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id="toc-Minor_second-sublist" class="vector-toc-list"> <li id="toc-In_other_temperaments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_other_temperaments"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>In other temperaments</span> </div> </a> <ul id="toc-In_other_temperaments-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Augmented_unison" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Augmented_unison"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Augmented unison</span> </div> </a> <ul id="toc-Augmented_unison-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semitones_in_different_tunings" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Semitones_in_different_tunings"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Semitones in different tunings</span> </div> </a> <button aria-controls="toc-Semitones_in_different_tunings-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Semitones in different tunings subsection</span> </button> <ul id="toc-Semitones_in_different_tunings-sublist" class="vector-toc-list"> <li id="toc-Meantone_temperament" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Meantone_temperament"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Meantone temperament</span> </div> </a> <ul id="toc-Meantone_temperament-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equal_temperament" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equal_temperament"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Equal temperament</span> </div> </a> <ul id="toc-Equal_temperament-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Well_temperament" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Well_temperament"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Well temperament</span> </div> </a> <ul id="toc-Well_temperament-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pythagorean_tuning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pythagorean_tuning"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Pythagorean tuning</span> </div> </a> <ul id="toc-Pythagorean_tuning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Just_5-limit_intonation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Just_5-limit_intonation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Just 5-limit intonation</span> </div> </a> <ul id="toc-Just_5-limit_intonation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extended_just_intonations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extended_just_intonations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Extended just intonations</span> </div> </a> <ul id="toc-Extended_just_intonations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_equal_temperaments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_equal_temperaments"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Other equal temperaments</span> </div> </a> <ul id="toc-Other_equal_temperaments-sublist" class="vector-toc-list"> </ul> </li> </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" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon 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Available in 36 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-36" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">36 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Semit%C3%B2" title="Semitò – Catalan" lang="ca" hreflang="ca" data-title="Semitò" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/P%C5%AFlt%C3%B3n" title="Půltón – Czech" lang="cs" hreflang="cs" data-title="Půltón" 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/Halbton" title="Halbton – German" lang="de" hreflang="de" data-title="Halbton" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pooltoon" title="Pooltoon – Estonian" lang="et" hreflang="et" data-title="Pooltoon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%97%CE%BC%CE%B9%CF%84%CF%8C%CE%BD%CE%B9%CE%BF" title="Ημιτόνιο – Greek" lang="el" hreflang="el" data-title="Ημιτόνιο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Semitono" title="Semitono – Spanish" lang="es" hreflang="es" data-title="Semitono" 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/Duonduto" title="Duonduto – Esperanto" lang="eo" hreflang="eo" data-title="Duonduto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Tonuerdi" title="Tonuerdi – Basque" lang="eu" hreflang="eu" data-title="Tonuerdi" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%DB%8C%D9%85%E2%80%8C%D9%BE%D8%B1%D8%AF%D9%87" title="نیم‌پرده – Persian" lang="fa" hreflang="fa" data-title="نیم‌پرده" 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/Demi-ton" title="Demi-ton – French" lang="fr" hreflang="fr" data-title="Demi-ton" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Semit%C3%B3n" title="Semitón – Galician" lang="gl" hreflang="gl" data-title="Semitón" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%98%EC%9D%8C" title="반음 – Korean" lang="ko" hreflang="ko" data-title="반음" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%AB%D5%BD%D5%A1%D5%BF%D5%B8%D5%B6" title="Կիսատոն – Armenian" lang="hy" hreflang="hy" data-title="Կիսատոն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Setengah_nada" title="Setengah nada – Indonesian" lang="id" hreflang="id" data-title="Setengah nada" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Semitono" title="Semitono – Italian" lang="it" hreflang="it" data-title="Semitono" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Pustonis" title="Pustonis – Latvian" lang="lv" hreflang="lv" data-title="Pustonis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pustonis" title="Pustonis – Lithuanian" lang="lt" hreflang="lt" data-title="Pustonis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Semiton" title="Semiton – Malay" lang="ms" hreflang="ms" data-title="Semiton" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Toonafstand" title="Toonafstand – Dutch" lang="nl" hreflang="nl" data-title="Toonafstand" 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/%E5%8D%8A%E9%9F%B3" title="半音 – Japanese" lang="ja" hreflang="ja" data-title="半音" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Halvtone" title="Halvtone – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Halvtone" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Halvtone" title="Halvtone – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Halvtone" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Miei_ton_(solfegi)" title="Miei ton (solfegi) – Occitan" lang="oc" hreflang="oc" data-title="Miei ton (solfegi)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/P%C3%B3%C5%82ton" title="Półton – Polish" lang="pl" hreflang="pl" data-title="Półton" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Semitom" title="Semitom – Portuguese" lang="pt" hreflang="pt" data-title="Semitom" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Semiton" title="Semiton – Romanian" lang="ro" hreflang="ro" data-title="Semiton" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%83%D1%82%D0%BE%D0%BD" title="Полутон – Russian" lang="ru" hreflang="ru" data-title="Полутон" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Semitone" title="Semitone – Simple English" lang="en-simple" hreflang="en-simple" data-title="Semitone" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Polton" title="Polton – Slovenian" lang="sl" hreflang="sl" data-title="Polton" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%83%D1%81%D1%82%D0%B5%D0%BF%D0%B5%D0%BD" title="Полустепен – Serbian" lang="sr" hreflang="sr" data-title="Полустепен" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Polustepen" title="Polustepen – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Polustepen" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Halvton" title="Halvton – Swedish" lang="sv" hreflang="sv" data-title="Halvton" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Yar%C4%B1mses" title="Yarımses – Turkish" lang="tr" hreflang="tr" data-title="Yarımses" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%B2%D1%82%D0%BE%D0%BD" title="Півтон – Ukrainian" lang="uk" hreflang="uk" data-title="Півтон" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8D%8A%E9%9F%B3" title="半音 – Cantonese" lang="yue" hreflang="yue" data-title="半音" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%8A%E9%9F%B3" title="半音 – Chinese" lang="zh" hreflang="zh" data-title="半音" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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/Q738785#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 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Pythagorean_limma&redirect=no" class="mw-redirect" title="Pythagorean limma">Pythagorean limma</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the musical interval. For the printing method, see <a href="/wiki/Halftone" title="Halftone">Halftone</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Musical interval</div><style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><caption class="infobox-title">semitone</caption><tbody><tr><th scope="row" class="infobox-label"><a href="/wiki/Inversion_(interval)" class="mw-redirect" title="Inversion (interval)">Inverse</a></th><td class="infobox-data"><a href="/wiki/Major_seventh" title="Major seventh">major seventh</a> (for minor second); <a href="/wiki/Diminished_octave" title="Diminished octave">diminished octave</a> (for augmented unison); <a href="/wiki/Augmented_octave" title="Augmented octave">augmented octave</a> (for diminished unison)</td></tr><tr><th colspan="2" class="infobox-header" style="background-color:palegoldenrod;">Name</th></tr><tr><th scope="row" class="infobox-label">Other names</th><td class="infobox-data">minor second,<br />diatonic semitone,<br />augmented unison,<br />diminished unison,<br />chromatic semitone</td></tr><tr><th scope="row" class="infobox-label">Abbreviation</th><td class="infobox-data">m2; A1</td></tr><tr><th colspan="2" class="infobox-header" style="background-color:palegoldenrod;">Size</th></tr><tr><th scope="row" class="infobox-label"><a class="mw-selflink selflink">Semitones</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Interval_class" title="Interval class">Interval class</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Just_intonation" title="Just intonation">Just interval</a></th><td class="infobox-data">16:15,<sup id="cite_ref-Duffin_1-0" class="reference"><a href="#cite_note-Duffin-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 17:16,<sup id="cite_ref-Haluska_2-0" class="reference"><a href="#cite_note-Haluska-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 27:25, 135:128,<sup id="cite_ref-Duffin_1-1" class="reference"><a href="#cite_note-Duffin-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 25:24,<sup id="cite_ref-Duffin_1-2" class="reference"><a href="#cite_note-Duffin-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 256:243</td></tr><tr><th colspan="2" class="infobox-header" style="background-color:palegoldenrod;"><a href="/wiki/Cent_(music)" title="Cent (music)">Cents</a></th></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Equal_temperament#Twelve-tone_equal_temperament" title="Equal temperament">12-Tone equal temperament</a></th><td class="infobox-data">100<sup id="cite_ref-Duffin_1-3" class="reference"><a href="#cite_note-Duffin-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Just_intonation" title="Just intonation">Just intonation</a></th><td class="infobox-data">112,<sup id="cite_ref-Duffin_1-4" class="reference"><a href="#cite_note-Duffin-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 105, 133, 92,<sup id="cite_ref-Duffin_1-5" class="reference"><a href="#cite_note-Duffin-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 71,<sup id="cite_ref-Duffin_1-6" class="reference"><a href="#cite_note-Duffin-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 90</td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Minor_second_on_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Minor_second_on_C.png/220px-Minor_second_on_C.png" decoding="async" width="220" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Minor_second_on_C.png/330px-Minor_second_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Minor_second_on_C.png/440px-Minor_second_on_C.png 2x" data-file-width="542" data-file-height="252" /></a><figcaption>Minor second<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-mwtitle="Minor_second_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/8/8a/Minor_second_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8a/Minor_second_on_C.mid/Minor_second_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/8/8a/Minor_second_on_C.mid/Minor_second_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <p>A <b>semitone</b>, also called a <b>minor second</b>, <b>half step</b>, or a <b>half tone</b>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> is the smallest <a href="/wiki/Interval_(music)" title="Interval (music)">musical interval</a> commonly used in Western tonal music,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and it is considered the most <a href="/wiki/Consonance_and_dissonance#Dissonance" title="Consonance and dissonance">dissonant</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> when sounded harmonically. It is defined as the interval between two adjacent notes in a <a href="/wiki/Chromatic_scale" title="Chromatic scale">12-tone scale</a> (or half of a <a href="/wiki/Whole_step" class="mw-redirect" title="Whole step">whole step</a>), visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>; the interval between them is a semitone.