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<span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diatonic_scale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diatonic_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Diatonic scale</span> </div> </a> <ul id="toc-Diatonic_scale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Twelve-tone_scale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Twelve-tone_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Twelve-tone scale</span> </div> </a> <button aria-controls="toc-Twelve-tone_scale-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Twelve-tone scale subsection</span> </button> <ul id="toc-Twelve-tone_scale-sublist" class="vector-toc-list"> <li id="toc-Pythagorean_tuning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pythagorean_tuning"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Pythagorean tuning</span> </div> </a> <ul id="toc-Pythagorean_tuning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Five-limit_tuning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Five-limit_tuning"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Five-limit tuning</span> </div> </a> <ul id="toc-Five-limit_tuning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_of_the_twelve-tone_scale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extension_of_the_twelve-tone_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Extension of the twelve-tone scale</span> </div> </a> <ul id="toc-Extension_of_the_twelve-tone_scale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Indian_scales" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Indian_scales"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Indian scales</span> </div> </a> <ul id="toc-Indian_scales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Practical_difficulties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Practical_difficulties"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Practical difficulties</span> </div> </a> <ul id="toc-Practical_difficulties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Singing_and_scale-free_instruments" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Singing_and_scale-free_instruments"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Singing and scale-free instruments</span> </div> </a> <ul id="toc-Singing_and_scale-free_instruments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Western_composers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Western_composers"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Western composers</span> </div> </a> <ul id="toc-Western_composers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Staff_notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Staff_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Staff notation</span> </div> </a> <ul id="toc-Staff_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Audio_examples" 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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">13</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" title="Table of Contents" > <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" 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href="https://cs.wikipedia.org/wiki/Didymick%C3%A9_lad%C4%9Bn%C3%AD" title="Didymické ladění – Czech" lang="cs" hreflang="cs" data-title="Didymické ladění" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ren_stemning" title="Ren stemning – Danish" lang="da" hreflang="da" data-title="Ren stemning" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Reine_Stimmung" title="Reine Stimmung – German" lang="de" hreflang="de" data-title="Reine Stimmung" 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/Puhas_h%C3%A4%C3%A4lestus" title="Puhas häälestus – Estonian" lang="et" hreflang="et" data-title="Puhas häälestus" 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/Temperamento_justo" title="Temperamento justo – Spanish" lang="es" hreflang="es" data-title="Temperamento justo" 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/Pura_agordo" title="Pura agordo – Esperanto" lang="eo" hreflang="eo" data-title="Pura agordo" 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/Intonazio_doi" title="Intonazio doi – Basque" lang="eu" hreflang="eu" data-title="Intonazio doi" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%A7%D9%85_%DA%A9%D9%88%DA%A9_%D8%AE%D8%A7%D9%84%D8%B5" title="نظام کوک خالص – Persian" lang="fa" hreflang="fa" data-title="نظام کوک خالص" 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/Intonation_juste" title="Intonation juste – French" lang="fr" hreflang="fr" data-title="Intonation juste" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%9C%EC%A0%95%EB%A5%A0" title="순정률 – Korean" lang="ko" hreflang="ko" data-title="순정률" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Nat%C5%ABralioji_darna" title="Natūralioji darna – Lithuanian" lang="lt" hreflang="lt" data-title="Natūralioji darna" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Tiszta_hangol%C3%A1s" title="Tiszta hangolás – Hungarian" lang="hu" hreflang="hu" data-title="Tiszta hangolás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Reine_stemming" title="Reine stemming – Dutch" lang="nl" hreflang="nl" data-title="Reine stemming" 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/%E7%B4%94%E6%AD%A3%E5%BE%8B" title="純正律 – Japanese" lang="ja" hreflang="ja" data-title="純正律" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Entona%C3%A7%C3%A3o_justa" title="Entonação justa – Portuguese" lang="pt" hreflang="pt" data-title="Entonação justa" 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%A7%D0%B8%D1%81%D1%82%D1%8B%D0%B9_%D1%81%D1%82%D1%80%D0%BE%D0%B9" title="Чистый строй – Russian" lang="ru" hreflang="ru" data-title="Чистый строй" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ren_st%C3%A4mning" title="Ren stämning – Swedish" lang="sv" hreflang="sv" data-title="Ren stämning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D1%82%D0%B8%D0%B9_%D1%81%D1%82%D1%80%D1%96%D0%B9" title="Чистий стрій – Ukrainian" lang="uk" hreflang="uk" data-title="Чистий стрій" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B4%94%E5%BE%8B" title="純律 – Cantonese" lang="yue" hreflang="yue" data-title="純律" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BA%AF%E5%BE%8B" title="纯律 – Chinese" lang="zh" hreflang="zh" data-title="纯律" 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/Q752518#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> 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Statements consisting only of original research should be removed.</span> <span class="date-container"><i>(<span class="date">April 2021</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> </div> </div><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 class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Harmonic_series_klang.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Harmonic_series_klang.png/220px-Harmonic_series_klang.png" decoding="async" width="220" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Harmonic_series_klang.png/330px-Harmonic_series_klang.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Harmonic_series_klang.png/440px-Harmonic_series_klang.png 2x" data-file-width="1089" data-file-height="858" /></a><figcaption>Harmonic series, partials 1–5 numbered<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_0" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-mwtitle="Harmonic_series_klang.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/6/65/Harmonic_series_klang.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/65/Harmonic_series_klang.mid/Harmonic_series_klang.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/65/Harmonic_series_klang.mid/Harmonic_series_klang.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <p>In music, <b>just intonation</b> or <b>pure intonation</b> is the <a href="/wiki/Musical_tuning" title="Musical tuning">tuning</a> of <a href="/wiki/Interval_(music)" title="Interval (music)">musical intervals</a> as <a href="/wiki/Natural_number" title="Natural number">whole number</a> <a href="/wiki/Ratio" title="Ratio">ratios</a> (such as 3:2 or 4:3) of <a href="/wiki/Frequency" title="Frequency">frequencies</a>. An <a href="/wiki/Interval_(music)" title="Interval (music)">interval</a> tuned in this way is said to be pure, and is called a <b>just interval</b>. Just intervals (and chords created by combining them) consist of tones from a single <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonic series</a> of an implied <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental</a>. For example, in the diagram, if the notes G3 and C4 (labelled 3 and 4) are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just <a href="/wiki/Fourth_(music)" title="Fourth (music)">fourth</a>. </p><p>In Western musical practice, bowed instruments such as violins, violas, cellos, and double basses are tuned using pure fifths or fourths. In contrast, keyboard instruments are rarely tuned using only pure intervals—the desire for different keys to have identical intervals in Western music makes this impractical. Some instruments of fixed pitch, such as electric pianos, are commonly tuned using <a href="/wiki/Equal_temperament" title="Equal temperament">equal temperament</a>, in which all intervals other than octaves consist of irrational-number frequency ratios. Acoustic pianos are usually <a href="/wiki/Piano_tuning" title="Piano tuning">tuned</a> with the octaves <a href="/wiki/Piano_acoustics#The_Railsback_curve" title="Piano acoustics">slightly widened</a>, and thus with no pure intervals at all. </p><p>The phrase "just intonation" is used both to refer to one specific version of a <a href="/wiki/5-limit_tuning" class="mw-redirect" title="5-limit tuning">5-limit diatonic intonation</a>, that is, <a href="/wiki/Ptolemy%27s_intense_diatonic_scale" title="Ptolemy's intense diatonic scale">Ptolemy's intense diatonic</a>, as well to a whole class of tunings which use <a href="/wiki/Fraction" title="Fraction">whole number intervals</a> derived from the <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonic series</a>. In this sense, "just intonation" is differentiated from <a href="/wiki/Equal_temperament" title="Equal temperament">equal temperaments</a> and the "<a href="/wiki/Tempered_tuning" class="mw-redirect" title="Tempered tuning">tempered</a>" tunings of the early <a href="/wiki/Renaissance" title="Renaissance">renaissance</a> and <a href="/wiki/Baroque_music" title="Baroque music">baroque</a>, such as <a href="/wiki/Well_temperament" title="Well temperament">Well temperament</a>, or <a href="/wiki/Meantone_temperament" title="Meantone temperament">Meantone temperament</a>. Since 5-limit has been the most prevalent just intonation used in western music, western musicians have subsequently tended to consider this scale to be the only version of just intonation. In principle, there are an infinite number of possible "just intonations", since the harmonic series is infinite. </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=Just_intonation&action=edit&section=1" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Just intonations are categorized by the notion of <a href="/wiki/Limit_(music)" title="Limit (music)">limits</a>. The limit refers to the highest prime number fraction included in the intervals of a scale. All the intervals of any 3 limit just intonation will be multiples of 3. So <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"> 6 </span><span class="sr-only">/</span><span class="den"> 5 </span></span>⁠</span> is included in 5 limit, because it has 5 in the denominator. If a scale uses an interval of 21:20, it is a 7 limit just intonation, since 21 is a multiple of 7. The interval <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 9 </span><span class="sr-only">/</span><span class="den"> 8 </span></span>⁠</span> is a 3 limit interval because the numerator and denominator are multiples of 3 and 2, respectively. It is possible to have a scale that uses 5 limit intervals but not 2 limit intervals, i.e. no octaves, such as <a href="/wiki/Wendy_Carlos" title="Wendy Carlos">Wendy Carlos</a>'s <a href="/wiki/Alpha_scale" title="Alpha scale">alpha</a> and <a href="/wiki/Beta_scale" title="Beta scale">beta</a> scales. It is also possible to make diatonic scales that do not use fourths or fifths (3 limit), but use 5 and 7 limit intervals only. Thus, the notion of limit is a helpful distinction, but certainly does not tell us everything there is to know about a particular scale. </p><p><a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a>, or 3 limit tuning, allows ratios including the numbers 2 and 3 and their powers, such as 3:2, a <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a>, and 9:4, a <a href="/wiki/Ninth" title="Ninth">major ninth</a>. Although the interval from C to G is called a perfect fifth for purposes of <a href="/wiki/Musical_analysis" title="Musical analysis">music analysis</a> regardless of its tuning method, for purposes of discussing tuning systems musicologists may distinguish between a <i>perfect fifth</i> created using the 3:2 ratio and a <i>tempered fifth</i> using some other system, such as <a href="/wiki/Meantone_temperament" title="Meantone temperament">meantone</a> or <a href="/wiki/Equal_temperament" title="Equal temperament">equal temperament</a>. </p><p><a href="/wiki/Five-limit_tuning" title="Five-limit tuning">5-limit tuning</a> encompasses ratios additionally using the number 5 and its powers, such as 5:4, a <a href="/wiki/Major_third" title="Major third">major third</a>, and 15:8, a <a href="/wiki/Major_seventh" title="Major seventh">major seventh</a>. The specialized term <i>perfect third</i> is occasionally used to distinguish the 5:4 ratio from major thirds created using other tuning methods. <a href="/wiki/7-limit_tuning" title="7-limit tuning">7 limit</a> and higher systems use higher <a href="/wiki/Prime_number" title="Prime number">prime number</a> partials in the <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">overtone series</a> (e.g. 11, 13, 17, etc.) </p><p><a href="/wiki/Comma_(music)" title="Comma (music)">Commas</a> are very small intervals that result from minute differences between pairs of just intervals. For example, the (5 limit) 5:4 ratio is different from the Pythagorean (3 limit) major third (81:64) by a difference of 81:80, called the <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a>. The <a href="/wiki/Septimal_comma" title="Septimal comma">septimal comma</a>, the ratio of 64:63, is a 7 limit interval which is the distance between the Pythagorean <a href="/wiki/Minor_third" title="Minor third">semi-ditone</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 32 </span><span class="sr-only">/</span><span class="den"> 27 </span></span>⁠</span>, and the <a href="/wiki/Septimal_minor_third" title="Septimal minor third">septimal minor third</a>, 7:6 , since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left({\tfrac {\ 32\ }{27}}\right)\div \left({\tfrac {\ 7\ }{6}}\right)={\tfrac {\ 64\ }{63}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mn>32</mn> <mtext> </mtext> </mrow> <mn>27</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>÷</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mn>7</mn> <mtext> </mtext> </mrow> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mtext> </mtext> <mn>64</mn> <mtext> </mtext> </mrow> <mn>63</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left({\tfrac {\ 32\ }{27}}\right)\div \left({\tfrac {\ 7\ }{6}}\right)={\tfrac {\ 64\ }{63}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63ded73ef3c2cf534852810255d85ac586041f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.787ex; height:4.843ex;" alt="{\displaystyle \ \left({\tfrac {\ 32\ }{27}}\right)\div \left({\tfrac {\ 7\ }{6}}\right)={\tfrac {\ 64\ }{63}}~.}"></span> </p><p>A <a href="/wiki/Cent_(music)" title="Cent (music)">cent</a> is a measure of interval size. It is logarithmic in the musical frequency ratios. The octave is divided into 1200 steps, 100 cents for each semitone. Cents are often used to describe how much a just interval deviates from <span class="nowrap"><a href="/wiki/12_equal_temperament" title="12 equal temperament">12 <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">TET</span></span></a>.</span> For example, the <a href="/wiki/Major_third" title="Major third">major third</a> is 400 cents in 12 TET, but the 5th harmonic, 5:4 is 386.314 cents. Thus, the just major third deviates by −13.686 cents. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:12-tet_scale_compared_to_just_in_the_chromatic_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/12-tet_scale_compared_to_just_in_the_chromatic_circle.svg/220px-12-tet_scale_compared_to_just_in_the_chromatic_circle.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/12-tet_scale_compared_to_just_in_the_chromatic_circle.svg/330px-12-tet_scale_compared_to_just_in_the_chromatic_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/12-tet_scale_compared_to_just_in_the_chromatic_circle.svg/440px-12-tet_scale_compared_to_just_in_the_chromatic_circle.svg.png 2x" data-file-width="585" data-file-height="585" /></a><figcaption>Just (black) major and parallel minor triad, compared to its equal temperament (gray) approximations, within the <a href="/wiki/Chromatic_circle" title="Chromatic circle">chromatic circle</a></figcaption></figure> <p><a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a> has been attributed to both <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> and <a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a> by later writers, but may have been analyzed by other early Greeks or other early cultures as well. The oldest known description of the Pythagorean tuning system appears in Babylonian artifacts.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>During the second century AD, <a href="/wiki/Ptolemy" title="Ptolemy">Claudius Ptolemy</a> described a 5-limit diatonic scale in his influential text on music theory <i>Harmonics</i>, which he called "intense diatonic".<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Given ratios of string lengths 120, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠112<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, 100, 90, 80, 75, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠66<span class="sr-only">+</span><span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>, and 60,<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Ptolemy quantified the tuning of what would later be called the <a href="/wiki/Phrygian_mode" title="Phrygian mode">Phrygian scale</a> (equivalent to the <a href="/wiki/Major_scale" title="Major scale">major scale</a> beginning and ending on the third note) – 16:15, 9:8, 10:9, 9:8, 16:15, 9:8, and 10:9. </p><p>Ptolemy describes a variety of other just intonations derived from history (<a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a>, <a href="/wiki/Philolaus" title="Philolaus">Philolaus</a>, <a href="/wiki/Archytas" title="Archytas">Archytas</a>, <a href="/wiki/Aristoxenus" title="Aristoxenus">Aristoxenus</a>, <a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a>, and <a href="/wiki/Didymus_the_Musician" title="Didymus the Musician">Didymus</a>) and several of his own discovery / invention, including many interval patterns in <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">3-limit</a>, <a href="/wiki/Five-limit_tuning" title="Five-limit tuning">5-limit</a>, <a href="/wiki/7-limit_tuning" title="7-limit tuning">7-limit</a>, and even an 11-limit diatonic. </p><p>Non-Western music, particularly that built on pentatonic scales, is largely tuned using just intonation. In China, the <a href="/wiki/Guqin" title="Guqin">guqin</a> has a musical scale based on <a href="/wiki/Harmonic" title="Harmonic">harmonic</a> <a href="/wiki/Overtone" title="Overtone">overtone</a> positions. The dots on its soundboard indicate the harmonic positions: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">4</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">7</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span>.<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> Indian music has an extensive <a href="/wiki/Svara" title="Svara">theoretical framework</a> for tuning in just intonation.<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. (October 2022)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Diatonic_scale">Diatonic scale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=3" title="Edit section: Diatonic scale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Five-limit_tuning" title="Five-limit tuning">Five-limit tuning</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Primary_triads_in_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Primary_triads_in_C.png/220px-Primary_triads_in_C.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Primary_triads_in_C.png/330px-Primary_triads_in_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Primary_triads_in_C.png/440px-Primary_triads_in_C.png 2x" data-file-width="1098" data-file-height="627" /></a><figcaption>Primary triads in C<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_1" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-mwtitle="Primary_triads_in_C_just.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/7/73/Primary_triads_in_C_just.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/73/Primary_triads_in_C_just.mid/Primary_triads_in_C_just.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/73/Primary_triads_in_C_just.mid/Primary_triads_in_C_just.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Just_intonation_diatonic_scale_derivation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Just_intonation_diatonic_scale_derivation.png/220px-Just_intonation_diatonic_scale_derivation.png" decoding="async" width="220" height="89" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Just_intonation_diatonic_scale_derivation.png/330px-Just_intonation_diatonic_scale_derivation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/74/Just_intonation_diatonic_scale_derivation.png 2x" data-file-width="362" data-file-height="147" /></a><figcaption>Just tuned diatonic scale derivation.