant, since only E has been flattened (C-E × E-G = 5/4 × 6/5 = 3/2), and can be used together with C-E to produce a C-<a href="/wiki/Major_chord" title="Major chord">major</a> triad (C-E-G). These experiments eventually brought to the creation of a new <a href="/wiki/Tuning_system" class="mw-redirect" title="Tuning system">tuning system</a>, known as <a href="/wiki/Quarter-comma_meantone" title="Quarter-comma meantone">quarter-comma meantone</a>, in which the number of major thirds was maximized, and most minor thirds were tuned to a ratio which was very close to the just 6:5. This result was obtained by narrowing each fifth by a quarter of a syntonic comma, an amount which was considered negligible, and permitted the full development of music with complex <a href="/wiki/Texture_(music)" title="Texture (music)">texture</a>, such as <a href="/wiki/Polyphony" title="Polyphony">polyphonic music</a>, or melody with <a href="/wiki/Homophony" title="Homophony">instrumental accompaniment</a>. Since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths in an entire family of them. Namely, in the family belonging to the <a href="/wiki/Syntonic_temperament" class="mw-redirect" title="Syntonic temperament">syntonic temperament</a> continuum, including <a href="/wiki/Meantone_temperament" title="Meantone temperament">meantone temperaments</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Comma_pump">Comma pump</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syntonic_comma&action=edit&section=3" title="Edit section: Comma pump"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Comma_pump_Benedetti.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Comma_pump_Benedetti.png/300px-Comma_pump_Benedetti.png" decoding="async" width="300" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Comma_pump_Benedetti.png/450px-Comma_pump_Benedetti.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Comma_pump_Benedetti.png/600px-Comma_pump_Benedetti.png 2x" data-file-width="1077" data-file-height="555" /></a><figcaption>Giovanni Benedetti's 1563 example of a comma "pump" or drift by a comma during a progression.<sup id="cite_ref-Historically_5-0" class="reference"><a href="#cite_note-Historically-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/4c\/Comma_pump_Benedetti.mid\/Comma_pump_Benedetti.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Comma pump Benedetti.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4c/Comma_pump_Benedetti.mid/Comma_pump_Benedetti.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Play</span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:Comma_pump_Benedetti.mid" title="File:Comma pump Benedetti.mid">ⓘ</a></sup></span></span> Common tones between chords are the same pitch, with the other notes tuned in pure intervals to the common tones. <span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-2" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/fa\/Comma_pump_Benedetti_first_last.mid\/Comma_pump_Benedetti_first_last.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play first and last chords"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Comma pump Benedetti first last.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/fa/Comma_pump_Benedetti_first_last.mid/Comma_pump_Benedetti_first_last.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Play first and last chords</span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:Comma_pump_Benedetti_first_last.mid" title="File:Comma pump Benedetti first last.mid">ⓘ</a></sup></span></span></figcaption></figure> <p>The syntonic comma arises in <i><a href="/wiki/Comma_pump" title="Comma pump">comma pump</a></i> (<i>comma drift</i>) sequences such as C G D A E C, when each interval from one note to the next is played with certain specific intervals in <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a> tuning. If we use the <a href="/wiki/Frequency_ratio" class="mw-redirect" title="Frequency ratio">frequency ratio</a> 3/2 for the <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifths</a> (C-G and D-A), 3/4 for the descending <a href="/wiki/Perfect_fourth" title="Perfect fourth">perfect fourths</a> (G-D and A-E), and 4/5 for the descending <a href="/wiki/Major_third" title="Major third">major third</a> (E-C), then the sequence of intervals from one note to the next in that sequence goes 3/2, 3/4, 3/2, 3/4, 4/5. These multiply together to give </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {3 \over 2}\cdot {3 \over 4}\cdot {3 \over 2}\cdot {3 \over 4}\cdot {4 \over 5}={81 \over 80}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mn>80</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {3 \over 2}\cdot {3 \over 4}\cdot {3 \over 2}\cdot {3 \over 4}\cdot {4 \over 5}={81 \over 80}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b4caa0dcfc2f4c0bbd5bff3a341e91448f52d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.969ex; height:5.176ex;" alt="{\displaystyle {3 \over 2}\cdot {3 \over 4}\cdot {3 \over 2}\cdot {3 \over 4}\cdot {4 \over 5}={81 \over 80}}"></span></dd></dl></dd></dl> <p>which is the syntonic comma (musical intervals stacked in this way are multiplied together). The "drift" is created by the combination of Pythagorean and 5-limit intervals in just intonation, and would not occur in Pythagorean tuning due to the use only of the Pythagorean major third (64/81) which would thus return the last step of the sequence to the original pitch. </p><p>So in that sequence, the second C is sharper than the first C by a syntonic comma <span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-3" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/e\/e7\/Comma_pump_on_C.mid\/Comma_pump_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Comma pump on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e7/Comma_pump_on_C.mid/Comma_pump_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Play</span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:Comma_pump_on_C.mid" title="File:Comma pump on C.mid">ⓘ</a></sup></span></span>. That sequence, or any <a href="/wiki/Transposition_(music)" title="Transposition (music)">transposition</a> of it, is known as the comma pump. If a line of music follows that sequence, and if each of the intervals between adjacent notes is justly tuned, then every time the sequence is followed, the pitch of the piece rises by a syntonic comma (about a fifth of a semitone). </p><p>Study of the comma pump dates back at least to the sixteenth century when the Italian scientist <a href="/wiki/Giambattista_Benedetti" title="Giambattista Benedetti">Giovanni Battista Benedetti</a> composed a piece of music to illustrate syntonic comma drift.<sup id="cite_ref-Historically_5-1" class="reference"><a href="#cite_note-Historically-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Note that a descending perfect fourth (3/4) is the same as a descending <a href="/wiki/Octave" title="Octave">octave</a> (1/2) followed by an ascending perfect fifth (3/2). Namely, (3/4) = (1/2) × (3/2). Similarly, a descending major third (4/5) is the same as a descending octave (1/2) followed by an ascending <a href="/wiki/Minor_sixth" title="Minor sixth">minor sixth</a> (8/5). Namely, (4/5) = (1/2) × (8/5). Therefore, the above-mentioned sequence is equivalent to: </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {3 \over 2}\cdot {1 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {1 \over 2}\cdot {3 \over 2}\cdot {1 \over 2}\cdot {8 \over 5}={81 \over 80}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mn>80</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {3 \over 2}\cdot {1 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {1 \over 2}\cdot {3 \over 2}\cdot {1 \over 2}\cdot {8 \over 5}={81 \over 80}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33d476c68607b24480090bcb22b5e88b8facbda6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.002ex; height:5.176ex;" alt="{\displaystyle {3 \over 2}\cdot {1 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {1 \over 2}\cdot {3 \over 2}\cdot {1 \over 2}\cdot {8 \over 5}={81 \over 80}}"></span></dd></dl></dd></dl> <p>or, by grouping together similar intervals, </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {3 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {8 \over 5}\cdot {1 \over 2}\cdot {1 \over 2}\cdot {1 \over 2}={81 \over 80}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>5</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mn>80</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {3 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {8 \over 5}\cdot {1 \over 2}\cdot {1 \over 2}\cdot {1 \over 2}={81 \over 80}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd8de8312472503e1ad76edb22154db3995d8c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.002ex; height:5.176ex;" alt="{\displaystyle {3 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {3 \over 2}\cdot {8 \over 5}\cdot {1 \over 2}\cdot {1 \over 2}\cdot {1 \over 2}={81 \over 80}}"></span></dd></dl></dd></dl> <p>This means that, if all intervals are justly tuned, a syntonic comma can be obtained with a stack of four perfect fifths plus one minor sixth, followed by three descending octaves (in other words, four <b>P5</b> plus one <b>m6</b> minus three <b>P8</b>). </p> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syntonic_comma&action=edit&section=4" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:204px;max-width:204px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Major_chord_on_C.