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In a 12-note approximately equally divided scale, any interval can be defined in terms of an appropriate number of semitones (e.g. a <a href="/wiki/Whole_tone" class="mw-redirect" title="Whole tone">whole tone</a> or major second is 2 semitones wide, a <a href="/wiki/Major_third" title="Major third">major third</a> 4 semitones, and a <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> 7 semitones). </p><p>In <a href="/wiki/Music_theory" title="Music theory">music theory</a>, a distinction is made<sup id="cite_ref-Wharram_7-0" class="reference"><a href="#cite_note-Wharram-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> between a <b>diatonic semitone</b>, or <b>minor second</b> (an interval encompassing two different <a href="/wiki/Staff_position" class="mw-redirect" title="Staff position">staff positions</a>, e.g. from C to D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>) and a <b>chromatic semitone</b> or <b>augmented unison</b> (an interval between two notes at the same staff position, e.g. from C to C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>). These are <a href="/wiki/Enharmonic" class="mw-redirect" title="Enharmonic">enharmonically equivalent</a> if and only if <a href="/wiki/Equal_temperament" title="Equal temperament">twelve-tone equal temperament</a> is used; for example, they are not the same thing in <a href="/wiki/Meantone_temperament" title="Meantone temperament">meantone temperament</a>, where the diatonic semitone is distinguished from and larger than the chromatic semitone (augmented unison), or in <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a>, where the diatonic semitone is smaller instead. See <a href="/wiki/Interval_(music)#Number" title="Interval (music)">Interval (music) § Number</a> for more details about this terminology. </p><p>In <a href="/wiki/Twelve-tone_equal_temperament" class="mw-redirect" title="Twelve-tone equal temperament">twelve-tone equal temperament</a> all semitones are equal in size (100 cents). In other tuning systems, "semitone" refers to a family of intervals that may vary both in size and name. In <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a>, seven semitones out of twelve are diatonic, with ratio 256:243 or 90.2 cents (<a href="#Pythagorean_tuning">Pythagorean limma</a>), and the other five are chromatic, with ratio 2187:2048 or 113.7 cents (<a href="#Pythagorean_tuning">Pythagorean apotome</a>); they differ by the <a href="/wiki/Pythagorean_comma" title="Pythagorean comma">Pythagorean comma</a> of ratio 531441:524288 or 23.5 cents. In <a href="/wiki/Quarter-comma_meantone" title="Quarter-comma meantone">quarter-comma meantone</a>, seven of them are diatonic, and 117.1 cents wide, while the other five are chromatic, and 76.0 cents wide; they differ by the lesser <a href="/wiki/Diesis" title="Diesis">diesis</a> of ratio 128:125 or 41.1 cents. 12-tone scales tuned in <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a> typically define three or four kinds of semitones. For instance, <a href="/wiki/Five-limit_tuning#Size_of_intervals" title="Five-limit tuning">Asymmetric</a> <a href="/wiki/Five-limit_tuning" title="Five-limit tuning">five-limit tuning</a> yields chromatic semitones with ratios 25:24 (70.7 cents) and 135:128 (92.2 cents), and diatonic semitones with ratios 16:15 (111.7 cents) and 27:25 (133.2 cents). For further details, see <a class="mw-selflink-fragment" href="#Just_intonation">below</a>. </p><p>The condition of having semitones is called hemitonia; that of having no semitones is <a href="/wiki/Anhemitonic_scale" title="Anhemitonic scale">anhemitonia</a>. A <a href="/wiki/Scale_(music)" title="Scale (music)">musical scale</a> or <a href="/wiki/Chord_(music)" title="Chord (music)">chord</a> containing semitones is called hemitonic; one without semitones is anhemitonic. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Minor_second">Minor second</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=1" title="Edit section: Minor second"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cadence_minor_second.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Cadence_minor_second.png/220px-Cadence_minor_second.png" decoding="async" width="220" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/8f/Cadence_minor_second.png 1.5x" data-file-width="290" data-file-height="338" /></a><figcaption>The melodic minor second is an integral part of most cadences of the <a href="/wiki/Common_practice_period" title="Common practice period">Common practice period</a>.<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="Cadence_minor_second_V65-I.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/6/6f/Cadence_minor_second_V65-I.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6f/Cadence_minor_second_V65-I.mid/Cadence_minor_second_V65-I.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6f/Cadence_minor_second_V65-I.mid/Cadence_minor_second_V65-I.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Musical interval</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><table class="infobox"><caption class="infobox-title">minor second</caption><tbody><tr><th scope="row" class="infobox-label"><a href="/wiki/Inversion_(interval)" class="mw-redirect" title="Inversion (interval)">Inverse</a></th><td class="infobox-data">major seventh</td></tr><tr><th colspan="2" class="infobox-header" style="background-color:palegoldenrod;">Name</th></tr><tr><th scope="row" class="infobox-label">Other names</th><td class="infobox-data">semitone, half step</td></tr><tr><th scope="row" class="infobox-label">Abbreviation</th><td class="infobox-data">m2</td></tr><tr><th colspan="2" class="infobox-header" style="background-color:palegoldenrod;">Size</th></tr><tr><th scope="row" class="infobox-label"><a class="mw-selflink selflink">Semitones</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Interval_class" title="Interval class">Interval class</a></th><td class="infobox-data">1</td></tr><tr><th colspan="2" class="infobox-header" style="background-color:palegoldenrod;"><a href="/wiki/Cent_(music)" title="Cent (music)">Cents</a></th></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Equal_temperament#Twelve-tone_equal_temperament" title="Equal temperament">12-Tone equal temperament</a></th><td class="infobox-data">100.0</td></tr></tbody></table> <p>The <i>minor second</i> occurs in the <a href="/wiki/Major_scale" title="Major scale">major scale</a>, between the third and fourth degree, (<i>mi</i> (E) and <i>fa</i> (F) in C major), and between the seventh and eighth degree (<i>ti</i> (B) and <i>do</i> (C) in C major). It is also called the <i>diatonic semitone</i> because it occurs between <a href="/wiki/Step_(music)" class="mw-redirect" title="Step (music)">steps</a> in the <a href="/wiki/Diatonic_scale" title="Diatonic scale">diatonic scale</a>. The minor second is abbreviated <b>m2</b> (or <b>−2</b>). Its inversion is the <i><a href="/wiki/Major_seventh" title="Major seventh">major seventh</a></i> (<i>M7</i> or <i>Ma7</i>). </p><p><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\/a\/a2\/Minor_Second_ET.ogg\/Minor_Second_ET.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Listen to a minor second in equal temperament"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Minor Second ET.ogg"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a2/Minor_Second_ET.ogg/Minor_Second_ET.ogg.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">Listen to a minor second in equal temperament</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:Minor_Second_ET.ogg" title="File:Minor Second ET.ogg">ⓘ</a></sup></span></span>. Here, <a href="/wiki/Middle_C" class="mw-redirect" title="Middle C">middle C</a> is followed by D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, which is a tone 100 <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a> sharper than C, and then by both tones together. </p><p><a href="/wiki/Melody" title="Melody">Melodically</a>, this interval is very frequently used, and is of particular importance in <a href="/wiki/Cadence_(music)" class="mw-redirect" title="Cadence (music)">cadences</a>. In the <a href="/wiki/Interval_(music)" title="Interval (music)">perfect</a> and <a href="/wiki/Cadence_(music)#Deceptive_cadence" class="mw-redirect" title="Cadence (music)">deceptive cadences</a> it appears as a resolution of the <a href="/wiki/Leading-tone" class="mw-redirect" title="Leading-tone">leading-tone</a> to the <a href="/wiki/Tonic_(music)" title="Tonic (music)">tonic</a>. In the <a href="/wiki/Plagal_cadence" class="mw-redirect" title="Plagal cadence">plagal cadence</a>, it appears as the falling of the <a href="/wiki/Subdominant" title="Subdominant">subdominant</a> to the <a href="/wiki/Mediant" title="Mediant">mediant</a>. It also occurs in many forms of the <a href="/wiki/Imperfect_cadence" class="mw-redirect" title="Imperfect cadence">imperfect cadence</a>, wherever the tonic falls to the leading-tone. </p><p><a href="/wiki/Harmony" title="Harmony">Harmonically</a>, the interval usually occurs as some form of <a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">dissonance</a> or a <a href="/wiki/Nonchord_tone" title="Nonchord tone">nonchord tone</a> that is not part of the <a href="/wiki/Diatonic_function" class="mw-redirect" title="Diatonic function">functional harmony</a>. It may also appear in inversions of a <a href="/wiki/Major_seventh_chord" title="Major seventh chord">major seventh chord</a>, and in many <a href="/wiki/Added_tone_chord" title="Added tone chord">added tone chords</a>. </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Bach_minor_second_smaller.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Bach_minor_second_smaller.png/400px-Bach_minor_second_smaller.png" decoding="async" width="400" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Bach_minor_second_smaller.png/600px-Bach_minor_second_smaller.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Bach_minor_second_smaller.png/800px-Bach_minor_second_smaller.png 2x" data-file-width="1654" data-file-height="333" /></a><figcaption>Harmonic minor second in <a href="/wiki/Johann_Sebastian_Bach" title="Johann Sebastian Bach">J. S. Bach</a>'s Prelude in C major from the <a href="/wiki/The_Well-Tempered_Clavier" title="The Well-Tempered Clavier">WTC</a>, book 1, mm. 7–9.<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="Bach_minor_second_smaller_BWV_846.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/1/14/Bach_minor_second_smaller_BWV_846.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/14/Bach_minor_second_smaller_BWV_846.mid/Bach_minor_second_smaller_BWV_846.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/14/Bach_minor_second_smaller_BWV_846.mid/Bach_minor_second_smaller_BWV_846.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span> The minor second may be viewed as a <a href="/wiki/Suspension_(music)" class="mw-redirect" title="Suspension (music)">suspension</a> of the <i>B</i> resolving into the following <i>A minor seventh</i> chord.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1096940132">.mw-parser-output .listen .side-box-text{line-height:1.1em}.mw-parser-output .listen-plain{border:none;background:transparent}.mw-parser-output .listen-embedded{width:100%;margin:0;border-width:1px 0 0 0;background:transparent}.mw-parser-output .listen-header{padding:2px}.mw-parser-output .listen-embedded .listen-header{padding:2px 0}.mw-parser-output .listen-file-header{padding:4px 0}.mw-parser-output .listen .description{padding-top:2px}.mw-parser-output .listen .mw-tmh-player{max-width:100%}@media(max-width:719px){.mw-parser-output .listen{clear:both}}@media(min-width:720px){.mw-parser-output .listen:not(.listen-noimage){width:320px}.mw-parser-output .listen-left{overflow:visible;float:left}.mw-parser-output .listen-center{float:none;margin-left:auto;margin-right:auto}}</style><div class="side-box side-box-left listen noprint listen-left"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><figure class="mw-halign-center" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Gnome-mime-audio-openclipart.svg/50px-Gnome-mime-audio-openclipart.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Gnome-mime-audio-openclipart.svg/75px-Gnome-mime-audio-openclipart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Gnome-mime-audio-openclipart.svg/100px-Gnome-mime-audio-openclipart.svg.png 2x" data-file-width="160" data-file-height="160" /></span><figcaption></figcaption></figure></div> <div class="side-box-text plainlist"><div class="listen-header"><b><a href="/wiki/Fr%C3%A9d%C3%A9ric_Chopin" title="Frédéric Chopin">Frédéric Chopin</a>'s "wrong note" <a href="/wiki/%C3%89tudes_(Chopin)" title="Études (Chopin)">Étude</a></b></div> <div class="haudio"> <div class="listen-file-header"><a href="/wiki/File:Frederic_Chopin_-_Opus_25_-_Twelve_Grand_Etudes_-_e_minor.ogg" title="File:Frederic Chopin - Opus 25 - Twelve Grand Etudes - e minor.ogg">Étude Op. 25, No. 5</a></div> <div><span typeof="mw:File"><span><audio id="mwe_player_3" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="232" style="width:232px;" data-durationhint="188" data-mwtitle="Frederic_Chopin_-_Opus_25_-_Twelve_Grand_Etudes_-_e_minor.ogg" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/5/5b/Frederic_Chopin_-_Opus_25_-_Twelve_Grand_Etudes_-_e_minor.ogg" type="audio/ogg; codecs="vorbis"" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5b/Frederic_Chopin_-_Opus_25_-_Twelve_Grand_Etudes_-_e_minor.ogg/Frederic_Chopin_-_Opus_25_-_Twelve_Grand_Etudes_-_e_minor.ogg.