<sup id="cite_ref-C&G_4-0" class="reference"><a href="#cite_note-C&G-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The prominent notes of a given scale may be tuned so that their frequencies form (relatively) small whole number ratios. </p><p>The 5 limit diatonic major scale is tuned in such a way that major triads on the <a href="/wiki/Tonic_(music)" title="Tonic (music)">tonic</a>, <a href="/wiki/Subdominant" title="Subdominant">subdominant</a>, and <a href="/wiki/Dominant_(music)" title="Dominant (music)">dominant</a> are tuned in the proportion 4:5:6, and minor triads on the <a href="/wiki/Mediant" title="Mediant">mediant</a> and <a href="/wiki/Submediant" title="Submediant">submediant</a> are tuned in the proportion 10:12:15. Because of the two sizes of <a href="/wiki/Major_second" title="Major second">wholetone</a> – 9:8 (greater wholetone) and 10:9 (lesser wholetone) – the <a href="/wiki/Supertonic" title="Supertonic">supertonic</a> must be microtonally lowered by a <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a> to form a pure minor triad. </p><p>The 5 limit diatonic major scale (<a href="/wiki/Ptolemy%27s_intense_diatonic_scale" title="Ptolemy's intense diatonic scale">Ptolemy's intense diatonic scale</a>) on C is shown in the table below:<sup id="cite_ref-C&G_4-1" class="reference"><a href="#cite_note-C&G-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Johnson2006_6-0" class="reference"><a href="#cite_note-Johnson2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 78">: 78 </span></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><table class="wikitable" style="text-align:center;"> <caption><a href="/wiki/Ptolemy%27s_intense_diatonic_scale" title="Ptolemy's intense diatonic scale">Diatonic, justly tuned, major scale</a> </caption> <tbody><tr> <th rowspan="4">Note </th> <th style="width:6em;">Name </th> <th colspan="2" style="width: 3em">C </th> <th colspan="2" style="width: 3em">D </th> <th colspan="2" style="width: 3em">E </th> <th colspan="2" style="width: 3em">F </th> <th colspan="2" style="width: 3em">G </th> <th colspan="2" style="width: 3em">A </th> <th colspan="2" style="width: 3em">B </th> <th colspan="2" style="width: 3em">C </th></tr> <tr> <th>Ratio from C </th> <td colspan="2" style="background: #fbb">1:1 </td> <td colspan="2" style="background: #8f6">9:8 </td> <td colspan="2" style="background: #aeb">5:4 </td> <td colspan="2" style="background: #fe8">4:3 </td> <td colspan="2" style="background: #fc8">3:2 </td> <td colspan="2" style="background: #fe8">5:3 </td> <td colspan="2" style="background: #8f6">15:8 </td> <td colspan="2" style="background: #fe8">2:1 </td></tr> <tr> <th><span style="font-size:85%;">Harmonic of nominal fundamental F</span> </th> <td colspan="2">24 </td> <td colspan="2">27 </td> <td colspan="2">30 </td> <td colspan="2">32 </td> <td colspan="2">36 </td> <td colspan="2">40 </td> <td colspan="2">45 </td> <td colspan="2">48 </td></tr> <tr> <th><a href="/wiki/Cents_(music)" class="mw-redirect" title="Cents (music)">Cents</a> </th> <td colspan="2">0 </td> <td colspan="2">204 </td> <td colspan="2">386 </td> <td colspan="2">498 </td> <td colspan="2">702 </td> <td colspan="2">884 </td> <td colspan="2">1088 </td> <td colspan="2">1200 </td></tr> <tr> <td colspan="18"> </td></tr> <tr> <th rowspan="3">Step </th> <th>Symbol </th> <td rowspan="3" style="background-color: white; width:1.5em"> </td> <td colspan="2" style="width:3em"><span class="texhtml">T</span> </td> <td colspan="2" style="width:3em"><span class="texhtml">t</span> </td> <td colspan="2" style="width:3em"><span class="texhtml">s</span> </td> <td colspan="2" style="width:3em"><span class="texhtml">T</span> </td> <td colspan="2" style="width:3em"><span class="texhtml">t</span> </td> <td colspan="2" style="width:3em"><span class="texhtml">T</span> </td> <td colspan="2" style="width:3em"><span class="texhtml">s</span> </td> <td rowspan="3" style="background-color: white; width: 1.5em"> </td></tr> <tr> <th>Ratio </th> <td colspan="2" style="background: #8f6">9:8 </td> <td colspan="2" style="background: #5f3">10:9 </td> <td colspan="2" style="background: #2b2">16:15 </td> <td colspan="2" style="background: #8f6">9:8 </td> <td colspan="2" style="background: #5f3">10:9 </td> <td colspan="2" style="background: #8f6">9:8 </td> <td colspan="2" style="background: #2a2">16:15 </td></tr> <tr> <th><a href="/wiki/Cents_(music)" class="mw-redirect" title="Cents (music)">Cents</a> </th> <td colspan="2">204 </td> <td colspan="2">182 </td> <td colspan="2">112 </td> <td colspan="2">204 </td> <td colspan="2">182 </td> <td colspan="2">204 </td> <td colspan="2">112 </td></tr></tbody></table></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:C_major_just_intonation_vs_equal_temperament.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/C_major_just_intonation_vs_equal_temperament.svg/220px-C_major_just_intonation_vs_equal_temperament.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/C_major_just_intonation_vs_equal_temperament.svg/330px-C_major_just_intonation_vs_equal_temperament.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/C_major_just_intonation_vs_equal_temperament.svg/440px-C_major_just_intonation_vs_equal_temperament.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Intervals of the C major scale in just intonation, compared with <a href="/wiki/12_equal_temperament" title="12 equal temperament">equal temperament</a> (blue)</figcaption></figure> <p>In this example the interval from D up to A would be a <a href="/wiki/Wolf_interval" title="Wolf interval">wolf fifth</a> with the ratio <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 40 </span><span class="sr-only">/</span><span class="den"> 27 </span></span>⁠</span>, about 680 <a href="/wiki/Cents_(music)" class="mw-redirect" title="Cents (music)">cents</a>; noticeably smaller than the 702 cents of the pure <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 3 </span><span class="sr-only">/</span><span class="den"> 2 </span></span>⁠</span> ratio. This is mentioned by <a href="/wiki/Heinrich_Schenker" title="Heinrich Schenker">Schenker</a> in reference to the teaching of Bruckner.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>For a justly tuned diatonic minor scale, the mediant is tuned 6:5 and the submediant is tuned 8:5. It would include a tuning of 9:5 for the <a href="/wiki/Subtonic" title="Subtonic">subtonic</a>. For example, on A: </p> <dl><dd><table class="wikitable" style="text-align: center"> <caption><a href="/wiki/Ptolemy%27s_intense_diatonic_scale" title="Ptolemy's intense diatonic scale">Diatonic, justly tuned, minor scale</a> </caption> <tbody><tr> <th rowspan="4">Note </th> <th style="width:6em;">Name </th> <th colspan="2" style="width: 3em">A </th> <th colspan="2" style="width: 3em">B </th> <th colspan="2" style="width: 3em">C </th> <th colspan="2" style="width: 3em">D </th> <th colspan="2" style="width: 3em">E </th> <th colspan="2" style="width: 3em">F </th> <th colspan="2" style="width: 3em">G </th> <th colspan="2" style="width: 3em">A </th></tr> <tr> <th>Ratio from A </th> <td colspan="2" style="background: #fbb">1:1 </td> <td colspan="2" style="background: #8f6">9:8 </td> <td colspan="2" style="background: #fe4">6:5 </td> <td colspan="2" style="background: #fe8">4:3 </td> <td colspan="2" style="background: #fc8">3:2 </td> <td colspan="2" style="background: #fe4">8:5 </td> <td colspan="2" style="background: #fe4">9:5 </td> <td colspan="2" style="background: #fbb">2:1 </td></tr> <tr> <th><span style="font-size:85%;">Harmonic of nominal fundamental B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </span> </th> <td colspan="2">120 </td> <td colspan="2">135 </td> <td colspan="2">144 </td> <td colspan="2">160 </td> <td colspan="2">180 </td> <td colspan="2">192 </td> <td colspan="2">216 </td> <td colspan="2">240 </td></tr> <tr> <th><a href="/wiki/Cents_(music)" class="mw-redirect" title="Cents (music)">Cents</a> </th> <td colspan="2">0 </td> <td colspan="2">204 </td> <td colspan="2">316 </td> <td colspan="2">498 </td> <td colspan="2">702 </td> <td colspan="2">814 </td> <td colspan="2">1018 </td> <td colspan="2">1200 </td></tr> <tr> <td colspan="18"> </td></tr> <tr> <th rowspan="3">Step </th> <th>Name </th> <td rowspan="3" style="background-color: white; width:1.5em"> </td> <td colspan="2" style="width:3em">T </td> <td colspan="2" style="width:3em">s </td> <td colspan="2" style="width:3em">t </td> <td colspan="2" style="width:3em">T </td> <td colspan="2" style="width:3em">s </td> <td colspan="2" style="width:3em">T </td> <td colspan="2" style="width:3em">t </td> <td rowspan="3" style="background-color: white; width: 1.5em"> </td></tr> <tr> <th>Ratio </th> <td colspan="2" style="background: #8f6">9:8 </td> <td colspan="2" style="background: #2a2">16:15 </td> <td colspan="2" style="background: #5f3">10:9 </td> <td colspan="2" style="background: #8f6">9:8 </td> <td colspan="2" style="background: #2a2">16:15 </td> <td colspan="2" style="background: #8f6">9:8 </td> <td colspan="2" style="background: #5f3">10:9 </td></tr> <tr> <th><a href="/wiki/Cents_(music)" class="mw-redirect" title="Cents (music)">Cents</a> </th> <td colspan="2">204 </td> <td colspan="2">112 </td> <td colspan="2">182 </td> <td colspan="2">204 </td> <td colspan="2">112 </td> <td colspan="2">204 </td> <td colspan="2">182 </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Twelve-tone_scale">Twelve-tone scale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=4" title="Edit section: Twelve-tone scale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Chromatic_scale"></span> There are several ways to create a just tuning of the twelve-tone scale. </p> <div class="mw-heading mw-heading3"><h3 id="Pythagorean_tuning">Pythagorean tuning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=5" title="Edit section: Pythagorean tuning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a></div> <p><a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a> can produce a twelve-tone scale, but it does so by involving ratios of very large numbers, corresponding to natural harmonics very high in the harmonic series that do not occur widely in physical phenomena. This tuning uses ratios involving only powers of 3 and 2, creating a sequence of just <a href="/wiki/Perfect_fifth" title="Perfect fifth">fifths</a> or <a href="/wiki/Perfect_fourth" title="Perfect fourth">fourths</a>, as follows: </p> <table class="wikitable" style="text-align: center"> <tbody><tr> <th style="width:5em;">Note </th> <th style="width:5em;">G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> </th> <th style="width:5em;">D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> </th> <th style="width:5em;">A<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> </th> <th style="width:5em;">E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> </th> <th style="width:5em;">B<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> </th> <th style="width:5em;">F </th> <th style="width:5em;">C </th> <th style="width:5em;">G </th> <th style="width:5em;">D </th> <th style="width:5em;">A </th> <th style="width:5em;">E </th> <th style="width:5em;">B </th> <th style="width:5em;">F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> </th></tr> <tr> <th>Ratio </th> <td>1024:729 </td> <td>256:243 </td> <td>128:81 </td> <td>32:27 </td> <td>16:9 </td> <td>4:3 </td> <td>1:1 </td> <td>3:2 </td> <td>9:8 </td> <td>27:16 </td> <td>81:64 </td> <td>243:128 </td> <td>729:512 </td></tr> <tr> <th>Cents </th> <td>588 </td> <td>90 </td> <td>792 </td> <td>294 </td> <td>996 </td> <td>498 </td> <td>0 </td> <td>702 </td> <td>204 </td> <td>906 </td> <td>408 </td> <td>1110 </td> <td>612 </td></tr></tbody></table> <p>The ratios are computed with respect to C (the <i>base note</i>). Starting from C, they are obtained by moving six steps (around the <a href="/wiki/Circle_of_fifths" title="Circle of fifths">circle of fifths</a>) to the left and six to the right. Each step consists of a multiplication of the previous pitch by <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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="frac"><span class="num">2</span>⁄<span