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Major_chord_on_C.png/200px-Major_chord_on_C.png" decoding="async" width="200" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Major_chord_on_C.png/300px-Major_chord_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Major_chord_on_C.png/400px-Major_chord_on_C.png 2x" data-file-width="539" data-file-height="252" /></a></span></div><div class="thumbcaption">Just major chord on C in Ben Johnston's notation. <span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-4" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/d\/df\/Major_chord_on_C_in_just_intonation.mid\/Major_chord_on_C_in_just_intonation.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major chord on C in just intonation.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/d/df/Major_chord_on_C_in_just_intonation.mid/Major_chord_on_C_in_just_intonation.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Play</span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:Major_chord_on_C_in_just_intonation.mid" title="File:Major chord on C in just intonation.mid">ⓘ</a></sup></span></span> Pythagorean major chord on C in Helmholtz-Ellis notation. <span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-5" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/6\/63\/Pythagorean_major_chord_on_C.mid\/Pythagorean_major_chord_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean major chord on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/6/63/Pythagorean_major_chord_on_C.mid/Pythagorean_major_chord_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Play</span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:Pythagorean_major_chord_on_C.mid" title="File:Pythagorean major chord on C.mid">ⓘ</a></sup></span></span></div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Pythagorean_major_chord_on_C.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Pythagorean_major_chord_on_C.png/200px-Pythagorean_major_chord_on_C.png" decoding="async" width="200" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Pythagorean_major_chord_on_C.png/300px-Pythagorean_major_chord_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Pythagorean_major_chord_on_C.png/400px-Pythagorean_major_chord_on_C.png 2x" data-file-width="539" data-file-height="252" /></a></span></div><div class="thumbcaption">Pythagorean major chord, Ben Johnston's notation.</div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Just_major_chord_on_C_HE_notation.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Just_major_chord_on_C_HE_notation.png/200px-Just_major_chord_on_C_HE_notation.png" decoding="async" width="200" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Just_major_chord_on_C_HE_notation.png/300px-Just_major_chord_on_C_HE_notation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Just_major_chord_on_C_HE_notation.png/400px-Just_major_chord_on_C_HE_notation.png 2x" data-file-width="539" data-file-height="262" /></a></span></div><div class="thumbcaption">Just major chord, in Helmholtz-Ellis notation.</div></div></div></div></div> <p><a href="/wiki/Moritz_Hauptmann" title="Moritz Hauptmann">Moritz Hauptmann</a> developed a method of notation used by <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a>. Based on Pythagorean tuning, subscript numbers are then added to indicate the number of syntonic commas to lower a note by. Thus a Pythagorean scale is C D E F G A B, while a just scale is C D E<sub>1</sub> F G A<sub>1</sub> B<sub>1</sub>. <a href="/wiki/Carl_Eitz" title="Carl Eitz">Carl Eitz</a> developed a similar system used by <a href="/wiki/J._Murray_Barbour" class="mw-redirect" title="J. Murray Barbour">J. Murray Barbour</a>. Superscript positive and negative numbers are added, indicating the number of syntonic commas to raise or lower from Pythagorean tuning. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E<sup>−1</sup> F G A<sup>−1</sup> B<sup>−1</sup>. </p><p>In <a href="/wiki/Helmholtz-Ellis_notation" class="mw-redirect" title="Helmholtz-Ellis notation">Helmholtz-Ellis notation</a>, a syntonic comma is indicated with up and down arrows added to the traditional accidentals. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E<span typeof="mw:File"><a href="/wiki/File:HE_syntonic_comma_-_natural_down.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/9px-HE_syntonic_comma_-_natural_down.png" decoding="async" width="9" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/14px-HE_syntonic_comma_-_natural_down.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/18px-HE_syntonic_comma_-_natural_down.png 2x" data-file-width="52" data-file-height="113" /></a></span> F G A <span typeof="mw:File"><a href="/wiki/File:HE_syntonic_comma_-_natural_down.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/9px-HE_syntonic_comma_-_natural_down.png" decoding="async" width="9" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/14px-HE_syntonic_comma_-_natural_down.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/18px-HE_syntonic_comma_-_natural_down.png 2x" data-file-width="52" data-file-height="113" /></a></span> B<span typeof="mw:File"><a href="/wiki/File:HE_syntonic_comma_-_natural_down.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/9px-HE_syntonic_comma_-_natural_down.png" decoding="async" width="9" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/14px-HE_syntonic_comma_-_natural_down.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/18px-HE_syntonic_comma_-_natural_down.png 2x" data-file-width="52" data-file-height="113" /></a></span>. </p><p>Composer <a href="/wiki/Ben_Johnston_(composer)" title="Ben Johnston (composer)">Ben Johnston</a> uses a "−" as an accidental to indicate a note is lowered by a syntonic comma, or a "+" to indicate a note is raised by a syntonic comma.<sup id="cite_ref-Fonville_1-1" class="reference"><a href="#cite_note-Fonville-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Thus a Pythagorean scale is C D E+ F G A+ B+, while the 5-limit Ptolemaic scale is C D E F G A B. </p> <table class="wikitable"> <tbody><tr> <th> </th> <th>5-limit just </th> <th>Pythagorean </th></tr> <tr> <td>HE </td> <td>C D E<span typeof="mw:File"><a href="/wiki/File:HE_syntonic_comma_-_natural_down.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/9px-HE_syntonic_comma_-_natural_down.png" decoding="async" width="9" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/14px-HE_syntonic_comma_-_natural_down.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/18px-HE_syntonic_comma_-_natural_down.png 2x" data-file-width="52" data-file-height="113" /></a></span> F G A<span typeof="mw:File"><a href="/wiki/File:HE_syntonic_comma_-_natural_down.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/9px-HE_syntonic_comma_-_natural_down.png" decoding="async" width="9" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/14px-HE_syntonic_comma_-_natural_down.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/18px-HE_syntonic_comma_-_natural_down.png 2x" data-file-width="52" data-file-height="113" /></a></span> B<span typeof="mw:File"><a href="/wiki/File:HE_syntonic_comma_-_natural_down.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/9px-HE_syntonic_comma_-_natural_down.png" decoding="async" width="9" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/14px-HE_syntonic_comma_-_natural_down.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/HE_syntonic_comma_-_natural_down.png/18px-HE_syntonic_comma_-_natural_down.png 2x" data-file-width="52" data-file-height="113" /></a></span> </td> <td>C D E F G A B </td></tr> <tr> <td>Johnston </td> <td>C D E F G A B </td> <td>C D E+ F G A+ B+ </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syntonic_comma&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/F%2B_(pitch)" title="F+ (pitch)">F+ (pitch)</a></li> <li><a href="/wiki/Holdrian_comma" class="mw-redirect" title="Holdrian comma">Holdrian comma</a></li> <li><a href="/wiki/Comma_(music)" title="Comma (music)">Comma (music)</a></li> <li><a href="/wiki/Pythagorean_comma" title="Pythagorean comma">Pythagorean