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div> <div class="description"><a href="/wiki/Martha_Goldstein" title="Martha Goldstein">Martha Goldstein</a> playing on an <a href="/wiki/S%C3%A9bastien_%C3%89rard" title="Sébastien Érard">Érard</a> (1851)<br /><small>Opening bars</small><span typeof="mw:File"><a href="/wiki/File:Chopin_minor_second.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Chopin_minor_second.png/250px-Chopin_minor_second.png" decoding="async" width="250" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Chopin_minor_second.png/375px-Chopin_minor_second.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Chopin_minor_second.png/500px-Chopin_minor_second.png 2x" data-file-width="876" data-file-height="411" /></a></span></div></div></div></div> <div class="side-box-abovebelow"><hr /><i class="selfreference">Problems playing this file? See <a href="/wiki/Help:Media" title="Help:Media">media help</a>.</i></div> </div> <p>In unusual situations, the minor second can add a great deal of character to the music. For instance, <a href="/wiki/Fr%C3%A9d%C3%A9ric_Chopin" title="Frédéric Chopin">Frédéric Chopin</a>'s <a href="/wiki/%C3%89tude_Op._25,_No._5_(Chopin)" title="Étude Op. 25, No. 5 (Chopin)">Étude Op. 25, No. 5</a> opens with a melody accompanied by a line that plays fleeting minor seconds. These are used to humorous and whimsical effect, which contrasts with its more lyrical middle section. This eccentric dissonance has earned the piece its nickname: the "wrong note" étude. This kind of usage of the minor second appears in many other works of the <a href="/wiki/Romantic_music" title="Romantic music">Romantic</a> period, such as <a href="/wiki/Modest_Mussorgsky" title="Modest Mussorgsky">Modest Mussorgsky</a>'s <i><a href="/wiki/Pictures_at_an_Exhibition" title="Pictures at an Exhibition">Ballet of the Unhatched Chicks</a></i>. More recently, the music to the movie <i><a href="/wiki/Jaws_(film)#Music" title="Jaws (film)">Jaws</a></i> exemplifies the minor second. </p> <div class="mw-heading mw-heading3"><h3 id="In_other_temperaments">In other temperaments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=2" title="Edit section: In other temperaments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a> a 16:15 minor second arises in the C <a href="/wiki/Major_scale" title="Major scale">major scale</a> between B & C and E & F, and is "the sharpest <a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">dissonance</a> found in the [major] <a href="/wiki/Musical_scale" class="mw-redirect" title="Musical scale">scale</a>."<sup id="cite_ref-books.google.com_8-0" class="reference"><a href="#cite_note-books.google.com-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <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\/d\/d8\/Just_minor_second_in_scale.mid\/Just_minor_second_in_scale.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play B &amp; C"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just minor second in scale.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d8/Just_minor_second_in_scale.mid/Just_minor_second_in_scale.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 B & C</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:Just_minor_second_in_scale.mid" title="File:Just minor second in scale.mid">ⓘ</a></sup></span></span> </p> <div class="thumb tnone" style="margin-left:auto;margin-right:auto;overflow:hidden;width:400px;max-width:508px"><div class="thumbinner"><div class="noresize" style="overflow:auto"><span typeof="mw:File"><a href="/wiki/File:Comparison_of_minor_seconds.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Comparison_of_minor_seconds.png/500px-Comparison_of_minor_seconds.png" decoding="async" width="500" height="254" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Comparison_of_minor_seconds.png/750px-Comparison_of_minor_seconds.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Comparison_of_minor_seconds.png/1000px-Comparison_of_minor_seconds.png 2x" data-file-width="1705" data-file-height="865" /></a></span></div></div></div><div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Augmented_unison">Augmented unison</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=3" title="Edit section: Augmented unison"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Augmented_unison" title="Augmented unison">Augmented unison</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Augmented_unison_on_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Augmented_unison_on_C.png/220px-Augmented_unison_on_C.png" decoding="async" width="220" height="56" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Augmented_unison_on_C.png/330px-Augmented_unison_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Augmented_unison_on_C.png/440px-Augmented_unison_on_C.png 2x" data-file-width="539" data-file-height="137" /></a><figcaption>Augmented unison on C</figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Mendelssohn_dominants.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Mendelssohn_dominants.png/220px-Mendelssohn_dominants.png" decoding="async" width="220" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Mendelssohn_dominants.png/330px-Mendelssohn_dominants.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Mendelssohn_dominants.png/440px-Mendelssohn_dominants.png 2x" data-file-width="984" data-file-height="424" /></a><figcaption>Augmented unisons often appear as a consequence of <a href="/wiki/Secondary_dominant" class="mw-redirect" title="Secondary dominant">secondary dominants</a>, such as those in the soprano voice of this <a href="/wiki/Sequence_(music)" title="Sequence (music)">sequence</a> from <a href="/wiki/Felix_Mendelssohn" title="Felix Mendelssohn">Felix Mendelssohn</a>'s <i><a href="/wiki/Songs_without_Words" class="mw-redirect" title="Songs without Words">Song Without Words</a></i> Op. 102 No. 3, mm. 47–49.<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_4" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-durationhint="4" data-mwtitle="Mendelssohn_dominants.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/0/08/Mendelssohn_dominants.mid/Mendelssohn_dominants.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/0/08/Mendelssohn_dominants.mid/Mendelssohn_dominants.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/0/08/Mendelssohn_dominants.mid" type="audio/midi" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <p>The <b>augmented unison</b>, the interval produced by the <a href="/wiki/Augmentation_(music)" title="Augmentation (music)">augmentation</a>, or widening by one half step, of the perfect unison,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> does not occur between diatonic scale steps, but instead between a scale step and a <a href="/wiki/Chromatic" class="mw-redirect" title="Chromatic">chromatic</a> alteration of the same step. It is also called a <i>chromatic semitone</i>. The augmented unison is abbreviated <b>A1</b>, or <b>aug 1</b>. Its inversion is the <i><a href="/wiki/Diminished_octave" title="Diminished octave">diminished octave</a></i> (<i>d8</i>, or <i>dim 8</i>). The augmented unison is also the inversion of the <a href="/wiki/Augmented_octave" title="Augmented octave">augmented octave</a>, because the interval of the diminished unison does not exist.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This is because a unison is always made larger when one note of the interval is changed with an accidental.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Melody" title="Melody">Melodically</a>, an augmented unison very frequently occurs when proceeding to a chromatic chord, such as a <a href="/wiki/Secondary_dominant" class="mw-redirect" title="Secondary dominant">secondary dominant</a>, a <a href="/wiki/Diminished_seventh_chord" title="Diminished seventh chord">diminished seventh chord</a>, or an <a href="/wiki/Augmented_sixth_chord" title="Augmented sixth chord">augmented sixth chord</a>. Its use is also often the consequence of a melody proceeding in semitones, regardless of harmonic underpinning, e.g. <b>D</b>, <b>D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></b>, <b>E</b>, <b>F</b>, <b>F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></b>. (Restricting the notation to only minor seconds is impractical, as the same example would have a rapidly increasing number of accidentals, written enharmonically as <b>D</b>, <b>E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></b>, <b>F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></b>, <b>G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="mw-default-size skin-invert" typeof="mw:File"><a href="/wiki/File:Doubleflat.svg" class="mw-file-description" title="double flat"><img alt="double flat" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Doubleflat.svg/8px-Doubleflat.svg.png" decoding="async" width="8" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Doubleflat.svg/12px-Doubleflat.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Doubleflat.svg/16px-Doubleflat.svg.png 2x" data-file-width="133" data-file-height="200" /></a></span></span></b>, <b>A<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="mw-default-size skin-invert" typeof="mw:File"><a href="/wiki/File:Llpd-3.svg" class="mw-file-description" title="triple flat"><img alt="triple flat" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Llpd-3.svg/12px-Llpd-3.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Llpd-3.svg/18px-Llpd-3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Llpd-3.svg/24px-Llpd-3.svg.png 2x" data-file-width="196" data-file-height="200" /></a></span></span></b>). </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Liszt_augmented_unison.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Liszt_augmented_unison.png/220px-Liszt_augmented_unison.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Liszt_augmented_unison.png/330px-Liszt_augmented_unison.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Liszt_augmented_unison.png/440px-Liszt_augmented_unison.png 2x" data-file-width="701" data-file-height="432" /></a><figcaption><a href="/wiki/Franz_Liszt" title="Franz Liszt">Franz Liszt</a>'s second <a href="/wiki/Transcendental_%C3%89tudes" title="Transcendental Études">Transcendental Étude</a>, measure 63</figcaption></figure> <p><a href="/wiki/Harmony" title="Harmony">Harmonically</a>, augmented unisons are quite rare in tonal repertoire. In the example to the right, <a href="/wiki/Franz_Liszt" title="Franz Liszt">Liszt</a> had written an <b>E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></b> against an <b>E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-natural">♮</span></span></b> in the bass. Here <b>E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></b> was preferred to a <b>D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></b> to make the tone's function clear as part of an <b>F</b> <a href="/wiki/Dominant_seventh" class="mw-redirect" title="Dominant seventh">dominant seventh</a> chord, and the augmented unison is the result of superimposing this harmony upon an <b>E</b> <a href="/wiki/Pedal_point" title="Pedal point">pedal point</a>. </p><p>In addition to this kind of usage, harmonic augmented unisons are frequently written in modern works involving <a href="/wiki/Tone_clusters" class="mw-redirect" title="Tone clusters">tone clusters</a>, such as <a href="/wiki/Iannis_Xenakis" title="Iannis Xenakis">Iannis Xenakis</a>' <i>Evryali</i> for piano solo. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=4" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The semitone appeared in the music theory of Greek antiquity as part of a diatonic or chromatic <a href="/wiki/Tetrachord" title="Tetrachord">tetrachord</a>, and it has always had a place in the diatonic scales of Western music since. The various <a href="/wiki/Musical_mode" class="mw-redirect" title="Musical mode">modal</a> scales of <a href="/wiki/Medieval_music" title="Medieval music">medieval music</a> theory were all based upon this diatonic pattern of <a href="/wiki/Whole_tone" class="mw-redirect" title="Whole tone">tones</a> and semitones. </p><p>Though it would later become an integral part of the musical <a href="/wiki/Cadence_(music)" class="mw-redirect" title="Cadence (music)">cadence</a>, in the early polyphony of the 11th century this was not the case. <a href="/wiki/Guido_of_Arezzo" title="Guido of Arezzo">Guido of Arezzo</a> suggested instead in his <i><a href="/wiki/Micrologus" title="Micrologus">Micrologus</a></i> other alternatives: either proceeding by whole tone from a <a href="/wiki/Major_second" title="Major second">major second</a> to a unison, or an <i>occursus</i> having two notes at a <a href="/wiki/Major_third" title="Major third">major third</a> move by contrary motion toward a unison, each having moved a whole tone. </p><p>"As late as the 13th century the half step was experienced as a problematic interval not easily understood, as the irrational  [<i><a href="/wiki/Sic" title="Sic">sic</a></i>] remainder between the perfect fourth and the <a href="/wiki/Ditone" title="Ditone">ditone</a> <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 \left({\begin{matrix}{\frac {4}{3}}\end{matrix}}/{{\begin{matrix}({\frac {9}{8}})\end{matrix}}^{2}}={\begin{matrix}{\frac {256}{243}}\end{matrix}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>8</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>256</mn> <mn>243</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\begin{matrix}{\frac {4}{3}}\end{matrix}}/{{\begin{matrix}({\frac {9}{8}})\end{matrix}}^{2}}={\begin{matrix}{\frac {256}{243}}\end{matrix}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b476d93b3a12fabd9470850ee65e99f85af644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.773ex; height:4.843ex;" alt="{\displaystyle \left({\begin{matrix}{\frac {4}{3}}\end{matrix}}/{{\begin{matrix}({\frac {9}{8}})\end{matrix}}^{2}}={\begin{matrix}{\frac {256}{243}}\end{matrix}}\right)}"></span>." In a melodic half step, no "tendency was perceived of the lower tone toward the upper, or of the upper toward the lower. The second tone was not taken to be the 'goal' of the first. Instead, the half step was avoided in <a href="/wiki/Clausula_(music)" title="Clausula (music)">clausulae</a> because it lacked clarity as an interval."