class="den">3</span></span> (descending fifth), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">3</span>⁄<span class="den">2</span></span> (ascending fifth), or their <a href="/wiki/Inversion_(music)" title="Inversion (music)">inversions</a> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">3</span>⁄<span class="den">4</span></span> or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">4</span>⁄<span class="den">3</span></span>). </p><p>Between the <a href="/wiki/Enharmonic" class="mw-redirect" title="Enharmonic">enharmonic</a> notes at both ends of this sequence is a <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitch</a> ratio of <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3<sup>12</sup></span><span class="sr-only">/</span><span class="den">2<sup>19</sup></span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">531441</span><span class="sr-only">/</span><span class="den">524288</span></span>⁠</span></span>, or about 23 <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>, known as the <a href="/wiki/Pythagorean_comma" title="Pythagorean comma">Pythagorean comma</a>. To produce a twelve-tone scale, one of them is arbitrarily discarded. The twelve remaining notes are repeated by increasing or decreasing their frequencies by a power of 2 (the size of one or more <a href="/wiki/Octave" title="Octave">octaves</a>) to build scales with multiple octaves (such as the keyboard of a piano). A drawback of Pythagorean tuning is that one of the twelve fifths in this scale is badly tuned and hence unusable (the <a href="/wiki/Wolf_fifth" class="mw-redirect" title="Wolf fifth">wolf fifth</a>, either F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>–D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> if G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> is discarded, or B–G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> if F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> is discarded). This twelve-tone scale is fairly close to <a href="/wiki/Equal_temperament" title="Equal temperament">equal temperament</a>, but it does not offer much advantage for <a href="/wiki/Tonality" title="Tonality">tonal</a> harmony because only the perfect intervals (fourth, fifth, and octave) are simple enough to sound pure. Major thirds, for instance, receive the rather unstable interval of 81:64, sharp of the preferred 5:4 by an 81:80 ratio.<sup id="cite_ref-autogenerated1968_9-0" class="reference"><a href="#cite_note-autogenerated1968-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The primary reason for its use is that it is extremely easy to tune, as its building block, the perfect fifth, is the simplest and consequently the most <a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">consonant</a> interval after the octave and unison. </p><p>Pythagorean tuning may be regarded as a "three-limit" tuning system, because the ratios can be expressed as a product of integer powers of only whole numbers less than or equal to 3. </p> <div class="mw-heading mw-heading3"><h3 id="Five-limit_tuning">Five-limit tuning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=6" title="Edit section: Five-limit tuning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Five-limit_tuning" title="Five-limit tuning">Five-limit tuning</a></div> <p>A twelve-tone scale can also be created by compounding harmonics up to the fifth: namely, by multiplying the frequency of a given reference note (the base note) by powers of 2, 3, or 5, or a combination of them. This method is called five-limit tuning. </p><p>To build such a twelve-tone scale (using C as the base note), we may start by constructing a table containing fifteen pitches: </p> <dl><dd><table class="wikitable" style="text-align:center"> <tbody><tr> <th>Factor </th> <th style="width:4em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 9 </span></span>⁠</span> </th> <th style="width:4em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span> </th> <th style="width:4em;">1 </th> <th style="width:4em;">3 </th> <th style="width:4em;">9 </th> <th> </th></tr> <tr> <th rowspan="3">5 </th> <td style="background:coral;"><b>D</b> </td> <td><b>A</b> </td> <td><b>E</b> </td> <td><b>B</b> </td> <td style="background:springgreen;"><b>F</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> </td> <th>note </th></tr> <tr> <td style="background:coral;">10:9 </td> <td>5:3 </td> <td>5:4 </td> <td>15:8 </td> <td style="background:springgreen;">45:32 </td> <th>ratio </th></tr> <tr> <td style="background:coral;">182 ¢ </td> <td>884 ¢ </td> <td>386 ¢ </td> <td>1088 ¢ </td> <td style="background:springgreen;">590 ¢ </td> <th>cents </th></tr> <tr> <th rowspan="3">1 </th> <td style="background:gold;"><b>B</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td><b>F</b> </td> <td><span style="color: red;"><b><span style="font-size:120%">C</span></b></span> </td> <td><b>G</b> </td> <td style="background:coral;"><b>D</b> </td> <th>note </th></tr> <tr> <td style="background:gold;">16:9 </td> <td>4:3 </td> <td>1:1 </td> <td>3:2 </td> <td style="background:coral;">9:8 </td> <th>ratio </th></tr> <tr> <td style="background:gold;">996 ¢ </td> <td>498 ¢ </td> <td>0 ¢ </td> <td>702 ¢ </td> <td style="background:coral;">204 ¢ </td> <th>cents </th></tr> <tr> <th rowspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 5 </span></span>⁠</span> </th> <td style="background:springgreen;"><b>G</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td><b>D</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td><b>A</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td><b>E</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td style="background:gold;"><b>B</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <th>note </th></tr> <tr> <td style="background:springgreen;">64:45 </td> <td>16:15 </td> <td>8:5 </td> <td>6:5 </td> <td style="background:gold;">9:5 </td> <th>ratio </th></tr> <tr> <td style="background:springgreen;">610 ¢ </td> <td>112 ¢ </td> <td>814 ¢ </td> <td>316 ¢ </td> <td style="background:gold;">1018 ¢ </td> <th>cents </th></tr></tbody></table></dd></dl> <p>The factors listed in the first row and column are powers of 3 and 5, respectively (e.g., <span class="nowrap"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 9 </span></span>⁠</span> = 3<sup>−2</sup> ).</span> Colors indicate couples of <a href="/wiki/Enharmonic" class="mw-redirect" title="Enharmonic">enharmonic</a> notes with almost identical pitch. The ratios are all expressed relative to C in the centre of this diagram (the base note for this scale). They are computed in two steps: </p> <ol><li>For each cell of the table, a <i>base ratio</i> is obtained by multiplying the corresponding factors. For instance, the base ratio for the lower-left cell is <span class="nowrap"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 9 </span></span>⁠</span> × <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 5 </span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 45 </span></span>⁠</span> .</span></li> <li>The base ratio is then multiplied by a negative or positive power of 2, as large as needed to bring it within the range of the octave starting from C (from 1:1 to 2:1). For instance, the base ratio for the lower left cell (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 45 </span></span>⁠</span>) is multiplied by 2<sup>6</sup>, and the resulting ratio is 64:45, which is a number between 1:1 and 2:1.</li></ol> <p>Note that the powers of 2 used in the second step may be interpreted as ascending or descending <a href="/wiki/Octave" title="Octave">octaves</a>. For instance, multiplying the frequency of a note by 2<sup>6</sup> means increasing it by 6 octaves. Moreover, each row of the table may be considered to be a sequence of fifths (ascending to the right), and each column a sequence of major thirds (ascending upward). For instance, in the first row of the table, there is an ascending fifth from D and A, and another one (followed by a descending octave) from A to E. This suggests an alternative but equivalent method for computing the same ratios. For instance, one can obtain A, starting from C, by moving one cell to the left and one upward in the table, which means descending by a fifth and ascending by a major third: </p> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> × <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">10</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> .</dd></dl> <p>Since this is below C, one needs to move up by an octave to end up within the desired range of ratios (from 1:1 to 2:1): </p> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> × <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">1</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">10</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> .</dd></dl> <p>A 12 tone scale is obtained by removing one note for each couple of enharmonic notes. This can be done in four ways that have in common the removal of G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, according to a convention which was valid even for C-based Pythagorean and <a href="/wiki/Quarter-comma_meantone" title="Quarter-comma meantone">quarter-comma meantone</a> scales. Note that it is a <a href="/wiki/Diminished_fifth" class="mw-redirect" title="Diminished fifth">diminished fifth</a>, close to half an octave, above the tonic C, which is a discordant interval; also its ratio has the largest values in its numerator and denominator of all tones in the scale, which make it least harmonious: All are reasons to avoid it. </p><p>The following chart shows one way to obtain a 12 tone scale by removing one note for each pair of enharmonic notes. In this method one discards the first column of the table (labeled "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 9 </span></span>⁠</span>"). </p> <dl><dd><table class="wikitable" style="text-align: center"> <tbody><tr> <th colspan="7">Asymmetric scale </th></tr> <tr> <th>Factor </th> <th style="width: 3em"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span> </th> <th style="width: 3em">1 </th> <th style="width: 3em">3 </th> <th style="width: 3em">9 </th></tr> <tr> <th rowspan="2">5 </th> <td><b>A</b> </td> <td><b>E</b> </td> <td><b>B</b> </td> <td style="background:springgreen;"><b>F</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> </td></tr> <tr> <td>5:3 </td> <td>5:4 </td> <td>15:8 </td> <td style="background:springgreen;">45:32 </td></tr> <tr> <th rowspan="2">1 </th> <td><b>F</b> </td> <td><span style="color: red;"><b><span style="font-size:120%">C</span></b></span> </td> <td><b>G</b> </td> <td style="background:coral;"><b>D</b> </td></tr> <tr> <td>4:3 </td> <td>1:1 </td> <td>3:2 </td> <td style="background:coral;">9:8 </td></tr> <tr> <th rowspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 5 </span></span>⁠</span> </th> <td><b>D</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td><b>A</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td><b>E</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td style="background:gold;"><b>B</b><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td></tr> <tr style="text-align:center;"> <td>16:15 </td> <td>8:5 </td> <td>6:5 </td> <td style="background:gold;">9:5 </td></tr></tbody></table></dd></dl> <p>This scale is "asymmetric" in the sense that going up from the tonic two semitones we multiply the frequency by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 9 </span><span class="sr-only">/</span><span class="den"> 8 </span></span>⁠</span>, while going down from the tonic two semitones we do not divide the frequency by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 9 </span><span class="sr-only">/</span><span class="den"> 8 </span></span>⁠</span>. For two methods that give "symmetric" scales, see <a href="/wiki/Five-limit_tuning#Twelve-tone_scale" title="Five-limit tuning">Five-limit tuning: Twelve-tone scale</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Extension_of_the_twelve-tone_scale">Extension of the twelve-tone scale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=7" title="Edit section: Extension of the twelve-tone scale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The table above uses only low powers of 3 and 5 to build the base ratios. However, it can be easily extended by using higher positive and negative powers of the same numbers, such as 5<sup>2</sup> = 25, 5<sup>−2</sup> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">25</span></span>, 3<sup>3</sup> = 27, or 3<sup>−3</sup> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">27</span></span>. A scale with 25, 35 or even more pitches can be obtained by combining these base ratios. </p> <div class="mw-heading mw-heading2"><h2 id="Indian_scales">Indian scales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=8" title="Edit section: Indian scales"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Indian_music" class="mw-redirect" title="Indian music">Indian music</a>, the just diatonic scale described above is used, though there are different possibilities, for instance for the sixth pitch (<i>dha</i>), and further modifications may be made to all pitches excepting <i>sa</i> and <i>pa</i>.