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class="vector-toc-numb">2</span> <span>Characteristics</span> </div> </a> <ul id="toc-Characteristics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Partials,_overtones,_and_harmonics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Partials,_overtones,_and_harmonics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Partials, overtones, and harmonics</span> </div> </a> <ul id="toc-Partials,_overtones,_and_harmonics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-On_stringed_instruments" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#On_stringed_instruments"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>On stringed instruments</span> </div> </a> <button aria-controls="toc-On_stringed_instruments-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only 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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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 mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Harmonic</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 32 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-32" 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">32 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%A7%D9%81%D9%82" title="توافق – Arabic" lang="ar" hreflang="ar" data-title="توافق" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Harm%C3%B2nic" title="Harmònic – Catalan" lang="ca" hreflang="ca" data-title="Harmònic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Harmonische" title="Harmonische – German" lang="de" hreflang="de" data-title="Harmonische" 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/Osahelid" title="Osahelid – Estonian" lang="et" hreflang="et" data-title="Osahelid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Arm%C3%B3nico" title="Armónico – Spanish" lang="es" hreflang="es" data-title="Armónico" 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/Harmono" title="Harmono – Esperanto" lang="eo" hreflang="eo" data-title="Harmono" 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/Harmoniko" title="Harmoniko – Basque" lang="eu" hreflang="eu" data-title="Harmoniko" 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%87%D8%A7%D8%B1%D9%85%D9%88%D9%86%DB%8C%DA%A9" title="هارمونیک – Persian" lang="fa" hreflang="fa" data-title="هارمونیک" 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/Harmonique_(musique)" title="Harmonique (musique) – French" lang="fr" hreflang="fr" data-title="Harmonique (musique)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Arm%C3%B3nach" title="Armónach – Irish" lang="ga" hreflang="ga" data-title="Armónach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Harm%C3%B3nico" title="Harmónico – Galician" lang="gl" hreflang="gl" data-title="Harmónico" 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/%EA%B3%A0%EC%A1%B0%ED%8C%8C" title="고조파 – Korean" lang="ko" hreflang="ko" data-title="고조파" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Pregib_(akustika)" title="Pregib (akustika) – Croatian" lang="hr" hreflang="hr" data-title="Pregib (akustika)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Harmonisa" title="Harmonisa – Indonesian" lang="id" hreflang="id" data-title="Harmonisa" 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/Armonica_(fisica)" title="Armonica (fisica) – Italian" lang="it" hreflang="it" data-title="Armonica (fisica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Amoni" title="Amoni – Haitian Creole" lang="ht" hreflang="ht" data-title="Amoni" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Harmonische" title="Harmonische – Dutch" lang="nl" hreflang="nl" data-title="Harmonische" 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/%E9%AB%98%E8%AA%BF%E6%B3%A2" title="高調波 – Japanese" lang="ja" hreflang="ja" data-title="高調波" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Harmonisk_b%C3%B8lgje" title="Harmonisk bølgje – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Harmonisk bølgje" 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/Armonica_(musica)" title="Armonica (musica) – Occitan" lang="oc" hreflang="oc" data-title="Armonica (musica)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl badge-Q70893996 mw-list-item" title=""><a href="https://pl.wikipedia.org/wiki/Sk%C5%82adowa_harmoniczna" title="Składowa harmoniczna – Polish" lang="pl" hreflang="pl" data-title="Składowa harmoniczna" 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/Harm%C3%B4nica" title="Harmônica – Portuguese" lang="pt" hreflang="pt" data-title="Harmônica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D0%B8%D0%BA%D0%B0_(%D0%BC%D1%83%D0%B7%D1%8B%D0%BA%D0%B0)" title="Гармоника (музыка) – Russian" lang="ru" hreflang="ru" data-title="Гармоника (музыка)" 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/Harmonic" title="Harmonic – Simple English" lang="en-simple" hreflang="en-simple" data-title="Harmonic" 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/Fla%C5%BEolet" title="Flažolet – Slovenian" lang="sl" hreflang="sl" data-title="Flažolet" 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%90%D0%BB%D0%B8%D0%BA%D0%B2%D0%BE%D1%82%D0%BD%D0%B8_%D1%82%D0%BE%D0%BD%D0%BE%D0%B2%D0%B8" title="Аликвотни тонови – Serbian" lang="sr" hreflang="sr" data-title="Аликвотни тонови" 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/Alikvotni_tonovi" title="Alikvotni tonovi – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Alikvotni tonovi" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Huilu%C3%A4%C3%A4ni" title="Huiluääni – Finnish" lang="fi" hreflang="fi" data-title="Huiluääni" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Harmonisk_v%C3%A5g" title="Harmonisk våg – Swedish" lang="sv" hreflang="sv" data-title="Harmonisk våg" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AE%E0%B8%B2%E0%B8%A3%E0%B9%8C%E0%B8%A1%E0%B8%AD%E0%B8%99%E0%B8%B4%E0%B8%81" title="ฮาร์มอนิก – Thai" lang="th" hreflang="th" data-title="ฮาร์มอนิก" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B0%D1%80%D0%BC%D0%BE%D0%BD%D1%96%D0%BA%D0%B8" title="Гармоніки – Ukrainian" lang="uk" hreflang="uk" data-title="Гармоніки" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B0%90%E6%B3%A2" title="谐波 – Chinese" lang="zh" hreflang="zh" data-title="谐波" 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/Q1148098#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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For other uses, see <a href="/wiki/Harmonic_(disambiguation)" class="mw-disambig" title="Harmonic (disambiguation)">Harmonic (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Harmonic" title="Special:EditPage/Harmonic">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Harmonic%22">"Harmonic"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Harmonic%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Harmonic%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Harmonic%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Harmonic%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Harmonic%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Moodswingerscale.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Moodswingerscale.svg/250px-Moodswingerscale.svg.png" decoding="async" width="250" height="277" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Moodswingerscale.svg/375px-Moodswingerscale.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Moodswingerscale.svg/500px-Moodswingerscale.svg.png 2x" data-file-width="512" data-file-height="568" /></a><figcaption>The <a href="/wiki/Node_(physics)" title="Node (physics)">nodes</a> of a vibrating string are harmonics.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cello_natural_harmonics.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Cello_natural_harmonics.png/220px-Cello_natural_harmonics.png" decoding="async" width="220" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Cello_natural_harmonics.png/330px-Cello_natural_harmonics.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Cello_natural_harmonics.png/440px-Cello_natural_harmonics.png 2x" data-file-width="2204" data-file-height="1406" /></a><figcaption>Two different notations of natural harmonics on the cello. First as sounded (more common), then as fingered (easier to sightread).