<sup id="cite_ref-Dahlhaus_13-0" class="reference"><a href="#cite_note-Dahlhaus-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Marenzio_solo_e_pensoso_chromatic.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Marenzio_solo_e_pensoso_chromatic.png/220px-Marenzio_solo_e_pensoso_chromatic.png" decoding="async" width="220" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Marenzio_solo_e_pensoso_chromatic.png/330px-Marenzio_solo_e_pensoso_chromatic.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Marenzio_solo_e_pensoso_chromatic.png/440px-Marenzio_solo_e_pensoso_chromatic.png 2x" data-file-width="1366" data-file-height="479" /></a><figcaption>Dramatic chromatic scale in the opening measures of <a href="/wiki/Luca_Marenzio" title="Luca Marenzio">Luca Marenzio</a>'s <i>Solo e pensoso</i>, ca. 1580.<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_5" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-mwtitle="Marenzio_solo_e_pensoso_opening.MID" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/1/19/Marenzio_solo_e_pensoso_opening.MID" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/19/Marenzio_solo_e_pensoso_opening.MID/Marenzio_solo_e_pensoso_opening.MID.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/19/Marenzio_solo_e_pensoso_opening.MID/Marenzio_solo_e_pensoso_opening.MID.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <p>However, beginning in the 13th century <a href="/wiki/Cadence_(music)" class="mw-redirect" title="Cadence (music)">cadences</a> begin to require motion in one voice by half step and the other a whole step in contrary motion.<sup id="cite_ref-Dahlhaus_13-1" class="reference"><a href="#cite_note-Dahlhaus-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> These cadences would become a fundamental part of the musical language, even to the point where the usual accidental accompanying the minor second in a cadence was often omitted from the written score (a practice known as <a href="/wiki/Musica_ficta" title="Musica ficta">musica ficta</a>). By the 16th century, the semitone had become a more versatile interval, sometimes even appearing as an augmented unison in very <a href="/wiki/Chromatic" class="mw-redirect" title="Chromatic">chromatic</a> passages. <a href="/wiki/Music_semiotics" class="mw-redirect" title="Music semiotics">Semantically</a>, in the 16th century the repeated melodic semitone became associated with weeping, see: <a href="/wiki/Chromatic_fourth" title="Chromatic fourth">passus duriusculus</a>, <a href="/wiki/Lament_bass" title="Lament bass">lament bass</a>, and <a href="/wiki/Pianto" title="Pianto">pianto</a>. </p><p>By the <a href="/wiki/Baroque_music" title="Baroque music">Baroque era</a> (1600 to 1750), the <a href="/wiki/Tonality" title="Tonality">tonal</a> harmonic framework was fully formed, and the various musical functions of the semitone were rigorously understood. Later in this period the adoption of <a href="/wiki/Well_temperament" title="Well temperament">well temperaments</a> for instrumental tuning and the more frequent use of <a href="/wiki/Enharmonic" class="mw-redirect" title="Enharmonic">enharmonic</a> equivalences increased the ease with which a semitone could be applied. Its function remained similar through the <a href="/wiki/Classical_music" title="Classical music">Classical</a> period, and though it was used more frequently as the language of tonality became more chromatic in the <a href="/wiki/Romantic_music" title="Romantic music">Romantic</a> period, the musical function of the semitone did not change. </p><p>In the 20th century, however, composers such as <a href="/wiki/Arnold_Schoenberg" title="Arnold Schoenberg">Arnold Schoenberg</a>, <a href="/wiki/B%C3%A9la_Bart%C3%B3k" title="Béla Bartók">Béla Bartók</a>, and <a href="/wiki/Igor_Stravinsky" title="Igor Stravinsky">Igor Stravinsky</a> sought alternatives or extensions of tonal harmony, and found other uses for the semitone. Often the semitone was exploited harmonically as a caustic dissonance, having no resolution. Some composers would even use large collections of harmonic semitones (<a href="/wiki/Tone_clusters" class="mw-redirect" title="Tone clusters">tone clusters</a>) as a source of cacophony in their music (e.g. the early piano works of <a href="/wiki/Henry_Cowell" title="Henry Cowell">Henry Cowell</a>). By now, enharmonic equivalence was a commonplace property of <a href="/wiki/Equal_temperament" title="Equal temperament">equal temperament</a>, and instrumental use of the semitone was not at all problematic for the performer. The composer was free to write semitones wherever he wished. </p> <div class="mw-heading mw-heading2"><h2 id="Semitones_in_different_tunings">Semitones in different tunings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=5" title="Edit section: Semitones in different tunings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The exact size of a semitone depends on the <a href="/wiki/Musical_tuning" title="Musical tuning">tuning</a> system used. <a href="/wiki/Meantone_temperament" title="Meantone temperament">Meantone temperaments</a> have two distinct types of semitones, but in the exceptional case of <a href="/wiki/Equal_temperament" title="Equal temperament">equal temperament</a>, there is only one. The unevenly distributed <a href="/wiki/Well_temperament" title="Well temperament">well temperaments</a> contain many different semitones. <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a>, similar to meantone tuning, has two, but in other systems of just intonation there are many more possibilities. </p> <div class="mw-heading mw-heading3"><h3 id="Meantone_temperament">Meantone temperament</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=6" title="Edit section: Meantone temperament"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Meantone_temperament" title="Meantone temperament">meantone</a> systems, there are two different semitones. This results because of the break in the <a href="/wiki/Circle_of_fifths" title="Circle of fifths">circle of fifths</a> that occurs in the tuning system: diatonic semitones derive from a chain of five fifths that does not cross the break, and chromatic semitones come from one that does. </p><p>The chromatic semitone is usually smaller than the diatonic. In the common <a href="/wiki/Quarter-comma_meantone" title="Quarter-comma meantone">quarter-comma meantone</a>, tuned as a cycle of <a href="/wiki/Musical_temperament" title="Musical temperament">tempered</a> <a href="/wiki/Perfect_fifth" title="Perfect fifth">fifths</a> from E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> to G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>, the chromatic and diatonic semitones are 76.0 and 117.1 cents wide respectively. </p> <table class="wikitable" style="text-align:center" align="center"> <tbody><tr> <td bgcolor="#ffeeee"><b>Chromatic semitone</b> </td> <td> </td> <td colspan="2" bgcolor="#ffeeee"><small>76.0</small> </td> <td colspan="2"> </td> <td colspan="2"> </td> <td colspan="2" bgcolor="#ffeeee"><small>76.0</small> </td> <td colspan="2"> </td> <td colspan="2" bgcolor="#ffeeee"><small>76.0</small> </td> <td colspan="2"> </td> <td colspan="2" bgcolor="#ffeeee"><small>76.0</small> </td> <td colspan="2"> </td> <td colspan="2"> </td> <td colspan="2" bgcolor="#ffeeee"><small>76.0</small> </td> <td colspan="2"> </td></tr> <tr> <td bgcolor="#fffbee"><b>Pitch</b> </td> <td colspan="2" bgcolor="#fffbee">C </td> <td colspan="2" bgcolor="#fffbee">C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> </td> <td colspan="2" bgcolor="#fffbee">D </td> <td colspan="2" bgcolor="#fffbee">E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> </td> <td colspan="2" bgcolor="#fffbee">E </td> <td colspan="2" bgcolor="#fffbee">F </td> <td colspan="2" bgcolor="#fffbee">F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> </td> <td colspan="2" bgcolor="#fffbee">G </td> <td colspan="2" bgcolor="#fffbee">G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> </td> <td colspan="2" bgcolor="#fffbee">A </td> <td colspan="2" bgcolor="#fffbee">B<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> </td> <td colspan="2" bgcolor="#fffbee">B </td> <td colspan="2" bgcolor="#fffbee">C </td></tr> <tr> <td bgcolor="#fffbee"><b>Cents</b> </td> <td colspan="2" bgcolor="#fffbee"><small>0.0</small> </td> <td colspan="2" bgcolor="#fffbee"><small>76.0</small> </td> <td colspan="2" bgcolor="#fffbee"><small>193.2</small> </td> <td colspan="2" bgcolor="#fffbee"><small>310.3</small> </td> <td colspan="2" bgcolor="#fffbee"><small>386.3</small> </td> <td colspan="2" bgcolor="#fffbee"><small>503.4</small> </td> <td colspan="2" bgcolor="#fffbee"><small>579.5</small> </td> <td colspan="2" bgcolor="#fffbee"><small>696.6</small> </td> <td colspan="2" bgcolor="#fffbee"><small>772.6</small> </td> <td colspan="2" bgcolor="#fffbee"><small>889.7</small> </td> <td colspan="2" bgcolor="#fffbee"><small>1006.8</small> </td> <td colspan="2" bgcolor="#fffbee"><small>1082.9</small> </td> <td colspan="2" bgcolor="#fffbee"><small>1200.0</small> </td></tr> <tr> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td bgcolor="#eeeeff"><b>Diatonic semitone</b> </td> <td colspan="3"> </td> <td colspan="2" bgcolor="#eeeeff"><small>117.1</small> </td> <td colspan="2" bgcolor="#eeeeff"><small>117.1</small> </td> <td colspan="2"> </td> <td colspan="2" bgcolor="#eeeeff"><small>117.1</small> </td> <td colspan="2"> </td> <td colspan="2" bgcolor="#eeeeff"><small>117.1</small> </td> <td colspan="2"> </td> <td colspan="2" bgcolor="#eeeeff"><small>117.1</small> </td> <td colspan="2" bgcolor="#eeeeff"><small>117.1</small> </td> <td colspan="2"> </td> <td colspan="2" bgcolor="#eeeeff"><small>117.1</small> </td> <td> </td></tr></tbody></table> <p>Extended meantone temperaments with more than 12 notes still retain the same two semitone sizes, but there is more flexibility for the musician about whether to use an augmented unison or minor second. <a href="/wiki/31-TET" class="mw-redirect" title="31-TET">31-tone equal temperament</a> is the most flexible of these, which makes an unbroken circle of 31 fifths, allowing the choice of semitone to be made for any pitch. </p> <div class="mw-heading mw-heading3"><h3 id="Equal_temperament">Equal temperament</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=7" title="Edit section: Equal temperament"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/12-tone_equal_temperament" class="mw-redirect" title="12-tone equal temperament">12-tone equal temperament</a> is a form of meantone tuning in which the diatonic and chromatic semitones are exactly the same, because its circle of fifths has no break. Each semitone is equal to one twelfth of an octave. This is a ratio of <a href="/wiki/Twelfth_root_of_two" title="Twelfth root of two">2<sup>1/12</sup></a> (approximately 1.05946), or 100 cents, and is 11.7 cents narrower than the 16:15 ratio (its most common form in <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a>, <a href="#Just_intonation">discussed below</a>). </p><p>All diatonic intervals can be expressed as an equivalent number of semitones. For instance a <a href="/wiki/Major_sixth" title="Major sixth">major sixth</a> equals nine semitones. </p><p>There are many approximations, <a href="/wiki/Rational_number" title="Rational number">rational</a> or otherwise, to the equal-tempered semitone. To cite a few: </p> <dl><dd><ul><li><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 18/17\approx 99.0{\text{ cents,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>17</mn> <mo>≈</mo> <mn>99.0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents,</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 18/17\approx 99.0{\text{ cents,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e1db81cf773940ebab5b9403217032b591125c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.451ex; height:2.843ex;" alt="{\displaystyle 18/17\approx 99.0{\text{ cents,}}}"></span><br />suggested by <a href="/wiki/Vincenzo_Galilei" title="Vincenzo Galilei">Vincenzo Galilei</a> and used by <a href="/wiki/Luthier" title="Luthier">luthiers</a> of the <a href="/wiki/Renaissance_music" title="Renaissance music">Renaissance</a>,</li></ul></dd></dl> <dl><dd><ul><li><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 {\sqrt[{4}]{\frac {2}{3-{\sqrt {2}}}}}\approx 100.4{\text{ cents,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>≈</mo> <mn>100.4</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents,</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{4}]{\frac {2}{3-{\sqrt {2}}}}}\approx 100.4{\text{ cents,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237847316a3aea533ae093273132eda8e4d25250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.062ex; height:7.676ex;" alt="{\displaystyle {\sqrt[{4}]{\frac {2}{3-{\sqrt {2}}}}}\approx 100.4{\text{ cents,}}}"></span><br />suggested by <a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a> as a <a href="/wiki/Constructible_number" title="Constructible number">constructible</a> and more accurate alternative,</li></ul></dd></dl> <dl><dd><ul><li><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 (139/138)^{8}\approx 99.9995{\text{ cents,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>139</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>138</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>≈</mo> <mn>99.9995</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents,</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (139/138)^{8}\approx 99.9995{\text{ cents,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4746e517860da437e1f9dc5296109bcb0be380d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.126ex; height:3.176ex;" alt="{\displaystyle (139/138)^{8}\approx 99.9995{\text{ cents,}}}"></span><br />used by <a href="/wiki/Juli%C3%A1n_Carrillo" title="Julián Carrillo">Julián Carrillo</a> as part of a sixteenth-tone system.