<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> <dl><dd><table class="wikitable" style="text-align:center;"> <tbody><tr> <th style="width:5em;">Note </th> <th style="width:5em;"><i>sa</i> </th> <th style="width:5em;"><i>re</i> </th> <th style="width:5em;"><i>ga</i> </th> <th style="width:5em;"><i>ma</i> </th> <th style="width:5em;"><i>pa</i> </th> <th style="width:5em;"><i>dha</i> </th> <th style="width:5em;"><i>ni</i> </th> <th style="width:5em;"><i>sa</i> </th></tr> <tr> <th>Ratio </th> <td>1:1 </td> <td>9:8 </td> <td>5:4 </td> <td>4:3 </td> <td>3:2 </td> <td style="background:#efcfcf;">5:3 or 27:16 </td> <td>15:8 </td> <td>2:1 </td></tr> <tr> <th>Cents </th> <td>0 </td> <td>204 </td> <td>386 </td> <td>498 </td> <td>702 </td> <td style="background:#efcfcf;">884 or 906 </td> <td>1088 </td> <td>1200 </td></tr></tbody></table></dd></dl> <p>Some accounts of Indian intonation system cite a given 12 swaras being divided into 22 <a href="/wiki/Shruti_(music)" title="Shruti (music)">shrutis</a>.<sup id="cite_ref-autogenerated1995_11-0" class="reference"><a href="#cite_note-autogenerated1995-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-autogenerated1999_12-0" class="reference"><a href="#cite_note-autogenerated1999-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> According to some musicians, one has a scale of a given 12 pitches and ten in addition (the tonic, shadja (<i>sa</i>), and the pure fifth, pancham (<i>pa</i>), are inviolate (known as achala<sup id="cite_ref-shruti-webst_13-0" class="reference"><a href="#cite_note-shruti-webst-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> in Indian music theory): </p> <table class="wikitable" style="text-align:center;width:60em;"> <tbody><tr> <th style="width:5em;">Note </th> <th>C </th> <th>D<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>↓</sup> </th> <th>D<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </th> <th>D<sup>↓</sup> </th> <th>D </th> <th>E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>↓</sup> </th> <th>E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </th> <th>E </th> <th>E<sup>↑</sup> </th> <th>F </th> <th>F<sup>↑</sup> </th> <th>F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> </th> <th>F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>↑</sup> </th> <th>G </th> <th>A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>↓</sup> </th> <th>A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </th> <th>A </th> <th>A<sup>↑</sup> </th> <th>B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>↓</sup> </th> <th>B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </th> <th>B </th> <th>B<sup>↑</sup> </th> <th>C </th></tr> <tr> <th>Ratio </th> <td>1:1 </td> <td>256:243 </td> <td>16:15 </td> <td>10:9 </td> <td>9:8 </td> <td>32:27 </td> <td>6:5 </td> <td>5:4 </td> <td>81:64 </td> <td>4:3 </td> <td>27:20 </td> <td>45:32 </td> <td>729:512 </td> <td>3:2 </td> <td>128:81 </td> <td>8:5 </td> <td>5:3 </td> <td>27:16 </td> <td>16:9 </td> <td>9:5 </td> <td>15:8 </td> <td>243:128 </td> <td>2:1 </td></tr> <tr> <th>Cents </th> <td>0 </td> <td>90 </td> <td>112 </td> <td>182 </td> <td>204 </td> <td>294 </td> <td>316 </td> <td>386 </td> <td>408 </td> <td>498 </td> <td>520 </td> <td>590 </td> <td>612 </td> <td>702 </td> <td>792 </td> <td>814 </td> <td>884 </td> <td>906 </td> <td>996 </td> <td>1018 </td> <td>1088 </td> <td>1110 </td> <td>1200 </td></tr></tbody></table> <p>Where we have <i>two</i> ratios for a given letter name or swara, we have a difference of 81:80 (22 cents), which is the <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a><sup id="cite_ref-autogenerated1968_9-1" class="reference"><a href="#cite_note-autogenerated1968-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> or the praman<sup id="cite_ref-shruti-webst_13-1" class="reference"><a href="#cite_note-shruti-webst-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> in Indian music theory. These notes are known as <i>chala</i>.<sup id="cite_ref-shruti-webst_13-2" class="reference"><a href="#cite_note-shruti-webst-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The distance between two letter names comes in to sizes, <i>poorna</i> (256:243) and <i>nyuna</i> (25:24).<sup id="cite_ref-shruti-webst_13-3" class="reference"><a href="#cite_note-shruti-webst-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> One can see the symmetry, looking at it from the tonic, then the octave. </p><p>(This is just one example of explaining a 22 Śhruti scale of tones. There are many different explanations.) </p> <div class="mw-heading mw-heading2"><h2 id="Practical_difficulties">Practical difficulties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=9" title="Edit section: Practical difficulties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" 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 section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Just_intonation" title="Special:EditPage/Just intonation">improve this section</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 <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">April 2016</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> <p>Some fixed just intonation scales and systems, such as the diatonic scale above, produce <a href="/wiki/Wolf_interval" title="Wolf interval">wolf intervals</a> when the approximately equivalent flat note is substituted for a sharp note not available in the scale, or vice versa. The above scale allows a minor tone to occur next to a semitone which produces the awkward ratio 32:27 for D→F, and still worse, a minor tone next to a fourth giving 40:27 for D→A. Flattening D by a comma to 10:9 alleviates these difficulties but creates new ones: D→G becomes 27:20, and D→B becomes 27:16. This fundamental problem arises in any system of tuning using a limited number of notes. </p><p>One can have more <a href="/wiki/Fret" title="Fret">frets</a> on a <a href="/wiki/Guitar" title="Guitar">guitar</a> (or keys on a piano) to handle both As, 9:8 with respect to G and 10:9 with respect to G so that A→C can be played as 6:5 while A→D can still be played as 3:2. 9:8 and 10:9 are less than <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">53</span></span>⁠</span> of an octave apart, so mechanical and performance considerations have made this approach extremely rare. And the problem of how to tune complex chords such as <a href="/wiki/6/9_chord" class="mw-redirect" title="6/9 chord">C<sup>6 add 9</sup></a> <span class="nowrap">(C→E→G→A→D),</span> in typical 5 limit just intonation, is left unresolved (for instance, A could be 4:3 below D (making it 9:8, if G is 1) or 4:3 above E (making it 10:9, if G is 1) but not both at the same time, so one of the fourths in the chord will have to be an out-of-tune wolf interval). Most complex (added-tone and extended) chords usually require intervals beyond common 5 limit ratios in order to sound harmonious (for instance, the previous chord could be tuned to 8:10:12:13:18, using the A note from the 13th harmonic), which implies even more keys or frets. However the frets may be removed entirely – unfortunately, this makes in-tune fingering of many chords exceedingly difficult, due to the construction and mechanics of the human hand – and the tuning of most complex chords in just intonation is generally ambiguous. </p><p>Some composers deliberately use these wolf intervals and other dissonant intervals as a way to expand the tone color palette of a piece of music. For example, the extended piano pieces <i><a href="/wiki/The_Well-Tuned_Piano" title="The Well-Tuned Piano">The Well-Tuned Piano</a></i> by La Monte Young and <i>The Harp of New Albion</i> by Terry Riley use a combination of very consonant and dissonant intervals for musical effect. In "Revelation", <a href="/wiki/Michael_Harrison_(musician)" title="Michael Harrison (musician)">Michael Harrison</a> goes even further, and uses the tempo of beat patterns produced by some dissonant intervals as an integral part of several movements. </p><p>When tuned in just intonation, many fixed-pitch instruments cannot be played in a new <a href="/wiki/Key_(music)" title="Key (music)">key</a> without retuning the instrument. For instance, if a piano is tuned in just intonation intervals and a minimum of wolf intervals for the key of G, then only one other key (typically E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>) can have the same intervals, and many of the keys have a very dissonant and unpleasant sound. This makes <a href="/wiki/Modulation_(music)" title="Modulation (music)">modulation</a> within a piece, or playing a repertoire of pieces in different keys, impractical to impossible. </p><p><a href="/wiki/Synthesizer" title="Synthesizer">Synthesizers</a> have proven a valuable tool for composers wanting to experiment with just intonation. They can be easily retuned with a <a href="/wiki/Microtuner" title="Microtuner">microtuner</a>. Many commercial synthesizers provide the ability to use built-in just intonation scales or to create them manually. <a href="/wiki/Wendy_Carlos" title="Wendy Carlos">Wendy Carlos</a> used a system on her 1986 album <i><a href="/wiki/Beauty_in_the_Beast" title="Beauty in the Beast">Beauty in the Beast</a></i>, where one electronic keyboard was used to play the notes, and another used to instantly set the root note to which all intervals were tuned, which allowed for modulation. On her 1987 lecture album <i>Secrets of Synthesis</i> there are audible examples of the difference in sound between equal temperament and just intonation. </p><p>Many singers (especially barbershop quartets) and fretless instrument players naturally aim for a more just intonation when playing: </p> <dl><dd>"Don't be scared if your intonation differs from that of the piano. It is the piano