</figcaption></figure> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Acoustics" title="Acoustics">acoustics</a>, and <a href="/wiki/Telecommunications" title="Telecommunications">telecommunications</a>, a <b>harmonic</b> is a <a href="/wiki/Sinusoidal_wave" class="mw-redirect" title="Sinusoidal wave">sinusoidal wave</a> with a <a href="/wiki/Frequency" title="Frequency">frequency</a> that is a positive <a href="/wiki/Integer" title="Integer">integer</a> multiple of the <i><a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a></i> of a <a href="/wiki/Periodic_signal" class="mw-redirect" title="Periodic signal">periodic signal</a>. The fundamental frequency is also called the <i><b>1st harmonic</b></i>; the other harmonics are known as <i><b>higher harmonics</b></i>. As all harmonics are <a href="/wiki/Periodic_function" title="Periodic function">periodic</a> at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a <i><a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonic series</a></i>. </p><p>The term is employed in various disciplines, including music, physics, <a href="/wiki/Acoustics" title="Acoustics">acoustics</a>, electronic power transmission, radio technology, and other fields. For example, if the fundamental frequency is 50 <a href="/wiki/Hertz" title="Hertz">Hz</a>, a common <a href="/wiki/Alternating_current" title="Alternating current">AC</a> power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz. </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>An <i>n</i><sup>th</sup> characteristic mode, for <i>n</i> > 1, will have nodes that are not vibrating. For example, the 3rd characteristic mode will have nodes at <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 {\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9efc37d09854a3f8fb997e7de4331876bc49c2c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}}"></span><i>L</i> and <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 {\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}}"></span><i>L</i>, where <i>L</i> is the length of the string. In fact, each <i>n</i><sup>th</sup> characteristic mode, for <i>n</i> not a multiple of 3, will <b>not</b> have nodes at these points. These other characteristic modes will be <i>vibrating</i> at the positions <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 {\tfrac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9efc37d09854a3f8fb997e7de4331876bc49c2c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}}"></span><i>L</i> and <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 {\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a6ce6d697175e9e5e723b8c40eaa7efcfeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2}{3}}}"></span><i>L</i>. If the player <i>gently touches</i> one of these positions, then these other characteristic modes will be suppressed. The tonal harmonics from these other characteristic modes will then also be suppressed. Consequently, the tonal harmonics from the <i>n</i><sup>th</sup> characteristic modes, where <i>n</i> is a multiple of 3, will be made relatively more prominent.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>In music, harmonics are used on string instruments and wind instruments as a way of producing sound on the instrument, particularly to play higher notes and, with strings, obtain notes that have a unique sound quality or "tone colour". On strings, bowed harmonics have a "glassy", pure tone. On stringed instruments, harmonics are played by touching (but not fully pressing down the string) at an exact point on the string while sounding the string (plucking, bowing, etc.); this allows the harmonic to sound, a pitch which is always higher than the fundamental frequency of the string. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic&action=edit&section=1" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Harmonics may be called "overtones", "partials", or "upper partials", and in some music contexts, the terms "harmonic", "overtone" and "partial" are used fairly interchangeably. But more precisely, the term "harmonic" includes <i>all</i> pitches in a harmonic series (including the fundamental frequency) while the term "overtone" only includes pitches <i>above</i> the fundamental. </p> <div class="mw-heading mw-heading2"><h2 id="Characteristics">Characteristics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic&action=edit&section=2" title="Edit section: Characteristics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>A whizzing, whistling tonal character, distinguishes all the harmonics both natural and artificial from the firmly stopped intervals; therefore their application in connection with the latter must always be carefully considered.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2024)">citation needed</span></a></i>]</sup> </p><div style="float:right;">— Richard Scholz (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1888-1912</span>)<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></div> </blockquote> <p>Most acoustic instruments emit complex tones containing many individual partials (component simple tones or sinusoidal waves), but the untrained human ear typically does not perceive those partials as separate phenomena. Rather, a musical note is perceived as one sound, the quality or <a href="/wiki/Timbre" title="Timbre">timbre</a> of that sound being a result of the relative strengths of the individual partials. Many acoustic <a href="/wiki/Oscillator" class="mw-redirect" title="Oscillator">oscillators</a>, such as the <a href="/wiki/Human_voice" title="Human voice">human voice</a> or a <a href="/wiki/Bow_(music)" title="Bow (music)">bowed</a> <a href="/wiki/Violin" title="Violin">violin</a> string, produce complex tones that are more or less <a href="/wiki/Periodic_function" title="Periodic function">periodic</a>, and thus are composed of partials that are nearly matched to the integer multiples of fundamental frequency and therefore resemble the ideal harmonics and are called "harmonic partials" or simply "harmonics" for convenience (although it's not strictly accurate to call a  <i>partial</i>  a  <i>harmonic</i>,  the first being actual and the second being theoretical). </p><p>Oscillators that produce harmonic partials behave somewhat like one-dimensional <a href="/wiki/Resonator" title="Resonator">resonators</a>, and are often long and thin, such as a guitar string or a column of air open at both ends (as with the metallic modern orchestral <a href="/wiki/Transverse_flute" title="Transverse flute">transverse flute</a>). Wind instruments whose air column is open at only one end, such as <a href="/wiki/Trumpet" title="Trumpet">trumpets</a> and <a href="/wiki/Clarinet" title="Clarinet">clarinets</a>, also produce partials resembling harmonics. However they only produce partials matching the <i>odd</i> harmonics—at least in theory. In practical use, no real acoustic instrument behaves as perfectly as the simplified physical models predict; for example, instruments made of <a href="/wiki/Nonlinear_acoustics" title="Nonlinear acoustics">non-linearly</a> <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> wood, instead of metal, or strung with <a href="/wiki/Catgut" title="Catgut">gut</a> instead of <a href="/wiki/String_(music)#string_core_anchor" title="String (music)">brass or steel strings</a>, tend to have not-quite-integer partials. </p><p>Partials whose frequencies are not integer multiples of the fundamental are referred to as <a href="/wiki/Inharmonicity" title="Inharmonicity">inharmonic partials</a>. Some acoustic instruments emit a mix of harmonic and inharmonic partials but still produce an effect on the ear of having a definite fundamental pitch, such as <a href="/wiki/Piano" title="Piano">pianos</a>, strings plucked <a href="/wiki/Pizzicato" title="Pizzicato">pizzicato</a>, vibraphones, marimbas, and certain pure-sounding bells or chimes. Antique <a href="/wiki/Singing_bowl" class="mw-redirect" title="Singing bowl">singing bowls</a> are known for producing multiple harmonic partials or <a href="/wiki/Multiphonics" class="mw-redirect" title="Multiphonics">multiphonics</a>. <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><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> Other oscillators, such as <a href="/wiki/Cymbals" class="mw-redirect" title="Cymbals">cymbals</a>, drum heads, and most percussion instruments, naturally produce an abundance of inharmonic partials and do not imply any particular pitch, and therefore cannot be used melodically or harmonically in the same way other instruments can. </p><p>Building on of <a href="/wiki/William_Sethares" title="William Sethares">Sethares</a> (2004),<sup id="cite_ref-TTSS_5-0" class="reference"><a href="#cite_note-TTSS-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Dynamic_tonality" title="Dynamic tonality">dynamic tonality</a> introduces the notion of pseudo-harmonic partials, in which the frequency of each partial is aligned to match the pitch of a corresponding note in a pseudo-just tuning, thereby maximizing the <a href="/wiki/Consonance" class="mw-redirect" title="Consonance">consonance</a> of that pseudo-harmonic timbre with notes of that pseudo-just tuning.<sup id="cite_ref-Spectral_Tools_6-0" class="reference"><a href="#cite_note-Spectral_Tools-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Continua_7-0" class="reference"><a href="#cite_note-Continua-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Fingering_8-0" class="reference"><a href="#cite_note-Fingering-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-X_System_9-0" class="reference"><a href="#cite_note-X_System-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Partials,_overtones,_and_harmonics"><span id="Partials.2C_overtones.2C_and_harmonics"></span>Partials, overtones, and harmonics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic&action=edit&section=3" title="Edit section: Partials, overtones, and harmonics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Overtone" title="Overtone">overtone</a> is any partial higher than the lowest partial in a compound tone. The relative strengths and frequency relationships of the component partials determine the timbre of an instrument. The similarity between the terms overtone and partial sometimes leads to their being loosely used interchangeably in a <a href="/wiki/Music" title="Music">musical</a> context, but they are counted differently, leading to some possible confusion. In the special case of instrumental timbres whose component partials closely match a harmonic series (such as with most strings and winds) rather than being inharmonic partials (such as with most <a href="/wiki/Pitched_percussion" class="mw-redirect" title="Pitched percussion">pitched percussion</a> instruments), it is also convenient to call the component partials "harmonics", but not strictly correct, because harmonics are numbered the same even when missing, while partials and overtones are only counted when present. This chart demonstrates how the three types of names (partial, overtone, and harmonic) are counted (assuming that the harmonics are present): </p> <table class="wikitable"> <tbody><tr class="hintergrundfarbe5"> <th>Frequency </th> <th>Order<br />(<span class="texhtml mvar" style="font-style:italic;">n</span>) </th> <th>Name 1 </th> <th>Name 2 </th> <th>Name 3 </th> <th><a href="/wiki/Standing_wave" title="Standing wave">Standing wave</a> representation </th> <th><a href="/wiki/Longitudinal_wave" title="Longitudinal wave">Longitudinal wave</a> representation </th></tr> <tr> <td>1 × <span class="texhtml mvar" style="font-style:italic;">f</span> = <span style="visibility:hidden;color:transparent;">0</span>440 Hz </td> <td><span class="texhtml mvar" style="font-style:italic;">n</span> = 1 </td> <td>1st partial </td> <td><a href="/wiki/Fundamental_tone" class="mw-redirect" title="Fundamental tone">fundamental tone</a> </td> <td>1st harmonic </td> <td><span typeof="mw:File"><a href="/wiki/File:Pipe001.