</li></ul></dd></dl> <p>For more examples, see Pythagorean and Just systems of tuning below. </p> <div class="mw-heading mw-heading3"><h3 id="Well_temperament">Well temperament</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=8" title="Edit section: Well temperament"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are many forms of <a href="/wiki/Well_temperament" title="Well temperament">well temperament</a>, but the characteristic they all share is that their semitones are of an uneven size. Every semitone in a well temperament has its own interval (usually close to the equal-tempered version of 100 cents), and there is no clear distinction between a <i>diatonic</i> and <i>chromatic</i> semitone in the tuning. Well temperament was constructed so that <a href="/wiki/Enharmonic" class="mw-redirect" title="Enharmonic">enharmonic</a> equivalence could be assumed between all of these semitones, and whether they were written as a minor second or augmented unison did not effect a different sound. Instead, in these systems, each <a href="/wiki/Key_(music)" title="Key (music)">key</a> had a slightly different sonic color or character, beyond the limitations of conventional notation. </p> <div class="mw-heading mw-heading3"><h3 id="Pythagorean_tuning">Pythagorean tuning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=9" title="Edit section: Pythagorean tuning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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:408px;max-width:408px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Pythagorean_limma_on_C.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Pythagorean_limma_on_C.png/200px-Pythagorean_limma_on_C.png" decoding="async" width="200" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Pythagorean_limma_on_C.png/300px-Pythagorean_limma_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Pythagorean_limma_on_C.png/400px-Pythagorean_limma_on_C.png 2x" data-file-width="539" data-file-height="252" /></a></span></div><div class="thumbcaption">Pythagorean limma on C<span typeof="mw:File"><span><audio id="mwe_player_6" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="90" style="width:90px;" data-mwtitle="Pythagorean_minor_semitone_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/6/6e/Pythagorean_minor_semitone_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6e/Pythagorean_minor_semitone_on_C.mid/Pythagorean_minor_semitone_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/6/6e/Pythagorean_minor_semitone_on_C.mid/Pythagorean_minor_semitone_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Pythagorean_apotome_on_C.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Pythagorean_apotome_on_C.png/200px-Pythagorean_apotome_on_C.png" decoding="async" width="200" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Pythagorean_apotome_on_C.png/300px-Pythagorean_apotome_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Pythagorean_apotome_on_C.png/400px-Pythagorean_apotome_on_C.png 2x" data-file-width="539" data-file-height="273" /></a></span></div><div class="thumbcaption">Pythagorean apotome on C<span typeof="mw:File"><span><audio id="mwe_player_7" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="90" style="width:90px;" data-mwtitle="Pythagorean_apotome_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/3/30/Pythagorean_apotome_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/30/Pythagorean_apotome_on_C.mid/Pythagorean_apotome_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/3/30/Pythagorean_apotome_on_C.mid/Pythagorean_apotome_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></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:408px;max-width:408px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Pythagorean_limma.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Pythagorean_limma.png/200px-Pythagorean_limma.png" decoding="async" width="200" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Pythagorean_limma.png/300px-Pythagorean_limma.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Pythagorean_limma.png/400px-Pythagorean_limma.png 2x" data-file-width="588" data-file-height="556" /></a></span></div><div class="thumbcaption">Pythagorean limma as five descending just perfect fifths from C (the inverse is B+)</div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Pythagorean_apotome.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Pythagorean_apotome.png/200px-Pythagorean_apotome.png" decoding="async" width="200" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Pythagorean_apotome.png/300px-Pythagorean_apotome.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Pythagorean_apotome.png/400px-Pythagorean_apotome.png 2x" data-file-width="588" data-file-height="606" /></a></span></div><div class="thumbcaption">Pythagorean apotome as seven just perfect fifths</div></div></div></div></div> <p>Like meantone temperament, <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a> is a broken <a href="/wiki/Circle_of_fifths" title="Circle of fifths">circle of fifths</a>. This creates two distinct semitones, but because Pythagorean tuning is also a form of 3-limit <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a>, these semitones are rational. Also, unlike most meantone temperaments, the chromatic semitone is larger than the diatonic. </p><p>The <b>Pythagorean diatonic semitone</b> has a ratio of 256/243 (<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\/6\/6e\/Pythagorean_minor_semitone_on_C.mid\/Pythagorean_minor_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean minor semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6e/Pythagorean_minor_semitone_on_C.mid/Pythagorean_minor_semitone_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_minor_semitone_on_C.mid" title="File:Pythagorean minor semitone on C.mid">ⓘ</a></sup></span></span>), and is often called the <b>Pythagorean limma</b>. It is also sometimes called the <i>Pythagorean minor semitone</i>. It is about 90.2 cents. </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 {\frac {256}{243}}={\frac {2^{8}}{3^{5}}}\approx 90.2{\text{ cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>256</mn> <mn>243</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>≈</mo> <mn>90.2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {256}{243}}={\frac {2^{8}}{3^{5}}}\approx 90.2{\text{ cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998aede0b9880376137ab0db2f474172c4fe7662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.466ex; height:6.176ex;" alt="{\displaystyle {\frac {256}{243}}={\frac {2^{8}}{3^{5}}}\approx 90.2{\text{ cents}}}"></span></dd></dl> <p>It can be thought of as the difference between three <a href="/wiki/Octaves" class="mw-redirect" title="Octaves">octaves</a> and five <a href="/wiki/Perfect_fifth" title="Perfect fifth">just fifths</a>, and functions as a <a href="#Minor_second">diatonic semitone</a> in a <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a>. </p><p>The <b>Pythagorean chromatic semitone</b> has a ratio of 2187/2048 (<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\/3\/30\/Pythagorean_apotome_on_C.mid\/Pythagorean_apotome_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean apotome on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/3/30/Pythagorean_apotome_on_C.mid/Pythagorean_apotome_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_apotome_on_C.mid" title="File:Pythagorean apotome on C.mid">ⓘ</a></sup></span></span>). It is about 113.7 <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>. It may also be called the <b>Pythagorean apotome</b><sup id="cite_ref-Rashed_14-0" class="reference"><a href="#cite_note-Rashed-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> or the <i>Pythagorean major semitone</i>. (<i>See <a href="/wiki/Pythagorean_interval" title="Pythagorean interval">Pythagorean interval</a></i>.) </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 {\frac {2187}{2048}}={\frac {3^{7}}{2^{11}}}\approx 113.7{\text{ cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2187</mn> <mn>2048</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> </mfrac> </mrow> <mo>≈</mo> <mn>113.7</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2187}{2048}}={\frac {3^{7}}{2^{11}}}\approx 113.7{\text{ cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3477e268c03c6ad238fb37df41c7a34ebaf07264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.613ex; height:6.176ex;" alt="{\displaystyle {\frac {2187}{2048}}={\frac {3^{7}}{2^{11}}}\approx 113.7{\text{ cents}}}"></span></dd></dl> <p>It can be thought of as the difference between four perfect <a href="/wiki/Octave" title="Octave">octaves</a> and seven <a href="/wiki/Perfect_fifth" title="Perfect fifth">just fifths</a>, and functions as a <a href="/wiki/Chromatic_semitone" class="mw-redirect" title="Chromatic semitone">chromatic semitone</a> in a <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a>. </p><p>The Pythagorean limma and Pythagorean apotome are <a href="/wiki/Enharmonic" class="mw-redirect" title="Enharmonic">enharmonic</a> equivalents (chromatic semitones) and only a <a href="/wiki/Pythagorean_comma" title="Pythagorean comma">Pythagorean comma</a> apart, in contrast to diatonic and chromatic semitones in <a href="/wiki/Meantone_temperament" title="Meantone temperament">meantone temperament</a> and 5-limit <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Just_5-limit_intonation">Just 5-limit intonation <span class="anchor" id="Just_intonation"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=10" title="Edit section: Just 5-limit intonation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Just_diatonic_semitone.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Just_diatonic_semitone.png/220px-Just_diatonic_semitone.png" decoding="async" width="220" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Just_diatonic_semitone.png/330px-Just_diatonic_semitone.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Just_diatonic_semitone.png/440px-Just_diatonic_semitone.png 2x" data-file-width="1197" data-file-height="1140" /></a><figcaption>16:15 <a href="#Minor_second">diatonic semitone</a></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Just_diatonic_semitone_on_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Just_diatonic_semitone_on_C.png/220px-Just_diatonic_semitone_on_C.png" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Just_diatonic_semitone_on_C.png/330px-Just_diatonic_semitone_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Just_diatonic_semitone_on_C.png/440px-Just_diatonic_semitone_on_C.png 2x" data-file-width="539" data-file-height="252" /></a><figcaption>16:15 diatonic semitone<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_8" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-mwtitle="Just_diatonic_semitone_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/b/be/Just_diatonic_semitone_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/be/Just_diatonic_semitone_on_C.mid/Just_diatonic_semitone_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/b/be/Just_diatonic_semitone_on_C.mid/Just_diatonic_semitone_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:Major_limma_on_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Major_limma_on_C.png/220px-Major_limma_on_C.png" decoding="async" width="220" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Major_limma_on_C.png/330px-Major_limma_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Major_limma_on_C.png/440px-Major_limma_on_C.png 2x" data-file-width="539" data-file-height="273" /></a><figcaption>'Larger' or major limma on C<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_9" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-mwtitle="Greater_chromatic_semitone_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/1/11/Greater_chromatic_semitone_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/11/Greater_chromatic_semitone_on_C.mid/Greater_chromatic_semitone_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/1/11/Greater_chromatic_semitone_on_C.mid/Greater_chromatic_semitone_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Semitone_5-limit_diamond.