comma</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syntonic_comma&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Fonville-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fonville_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fonville_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/John_Fonville" title="John Fonville">John Fonville</a>. "Ben Johnston's Extended Just Intonation – A Guide for Interpreters", p. 109, <i><a href="/wiki/Perspectives_of_New_Music" title="Perspectives of New Music">Perspectives of New Music</a></i>, vol. 29, no. 2 (Summer 1991), pp. 106-137. and <a href="/wiki/Ben_Johnston_(composer)" title="Ben Johnston (composer)">Johnston, Ben</a> and <a href="/wiki/Bob_Gilmore" title="Bob Gilmore">Gilmore, Bob</a> (2006). "A Notation System for Extended Just Intonation" (2003), <i>"Maximum clarity" and Other Writings on Music</i>, p. 78. <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon 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.citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-252-03098-7" title="Special:BookSources/978-0-252-03098-7">978-0-252-03098-7</a>.</span> </li> <li id="cite_note-Johnston-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Johnston_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnston2006" class="citation cs2"><a href="/wiki/Ben_Johnston_(composer)" title="Ben Johnston (composer)">Johnston, B.</a> (2006), <a href="/wiki/Bob_Gilmore" title="Bob Gilmore">Gilmore, B.</a> (ed.), <i>"Maximum Clarity" and Other Writings on Music</i>, Urbana, IL: University of Illinois Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-252-03098-2" title="Special:BookSources/0-252-03098-2"><bdi>0-252-03098-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%22Maximum+Clarity%22+and+Other+Writings+on+Music&rft.place=Urbana%2C+IL&rft.pub=University+of+Illinois+Press&rft.date=2006&rft.isbn=0-252-03098-2&rft.aulast=Johnston&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASyntonic+comma" class="Z3988"></span></span> </li> <li id="cite_note-BBC-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-BBC_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20050208134843/http://www.bbc.co.uk/dna/collective/A1339076">"Sol-Fa – The Key to Temperament"</a>, <i>bbc.co.uk</i>, <a href="/wiki/BBC" title="BBC">BBC</a>, archived from <a rel="nofollow" class="external text" href="https://www.bbc.co.uk/dna/collective/A1339076">the original</a> on February 8, 2005</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=bbc.co.uk&rft.atitle=Sol-Fa+%E2%80%93+The+Key+to+Temperament&rft_id=https%3A%2F%2Fwww.bbc.co.uk%2Fdna%2Fcollective%2FA1339076&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASyntonic+comma" class="Z3988"></span> </span> </li> <li id="cite_note-Lloyd-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lloyd_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lloyd_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lloyd_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLloyd1937" class="citation cs2">Lloyd, Llewelyn Southworth (1937), <i>Music and Sound</i>, Books for Libraries Press, p. 12, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8369-5188-3" title="Special:BookSources/0-8369-5188-3"><bdi>0-8369-5188-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Music+and+Sound&rft.pages=12&rft.pub=Books+for+Libraries+Press&rft.date=1937&rft.isbn=0-8369-5188-3&rft.aulast=Lloyd&rft.aufirst=Llewelyn+Southworth&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASyntonic+comma" class="Z3988"></span></span> </li> <li id="cite_note-Historically-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Historically_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Historically_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWildSchubert2008" class="citation cs2">Wild, Jonathan; Schubert, Peter (Spring–Fall 2008), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100911051454/http://www.musicstudies.org/JIMS2008/articles/Wild_JIMS_0821208.pdf">"Historically Informed Retuning of Polyphonic Vocal Performance"</a> <span class="cs1-format">(PDF)</span>, <i>Journal of Interdisciplinary Music Studies</i>, <b>2</b> (1&2): 121–139 [127], archived from <a rel="nofollow" class="external text" href="http://www.musicstudies.org/JIMS2008/articles/Wild_JIMS_0821208.pdf">the original</a> <span class="cs1-format">(PDF)</span> on September 11, 2010<span class="reference-accessdate">, retrieved <span class="nowrap">April 5,</span> 2013</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Interdisciplinary+Music+Studies&rft.atitle=Historically+Informed+Retuning+of+Polyphonic+Vocal+Performance&rft.chron=spring%E2%80%93fall&rft.volume=2&rft.issue=1%262&rft.pages=121-139+127&rft.date=2008&rft.aulast=Wild&rft.aufirst=Jonathan&rft.au=Schubert%2C+Peter&rft_id=http%3A%2F%2Fwww.musicstudies.org%2FJIMS2008%2Farticles%2FWild_JIMS_0821208.