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Pipe001.gif/220px-Pipe001.gif" decoding="async" width="220" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Pipe001.gif/330px-Pipe001.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Pipe001.gif/440px-Pipe001.gif 2x" data-file-width="640" data-file-height="192" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Molecule1.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Molecule1.gif/220px-Molecule1.gif" decoding="async" width="220" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Molecule1.gif/330px-Molecule1.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Molecule1.gif/440px-Molecule1.gif 2x" data-file-width="576" data-file-height="304" /></a></span> </td></tr> <tr> <td>2 × <span class="texhtml mvar" style="font-style:italic;">f</span> = <span style="visibility:hidden;color:transparent;">0</span>880 Hz </td> <td><span class="texhtml mvar" style="font-style:italic;">n</span> = 2 </td> <td>2nd partial </td> <td>1st overtone </td> <td>2nd harmonic </td> <td><span typeof="mw:File"><a href="/wiki/File:Pipe002.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Pipe002.gif/220px-Pipe002.gif" decoding="async" width="220" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Pipe002.gif/330px-Pipe002.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Pipe002.gif/440px-Pipe002.gif 2x" data-file-width="640" data-file-height="192" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Molecule2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Molecule2.gif/220px-Molecule2.gif" decoding="async" width="220" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Molecule2.gif/330px-Molecule2.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Molecule2.gif/440px-Molecule2.gif 2x" data-file-width="576" data-file-height="304" /></a></span> </td></tr> <tr> <td>3 × <span class="texhtml mvar" style="font-style:italic;">f</span> = 1320 Hz </td> <td><span class="texhtml mvar" style="font-style:italic;">n</span> = 3 </td> <td>3rd partial </td> <td>2nd overtone </td> <td>3rd harmonic </td> <td><span typeof="mw:File"><a href="/wiki/File:Pipe003.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Pipe003.gif/220px-Pipe003.gif" decoding="async" width="220" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Pipe003.gif/330px-Pipe003.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Pipe003.gif/440px-Pipe003.gif 2x" data-file-width="640" data-file-height="192" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Molecule3.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Molecule3.gif/220px-Molecule3.gif" decoding="async" width="220" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Molecule3.gif/330px-Molecule3.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Molecule3.gif/440px-Molecule3.gif 2x" data-file-width="576" data-file-height="304" /></a></span> </td></tr> <tr> <td>4 × <span class="texhtml mvar" style="font-style:italic;">f</span> = 1760 Hz </td> <td><span class="texhtml mvar" style="font-style:italic;">n</span> = 4 </td> <td>4th partial </td> <td>3rd overtone </td> <td>4th harmonic </td> <td><span typeof="mw:File"><a href="/wiki/File:Pipe004.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Pipe004.gif/220px-Pipe004.gif" decoding="async" width="220" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Pipe004.gif/330px-Pipe004.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Pipe004.gif/440px-Pipe004.gif 2x" data-file-width="640" data-file-height="192" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Molecule4.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Molecule4.gif/220px-Molecule4.gif" decoding="async" width="220" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Molecule4.gif/330px-Molecule4.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Molecule4.gif/440px-Molecule4.gif 2x" data-file-width="576" data-file-height="304" /></a></span> </td></tr></tbody></table> <p>In many <a href="/wiki/Musical_instruments" class="mw-redirect" title="Musical instruments">musical instruments</a>, it is possible to play the upper harmonics without the fundamental note being present. In a simple case (e.g., <a href="/wiki/Recorder_(musical_instrument)" title="Recorder (musical instrument)">recorder</a>) this has the effect of making the note go up in pitch by an <a href="/wiki/Octave" title="Octave">octave</a>, but in more complex cases many other pitch variations are obtained. In some cases it also changes the <a href="/wiki/Timbre" title="Timbre">timbre</a> of the note. This is part of the normal method of obtaining higher notes in <a href="/wiki/Wind_instruments" class="mw-redirect" title="Wind instruments">wind instruments</a>, where it is called <i><a href="/wiki/Overblowing" title="Overblowing">overblowing</a></i>. The <a href="/wiki/Extended_technique" title="Extended technique">extended technique</a> of playing <a href="/wiki/Multiphonic" title="Multiphonic">multiphonics</a> also produces harmonics. On <a href="/wiki/String_instruments" class="mw-redirect" title="String instruments">string instruments</a> it is possible to produce very pure sounding notes, called <a href="/wiki/String_harmonic" title="String harmonic">harmonics</a> or <i>flageolets</i> by string players, which have an eerie quality, as well as being high in pitch. Harmonics may be used to check at a <a href="/wiki/Unison" title="Unison">unison</a> the tuning of strings that are not tuned to the unison. For example, lightly fingering the node found halfway down the highest string of a <a href="/wiki/Cello" title="Cello">cello</a> produces the same pitch as lightly fingering the node <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"> 1 </span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span> of the way down the second highest string. For the human voice see <a href="/wiki/Overtone_singing" title="Overtone singing">Overtone singing</a>, which uses harmonics. </p><p>While it is true that electronically produced periodic tones (e.g. square waves or other non-sinusoidal waves) have "harmonics" that are whole number multiples of the fundamental frequency, practical instruments do not all have this characteristic. For example, higher "harmonics" of piano notes are not true harmonics but are "overtones" and can be very sharp, i.e. a higher frequency than given by a pure <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonic series</a>. This is especially true of instruments other than <a href="/wiki/Stringed_instrument" class="mw-redirect" title="Stringed instrument">strings</a>, <a href="/wiki/Brass_instrument" title="Brass instrument">brass</a>, or <a href="/wiki/Woodwinds" class="mw-redirect" title="Woodwinds">woodwinds</a>. Examples of these "other" instruments are xylophones, drums, bells, chimes, etc.; not all of their overtone frequencies make a simple whole number ratio with the fundamental frequency. (The <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a> is the <a href="/wiki/Reciprocal_(mathematics)" class="mw-redirect" title="Reciprocal (mathematics)">reciprocal</a> of the longest <a href="/wiki/Period_(physics)" class="mw-redirect" title="Period (physics)">time period</a> of the collection of vibrations in some single periodic phenomenon.<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>) </p> <div class="mw-heading mw-heading2"><h2 id="On_stringed_instruments">On stringed instruments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic&action=edit&section=4" title="Edit section: On stringed instruments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Flageolette.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Flageolette.svg/200px-Flageolette.svg.png" decoding="async" width="200" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Flageolette.svg/300px-Flageolette.