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Semitone_5-limit_diamond.png/300px-Semitone_5-limit_diamond.png" decoding="async" width="300" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Semitone_5-limit_diamond.png/450px-Semitone_5-limit_diamond.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Semitone_5-limit_diamond.png/600px-Semitone_5-limit_diamond.png 2x" data-file-width="1032" data-file-height="744" /></a><figcaption>Relationship between the 4 common 5 limit semitones</figcaption></figure> <p>A minor second in <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a> typically corresponds to a pitch <a href="/wiki/Ratio" title="Ratio">ratio</a> of 16:15 (<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\/b\/be\/Just_diatonic_semitone_on_C.mid\/Just_diatonic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just diatonic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/be/Just_diatonic_semitone_on_C.mid/Just_diatonic_semitone_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:Just_diatonic_semitone_on_C.mid" title="File:Just diatonic semitone on C.mid">ⓘ</a></sup></span></span>) or 1.0666... (approximately 111.7 <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>), called the <b>just diatonic semitone</b>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This is a practical just semitone, since it is the interval that occurs twice within the diatonic scale between a: </p> <dl><dd><a href="/wiki/Major_third" title="Major third">major third</a> (5:4) and <a href="/wiki/Perfect_fourth" title="Perfect fourth">perfect fourth</a> (4:3) <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 \ \left(\ {\tfrac {4}{3}}\div {\tfrac {5}{4}}={\tfrac {16}{15}}\ \right)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>÷</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left(\ {\tfrac {4}{3}}\div {\tfrac {5}{4}}={\tfrac {16}{15}}\ \right)\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6833a76b293c6a67e766badde7aff69ad42506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.254ex; height:4.843ex;" alt="{\displaystyle \ \left(\ {\tfrac {4}{3}}\div {\tfrac {5}{4}}={\tfrac {16}{15}}\ \right)\ ,}"></span> and a</dd> <dd><a href="/wiki/Major_seventh" title="Major seventh">major seventh</a> (15:8) and the <a href="/wiki/Perfect_octave" class="mw-redirect" title="Perfect octave">perfect octave</a> (2:1) <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 \ \left(\ {\tfrac {2}{1}}\div {\tfrac {15}{8}}={\tfrac {16}{15}}\ \right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>1</mn> </mfrac> </mstyle> </mrow> <mo>÷</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>15</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left(\ {\tfrac {2}{1}}\div {\tfrac {15}{8}}={\tfrac {16}{15}}\ \right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4bd72832424963823c79ddc8d78a5d1a7de4862" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.076ex; height:4.843ex;" alt="{\displaystyle \ \left(\ {\tfrac {2}{1}}\div {\tfrac {15}{8}}={\tfrac {16}{15}}\ \right)~.}"></span></dd></dl> <p>The 16:15 just minor second arises in the C major scale between B & C and E & F, and is, "the sharpest dissonance found in the scale".<sup id="cite_ref-books.google.com_8-1" class="reference"><a href="#cite_note-books.google.com-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>An "augmented unison" (sharp) in just intonation is a different, smaller semitone, with frequency ratio 25:24 (<span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-6" 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\/1\/1d\/Just_chromatic_semitone_on_C.mid\/Just_chromatic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just chromatic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/1/1d/Just_chromatic_semitone_on_C.mid/Just_chromatic_semitone_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:Just_chromatic_semitone_on_C.mid" title="File:Just chromatic semitone on C.mid">ⓘ</a></sup></span></span>) or 1.0416... (approximately 70.7 cents). It is the interval between a <a href="/wiki/Major_third" title="Major third">major third</a> (5:4) and a minor third (6:5). In fact, it is the spacing between the minor and major thirds, sixths, and sevenths (but not necessarily the major and minor second). Composer <a href="/wiki/Ben_Johnston_(composer)" title="Ben Johnston (composer)">Ben Johnston</a> used a sharp (<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>) to indicate a note is raised 70.7 cents, or a flat (<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>) to indicate a note is lowered 70.7 cents.<sup id="cite_ref-Fonville_18-0" class="reference"><a href="#cite_note-Fonville-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> (This is the standard practice for just intonation, but not for all other microtunings.) </p><p>Two other kinds of semitones are produced by 5 limit tuning. A <a href="/wiki/Chromatic_scale" title="Chromatic scale">chromatic scale</a> defines 12 semitones as the 12 intervals between the 13 adjacent notes, spanning a full octave (e.g. from C<sub>4</sub> to C<sub>5</sub>). The 12 semitones produced by a <a href="/wiki/Five-limit_tuning#Size_of_intervals" title="Five-limit tuning">commonly used version</a> of 5 limit tuning have four different sizes, and can be classified as follows: </p> <dl><dt>Just chromatic semitone</dt> <dd><i>chromatic semitone</i>, or <i>smaller</i>, or <i>minor chromatic semitone</i> between harmonically related flats and sharps e.g. between E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> and E (6:5 and 5:4):</dd> <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 S_{1}={\tfrac {5}{4}}\div {\tfrac {6}{5}}={\tfrac {25}{24}}\approx 70.7\ {\hbox{cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>÷</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>6</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>25</mn> <mn>24</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>70.7</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>cents</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}={\tfrac {5}{4}}\div {\tfrac {6}{5}}={\tfrac {25}{24}}\approx 70.7\ {\hbox{cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d5c5468fc50145644c3a0a16698b8a4fd6e01f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:30.304ex; height:3.676ex;" alt="{\displaystyle S_{1}={\tfrac {5}{4}}\div {\tfrac {6}{5}}={\tfrac {25}{24}}\approx 70.7\ {\hbox{cents}}}"></span></dd> <dt>Larger chromatic semitone</dt> <dd>or <i>major chromatic semitone</i>, or <i>larger limma</i>, or <i>major chroma</i>,<sup id="cite_ref-Fonville_18-1" class="reference"><a href="#cite_note-Fonville-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> e.g. between C and an accute C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> (C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> raised by a <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a>) (1:1 and 135:128):</dd> <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 S_{2}={\tfrac {25}{24}}\times {\tfrac {81}{80}}={\tfrac {135}{128}}\approx 92.2\ {\hbox{cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>25</mn> <mn>24</mn> </mfrac> </mstyle> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>81</mn> <mn>80</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>135</mn> <mn>128</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>92.2</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>cents</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}={\tfrac {25}{24}}\times {\tfrac {81}{80}}={\tfrac {135}{128}}\approx 92.2\ {\hbox{cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6338101f2660286386079ddf74aece4f989cff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:32.77ex; height:3.676ex;" alt="{\displaystyle S_{2}={\tfrac {25}{24}}\times {\tfrac {81}{80}}={\tfrac {135}{128}}\approx 92.2\ {\hbox{cents}}}"></span></dd> <dt>Just diatonic semitone</dt> <dd>or <i>smaller</i>, or <i>minor diatonic semitone</i>, e.g. between E and F (5:4 to 4:3):</dd> <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 S_{3}={\tfrac {4}{3}}\div {\tfrac {5}{4}}={\tfrac {16}{15}}\approx 111.7\ {\hbox{cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>÷</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>111.7</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>cents</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{3}={\tfrac {4}{3}}\div {\tfrac {5}{4}}={\tfrac {16}{15}}\approx 111.7\ {\hbox{cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef3ddf4fdefdf203ca7072b343a7dccb5634304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.467ex; height:3.676ex;" alt="{\displaystyle S_{3}={\tfrac {4}{3}}\div {\tfrac {5}{4}}={\tfrac {16}{15}}\approx 111.7\ {\hbox{cents}}}"></span></dd> <dt>Larger diatonic semitone</dt> <dd>or <i>greater</i> or <i>major diatonic semitone</i>, e.g. between A and B<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> (5:3 to 9:5), or C and chromatic D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> (27:25), or F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> and G (25:18 and 3:2):</dd> <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 S_{4}={\tfrac {9}{5}}\div {\tfrac {5}{3}}={\tfrac {27}{25}}\approx 133.2\ {\hbox{cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>9</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mo>÷</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>27</mn> <mn>25</mn> </mfrac> </mstyle> </mrow> <mo>≈</mo> <mn>133.2</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>cents</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{4}={\tfrac {9}{5}}\div {\tfrac {5}{3}}={\tfrac {27}{25}}\approx 133.2\ {\hbox{cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce4941eea07e56f8a496e21168a3494d28675e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.467ex; height:3.843ex;" alt="{\displaystyle S_{4}={\tfrac {9}{5}}\div {\tfrac {5}{3}}={\tfrac {27}{25}}\approx 133.2\ {\hbox{cents}}}"></span></dd></dl> <p>The most frequently occurring semitones are the just ones (<span class="texhtml mvar" style="font-style:italic;">S</span><sub>3</sub>, 16:15, and <span class="texhtml mvar" style="font-style:italic;">S</span><sub>1</sub>, 25:24): S<sub>3</sub> occurs at 6 short intervals out of 12, <span class="texhtml mvar" style="font-style:italic;">S</span><sub>1</sub> 3 times, <span class="texhtml mvar" style="font-style:italic;">S</span><sub>2</sub> twice, and <span class="texhtml mvar" style="font-style:italic;">S</span><sub>4</sub> at only one interval (if diatonic D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> replaces chromatic D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> and sharp notes are not used). </p><p>The smaller chromatic and diatonic semitones differ from the larger by the <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a> (81:80 or 21.5 cents). The smaller and larger chromatic semitones differ from the respective diatonic semitones by the same 128:125 diesis as the above meantone semitones. Finally, while the inner semitones differ by the <a href="/wiki/Diaschisma" title="Diaschisma">diaschisma</a> (2048:2025 or 19.6 cents), the outer differ by the greater diesis (648:625 or 62.6 cents). </p> <div class="mw-heading mw-heading3"><h3 id="Extended_just_intonations">Extended just intonations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=11" title="Edit section: Extended just intonations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/7-limit" class="mw-redirect" title="7-limit">7 limit tuning</a> there is the <a href="/wiki/Septimal_diatonic_semitone" title="Septimal diatonic semitone">septimal diatonic semitone</a> of 15:14 (<span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-7" 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\/69\/Septimal_diatonic_semitone_on_C.mid\/Septimal_diatonic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal diatonic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/6/69/Septimal_diatonic_semitone_on_C.mid/Septimal_diatonic_semitone_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:Septimal_diatonic_semitone_on_C.mid" title="File:Septimal diatonic semitone on C.mid">ⓘ</a></sup></span></span>) available in between the 5 limit <a href="/wiki/Major_seventh" title="Major seventh">major seventh</a> (15:8) and the <a href="/wiki/Septimal_minor_seventh" class="mw-redirect" title="Septimal minor seventh">7 limit minor seventh</a> / <a href="/wiki/Harmonic_seventh" title="Harmonic seventh">harmonic seventh</a> (7:4). There is also a smaller <a href="/wiki/Septimal_chromatic_semitone" title="Septimal chromatic semitone">septimal chromatic semitone</a> of 21:20 (<span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-8" 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\/b\/ba\/Septimal_chromatic_semitone_on_C.mid\/Septimal_chromatic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal chromatic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/ba/Septimal_chromatic_semitone_on_C.mid/Septimal_chromatic_semitone_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:Septimal_chromatic_semitone_on_C.mid" title="File:Septimal chromatic semitone on C.mid">ⓘ</a></sup></span></span>) between a septimal minor seventh and a fifth (21:8) and an octave and a major third (5:2). Both are more rarely used than their 5 limit neighbours, although the former was often implemented by theorist <a href="/wiki/Henry_Cowell" title="Henry Cowell">Cowell</a>, while <a href="/wiki/Harry_Partch" title="Harry Partch">Partch</a> used the latter as part of <a