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASyntonic+comma" class="Z3988"></span>, art. #0821208.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Syntonic_comma&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://music.indiana.edu/departments/offices/piano-technology/temperaments/syntonic-comma.shtml">Indiana University School of Music: Piano Repair Shop: Harpsichord Tuning, Repair, and Temperaments: "What is the Syntonic Comma?"</a></li> <li><a rel="nofollow" class="external text" href="http://tonalsoft.com/enc/s/syntonic-comma.aspx">Tonalsoft: "Syntonic-comma"</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=TYhPAbsIqA8">Explanation of comma drift</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Intervals" style="padding:3px"><table class="nowraplinks hlist mw-collapsible expanded navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Intervals" title="Template:Intervals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Intervals" title="Template talk:Intervals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Intervals" title="Special:EditPage/Template:Intervals"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Intervals" style="font-size:114%;margin:0 4em"><a href="/wiki/Interval_(music)" title="Interval (music)">Intervals</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Equal_temperament" title="Equal temperament">Twelve-<br />semitone</a><br />(post-Bach<br />Western)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="(Numbers_in_bracketsare_the_number_ofsemitones_in_theinterval.)" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><i>(Numbers in brackets<br />are the number of<br /><a href="/wiki/Semitone" title="Semitone">semitones</a> in the<br />interval.)</i></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;">Perfect</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Unison" title="Unison">unison</a> (0)</li> <li><a href="/wiki/Perfect_fourth" title="Perfect fourth">fourth</a> (5)</li> <li><a href="/wiki/Perfect_fifth" title="Perfect fifth">fifth</a> (7)</li> <li><a href="/wiki/Octave" title="Octave">octave</a> (12)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="/wiki/Major_and_minor#Intervals_and_chords" title="Major and minor">Major</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Major_second" title="Major second">second</a> (2)</li> <li><a href="/wiki/Major_third" title="Major third">third</a> (4)</li> <li><a href="/wiki/Major_sixth" title="Major sixth">sixth</a> (9)</li> <li><a href="/wiki/Major_seventh" title="Major seventh">seventh</a> (11)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="/wiki/Major_and_minor#Intervals_and_chords" title="Major and minor">Minor</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Minor_second" class="mw-redirect" title="Minor second">second</a> (1)</li> <li><a href="/wiki/Minor_third" title="Minor third">third</a> (3)</li> <li><a href="/wiki/Minor_sixth" title="Minor sixth">sixth</a> (8)</li> <li><a href="/wiki/Minor_seventh" title="Minor seventh">seventh</a> (10)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="/wiki/Augmentation_(music)#Augmentation_of_intervals" title="Augmentation (music)">Augmented</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Augmented_unison" title="Augmented unison">unison</a> (1)</li> <li><a href="/wiki/Augmented_second" title="Augmented second">second</a> (3)</li> <li><a href="/wiki/Augmented_third" title="Augmented third">third</a> (5)</li> <li><a href="/wiki/Tritone" title="Tritone">fourth</a> (6)</li> <li><a href="/wiki/Augmented_fifth" title="Augmented fifth">fifth</a> (8)</li> <li><a href="/wiki/Augmented_sixth" title="Augmented sixth">sixth</a> (10)</li> <li><a href="/wiki/Augmented_seventh" title="Augmented seventh">seventh</a> (12)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="/wiki/Diminution#Diminution_of_intervals" title="Diminution">Diminished</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Diminished_second" title="Diminished second">second</a> (0)</li> <li><a href="/wiki/Diminished_third" title="Diminished third">third</a> (2)</li> <li><a