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Flageolette.svg/400px-Flageolette.svg.png 2x" data-file-width="600" data-file-height="680" /></a><figcaption>Playing a harmonic on a string</figcaption></figure><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/String_harmonic" title="String harmonic">String harmonic</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Harmonics may be singly produced [on stringed instruments] (1) by varying the point of contact with the bow, or (2) by slightly pressing the string at the nodes, or divisions of its aliquot parts (<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 {\tfrac {\ 1\ }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\ 1\ }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250f2826930a8694b03f7ed63bda807041fe0a95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.819ex; height:3.509ex;" alt="{\displaystyle {\tfrac {\ 1\ }{2}}}"></span>, <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 {\tfrac {\ 1\ }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\ 1\ }{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eff06a72f13ed8ec2ce4f76331ee43d3b5adca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.819ex; height:3.676ex;" alt="{\displaystyle {\tfrac {\ 1\ }{3}}}"></span>, <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 {\tfrac {\ 1\ }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> </mrow> <mn>4</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\ 1\ }{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17c774ddbd41d5f35c672b9fbc8ab4edf1dc6cf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.819ex; height:3.509ex;" alt="{\displaystyle {\tfrac {\ 1\ }{4}}}"></span>, etc.). <span class="nowrap">(1) In the first</span> case, advancing the bow from the usual place where the fundamental note is produced, towards the bridge, the whole scale of harmonics may be produced in succession, on an old and highly resonant instrument. The employment of this means produces the effect called '<a href="/wiki/Sul_ponticello" class="mw-redirect" title="Sul ponticello">sul ponticello</a>.' <span class="nowrap">(2) The production</span> of harmonics by the slight pressure of the finger on the open string is more useful. When produced by pressing slightly on the various nodes of the open strings they are called 'natural harmonics'. ... Violinists are well aware that the longer the string in proportion to its thickness, the greater the number of upper harmonics it can be made to yield.</p><div class="templatequotecite">— <cite><i><a href="/wiki/Grove%27s_Dictionary_of_Music_and_Musicians" class="mw-redirect" title="Grove's Dictionary of Music and Musicians">Grove's Dictionary of Music and Musicians</a></i> (1879)<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></cite></div></blockquote> <p>The following table displays the stop points on a stringed instrument at which gentle touching of a string will force it into a harmonic mode when vibrated. String harmonics (flageolet tones) are described as having a "flutelike, silvery quality" that can be highly effective as a special color or tone color (<a href="/wiki/Timbre" title="Timbre">timbre</a>) when used and heard in <a href="/wiki/Orchestration" title="Orchestration">orchestration</a>.<sup id="cite_ref-Marroco-2001_12-0" class="reference"><a href="#cite_note-Marroco-2001-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> It is unusual to encounter natural harmonics higher than the fifth partial on any stringed instrument except the double bass, on account of its much longer strings.<sup id="cite_ref-Marroco-2001_12-1" class="reference"><a href="#cite_note-Marroco-2001-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><dl><dd><table class="wikitable"> <tbody><tr> <th>Harmonic </th> <th><a href="/wiki/Stop_note" class="mw-redirect" title="Stop note">Stop note</a> </th> <th>Note sounded<br /><small>(relative to<br />open string)</small> </th> <th width="50"><a href="/wiki/Audio_frequency" title="Audio frequency">Audio frequency</a> (<a href="/wiki/Hertz_(unit)" class="mw-redirect" title="Hertz (unit)">Hz</a>) </th> <th width="50"><a href="/wiki/Cent_(music)" title="Cent (music)">Cents</a> above<br /> fundamental <small>(offset by octave)</small> </th> <th>Audio<br /> <small>(octave shifted)</small> </th></tr> <tr> <td style="text-align:center;">1 </td> <td><a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental</a>,<br /> perfect <a href="/wiki/Unison" title="Unison">unison</a> </td> <td style="text-align:center;">P1 </td> <td style="text-align:right;">600 </td> <td style="text-align:right;">0.0 </td> <td><span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/4c\/Unison_on_C.mid\/Unison_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Unison on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4c/Unison_on_C.mid/Unison_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:Unison_on_C.mid" title="File:Unison on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">2 </td> <td>first perfect <a href="/wiki/Octave" title="Octave">octave</a> </td> <td style="text-align:center;">P8 </td> <td style="text-align:right;">1,200 </td> <td style="text-align:right;">0.0 </td> <td><span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-2" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f0\/Perfect_octave_on_C.mid\/Perfect_octave_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Perfect octave on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f0/Perfect_octave_on_C.mid/Perfect_octave_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:Perfect_octave_on_C.mid" title="File:Perfect octave on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">3 </td> <td><a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> </td> <td style="text-align:center;">P8 + P5 </td> <td style="text-align:right;">1,800 </td> <td style="text-align:right;">702.0 </td> <td><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\/5\/5c\/Just_perfect_fifth_on_C.mid\/Just_perfect_fifth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just perfect fifth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5c/Just_perfect_fifth_on_C.mid/Just_perfect_fifth_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_perfect_fifth_on_C.mid" title="File:Just perfect fifth on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">4 </td> <td>doubled perfect <a href="/wiki/Octave" title="Octave">octave</a> </td> <td style="text-align:center;">2·P8 </td> <td style="text-align:right;">2,400 </td> <td style="text-align:right;">0.0 </td> <td><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\/b\/be\/Just_perfect_fourth_on_C.mid\/Just_perfect_fourth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just perfect fourth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/be/Just_perfect_fourth_on_C.mid/Just_perfect_fourth_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_perfect_fourth_on_C.mid" title="File:Just perfect fourth on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">5 </td> <td><a href="/wiki/Just_major_third" class="mw-redirect" title="Just major third">just major third</a>,<br /> <a href="/wiki/Major_third" title="Major third">major third</a> </td> <td style="text-align:center;">2·P8 + M3 </td> <td style="text-align:right;">3,000 </td> <td style="text-align:right;">386.3 </td> <td><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\/2\/2a\/Just_major_third_on_C.mid\/Just_major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/Just_major_third_on_C.mid/Just_major_third_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_major_third_on_C.mid" title="File:Just major third on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">6 </td> <td><a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> </td> <td style="text-align:center;">2·P8 + P5 </td> <td style="text-align:right;">3,600 </td> <td style="text-align:right;">702.0 </td> <td><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\/5\/5c\/Just_perfect_fifth_on_C.mid\/Just_perfect_fifth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just perfect fifth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5c/Just_perfect_fifth_on_C.mid/Just_perfect_fifth_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_perfect_fifth_on_C.mid" title="File:Just perfect fifth on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">7 </td> <td><a href="/wiki/Harmonic_seventh" title="Harmonic seventh">harmonic seventh</a>,<br /> <a href="/wiki/Harmonic_seventh" title="Harmonic seventh">septimal minor seventh</a> <br /> <small>(‘the lost chord’)</small> </td> <td style="text-align:center;">2·P8 + m7↓ </td> <td style="text-align:right;">4,200 </td> <td style="text-align:right;">968.8 </td> <td><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\/d\/dd\/Harmonic_seventh_on_C.mid\/Harmonic_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Harmonic seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/d/dd/Harmonic_seventh_on_C.mid/Harmonic_seventh_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:Harmonic_seventh_on_C.mid" title="File:Harmonic seventh on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">8 </td> <td>third perfect <a href="/wiki/Octave" title="Octave">octave</a> </td> <td style="text-align:center;">3·P8 </td> <td style="text-align:right;">4,800 </td> <td style="text-align:right;">0.0 </td> <td><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\/f\/f0\/Perfect_octave_on_C.mid\/Perfect_octave_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Perfect octave on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f0/Perfect_octave_on_C.mid/Perfect_octave_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:Perfect_octave_on_C.mid" title="File:Perfect octave on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">9 </td> <td><a href="/wiki/Pythagorean_interval" title="Pythagorean interval">Pythagorean</a> <a href="/wiki/Major_second" title="Major second">major second</a> <br /> harmonic ninth </td> <td style="text-align:center;">3·P8 + M2 </td> <td