that is out of tune. The piano with its tempered scale is a compromise in intonation." — <a href="/wiki/Pablo_Casals" title="Pablo Casals">Pablo Casals</a><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. (November 2024)">citation needed</span></a></i>]</sup></dd></dl> <p>In trying to get a more just system for instruments that is more adaptable like the human voice and fretless instruments, the tuning trade-offs between more consonant harmony versus easy transposability (between different keys) have traditionally been too complicated to solve mechanically, though there have been attempts throughout history with various drawbacks, including the <a href="/wiki/Archicembalo" title="Archicembalo">archicembalo</a>. </p><p>Since the advent of personal computing, there have been more attempts to solve the perceived problem by trying to algorithmically solve what many professional musicians have learned through practice and intuition. Four of the main problems are that consonance cannot be perfect for some complex chords, chords can have internal consistency but clash with the overall direction of the piece, and naively adjusting the tuning only taking into account chords in isolation can lead to a drift where the end of the piece is noticeably higher or lower in overall pitch rather than centered. </p><p>Software solutions like <a href="/w/index.php?title=Hermode_Tuning&action=edit&redlink=1" class="new" title="Hermode Tuning (page does not exist)">Hermode Tuning</a> often analyze solutions chord by chord instead of taking in the global context of the whole piece like it's theorized human players do. Since 2017, there has been research to address these problems algorithmically through dynamically adapted just intonation and machine learning.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Singing_and_scale-free_instruments">Singing and scale-free instruments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=10" title="Edit section: Singing and scale-free instruments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" 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 section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Just_intonation" title="Special:EditPage/Just intonation">improve this section</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 <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">May 2017</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> <p>The human voice is among the most pitch-flexible instruments in common use. Pitch can be varied with no restraints and adjusted in the midst of performance, without needing to retune. Although the explicit use of just intonation fell out of favour concurrently with the increasing use of instrumental accompaniment (with its attendant constraints on pitch), most <a href="/wiki/A_cappella" title="A cappella">a cappella</a> ensembles naturally tend toward just intonation because of the comfort of its stability. <a href="/wiki/Barbershop_music" title="Barbershop music">Barbershop quartets</a> are a good example of this. </p><p>The unfretted stringed instruments such as those from the violin family (the violin, the viola, and the cello), and the double bass are quite flexible in the way pitches can be adjusted. Stringed instruments that are not playing with fixed pitch instruments tend to adjust the pitch of key notes such as thirds and <a href="/wiki/Leading_tone" title="Leading tone">leading tones</a> so that the pitches differ from equal temperament. </p><p>Trombones have a slide that allows arbitrary tuning during performance. French horns can be tuned by shortening or lengthening the main tuning slide on the back of the instrument, with each individual rotary or piston slide for each rotary or piston valve, and by using the right hand inside the bell to adjust the pitch by pushing the hand in deeper to flatten the note, or pulling it out to sharpen the note while playing. Some natural horns also may adjust the tuning with the hand in the bell, and valved cornets, trumpets, Flugelhorns, Saxhorns, Wagner tubas, and tubas have overall and valve-by-valve tuning slides, like valved horns. </p><p>Wind instruments with valves are biased towards natural tuning and must be micro-tuned if equal temperament is required. </p><p>Other wind instruments, although built to a certain scale, can be micro-tuned to a certain extent by using the <a href="/wiki/Embouchure" title="Embouchure">embouchure</a> or adjustments to fingering. </p> <div class="mw-heading mw-heading2"><h2 id="Western_composers">Western composers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=11" title="Edit section: Western composers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Composers often impose a <a href="/wiki/Limit_(music)" title="Limit (music)">limit</a> on how complex the ratios may become.<sup id="cite_ref-:1_15-0" class="reference"><a href="#cite_note-:1-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (September 2022)">page needed</span></a></i>]</sup> For example, a composer who chooses to write in 7-limit just intonation will not employ ratios that use powers of prime numbers larger than 7. Under this scheme, ratios like 11:7 and 13:6 would not be permitted, because 11 and 13 cannot be expressed as powers of those prime numbers ≤ 7 (<i>i.e.</i> 2, 3, 5, and 7). </p> <div class="mw-heading mw-heading2"><h2 id="Staff_notation">Staff notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=12" title="Edit section: Staff notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Helmholtz-Ellis_JI_pitch_notation_microtonal_accidents_legend.gif" class="mw-file-description"><img alt="Legend of the HE Accidentals" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Helmholtz-Ellis_JI_pitch_notation_microtonal_accidents_legend.gif/375px-Helmholtz-Ellis_JI_pitch_notation_microtonal_accidents_legend.gif" decoding="async" width="375" height="407" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Helmholtz-Ellis_JI_pitch_notation_microtonal_accidents_legend.gif/563px-Helmholtz-Ellis_JI_pitch_notation_microtonal_accidents_legend.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/9/9f/Helmholtz-Ellis_JI_pitch_notation_microtonal_accidents_legend.gif 2x" data-file-width="662" data-file-height="719" /></a><figcaption><span class="anchor" id="fig1"></span>Fig. 1: Legend of the Helmholtz-Ellis accidentals within the 23-limit</figcaption></figure> <p>Originally a system of notation to describe scales was devised by <a href="/wiki/Moritz_Hauptmann" title="Moritz Hauptmann">Hauptmann</a> and modified by <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a> (1877); the starting note is presumed Pythagorean; a "+" is placed between if the next note is a just major third up, a "−" if it is a just minor third, among others; finally, subscript numbers are placed on the second note to indicate how many syntonic commas (81:80) to lower by.<sup id="cite_ref-Helmholtz1885_16-0" class="reference"><a href="#cite_note-Helmholtz1885-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> For example, the Pythagorean major third on C is C+E (<span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="noexcerpt ext-phonos-PhonosButton 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\/44\/Pythagorean_major_third_on_C.mid\/Pythagorean_major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean major third on C.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/44/Pythagorean_major_third_on_C.mid/Pythagorean_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:Pythagorean_major_third_on_C.mid" title="File:Pythagorean major third on C.mid">ⓘ</a></sup></span>) while the just major third is C+E<sub>1</sub> (<span class="ext-phonos"><span data-nosnippet="" id="ooui-php-2" class="noexcerpt ext-phonos-PhonosButton 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":["noexcerpt","ext-phonos-PhonosButton"]}"><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>). A similar system was devised by <a href="/wiki/Carl_Eitz" title="Carl Eitz">Carl Eitz</a> and used in <a href="/wiki/J._Murray_Barbour" class="mw-redirect" title="J. Murray Barbour">Barbour</a> (1951) in which Pythagorean notes are started with and positive or negative superscript numbers are added indicating how many commas (81:80, syntonic comma) to adjust by.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> For example, the Pythagorean major third on C is C−E<sup>0</sup> while the just major third is C−E<sup>−1</sup>. An extension of this Pythagorean-based notation to higher primes is the <i>Helmholtz / Ellis / Wolf / Monzo system</i><sup id="cite_ref-MonzoTonal_18-0" class="reference"><a href="#cite_note-MonzoTonal-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> of <a href="/wiki/ASCII" title="ASCII">ASCII</a> symbols and prime-factor-power vectors described in Monzo's <i>Tonalsoft Encyclopaedia</i>.<sup id="cite_ref-MonzoTonal_18-1" class="reference"><a href="#cite_note-MonzoTonal-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>While these systems allow precise indication of intervals and pitches in print, more recently some composers have been developing notation methods for Just Intonation using the conventional five-line staff. <a href="/wiki/James_Tenney" title="James Tenney">James Tenney</a>, amongst others, preferred to combine JI ratios with <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a> deviations from the <a href="/wiki/Equal_tempered" class="mw-redirect" title="Equal tempered">equal tempered</a> pitches, indicated in a legend or directly in the score, allowing performers to readily use electronic tuning devices if desired.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Beginning in the 1960s, <a href="/wiki/Ben_Johnston_(composer)" title="Ben Johnston (composer)">Ben Johnston</a> had proposed an alternative approach, redefining the understanding of conventional symbols (the seven "white" notes, the sharps and flats) and adding further accidentals, each designed to extend the notation into higher <a href="/wiki/Prime_limit" class="mw-redirect" title="Prime limit">prime limits</a>. His notation "begins with the 16th-century Italian definitions of intervals and continues from there."<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Johnston notation is based on a diatonic C Major scale tuned in JI <a href="#fig4">(Fig. 4)</a>, in which the interval between D (9:8 above C) and A (5:3 above C) is one <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a> less than a Pythagorean perfect fifth 3:2. To write a perfect fifth, Johnston introduces a pair of symbols, + and − again, to represent this comma. Thus, a series of perfect fifths beginning with F would proceed C G D A+ E+ B+. The three conventional white notes A E B are tuned as Ptolemaic major thirds (5:4) above F C G respectively. Johnston introduces new symbols for the septimal (<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:7_rightside_up.png" class="mw-file-description" title="7"><img alt="7" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/7_rightside_up.png/8px-7_rightside_up.png" decoding="async" width="8" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/7_rightside_up.png/13px-7_rightside_up.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/7_rightside_up.png/17px-7_rightside_up.png 2x" data-file-width="84" data-file-height="158" /></a></span></span> & <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:7_upside_down.png" class="mw-file-description" title="7 upside-down"><img alt="7 upside-down" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/7_upside_down.png/8px-7_upside_down.png" decoding="async" width="8" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/7_upside_down.png/13px-7_upside_down.