href="/wiki/Harry_Partch%27s_43-tone_scale" title="Harry Partch's 43-tone scale">his 43 tone scale</a>. </p><p>Under 11 limit tuning, there is a fairly common <i>undecimal <a href="/wiki/Neutral_second" class="mw-redirect" title="Neutral second">neutral second</a></i> (12:11) (<span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-9" 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\/2\/2d\/Neutral_second_on_C.mid\/Neutral_second_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Neutral second on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2d/Neutral_second_on_C.mid/Neutral_second_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:Neutral_second_on_C.mid" title="File:Neutral second on C.mid">ⓘ</a></sup></span></span>), but it lies on the boundary between the minor and <a href="/wiki/Major_second" title="Major second">major second</a> (150.6 cents). In just intonation there are infinitely many possibilities for intervals that fall within the range of the semitone (e.g. the Pythagorean semitones mentioned above), but most of them are impractical. </p><p>In 13 limit tuning, there is a tridecimal <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">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> tone (13:12 or 138.57 cents) and tridecimal <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">3</span></span>⁠</span> tone (27:26 or 65.34 cents). </p><p>In 17 limit just intonation, the major diatonic semitone is 15:14 or 119.4 cents (<span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-10" 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\/9\/91\/Major_diatonic_semitone_on_C.mid\/Major_diatonic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major diatonic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/91/Major_diatonic_semitone_on_C.mid/Major_diatonic_semitone_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:Major_diatonic_semitone_on_C.mid" title="File:Major diatonic semitone on C.mid">ⓘ</a></sup></span></span>), and the minor diatonic semitone is 17:16 or 105.0 cents,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> and septendecimal limma is 18:17 or 98.95 cents. </p><p>Though the names <i>diatonic</i> and <i>chromatic</i> are often used for these intervals, their musical function is not the same as the meantone semitones. For instance, 15:14 would usually be written as an augmented unison, functioning as the <i>chromatic</i> counterpart to a <i>diatonic</i> 16:15. These distinctions are highly dependent on the musical context, and just intonation is not particularly well suited to chromatic use (diatonic semitone function is more prevalent). </p> <div class="mw-heading mw-heading3"><h3 id="Other_equal_temperaments">Other equal temperaments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=12" title="Edit section: Other equal temperaments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/19_equal_temperament" title="19 equal temperament">19-tone equal temperament</a> distinguishes between the chromatic and diatonic semitones; in this tuning, the chromatic semitone is one step of the scale (<span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-11" 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\/3\/39\/1_step_in_19-et_on_C.mid\/1_step_in_19-et_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play 63.2 cents"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"1 step in 19-et on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/3/39/1_step_in_19-et_on_C.mid/1_step_in_19-et_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 63.2 cents</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:1_step_in_19-et_on_C.mid" title="File:1 step in 19-et on C.mid">ⓘ</a></sup></span></span>), and the diatonic semitone is two (<span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-12" 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\/9\/90\/2_steps_in_19-et_on_C.mid\/2_steps_in_19-et_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play 126.3 cents"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"2 steps in 19-et on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/90/2_steps_in_19-et_on_C.mid/2_steps_in_19-et_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 126.3 cents</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:2_steps_in_19-et_on_C.mid" title="File:2 steps in 19-et on C.mid">ⓘ</a></sup></span></span>). <a href="/wiki/31_equal_temperament" title="31 equal temperament">31-tone equal temperament</a> also distinguishes between these two intervals, which become 2 and 3 steps of the scale, respectively. <a href="/wiki/53_equal_temperament" title="53 equal temperament">53-ET</a> has an even closer match to the two semitones with 3 and 5 steps of its scale while <a href="/wiki/72_equal_temperament" title="72 equal temperament">72-ET</a> uses 4 (<span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-13" 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\/8\/80\/4_steps_in_72-et_on_C.mid\/4_steps_in_72-et_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play 66.7 cents"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"4 steps in 72-et on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/8/80/4_steps_in_72-et_on_C.mid/4_steps_in_72-et_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 66.7 cents</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:4_steps_in_72-et_on_C.mid" title="File:4 steps in 72-et on C.mid">ⓘ</a></sup></span></span>) and 7 (<span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-14" 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\/ee\/7_steps_in_72-et_on_C.mid\/7_steps_in_72-et_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play 116.7 cents"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"7 steps in 72-et on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/e/ee/7_steps_in_72-et_on_C.mid/7_steps_in_72-et_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 116.7 cents</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:7_steps_in_72-et_on_C.mid" title="File:7 steps in 72-et on C.mid">ⓘ</a></sup></span></span>) steps of its scale. </p><p>In general, because the smaller semitone can be viewed as the difference between a minor third and a major third, and the larger as the difference between a major third and a perfect fourth, tuning systems that closely match those just intervals (6/5, 5/4, and 4/3) will also distinguish between the two types of semitones and closely match their just intervals (25/24 and 16/15). </p> <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=Semitone&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/12_equal_temperament" title="12 equal temperament">12-tone equal temperament</a></li> <li><a href="/wiki/List_of_meantone_intervals" title="List of meantone intervals">List of meantone intervals</a></li> <li><a href="/wiki/Interval_(music)" title="Interval (music)">List of musical intervals</a></li> <li><a href="/wiki/List_of_pitch_intervals" title="List of pitch intervals">List of pitch intervals</a></li> <li><a href="/wiki/Approach_chord" title="Approach chord">Approach chord</a></li> <li><a href="/wiki/Major_second" title="Major second">Major second</a></li> <li><a href="/wiki/Neutral_second" class="mw-redirect" title="Neutral second">Neutral second</a></li> <li><a href="/wiki/Pythagorean_interval" title="Pythagorean interval">Pythagorean interval</a></li> <li><a href="/wiki/Regular_temperament" title="Regular temperament">Regular temperament</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=Semitone&action=edit&section=14" 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 mw-references-columns"><ol class="references"> <li id="cite_note-Duffin-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Duffin_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Duffin_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Duffin_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Duffin_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Duffin_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Duffin_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Duffin_1-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text"><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 a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .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><cite id="CITEREFDuffin2008" class="citation book cs1">Duffin, Ross W. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=i5LC7Csnw7UC&q=how+equal+temperament+ruined+harmony"><i>How equal temperament ruined harmony : (and why you should care)</i></a> (First published as a Norton paperback. ed.). New York: W. W. Norton. p. 163. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-33420-3" title="Special:BookSources/978-0-393-33420-3"><bdi>978-0-393-33420-3</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">28 June</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+equal+temperament+ruined+harmony+%3A+%28and+why+you+should+care%29&rft.place=New+York&rft.pages=163&rft.edition=First+published+as+a+Norton+paperback.&rft.pub=W.+W.+Norton&rft.date=2008&rft.isbn=978-0-393-33420-3&rft.aulast=Duffin&rft.aufirst=Ross+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Di5LC7Csnw7UC%26q%3Dhow%2Bequal%2Btemperament%2Bruined%2Bharmony&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemitone" class="Z3988"></span></span> </li> <li id="cite_note-Haluska-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Haluska_2-0">^</a></b></span> <span class="reference-text">Haluska, Jan (2003). <i>The Mathematical Theory of Tone Systems</i>, p. xxiv. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8247-4714-3" title="Special:BookSources/0-8247-4714-3">0-8247-4714-3</a>. Overtone semitone.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><i>Semitone</i>, <i>half step</i>, <i>half tone</i>, <i>halftone</i>, and <i>half-tone</i> are all variously used in sources.<a rel="nofollow" class="external autonumber" href="http://www.merriam-webster.com/dictionary/half+step">[1]</a><a rel="nofollow" class="external autonumber" href="http://www.merriam-webster.com/dictionary/half%20tone">[2]</a><a rel="nofollow" class="external autonumber" href="http://dictionary.reference.com/browse/half%20tone">[3]</a><a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=sTMbuSQdqPMC&q=a+half+step+is+called+a+semitone&pg=PA19">[4]</a><a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=iYgSJSxWW2sC">[5]</a> <br /><a href="/wiki/Aaron_Copland" title="Aaron Copland">Aaron Copland</a>, <a href="/wiki/Leonard_Bernstein" title="Leonard Bernstein">Leonard Bernstein</a>, and others use "half tone".<a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=dsyPycO3GfgC&q=the+twelve+chromatic+ones,+arranged+in+the+following+order:+two+whole+tones+followed+by+a+half+tone,+plus+three+whole+tones+followed+by+a+half+tone.&pg=PA41">[6]</a> <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=d-zwOLoDIcEC&q=Now,+if+you+remember+that+the+step+from+any+note+on+the+piano+to+the+note+just+next+to+it,+whether+it%27s+black+or+white,+is+a+step+of+a+half+tone,+you+can+see+that+the+entire+piano+keyboard+is+made+up+of+only+half+tones,+one+after+another.&pg=PA185">[7]</a><a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=PoM6AAAAMAAJ&q=almost+a+half-tone+sharp">[8]</a><a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=1nSqLzjZBKwC&q=This+is+easy+on+the+guitar,+since+from+one+fret+to+the+next+is+a+half+tone+(also+called+half+step+or+semitone).&pg=PA116">[9]</a> <br />One source says that <i>step</i> is "chiefly US",<a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20060301170305/http://www.askoxford.com/concise_oed/step">[10]</a> and that <i>half-tone</i> is "chiefly N. Amer."<a rel="nofollow" class="external autonumber" href="https://archive.today/20130117121046/http://www.askoxford.com/concise_oed/halftone">[11]</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Miller, Michael. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sTMbuSQdqPMC"><i>The Complete Idiot's Guide to Music Theory, 2nd ed</i></a>. [Indianapolis, Indiana]: Alpha, 2005. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-59257-437-8" title="Special:BookSources/1-59257-437-8">1-59257-437-8</a>. p. 19.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCapstick1913" class="citation book cs1">Capstick, John Walton (1913). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bwNJAAAAIAAJ&q=most+dissonant-interval+semitone+intitle:sound&pg=PA227"><i>Sound: An Elementary Text-book for Schools and Colleges</i></a>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sound%3A+An+Elementary+Text-book+for+Schools+and+Colleges&rft.pub=Cambridge+University+Press&rft.date=1913&rft.aulast=Capstick&rft.aufirst=John+Walton&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbwNJAAAAIAAJ%26q%3Dmost%2Bdissonant-interval%2Bsemitone%2Bintitle%3Asound%26pg%3DPA227&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemitone" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.musictheory.net/lessons/20">"musictheory.net"</a>. <i>www.musictheory.net</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-01-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.musictheory.net&rft.atitle=musictheory.net&rft_id=https%3A%2F%2Fwww.musictheory.net%2Flessons%2F20&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemitone" class="Z3988"></span></span> </li> <li id="cite_note-Wharram-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wharram_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWharram2010" class="citation book cs1">Wharram, Barbara (2010). <i>Elementary Rudiments of Music</i> (2nd ed.). Mississauga, Ontario: Frederick Harris Music. p. 17. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-55440-283-0" title="Special:BookSources/978-1-55440-283-0"><bdi>978-1-55440-283-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Rudiments+of+Music&rft.place=Mississauga%2C+Ontario&rft.pages=17&rft.edition=2nd&rft.pub=Frederick+Harris+Music&rft.date=2010&rft.isbn=978-1-55440-283-0&rft.aulast=Wharram&rft.aufirst=Barbara&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemitone" class="Z3988"></span></span> </li> <li id="cite_note-books.google.com-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-books.google.com_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-books.google.com_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Paul, Oscar (1885). <i><a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_4WEJAQAAMAAJ">A manual of harmony for use in music-schools and seminaries and for self-instruction</a></i>, p. 165. <a href="/wiki/Theodore_Baker" title="Theodore Baker">Theodore Baker</a>, trans. G. Schirmer.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Benward & Saker (2003). <i>Music: In Theory and Practice, Vol. I</i>, p. 54. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-294262-0" title="Special:BookSources/978-0-07-294262-0">978-0-07-294262-0</a>. Specific example of an A1 not given but general example of perfect intervals described.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Kostka and Payne (2003). <i>Tonal Harmony</i>, p. 21. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-285260-7" title="Special:BookSources/0-07-285260-7">0-07-285260-7</a>. "There is no such thing as a diminished unison."</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Day and Pilhofer (2007). <i>Music Theory for Dummies</i>, p. 113. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7645-7838-3" title="Special:BookSources/0-7645-7838-3">0-7645-7838-3</a>. "There is no such thing as a diminished unison, because no matter how you change the unisons with accidentals, you are adding half steps to the total interval."</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSurmaniKaren_Farnum_SurmaniMorton_Manus2009" class="citation book cs1"><a href="/wiki/Andrew_Surmani" title="Andrew Surmani">Surmani, Andrew</a>; Karen Farnum Surmani; Morton Manus (2009). <i>Alfred's Essentials of Music Theory: A Complete Self-Study Course for All Musicians</i>. Alfred Music Publishing. p. 135. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7390-3635-8" title="Special:BookSources/978-0-7390-3635-8"><bdi>978-0-7390-3635-8</bdi></a>. Since lowering either note of a perfect unison would actually increase its size, the perfect unison cannot be diminished, only augmented.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alfred%27s+Essentials+of+Music+Theory%3A+A+Complete+Self-Study+Course+for+All+Musicians&rft.pages=135&rft.pub=Alfred+Music+Publishing&rft.date=2009&rft.isbn=978-0-7390-3635-8&rft.aulast=Surmani&rft.aufirst=Andrew&rft.au=Karen+Farnum+Surmani&rft.au=Morton+Manus&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemitone" class="Z3988"></span></span> </li> <li id="cite_note-Dahlhaus-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dahlhaus_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dahlhaus_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Carl_Dahlhaus" title="Carl Dahlhaus">Dahlhaus, Carl</a>, trans. Gjerdingen, Robert O. <i>Studies in the Origin of Harmonic Tonality</i>. Princeton University Press: Princeton, 1990. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-09135-8" title="Special:BookSources/0-691-09135-8">0-691-09135-8</a>.</span> </li> <li id="cite_note-Rashed-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rashed_14-0">^</a></b></span> <span class="reference-text">Rashed, Roshdi (ed.) (1996). <i>Encyclopedia of the History of Arabic Science, Volume 2</i>, pp. 588, 608. Routledge. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-415-12411-5" title="Special:BookSources/0-415-12411-5">0-415-12411-5</a>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a> (1885). <i>On the Sensations of Tone as a Physiological Basis for the Theory of Music</i>, p. 454.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Benson, Dave (2006). <i>Music: A Mathematical Offering</i>, p. 369. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-85387-7" title="Special:BookSources/0-521-85387-7">0-521-85387-7</a>.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1">"<span style="color:gray">[no title cited]</span>". <i><a href="/wiki/Proceedings_of_the_Royal_Society_of_London" class="mw-redirect" title="Proceedings of the Royal Society of London">Proceedings of the Royal Society of London</a></i>. <b>30</b>. Great Britain: Royal Society: 531. 1880. <q>digitized 26 Feb 2008; Harvard University</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London&rft.atitle=%3Cspan+style%3D%22color%3Agray%22%3E%5Bno+title+cited%5D%3C%2Fspan%3E&rft.volume=30&rft.pages=531&rft.date=1880&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemitone" class="Z3988"></span></span> </li> <li id="cite_note-Fonville-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fonville_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fonville_18-1"><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="CITEREFFonville1991" class="citation journal cs1"><a href="/wiki/John_Fonville" title="John Fonville">Fonville, J.</a> (Summer 1991). "<a href="/wiki/Ben_Johnston_(composer)" title="Ben Johnston (composer)">Ben Johnston</a>'s extended just intonation – a guide for interpreters". <i><a href="/wiki/Perspectives_of_New_Music" title="Perspectives of New Music">Perspectives of New Music</a></i>. <b>29</b> (2): 106–137. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F833435">10.2307/833435</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/833435">833435</a>. <q>... the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">25</span><span class="sr-only">/</span><span class="den">24</span></span>⁠</span> ratio is the sharp (<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>) ratio ... this raises a note approximately 70.6 cents.<sup class="reference nowrap"><span title="Page: 109">(p109)</span></sup></q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Perspectives+of+New+Music&rft.atitle=Ben+Johnston%27s+extended+just+intonation+%E2%80%93+a+guide+for+interpreters&rft.ssn=summer&rft.volume=29&rft.issue=2&rft.pages=106-137&rft.date=1991&rft_id=info%3Adoi%2F10.2307%2F833435&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F833435%23id-name%3DJSTOR&rft.aulast=Fonville&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemitone" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProut2004" class="citation book cs1"><a href="/wiki/Ebenezer_Prout" title="Ebenezer Prout">Prout, E.</a> (2004). <i>Harmony</i>. University Press of the Pacific. p. 325. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4102-1920-8" title="Special:BookSources/1-4102-1920-8"><bdi>1-4102-1920-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Harmony&rft.pages=325&rft.pub=University+Press+of+the+Pacific&rft.date=2004&rft.isbn=1-4102-1920-8&rft.aulast=Prout&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASemitone" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Semitone&action=edit&section=15" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Donald_Jay_Grout" title="Donald Jay Grout">Grout, Donald Jay</a>, and <a href="/wiki/Claude_V._Palisca" title="Claude V. Palisca">Claude V. Palisca</a>. <i>A History of Western Music, 6th ed</i>. New York: Norton, 2001. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-393-97527-4" title="Special:BookSources/0-393-97527-4">0-393-97527-4</a>.</li> <li><a href="/wiki/Richard_Hoppin" title="Richard Hoppin">Hoppin, Richard H.</a> <i>Medieval Music</i>. New York: W. W. Norton, 1978. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-393-09090-6" title="Special:BookSources/0-393-09090-6">0-393-09090-6</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 dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .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 mw-collapsed 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 class="mw-selflink selflink">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 href="/wiki/Syntonic_comma" title="Syntonic comma">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 class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Twelve-tone_technique_and_serialism" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3" style="background:#dbc3a3;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Twelve-tone_technique" title="Template:Twelve-tone technique"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Twelve-tone_technique" title="Template talk:Twelve-tone technique"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Twelve-tone_technique" title="Special:EditPage/Template:Twelve-tone technique"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Twelve-tone_technique_and_serialism" style="font-size:114%;margin:0 4em"><a href="/wiki/Twelve-tone_technique" title="Twelve-tone technique">Twelve-tone technique</a> and <a href="/wiki/Serialism" title="Serialism">serialism</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatoriality" title="Combinatoriality">Combinatoriality</a></li> <li><a href="/wiki/Complement_(music)" title="Complement (music)">Complementation</a></li> <li><a href="/wiki/Derived_row" title="Derived row">Derivation</a></li> <li><a href="/wiki/Hexachord" title="Hexachord">Hexachord</a></li> <li><a href="/wiki/Interval_class" title="Interval class">Interval class</a></li> <li><a href="/wiki/Twelve-tone_technique#Invariance" title="Twelve-tone technique">Invariance</a></li> <li><a href="/wiki/Derived_row" title="Derived row">Partition</a> <ul><li><a href="/wiki/Cross_partition" class="mw-redirect" title="Cross partition">Cross partition</a></li></ul></li> <li><a href="/wiki/Tone_row" title="Tone row">Tone row</a> <ul><li><a href="/wiki/Tone_row#total_chromatic" title="Tone row">Aggregate</a></li> <li><a href="/wiki/List_of_tone_rows_and_series" title="List of tone rows and series">List</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Mutterakkord.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Mutterakkord.png/54px-Mutterakkord.png" decoding="async" width="54" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Mutterakkord.png/81px-Mutterakkord.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Mutterakkord.png/108px-Mutterakkord.png 2x" data-file-width="1575" data-file-height="3675" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Permutation_(music)" title="Permutation (music)">Permutations</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Twelve-tone_technique#Transformations" title="Twelve-tone technique">Prime row</a></li> <li><a href="/wiki/Retrograde_(music)" title="Retrograde (music)">Retrograde</a></li> <li><a href="/wiki/Melodic_inversion" class="mw-redirect" title="Melodic inversion">Inversion</a></li> <li><a href="/wiki/Retrograde_inversion" title="Retrograde inversion">Retrograde inversion</a></li> <li><a href="/wiki/Multiplication_(music)" title="Multiplication (music)">Multiplication</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Template:Twelve-tone_composers" title="Template:Twelve-tone composers">Notable<br /> composers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Milton_Babbitt" title="Milton Babbitt">Milton Babbitt</a></li> <li><a href="/wiki/Pierre_Boulez" title="Pierre Boulez">Pierre Boulez</a></li> <li><a href="/wiki/Josef_Matthias_Hauer" title="Josef Matthias Hauer">Josef Matthias Hauer</a></li> <li><a href="/wiki/Second_Viennese_School" title="Second Viennese School">Second Viennese School</a> <ul><li><a href="/wiki/Alban_Berg" title="Alban Berg">Alban Berg</a></li> <li><a href="/wiki/Arnold_Schoenberg" title="Arnold Schoenberg">Arnold Schoenberg</a></li> <li><a href="/wiki/Anton_Webern" title="Anton Webern">Anton Webern</a></li></ul></li> <li><a href="/wiki/Charles_Wuorinen" title="Charles Wuorinen">Charles Wuorinen</a></li> <li>...<i><a href="/wiki/Serialism#Notable_composers" title="Serialism">more</a></i>...</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Related articles</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/All-interval_twelve-tone_row" title="All-interval twelve-tone row">All-interval twelve-tone row</a></li> <li><a href="/wiki/All-trichord_hexachord" title="All-trichord hexachord">All-trichord hexachord</a></li> <li><a href="/wiki/Atonality" title="Atonality">Atonality</a></li> <li><a href="/wiki/Chromatic_scale" title="Chromatic scale">Chromatic scale</a></li> <li><a href="/wiki/Duration_series" title="Duration series">Duration series</a></li> <li><a href="/wiki/Equivalence_class_(music)" title="Equivalence class (music)">Equivalence</a></li> <li><a href="/wiki/Formula_composition" title="Formula composition">Formula composition</a></li> <li><a href="/wiki/Modernism_(music)" title="Modernism (music)">Modernism (music)</a></li> <li><a href="/wiki/Punctualism" title="Punctualism">Punctualism</a></li> <li><a class="mw-selflink selflink">Semitone</a></li> <li><a href="/wiki/Set_theory_(music)" title="Set theory (music)">Set theory</a></li> <li><a href="/wiki/Time_point" title="Time point">Time point</a></li> <li><a href="/wiki/Trope_(music)" title="Trope (music)">Trope</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" 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