style="text-align:right;">5,400 </td> <td style="text-align:right;">203.9 </td> <td><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\/7\/7b\/Major_tone_on_C.mid\/Major_tone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major tone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7b/Major_tone_on_C.mid/Major_tone_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_tone_on_C.mid" title="File:Major tone on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">10 </td> <td><a href="/wiki/Just_major_third" class="mw-redirect" title="Just major third">just major third</a> </td> <td style="text-align:center;">3·P8 + M3 </td> <td style="text-align:right;">6,000 </td> <td style="text-align:right;">386.3 </td> <td><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\/2\/2a\/Just_major_third_on_C.mid\/Just_major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/Just_major_third_on_C.mid/Just_major_third_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_major_third_on_C.mid" title="File:Just major third on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">11 </td> <td>lesser undecimal <a href="/wiki/Tritone" title="Tritone">tritone</a>,<br /> undecimal semi-<a href="/wiki/Augmented_fourth" class="mw-redirect" title="Augmented fourth">augmented fourth</a> </td> <td style="text-align:center;">3·P8 + a4<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-%C2%BD.svg" class="mw-file-description" title="half flat"><img alt="half flat" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Llpd-%C2%BD.svg/4px-Llpd-%C2%BD.svg.png" decoding="async" width="4" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Llpd-%C2%BD.svg/7px-Llpd-%C2%BD.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Llpd-%C2%BD.svg/9px-Llpd-%C2%BD.svg.png 2x" data-file-width="77" data-file-height="200" /></a></span></span> </td> <td style="text-align:right;">6,600 </td> <td style="text-align:right;">551.3 </td> <td><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\/2\/2a\/Eleventh_harmonic_on_C.mid\/Eleventh_harmonic_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Eleventh harmonic on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/Eleventh_harmonic_on_C.mid/Eleventh_harmonic_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:Eleventh_harmonic_on_C.mid" title="File:Eleventh harmonic on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">12 </td> <td><a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> </td> <td style="text-align:center;">3·P8 + P5 </td> <td style="text-align:right;">7,200 </td> <td style="text-align:right;">702.0 </td> <td><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\/5\/5c\/Just_perfect_fifth_on_C.mid\/Just_perfect_fifth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just perfect fifth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5c/Just_perfect_fifth_on_C.mid/Just_perfect_fifth_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_perfect_fifth_on_C.mid" title="File:Just perfect fifth on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">13 </td> <td>tridecimal <a href="/wiki/Neutral_sixth" title="Neutral sixth">neutral sixth</a> </td> <td style="text-align:center;">3·P8 + n6<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-%C2%BD.svg" class="mw-file-description" title="half flat"><img alt="half flat" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Llpd-%C2%BD.svg/4px-Llpd-%C2%BD.svg.png" decoding="async" width="4" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Llpd-%C2%BD.svg/7px-Llpd-%C2%BD.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Llpd-%C2%BD.svg/9px-Llpd-%C2%BD.svg.png 2x" data-file-width="77" data-file-height="200" /></a></span></span>↓ </td> <td style="text-align:right;">7,800 </td> <td style="text-align:right;">840.5 </td> <td><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\/9\/95\/Tridecimal_neutral_sixth_on_C.mid\/Tridecimal_neutral_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Tridecimal neutral sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/95/Tridecimal_neutral_sixth_on_C.mid/Tridecimal_neutral_sixth_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:Tridecimal_neutral_sixth_on_C.mid" title="File:Tridecimal neutral sixth on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">14 </td> <td><a href="/wiki/Harmonic_seventh" title="Harmonic seventh">harmonic seventh</a>,<br /> septimal <a href="/wiki/Minor_seventh" title="Minor seventh">minor seventh</a> <br /> <small>(‘the lost chord’)</small> </td> <td style="text-align:center;">3·P8 + m7⤈ </td> <td style="text-align:right;">8,400 </td> <td style="text-align:right;">968.8 </td> <td><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\/d\/dd\/Harmonic_seventh_on_C.mid\/Harmonic_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Harmonic seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/d/dd/Harmonic_seventh_on_C.mid/Harmonic_seventh_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:Harmonic_seventh_on_C.mid" title="File:Harmonic seventh on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">15 </td> <td><a href="/wiki/Just_intonation" title="Just intonation">just</a> <a href="/wiki/Major_seventh" title="Major seventh">major seventh</a> </td> <td style="text-align:center;">3·P8 + M7 </td> <td style="text-align:right;">9,000 </td> <td style="text-align:right;">1,088.3 </td> <td><span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-15" 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\/a4\/Just_major_seventh_on_C.mid\/Just_major_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a4/Just_major_seventh_on_C.mid/Just_major_seventh_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_major_seventh_on_C.mid" title="File:Just major seventh on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">16 </td> <td>fourth perfect <a href="/wiki/Octave" title="Octave">octave</a> </td> <td style="text-align:center;">4·P8 </td> <td style="text-align:right;">9,600 </td> <td style="text-align:right;">0.0 </td> <td><span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-16" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f0\/Perfect_octave_on_C.mid\/Perfect_octave_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Perfect octave on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f0/Perfect_octave_on_C.mid/Perfect_octave_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:Perfect_octave_on_C.mid" title="File:Perfect octave on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">17 </td> <td>septidecimal <a href="/wiki/Semitone" title="Semitone">semitone</a> </td> <td style="text-align:center;">4·P8 + m2⇟ </td> <td style="text-align:right;">10,200 </td> <td style="text-align:right;">105.0 </td> <td><span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-17" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/48\/Just_major_semitone_on_C.mid\/Just_major_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/48/Just_major_semitone_on_C.mid/Just_major_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_major_semitone_on_C.mid" title="File:Just major semitone on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">18 </td> <td><a href="/wiki/Pythagorean_interval" title="Pythagorean interval">Pythagorean</a> <a href="/wiki/Major_second" title="Major second">major second</a> </td> <td style="text-align:center;">4·P8 + M2 </td> <td style="text-align:right;">10,800 </td> <td style="text-align:right;">203.9 </td> <td><span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-18" 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\/7\/7b\/Major_tone_on_C.mid\/Major_tone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major tone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7b/Major_tone_on_C.mid/Major_tone_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_tone_on_C.mid" title="File:Major tone on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">19 </td> <td>nanodecimal <a href="/wiki/Minor_third" title="Minor third">minor third</a> </td> <td style="text-align:center;">4·P8 + m3<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-%C2%BD.svg" class="mw-file-description" title="half flat"><img alt="half flat" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Llpd-%C2%BD.svg/4px-Llpd-%C2%BD.svg.png" decoding="async" width="4" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Llpd-%C2%BD.svg/7px-Llpd-%C2%BD.