href="/wiki/Diminished_fourth" title="Diminished fourth">fourth</a> (4)</li> <li><a href="/wiki/Tritone" title="Tritone">fifth</a> (6)</li> <li><a href="/wiki/Diminished_sixth" title="Diminished sixth">sixth</a> (7)</li> <li><a href="/wiki/Diminished_seventh" title="Diminished seventh">seventh</a> (9)</li> <li><a href="/wiki/Diminished_octave" title="Diminished octave">octave</a> (11)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="/wiki/Interval_(music)#Simple_and_compound" title="Interval (music)">Compound</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ninth" title="Ninth">ninth</a> (13 or 14)</li> <li><a href="/wiki/Third_(chord)" title="Third (chord)">tenth</a> (15 or 16)</li> <li><a href="/wiki/Eleventh" title="Eleventh">eleventh</a> (17 or 18)</li> <li><a href="/wiki/Fifth_(chord)" title="Fifth (chord)">twelfth</a> (18 or 19)</li> <li><a href="/wiki/Thirteenth" title="Thirteenth">thirteenth</a> (20 or 21)</li> <li><a href="/wiki/Seventh_(chord)" title="Seventh (chord)">fourteenth</a> (22 or 23)</li> <li><a href="/wiki/Fifteenth" title="Fifteenth">fifteenth</a> (24)</li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other<br />tuning<br />systems</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><b><a href="/wiki/24-tone_equal_temperament" class="mw-redirect" title="24-tone equal temperament">24-tone equal temperament</a></b><br /><i>(Numbers in brackets refer<br />to fractional semitones.)</i></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Neutral" scope="row" class="navbox-group" style="width:5em;font-weight:normal;"><div><br /><a href="/wiki/Neutral_interval" title="Neutral interval">Neutral</a><br /><br /></div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quarter_tone" title="Quarter tone">quarter tone</a> (<span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>)</li> <li><a href="/wiki/Neutral_interval" title="Neutral interval">second</a> (<span class="frac">1<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li> <li><a href="/wiki/Neutral_third" title="Neutral third">third</a> (<span class="frac">3<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li> <li><a href="/wiki/Major_fourth_and_minor_fifth" title="Major fourth and minor fifth">major fourth</a> (<span class="frac">5<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li> <li><a href="/wiki/Major_fourth_and_minor_fifth" title="Major fourth and minor fifth">minor fifth</a> (<span class="frac">6<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li> <li><a href="/wiki/Neutral_sixth" title="Neutral sixth">sixth</a> (<span class="frac">8<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li> <li><a href="/wiki/Neutral_interval" title="Neutral interval">seventh</a> (<span class="frac">10<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><b><a href="/wiki/Just_intonation" title="Just intonation">Just intonations</a></b><br /><i>(Numbers in brackets<br />refer to pitch ratios.)</i></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/7-limit_tuning" title="7-limit tuning">7-limit</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="/wiki/Septimal_quarter_tone" title="Septimal quarter tone">septimal quarter tone</a> (36:35)</li> <li><a href="/wiki/Septimal_third_tone" title="Septimal third tone">septimal third tone</a> (28:27)</li> <li><a href="/wiki/Septimal_chromatic_semitone" title="Septimal chromatic semitone">septimal chromatic semitone</a> (21:20)</li> <li><a href="/wiki/Septimal_diatonic_semitone" title="Septimal diatonic semitone">septimal diatonic semitone</a> (15:14)</li> <li><a href="/wiki/Septimal_whole_tone" title="Septimal whole tone">supermajor second</a> (8:7)</li> <li><a href="/wiki/Septimal_minor_third" title="Septimal minor third">subminor third</a> (7:6)</li> <li><a href="/wiki/Septimal_major_third" title="Septimal major third">supermajor third</a> (9:7)</li> <li><a href="/wiki/Septimal_tritone" title="Septimal tritone">subminor fifth</a> (7:5)</li> <li><a href="/wiki/Septimal_tritone" title="Septimal tritone">supermajor fourth</a> (10:7)</li> <li><a href="/wiki/Harmonic_seventh" title="Harmonic seventh">subminor seventh</a> (7:4)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Limit_(music)" title="Limit (music)">Higher-limit</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="/wiki/Minor_diatonic_semitone" title="Minor diatonic semitone">minor diatonic semitone</a> (17-limit)</li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other<br />intervals</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><b>Groups</b></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Microtonal_music" class="mw-redirect" title="Microtonal music">Microtone</a></li> <li><a href="/wiki/List_of_intervals_in_5-limit_just_intonation" title="List of intervals in 5-limit just intonation">5-limit</a></li> <li><a href="/wiki/Comma_(music)" title="Comma (music)">Comma</a></li> <li><a href="/wiki/Pseudo-octave" title="Pseudo-octave">Pseudo-octave</a></li> <li><a href="/wiki/Pythagorean_interval" title="Pythagorean interval">Pythagorean interval</a></li> <li><a href="/wiki/Subminor_and_supermajor" title="Subminor and supermajor">Subminor and supermajor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><i><b>Semitones</b></i></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pythagorean_limma" class="mw-redirect" title="Pythagorean limma">Pythagorean limma</a></li> <li><a href="/wiki/Pythagorean_apotome" class="mw-redirect" title="Pythagorean apotome">Pythagorean apotome</a></li> <li><a href="/wiki/Major_limma" title="Major limma">Major limma</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><i><b>Quarter tones</b></i></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quarter_tone" title="Quarter tone">Quarter tone</a></li> <li><a href="/wiki/Septimal_quarter_tone" title="Septimal quarter tone">Septimal quarter tone</a></li> <li><a href="/wiki/Undecimal_quarter_tone" class="mw-redirect" title="Undecimal quarter tone">Undecimal quarter tone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><b><a href="/wiki/Comma_(music)" title="Comma (music)">Commas</a></b></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pythagorean_comma" title="Pythagorean comma">Pythagorean comma</a> (23.5 cents)</li> <li><a class="mw-selflink selflink">Syntonic comma</a> (21.5 cents)</li> <li><a href="/wiki/Holdrian_comma" class="mw-redirect" title="Holdrian comma">Holdrian comma</a> (22.6 cents)</li> <li><a href="/wiki/Septimal_comma" title="Septimal comma">Septimal comma</a> (27.3 cents)</li></ul> <ul><li><a href="/wiki/Diesis" title="Diesis">Lesser diesis</a> (41.1 cents)</li> <li><a href="/wiki/Diesis" title="Diesis">Greater diesis</a> (62.6 cents)</li> <li><a href="/wiki/Septimal_diesis" title="Septimal diesis">Septimal diesis</a> (35.7 cents)</li></ul> <ul><li><a href="/wiki/Diaschisma" title="Diaschisma">Diaschisma</a> (19.5 cents)</li> <li><a href="/wiki/Semicomma" title="Semicomma">Semicomma</a> (10.1 cents)</li> <li><a href="/wiki/Septimal_semicomma" title="Septimal semicomma">Septimal semicomma</a> (13.8 cents)</li> <li><a href="/wiki/Kleisma" title="Kleisma">Kleisma</a> (8.1 cents)</li> <li><a href="/wiki/Septimal_kleisma" title="Septimal kleisma">Septimal kleisma</a> (7.7 cents)</li></ul> <ul><li><a href="/wiki/Schisma" title="Schisma">Schisma</a> (1.95 cents)</li> <li><a href="/wiki/Breedsma" title="Breedsma">Breedsma</a> (0.72 cents)</li> <li><a href="/wiki/Ragisma" title="Ragisma">Ragisma</a> (0.4 cents)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><b>Measurement</b></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cent_(music)" title="Cent (music)">Cent</a></li> <li><a href="/wiki/Cent_(music)#Centitones" title="Cent (music)">Centitone</a></li> <li><a href="/wiki/Millioctave" title="Millioctave">Millioctave</a></li> <li><a href="/wiki/Savart" title="Savart">Savart</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><b>Others</b></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wolf_interval" title="Wolf interval">Wolf</a></li> <li><a href="/wiki/Ditone" title="Ditone">Ditone</a></li> <li><a href="/wiki/Semiditone" class="mw-redirect" title="Semiditone">Semiditone</a></li> <li><a href="/wiki/George_Secor#Secor" title="George Secor">Secor</a></li> <li><a href="/wiki/Incomposite_interval" title="Incomposite interval">Incomposite interval</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_pitch_intervals" title="List of pitch intervals">List of pitch intervals</a></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Syntonic_comma&oldid=1255262798">https://en.wikipedia.org/w/index.php?title=Syntonic_comma&oldid=1255262798</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:5-limit_tuning_and_intervals" title="Category:5-limit tuning and intervals">5-limit tuning and intervals</a></li><li><a href="/wiki/Category:Commas_(music)" 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