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/7_upside_down.png/17px-7_upside_down.png 2x" data-file-width="84" data-file-height="158" /></a></span></span>), undecimal (<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;">↑</span> & <span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;">↓</span>), tridecimal (<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:13_rightside_up.png" class="mw-file-description" title="13"><img alt="13" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/13_rightside_up.png/12px-13_rightside_up.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/13_rightside_up.png/18px-13_rightside_up.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/13_rightside_up.png/24px-13_rightside_up.png 2x" data-file-width="385" data-file-height="385" /></a></span></span> & <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:13_upside_down.png" class="mw-file-description" title="13 upside down"><img alt="13 upside down" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/13_upside_down.png/12px-13_upside_down.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/13_upside_down.png/18px-13_upside_down.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/13_upside_down.png/24px-13_upside_down.png 2x" data-file-width="385" data-file-height="385" /></a></span></span>), and further prime-number extensions to create an accidental based exact JI notation for what he has named "Extended Just Intonation" (<a href="#fig2">Fig. 2</a> & <a href="#fig3">Fig. 3</a>).<sup id="cite_ref-Johnson2006_6-1" class="reference"><a href="#cite_note-Johnson2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 77–88">: 77–88 </span></sup> For example, the Pythagorean major third on C is C-E+ while the just major third is C-E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-natural">♮</span></span> <a href="#fig4">(Fig. 4)</a>. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Notation_of_partials_1-19_for_1-1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Notation_of_partials_1-19_for_1-1.png/400px-Notation_of_partials_1-19_for_1-1.png" decoding="async" width="400" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Notation_of_partials_1-19_for_1-1.png/600px-Notation_of_partials_1-19_for_1-1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Notation_of_partials_1-19_for_1-1.png/800px-Notation_of_partials_1-19_for_1-1.png 2x" data-file-width="1030" data-file-height="266" /></a><figcaption><span class="anchor" id="fig2"></span>Fig. 2: Staff notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> using Johnston notation<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_2" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-mwtitle="Notation_of_partials_1-19_for_1-1.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/a/ad/Notation_of_partials_1-19_for_1-1.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/ad/Notation_of_partials_1-19_for_1-1.mid/Notation_of_partials_1-19_for_1-1.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/a/ad/Notation_of_partials_1-19_for_1-1.mid/Notation_of_partials_1-19_for_1-1.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Harmonic_seventh_chord_just_on_C.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Harmonic_seventh_chord_just_on_C.png/220px-Harmonic_seventh_chord_just_on_C.png" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Harmonic_seventh_chord_just_on_C.png/330px-Harmonic_seventh_chord_just_on_C.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Harmonic_seventh_chord_just_on_C.png/440px-Harmonic_seventh_chord_just_on_C.png 2x" data-file-width="539" data-file-height="252" /></a><figcaption><span class="anchor" id="fig3"></span>Fig. 3: Just harmonic seventh chord (4:5:6:7:8) on C in Johnston notation. The size of the 7th is 968.826 cents: <a href="/wiki/Septimal_quarter_tone" title="Septimal quarter tone">48.77 cents lower</a> than B<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> tuned 9:5 above C.<span class="mw-default-size" typeof="mw:File"><span><audio id="mwe_player_3" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" style="width:220px;" data-durationhint="5" data-mwtitle="Harmonic_seventh_chord_just_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/f/fe/Harmonic_seventh_chord_just_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/fe/Harmonic_seventh_chord_just_on_C.mid/Harmonic_seventh_chord_just_on_C.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/fe/Harmonic_seventh_chord_just_on_C.mid/Harmonic_seventh_chord_just_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></figcaption></figure> <p>In 2000–2004, <a href="/wiki/Marc_Sabat" title="Marc Sabat">Marc Sabat</a> and <a href="/wiki/Wolfgang_von_Schweinitz" title="Wolfgang von Schweinitz">Wolfgang von Schweinitz</a> worked in Berlin to develop a different accidental-based method, the Extended Helmholtz-Ellis JI Pitch Notation.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Following the method of notation suggested by <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a> in his classic <i><a href="/wiki/On_the_Sensations_of_Tone_as_a_Physiological_Basis_for_the_Theory_of_Music" class="mw-redirect" title="On the Sensations of Tone as a Physiological Basis for the Theory of Music">On the Sensations of Tone as a Physiological Basis for the Theory of Music</a></i>, incorporating Ellis' invention of cents, and continuing Johnston's step into "Extended JI", Sabat and Schweinitz propose unique symbols (accidentals) for each prime dimension of harmonic space. In particular, the conventional flats, naturals and sharps define a Pythagorean series of perfect fifths. The Pythagorean pitches are then paired with new symbols that commatically alter them to represent various other partials of the harmonic series <a href="#fig1">(Fig. 1)</a>. To facilitate quick estimation of pitches, cents indications may be added (e.g. downward deviations below and upward deviations above the respective accidental). A typically used convention is that cent deviations refer to the <i>tempered pitch</i> implied by the flat, natural, or sharp. A complete legend and fonts for the notation (see samples) are open source and available from the Plainsound Music Edition website.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> For example, the Pythagorean major third on C is C-E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-natural">♮</span></span> while the just major third is C-E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-natural">♮</span></span>↓ (see <a href="#fig4">Fig. 4</a> for "combined" symbol) </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:HEJI_Johnston.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/HEJI_Johnston.png/600px-HEJI_Johnston.png" decoding="async" width="600" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/HEJI_Johnston.png/900px-HEJI_Johnston.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/HEJI_Johnston.png/1200px-HEJI_Johnston.png 2x" data-file-width="1891" data-file-height="576" /></a><figcaption><span class="anchor" id="fig4"></span>Fig. 4: Comparison of Helmholtz-Ellis JI Pitch Notation and Johnston Notation. Unaltered naturals in Helmholtz-Ellis may be omitted if desired.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Just_harmonic_thirteenth_chord_on_G_Sagittal_notation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Just_harmonic_thirteenth_chord_on_G_Sagittal_notation.png/250px-Just_harmonic_thirteenth_chord_on_G_Sagittal_notation.png" decoding="async" width="250" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Just_harmonic_thirteenth_chord_on_G_Sagittal_notation.png/375px-Just_harmonic_thirteenth_chord_on_G_Sagittal_notation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/9/9d/Just_harmonic_thirteenth_chord_on_G_Sagittal_notation.png 2x" data-file-width="480" data-file-height="300" /></a><figcaption><span class="anchor" id="fig5"></span>Fig. 5: Just harmonic thirteenth chord (4:5:6:7:9:11:13) on G in Sagittal notation (with mnemonics)</figcaption></figure> <p>Sagittal notation (from Latin <i>sagitta</i>, "arrow") is a system of arrow-like accidentals that indicate prime-number comma alterations to tones in a Pythagorean series. It is used to notate both just intonation and equal temperaments. The size of the symbol indicates the size of the alteration.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>The great advantage of such notation systems is that they allow the natural harmonic series to be precisely notated. At the same time, they provide some degree of practicality through their extension of staff notation, as traditionally trained performers may draw on their intuition for roughly estimating pitch height. This may be contrasted with the more abstract use of ratios for representing pitches in which the amount by which two pitches differ and the "direction" of change may not be immediately obvious to most musicians. One caveat is the requirement for performers to learn and internalize a (large) number of new graphical symbols. However, the use of unique symbols reduces harmonic ambiguity and the potential confusion arising from indicating only cent deviations. </p> <div class="mw-heading mw-heading2"><h2 id="Audio_examples">Audio examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Just_intonation&action=edit&section=13" title="Edit section: Audio examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-3" class="noexcerpt ext-phonos-PhonosButton 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\/d2\/A_Major_Scale%2C_Triads%2C_and_Fifths_Just.ogg\/A_Major_Scale%2C_Triads%2C_and_Fifths_Just.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Just intonation"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"A Major Scale, Triads, and Fifths Just.ogg"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d2/A_Major_Scale%2C_Triads%2C_and_Fifths_Just.ogg/A_Major_Scale%2C_Triads%2C_and_Fifths_Just.ogg.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Just intonation</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:A_Major_Scale,_Triads,_and_Fifths_Just.ogg" title="File:A Major Scale, Triads, and Fifths Just.ogg">ⓘ</a></sup></span> An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.</li> <li><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-4" class="noexcerpt ext-phonos-PhonosButton 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\/A_Major_Scale%2C_Triads%2C_and_Fifths_Equal.ogg\/A_Major_Scale%2C_Triads%2C_and_Fifths_Equal.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Equal temperament"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"A Major Scale, Triads, and Fifths Equal.ogg"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/A_Major_Scale%2C_Triads%2C_and_Fifths_Equal.ogg/A_Major_Scale%2C_Triads%2C_and_Fifths_Equal.ogg.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Equal temperament</span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:A_Major_Scale,_Triads,_and_Fifths_Equal.ogg" title="File:A Major Scale, Triads, and Fifths Equal.ogg">ⓘ</a></sup></span> An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. The <a href="/wiki/Beat_(acoustics)" title="Beat (acoustics)">beating</a> in this file may be more noticeable after listening to the above file.