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Llpd-%C2%BD.svg/9px-Llpd-%C2%BD.svg.png 2x" data-file-width="77" data-file-height="200" /></a></span></span> </td> <td style="text-align:right;">11,400 </td> <td style="text-align:right;">297.5 </td> <td><span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-19" 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\/29\/19th_harmonic_on_C.mid\/19th_harmonic_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"19th harmonic on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/29/19th_harmonic_on_C.mid/19th_harmonic_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:19th_harmonic_on_C.mid" title="File:19th harmonic on C.mid">ⓘ</a></sup></span></span> </td></tr> <tr> <td style="text-align:center;">20 </td> <td>just <a href="/wiki/Major_third" title="Major third">major third</a> </td> <td style="text-align:center;">4·P8 + M3 </td> <td style="text-align:right;">12,000 </td> <td style="text-align:right;">386.3 </td> <td><span class="noprint"><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-20" 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\/2a\/Just_major_third_on_C.mid\/Just_major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/Just_major_third_on_C.mid/Just_major_third_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_major_third_on_C.mid" title="File:Just major third on C.mid">ⓘ</a></sup></span></span> </td></tr></tbody></table></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Artificial_harmonics">Artificial harmonics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic&action=edit&section=5" title="Edit section: Artificial harmonics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Occasionally a score will call for an <a href="/wiki/Artificial_harmonic" class="mw-redirect" title="Artificial harmonic">artificial harmonic</a>, produced by playing an overtone on an already stopped string. As a performance technique, it is accomplished by using two fingers on the fingerboard, the first to shorten the string to the desired fundamental, with the second touching the node corresponding to the appropriate harmonic. </p> <div class="mw-heading mw-heading2"><h2 id="Other_information">Other information</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic&action=edit&section=6" title="Edit section: Other information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Harmonics may be either used in or considered as the basis of <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a> systems. Composer <a href="/wiki/Arnold_Dreyblatt" title="Arnold Dreyblatt">Arnold Dreyblatt</a> is able to bring out different harmonics on the single string of his modified <a href="/wiki/Double_bass" title="Double bass">double bass</a> by slightly altering his unique <a href="/wiki/Bow_(music)" title="Bow (music)">bowing</a> technique halfway between hitting and bowing the strings. Composer <a href="/wiki/Lawrence_Ball" title="Lawrence Ball">Lawrence Ball</a> uses harmonics to generate music electronically. </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=Harmonic&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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-right listen noprint"><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/8/87/Gnome-mime-sound-openclipart.svg/50px-Gnome-mime-sound-openclipart.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/75px-Gnome-mime-sound-openclipart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/100px-Gnome-mime-sound-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="haudio"> <div class="listen-file-header"><a href="/wiki/File:Violin_harmonics.ogg" title="File:Violin harmonics.ogg">Violin harmonics</a></div> <div><span typeof="mw:File"><span><audio id="mwe_player_0" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="232" style="width:232px;" data-durationhint="10" data-mwtitle="Violin_harmonics.ogg" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/7/7d/Violin_harmonics.ogg" type="audio/ogg; codecs="vorbis"" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7d/Violin_harmonics.ogg/Violin_harmonics.ogg.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div> <div class="description">Violin natural harmonic stop points on the A string</div></div><hr /><div class="haudio"> <div class="listen-file-header"><a href="/wiki/File:Harmonics_110x16.ogg" title="File:Harmonics 110x16.ogg">Harmonics 110x16</a></div> <div><span typeof="mw:File"><span><audio id="mwe_player_1" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="232" style="width:232px;" data-durationhint="6" data-mwtitle="Harmonics_110x16.ogg" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/9/9d/Harmonics_110x16.ogg" type="audio/ogg; codecs="vorbis"" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9d/Harmonics_110x16.ogg/Harmonics_110x16.ogg.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div> <div class="description">Demonstration of 16 harmonics using electronic sine tones, starting with 110 Hz fundamental, 0.5s each. Note that each harmonic is presented at the same signal level as the fundamental; the sample tones sound louder as they increase in frequency</div></div></div></div> <div class="side-box-abovebelow"><hr /><i class="selfreference">Problems playing these files? See <a href="/wiki/Help:Media" title="Help:Media">media help</a>.</i></div> </div> <ul><li><a href="/wiki/Aristoxenus" title="Aristoxenus">Aristoxenus</a> – 4th century BC Greek Peripatetic philosopher</li> <li><a href="/wiki/Electronic_tuner" title="Electronic tuner">Electronic tuner</a> – Device used to tune musical instruments</li> <li><a href="/wiki/Formant" title="Formant">Formant</a> – Spectrum of phonetic resonance in speech production, or its peak</li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> – Decomposition of periodic functions into sums of simpler sinusoidal forms</li> <li><a href="/wiki/Guitar_harmonic" class="mw-redirect" title="Guitar harmonic">Guitar harmonic</a> – String instrument technique</li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a> – Study of superpositions in mathematics</li> <li><a href="/wiki/Harmonics_(electrical_power)" title="Harmonics (electrical power)">Harmonics (electrical power)</a> – Sinusoidal wave whose frequency is an integer multiple</li> <li><a href="/wiki/Harmonic_generation" title="Harmonic generation">Harmonic generation</a> – Nonlinear optical process</li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a> – Physical system that responds to a restoring force inversely proportional to displacement</li> <li><a href="/wiki/Harmony" title="Harmony">Harmony</a> – Aspect of music</li> <li><a href="/wiki/Pure_tone" title="Pure tone">Pure tone</a> – Sound with a sinusoidal waveform</li> <li><a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a> – Method of tuning a musical instrument</li> <li><a href="/wiki/Scale_of_harmonics" title="Scale of harmonics">Scale of harmonics</a></li> <li><a href="/wiki/Spherical_harmonics" title="Spherical harmonics">Spherical harmonics</a> – Special mathematical functions defined on the surface of a sphere</li> <li><a href="/wiki/Stretched_octave" class="mw-redirect" title="Stretched octave">Stretched octave</a></li> <li><a href="/wiki/Subharmonic" class="mw-redirect" title="Subharmonic">Subharmonic</a> – Having a frequency that is a fraction of a fundamental frequency</li> <li><a href="/wiki/Xenharmonic_music" title="Xenharmonic music">Xenharmonic music</a> – Music that uses a tuning system outside of 12-TET</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=Harmonic&action=edit&section=8" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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="CITEREFWalker2019" class="citation journal cs1">Walker, Russell (2019-06-14). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1287/7648739e-8e59-466e-82cb-3ded22bbebf6">"Russell Walker"</a></span>. <i>Authors group</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2F7648739e-8e59-466e-82cb-3ded22bbebf6">10.1287/7648739e-8e59-466e-82cb-3ded22bbebf6</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:241172832">241172832</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-12-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Authors+group&rft.atitle=Russell+Walker&rft.date=2019-06-14&rft_id=info%3Adoi%2F10.1287%2F7648739e-8e59-466e-82cb-3ded22bbebf6&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A241172832%23id-name%3DS2CID&rft.aulast=Walker&rft.aufirst=Russell&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1287%2F7648739e-8e59-466e-82cb-3ded22bbebf6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</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://imslp.org/wiki/Category:Scholz,_Richard">"Category:Scholz, Richard"</a>. <i>Petrucci Music Library / International Music Score Library Project (IMSLP) (imslp.org)</i> (site sub-index & mini-bio for Scholz). Canada<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-12-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Petrucci+Music+Library+%2F+International+Music+Score+Library+Project+%28IMSLP%29+%28imslp.org%29&rft.atitle=Category%3AScholz%2C+Richard&rft_id=https%3A%2F%2Fimslp.org%2Fwiki%2FCategory%3AScholz%2C_Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalemboCuddly1997" class="citation pressrelease cs1">Galembo, Alexander; Cuddly, Lola L. (2 December 1997). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120209050929/http://www.acoustics.org/press/134th/galembo.htm">"Large grand and small upright pianos"</a>. <i>acoustics.org</i> (Press release). <a href="/wiki/Acoustical_Society_of_America" title="Acoustical Society of America">Acoustical Society of America</a>. Archived from <a rel="nofollow" class="external text" href="http://www.acoustics.org/press/134th/galembo.htm">the original</a> on 2012-02-09<span class="reference-accessdate">. Retrieved <span class="nowrap">13 January</span> 2024</span>. <q>There are many ways to make matters worse, but very few to improve.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Large+grand+and+small+upright+pianos&rft.pub=Acoustical+Society+of+America&rft.date=1997-12-02&rft.aulast=Galembo&rft.aufirst=Alexander&rft.au=Cuddly%2C+Lola+L.&rft_id=http%3A%2F%2Fwww.acoustics.org%2Fpress%2F134th%2Fgalembo.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span> — Minimally technical summary of string acoustics research given at conference; discusses listeners' perceptions of pianos' inharmonic partials.</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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourt1927" class="citation journal cs1">Court, Sophie R.A. (April 1927). "<i>Golo und Genovefa</i> <span style="color:gray">[by]</span> Hanna Rademacher". <i>Books Abroad</i> (book review). <b>1</b> (2): 34–36. <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%2F40043442">10.2307/40043442</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0006-7431">0006-7431</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/40043442">40043442</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Books+Abroad&rft.atitle=Golo+und+Genovefa+%3Cspan+style%3D%22color%3Agray%22%3E%5Bby%5D%3C%2Fspan%3E+Hanna+Rademacher&rft.volume=1&rft.issue=2&rft.pages=34-36&rft.date=1927-04&rft.issn=0006-7431&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F40043442%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F40043442&rft.aulast=Court&rft.aufirst=Sophie+R.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></span> </li> <li id="cite_note-TTSS-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-TTSS_5-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSethares2004" class="citation book cs1"><a href="/wiki/William_Sethares" title="William Sethares">Sethares, W.A.</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KChoKKhjOb0C"><i>Tuning, Timbre, Spectrum, Scale</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1852337971" title="Special:BookSources/978-1852337971"><bdi>978-1852337971</bdi></a> – via Google books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tuning%2C+Timbre%2C+Spectrum%2C+Scale&rft.pub=Springer&rft.date=2004&rft.isbn=978-1852337971&rft.aulast=Sethares&rft.aufirst=W.A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKChoKKhjOb0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></span> </li> <li id="cite_note-Spectral_Tools-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Spectral_Tools_6-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSetharesMilneTiedjePrechtl2009" class="citation journal cs1"><a href="/wiki/William_Sethares" title="William Sethares">Sethares, W.A.</a>; Milne, A.; Tiedje, S.; Prechtl, A.; Plamondon, J. (2009). <a rel="nofollow" class="external text" href="http://muse.jhu.edu/journals/computer_music_journal/summary/v033/33.2.sethares.html">"Spectral tools for dynamic tonality and audio morphing"</a>. <i><a href="/wiki/Computer_Music_Journal" title="Computer Music Journal">Computer Music Journal</a></i>. <b>33</b> (2): 71–84. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1162%2Fcomj.2009.33.2.71">10.1162/comj.2009.33.2.71</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:216636537">216636537</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2009-09-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Music+Journal&rft.atitle=Spectral+tools+for+dynamic+tonality+and+audio+morphing&rft.volume=33&rft.issue=2&rft.pages=71-84&rft.date=2009&rft_id=info%3Adoi%2F10.1162%2Fcomj.2009.33.2.71&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A216636537%23id-name%3DS2CID&rft.aulast=Sethares&rft.aufirst=W.A.&rft.au=Milne%2C+A.&rft.au=Tiedje%2C+S.&rft.au=Prechtl%2C+A.&rft.au=Plamondon%2C+J.&rft_id=http%3A%2F%2Fmuse.jhu.edu%2Fjournals%2Fcomputer_music_journal%2Fsummary%2Fv033%2F33.2.sethares.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></span> </li> <li id="cite_note-Continua-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Continua_7-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilneSetharesPlamondon2008" class="citation journal cs1">Milne, Andrew; <a href="/wiki/William_Sethares" title="William Sethares">Sethares, William</a>; Plamondon, James (29 August 2008). <a rel="nofollow" class="external text" href="http://oro.open.ac.uk/21504/1/tuningcontinua.pdf">"Tuning continua and keyboard layouts"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/w/index.php?title=Journal_of_Mathematics_and_Music&action=edit&redlink=1" class="new" title="Journal of Mathematics and Music (page does not exist)">Journal of Mathematics and Music</a></i>. <b>2</b> (1): 1–19. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F17459730701828677">10.1080/17459730701828677</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1549755">1549755</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://oro.open.ac.uk/21504/1/tuningcontinua.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematics+and+Music&rft.atitle=Tuning+continua+and+keyboard+layouts&rft.volume=2&rft.issue=1&rft.pages=1-19&rft.date=2008-08-29&rft_id=info%3Adoi%2F10.1080%2F17459730701828677&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1549755%23id-name%3DS2CID&rft.aulast=Milne&rft.aufirst=Andrew&rft.au=Sethares%2C+William&rft.au=Plamondon%2C+James&rft_id=http%3A%2F%2Foro.open.ac.uk%2F21504%2F1%2Ftuningcontinua.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://sethares.engr.wisc.edu/paperspdf/tuningcontinua.pdf">"Alt URL"</a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/William_Sethares" title="William Sethares">Sethares</a> pers. academic site. <a href="/wiki/University_of_Wisconsin" class="mw-redirect" title="University of Wisconsin">University of Wisconsin</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Alt+URL&rft.series=Sethares+pers.+academic+site&rft.pub=University+of+Wisconsin&rft_id=http%3A%2F%2Fsethares.engr.wisc.edu%2Fpaperspdf%2Ftuningcontinua.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></span> </li> <li id="cite_note-Fingering-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Fingering_8-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilneSetharesPlamondon2007" class="citation journal cs1">Milne, A.; <a href="/wiki/William_Sethares" title="William Sethares">Sethares, W.A.</a>; Plamondon, J. (Winter 2007). <a rel="nofollow" class="external text" href="http://oro.open.ac.uk/21503/1/comj.2007.31.4.15">"Invariant fingerings across a tuning continuum"</a>. <i><a href="/wiki/Computer_Music_Journal" title="Computer Music Journal">Computer Music Journal</a></i>. <b>31</b> (4): 15–32. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1162%2Fcomj.2007.31.4.15">10.1162/comj.2007.31.4.15</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:27906745">27906745</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Music+Journal&rft.atitle=Invariant+fingerings+across+a+tuning+continuum&rft.ssn=winter&rft.volume=31&rft.issue=4&rft.pages=15-32&rft.date=2007&rft_id=info%3Adoi%2F10.1162%2Fcomj.2007.31.4.15&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A27906745%23id-name%3DS2CID&rft.aulast=Milne&rft.aufirst=A.&rft.au=Sethares%2C+W.A.&rft.au=Plamondon%2C+J.&rft_id=http%3A%2F%2Foro.open.ac.uk%2F21503%2F1%2Fcomj.2007.31.4.15&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></span> </li> <li id="cite_note-X_System-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-X_System_9-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilneSetharesPlamondon2006" class="citation report cs1">Milne, A.; <a href="/wiki/William_Sethares" title="William Sethares">Sethares, W.A.</a>; Plamondon, J. 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Retrieved <span class="nowrap">2020-05-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=X+System&rft.pub=Thumtronics+Inc.&rft.date=2006&rft.aulast=Milne&rft.aufirst=A.&rft.au=Sethares%2C+W.A.&rft.au=Plamondon%2C+J.&rft_id=http%3A%2F%2Foro.open.ac.uk%2F21510%2F1%2FX_System.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></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"><style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><span class="citation 1037C"><span class="noviewer" typeof="mw:File"><span><img alt="Public Domain" src="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/12px-PD-icon.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/18px-PD-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/24px-PD-icon.svg.png 2x" data-file-width="196" data-file-height="196" /></span></span> This article incorporates <a href="/wiki/Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220122224547/https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm"><i>Federal Standard 1037C</i></a>. <a href="/wiki/General_Services_Administration" title="General Services Administration">General Services Administration</a>. Archived from <a rel="nofollow" class="external text" href="https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm">the original</a> on 2022-01-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Federal+Standard+1037C&rft.pub=General+Services+Administration&rft_id=https%3A%2F%2Fwww.its.bldrdoc.gov%2Ffs-1037%2Ffs-1037c.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_alike2011" class="citation journal cs1">George alike (2011-01-21). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210214224413/http://www.scivee.tv/node/26742/">"Free Online Games for Kids"</a>. <i>MyFavoritegamez.com / SciVee.tv</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4016%2F26742.01">10.4016/26742.01</a>. Archived from <a rel="nofollow" class="external text" href="https://doi.org/10.4016/26742.01">the original</a> on 2021-02-14<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-12-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=MyFavoritegamez.com+%2F+SciVee.tv&rft.atitle=Free+Online+Games+for+Kids&rft.date=2011-01-21&rft_id=info%3Adoi%2F10.4016%2F26742.01&rft.au=George+alike&rft_id=http%3A%2F%2Fdoi.org%2F10.4016%2F26742.01&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></span> </li> <li id="cite_note-Marroco-2001-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-Marroco-2001_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Marroco-2001_12-1"><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="CITEREFMarrocco2001" class="citation encyclopaedia cs1">Marrocco, W. Thomas (2001). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1093/gmo/9781561592630.article.14882">"Kennan, Kent"</a>. <i>Oxford Music Online</i>. Oxford University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fgmo%2F9781561592630.article.14882">10.1093/gmo/9781561592630.article.14882</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-12-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Kennan%2C+Kent&rft.btitle=Oxford+Music+Online&rft.pub=Oxford+University+Press&rft.date=2001&rft_id=info%3Adoi%2F10.1093%2Fgmo%2F9781561592630.article.14882&rft.aulast=Marrocco&rft.aufirst=W.+Thomas&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1093%2Fgmo%2F9781561592630.article.14882&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Harmonic&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_50.html">The Feynman Lectures on Physics: Harmonics</a></li> <li><a rel="nofollow" class="external text" href="http://www.sengpielaudio.com/calculator-harmonics.htm">Harmonics, partials and overtones from fundamental frequency</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChisholm1911" class="citation encyclopaedia cs1"><a href="/wiki/Hugh_Chisholm" title="Hugh Chisholm">Chisholm, Hugh</a>, ed. (1911). <span class="cs1-ws-icon" title="s:1911 Encyclopædia Britannica/Harmonic"><a class="external text" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Harmonic">"Harmonic" </a></span>. <i><a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i> (11th ed.). Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Harmonic&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.edition=11th&rft.pub=Cambridge+University+Press&rft.date=1911&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHarmonic" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20091229052719/http://www.dranetz-bmi.com/pdf/harmonicsall.pdf">Harmonics</a></li> <li><a rel="nofollow" class="external text" href="http://harmonicviewerforpiano.com/">Hear and see harmonics on a Piano</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 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title="Monochord">Monochord</a></li> <li><a href="/wiki/Reverberation" title="Reverberation">Reverberation</a></li> <li><a href="/wiki/Soundproofing" title="Soundproofing">Soundproofing</a></li> <li><a href="/wiki/String_vibration" title="String vibration">String vibration</a></li> <li><a href="/wiki/Sympathetic_resonance" title="Sympathetic resonance">Sympathetic resonance</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><div class="plainlist"><ul><li><span typeof="mw:File"><a href="/wiki/File:Spectrogram_showing_shared_partials.png" class="mw-file-description"><img alt="A spectrogram of a violin playing a note and then a perfect fifth above it. The shared partials are highlighted by the white dashes." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Spectrogram_showing_shared_partials.png/80px-Spectrogram_showing_shared_partials.png" decoding="async" width="80" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Spectrogram_showing_shared_partials.png/120px-Spectrogram_showing_shared_partials.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Spectrogram_showing_shared_partials.png/160px-Spectrogram_showing_shared_partials.png 2x" data-file-width="300" data-file-height="250" /></a></span></li><li><span style="font-size:85%;"><a href="/wiki/Spectrogram" title="Spectrogram">Spectrogram</a></span></li></ul></div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustics</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/Pitch_(music)" title="Pitch (music)">Pitch</a> <ul><li><a href="/wiki/Bark_scale" title="Bark scale">Bark scale</a></li> <li><a href="/wiki/Mel_scale" title="Mel scale">Mel scale</a></li></ul></li> <li><a href="/wiki/Equal-loudness_contour" title="Equal-loudness contour">Equal-loudness contour</a> <ul><li><a href="/wiki/Fletcher%E2%80%93Munson_curves" class="mw-redirect" title="Fletcher–Munson curves">Fletcher–Munson curves</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Audio_frequency" title="Audio frequency">Audio frequency</a> and <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitch</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/Beat_(acoustics)" title="Beat (acoustics)">Beat</a></li> <li><a href="/wiki/Formant" title="Formant">Formant</a></li> <li><a href="/wiki/Fundamental_frequency" title="Fundamental frequency">Fundamental frequency</a></li> <li><a href="/wiki/Frequency_spectrum" class="mw-redirect" title="Frequency spectrum">Frequency spectrum</a> <ul><li><a href="/wiki/Harmonic_spectrum" title="Harmonic spectrum">harmonic spectrum</a></li></ul></li> <li><a class="mw-selflink selflink">Harmonic</a> <ul><li><a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">Series</a></li> <li><a href="/wiki/Inharmonicity" title="Inharmonicity">Inharmonicity</a></li> <li><a href="/wiki/Missing_fundamental" title="Missing fundamental">Missing fundamental</a></li></ul></li> <li><a href="/wiki/Combination_tone" title="Combination tone">Combination tone</a></li> <li><a href="/wiki/Mersenne%27s_laws" title="Mersenne's laws">Mersenne's laws</a></li> <li><a href="/wiki/Overtone" title="Overtone">Overtone</a></li> <li><a href="/wiki/Acoustic_resonance" title="Acoustic resonance">Resonance</a></li> <li><a href="/wiki/Standing_wave" title="Standing wave">Standing wave</a> <ul><li><a href="/wiki/Node_(physics)" title="Node (physics)">Node</a></li></ul></li> <li><a href="/wiki/Undertone_series" title="Undertone series">Subharmonic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Acousticians</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/John_Backus_(acoustician)" title="John Backus (acoustician)">John Backus</a></li> <li><a href="/wiki/Jens_Blauert" title="Jens Blauert">Jens Blauert</a></li> <li><a href="/wiki/Ernst_Chladni" title="Ernst Chladni">Ernst Chladni</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a></li> <li><a href="/wiki/Carleen_Hutchins" title="Carleen Hutchins">Carleen Hutchins</a></li> <li><a href="/wiki/Franz_Melde" title="Franz Melde">Franz Melde</a></li> <li><a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a></li> <li><a href="/wiki/Werner_Meyer-Eppler" title="Werner Meyer-Eppler">Werner Meyer-Eppler</a></li> <li><a href="/wiki/John_William_Strutt,_3rd_Baron_Rayleigh" title="John William Strutt, 3rd Baron Rayleigh">Lord Rayleigh</a></li> <li><a href="/wiki/Joseph_Sauveur" title="Joseph Sauveur">Joseph Sauveur</a></li> <li><a href="/wiki/D._Van_Holliday" title="D. Van Holliday">D. Van Holliday</a></li> <li><a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</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/Echo" title="Echo">Echo</a></li> <li><a href="/wiki/Infrasound" title="Infrasound">Infrasound</a></li> <li><a href="/wiki/Sound" title="Sound">Sound</a></li> <li><a href="/wiki/Ultrasound" title="Ultrasound">Ultrasound</a></li> <li><a href="/wiki/Musical_acoustics" title="Musical acoustics">Musical acoustics</a> <ul><li><a href="/wiki/Piano_acoustics" title="Piano acoustics">Piano</a></li> <li><a href="/wiki/Violin_acoustics" title="Violin acoustics">Violin</a></li></ul></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#efefef;"><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:Mental_processes" title="Template:Mental processes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mental_processes" title="Template talk:Mental processes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mental_processes" title="Special:EditPage/Template:Mental processes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mental_processes" style="font-size:114%;margin:0 4em">Mental processes</div></th></tr><tr><th scope="row" class="navbox-group" style="background:#efefef;;width:1%"><a href="/wiki/Cognition" title="Cognition">Cognition</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/Association_(psychology)" title="Association (psychology)">Association</a></li> <li><a href="/wiki/Awareness" title="Awareness">Awareness</a></li> <li><a href="/wiki/Cognitive_flexibility" title="Cognitive flexibility">Cognitive flexibility</a></li> <li><a href="/wiki/Cognitive_liberty" title="Cognitive liberty">Cognitive liberty</a></li> <li><a href="/wiki/Forecasting" title="Forecasting">Forecasting</a> <ul><li><a href="/wiki/Affective_forecasting" title="Affective forecasting">affective</a></li></ul></li> <li><a href="/wiki/Foresight_(psychology)" title="Foresight (psychology)">Foresight</a></li> <li><a href="/wiki/Understanding" title="Understanding">Comprehension</a></li> <li><a href="/wiki/Consciousness" title="Consciousness">Consciousness</a></li> <li><a href="/wiki/Critical_thinking" title="Critical thinking">Critical thinking</a></li> <li><a href="/wiki/Decision-making" title="Decision-making">Decision-making</a></li> <li><a href="/wiki/Imagination" title="Imagination">Imagination</a></li> <li><a href="/wiki/Intuition" title="Intuition">Intuition</a></li> <li><a href="/wiki/Problem_solving" title="Problem solving">Problem solving</a> <ul><li><a href="/wiki/Problem_solving#Problem-solving_methods" title="Problem solving">methods</a></li> <li><a href="/wiki/Problem_solving#Problem-solving_strategies" title="Problem solving">strategies</a></li></ul></li> <li><a href="/wiki/Prospection" title="Prospection">Prospection</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background:#efefef;;width:1%"><a href="/wiki/Perception" title="Perception">Perception</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div 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