</li> <li><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-5" class="noexcerpt ext-phonos-PhonosButton 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\/en\/transcoded\/9\/99\/Just_vs_equal.ogg\/Just_vs_equal.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Equal temperament and just intonation compared"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just vs equal.ogg"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/en/transcoded/9/99/Just_vs_equal.ogg/Just_vs_equal.ogg.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Equal temperament and just intonation compared</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_vs_equal.ogg" title="File:Just vs equal.ogg">ⓘ</a></sup></span> A pair of major thirds, followed by a pair of full major chords. The first in each pair is in equal temperament; the second is in just intonation. Piano sound.</li> <li><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-6" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/9\/90\/A-major-triad-equal-temperament-compared-to-just-intonation-6-2008C.ogg\/A-major-triad-equal-temperament-compared-to-just-intonation-6-2008C.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Equal temperament and just intonation compared with square waveform"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"A-major-triad-equal-temperament-compared-to-just-intonation-6-2008C.ogg"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/90/A-major-triad-equal-temperament-compared-to-just-intonation-6-2008C.ogg/A-major-triad-equal-temperament-compared-to-just-intonation-6-2008C.ogg.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Equal temperament and just intonation compared with square waveform</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:A-major-triad-equal-temperament-compared-to-just-intonation-6-2008C.ogg" title="File:A-major-triad-equal-temperament-compared-to-just-intonation-6-2008C.ogg">ⓘ</a></sup></span> A pair of major chords. The first is in equal temperament; the second is in just intonation. The pair of chords is repeated with a transition from equal temperament to just intonation between the two chords. In the equal temperament chords a roughness or <a href="/wiki/Beat_(acoustics)" title="Beat (acoustics)">beating</a> can be heard at about 4 <a href="/wiki/Hertz" title="Hertz">Hz</a> and about 0.8 Hz. In the just intonation triad, this roughness is absent. The <a href="/wiki/Square_wave" title="Square wave">square waveform</a> makes the difference between equal temperament and just intonation more obvious.</li></ul> <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=Just_intonation&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Lists</dt> <dd></dd></dl> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 16em;"> <ul><li><a href="/wiki/List_of_compositions_in_just_intonation" title="List of compositions in just intonation">List of compositions in just intonation</a></li> <li><a href="/wiki/List_of_intervals_in_5-limit_just_intonation" title="List of intervals in 5-limit just intonation">List of intervals in 5-limit just intonation</a></li> <li><a href="/wiki/List_of_meantone_intervals" title="List of meantone intervals">List of meantone intervals</a></li> <li><a href="/wiki/List_of_pitch_intervals" title="List of pitch intervals">List of pitch intervals</a></li></ul> </div> <dl><dt>Article topics</dt> <dd></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 12em;"> <ul><li><a href="/wiki/Dynamic_tonality" title="Dynamic tonality">Dynamic tonality</a></li> <li><a href="/wiki/Electronic_tuner" title="Electronic tuner">Electronic tuner</a></li> <li><a href="/wiki/Hexany" title="Hexany">Hexany</a></li> <li><a href="/wiki/Microtonal_music" class="mw-redirect" title="Microtonal music">Microtonal music</a></li> <li><a href="/wiki/Microtuner" title="Microtuner">Microtuner</a></li> <li><a href="/wiki/Music_and_mathematics" title="Music and mathematics">Music and mathematics</a></li> <li><a href="/wiki/Interval_(music)" title="Interval (music)">Musical interval</a></li> <li><a href="/wiki/Pythagorean_interval" title="Pythagorean interval">Pythagorean interval</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular number</a></li> <li><a href="/wiki/Superparticular_ratio" title="Superparticular ratio">Superparticular ratio</a></li> <li><a href="/wiki/Whole-tone_scale" title="Whole-tone scale">Whole-tone scale</a></li></ul> </div> <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=Just_intonation&action=edit&section=15" 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 reflist-columns references-column-width" style="column-width: 25em;"> <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="CITEREFWest1994" class="citation journal cs1"><a href="/wiki/Martin_Litchfield_West" title="Martin Litchfield West">West, M. L.</a> (May 1994). "The Babylonian musical notation and the Hurrian melodic texts". <i><a href="/wiki/Music_%26_Letters" title="Music & Letters">Music & Letters</a></i>. <b>75</b> (2): <span class="nowrap">161–</span>179. <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%2Fml%2F75.2.161">10.1093/ml/75.2.161</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/737674">737674</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Music+%26+Letters&rft.atitle=The+Babylonian+musical+notation+and+the+Hurrian+melodic+texts&rft.volume=75&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E161-%3C%2Fspan%3E179&rft.date=1994-05&rft_id=info%3Adoi%2F10.1093%2Fml%2F75.2.161&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F737674%23id-name%3DJSTOR&rft.aulast=West&rft.aufirst=M.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><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="CITEREFBarker1989" class="citation book cs1"><a href="/wiki/Andrew_Barker_(classicist)" title="Andrew Barker (classicist)">Barker, Andrew</a> (1989). <i>Greek musical writings</i>. Cambridge: Cambridge University Press. p. 350. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-23593-6" title="Special:BookSources/0-521-23593-6"><bdi>0-521-23593-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/10022960">10022960</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Greek+musical+writings&rft.place=Cambridge&rft.pages=350&rft.pub=Cambridge+University+Press&rft.date=1989&rft_id=info%3Aoclcnum%2F10022960&rft.isbn=0-521-23593-6&rft.aulast=Barker&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.silkqin.com/08anal/tunings.htm">"Qin tunings, some theoretical concepts"</a>. <i>silkqin.com</i>. Table 2: Relative positions of studs on the <i>qin</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=silkqin.com&rft.atitle=Qin+tunings%2C+some+theoretical+concepts&rft.pages=Table-2%3A+Relative+positions+of+studs+on+the+%27%27qin%27%27&rft_id=http%3A%2F%2Fwww.silkqin.com%2F08anal%2Ftunings.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-C&G-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-C&G_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-C&G_4-1"><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="CITEREFCampbellGreated2001" class="citation book cs1">Campbell, Murray; Greated, Clive (2001) [1987]. <i>The Musician's Guide to Acoustics</i> (reprint of 1st ed.). London, UK & New York, NY: Oxford University Press. pp. <span class="nowrap">172–</span>173. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-816505-7" title="Special:BookSources/978-0-19-816505-7"><bdi>978-0-19-816505-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Musician%27s+Guide+to+Acoustics&rft.place=London%2C+UK+%26+New+York%2C+NY&rft.pages=%3Cspan+class%3D%22nowrap%22%3E172-%3C%2Fspan%3E173&rft.edition=reprint+of+1st&rft.pub=Oxford+University+Press&rft.date=2001&rft.isbn=978-0-19-816505-7&rft.aulast=Campbell&rft.aufirst=Murray&rft.au=Greated%2C+Clive&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWright2009" class="citation book cs1">Wright, David (2009). <i>Mathematics and Music</i>. Mathematical World. Vol. 28. Providence, RI: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. pp. <span class="nowrap">140–</span>141. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4873-9" title="Special:BookSources/978-0-8218-4873-9"><bdi>978-0-8218-4873-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Music&rft.place=Providence%2C+RI&rft.series=Mathematical+World&rft.pages=%3Cspan+class%3D%22nowrap%22%3E140-%3C%2Fspan%3E141&rft.pub=American+Mathematical+Society&rft.date=2009&rft.isbn=978-0-8218-4873-9&rft.aulast=Wright&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-Johnson2006-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Johnson2006_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Johnson2006_6-1"><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="CITEREFJohnston2006" class="citation book cs1"><a href="/wiki/Ben_Johnston_(composer)" title="Ben Johnston (composer)">Johnston, Ben</a> (2006) [2003]. "A notation system for extended just intonation". In <a href="/wiki/Bob_Gilmore" title="Bob Gilmore">Gilmore, Bob</a> (ed.). <i><span></span></i>Maximum Clarity<i> and Other Writings on Music</i>. Urbana and Chicago, IL: University of Illinois Press. pp. <span class="nowrap">77–</span>88. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-252-03098-7" title="Special:BookSources/978-0-252-03098-7"><bdi>978-0-252-03098-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+notation+system+for+extended+just+intonation&rft.btitle=Maximum+Clarity+and+Other+Writings+on+Music&rft.place=Urbana+and+Chicago%2C+IL&rft.pages=%3Cspan+class%3D%22nowrap%22%3E77-%3C%2Fspan%3E88&rft.pub=University+of+Illinois+Press&rft.date=2006&rft.isbn=978-0-252-03098-7&rft.aulast=Johnston&rft.aufirst=Ben&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPartch1979" class="citation book cs1"><a href="/wiki/Harry_Partch" title="Harry Partch">Partch, Harry</a> (1979). <a href="/wiki/Genesis_of_a_Music" title="Genesis of a Music"><i>Genesis of a Music</i></a>. pp. 165, 73. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-306-80106-8" title="Special:BookSources/978-0-306-80106-8"><bdi>978-0-306-80106-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Genesis+of+a+Music&rft.pages=165%2C+73&rft.date=1979&rft.isbn=978-0-306-80106-8&rft.aulast=Partch&rft.aufirst=Harry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchenker2001" class="citation book cs1"><a href="/wiki/Heinrich_Schenker" title="Heinrich Schenker">Schenker, Heinrich</a> (2001) [1935]. <a href="/wiki/Free_Composition" title="Free Composition"><i>Free Composition</i></a>. Pendragon Press. p. 8, footnote 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781576470749" title="Special:BookSources/9781576470749"><bdi>9781576470749</bdi></a>. <q>Bruckner used to teach that even the sixth tone of the diatonic scale was dissonant and therefore had to be resolved downward.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Free+Composition&rft.pages=p.-8%2C+footnote-14&rft.pub=Pendragon+Press&rft.date=2001&rft.isbn=9781576470749&rft.aulast=Schenker&rft.aufirst=Heinrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-autogenerated1968-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-autogenerated1968_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-autogenerated1968_9-1"><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="CITEREFDaniélou1968" class="citation book cs1"><a href="/wiki/Alain_Dani%C3%A9lou" title="Alain Daniélou">Daniélou, Alain</a> (1968). <i>The Ragas of Northern Indian Music</i>. London: Barrie & Rockliff. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-214-15689-3" title="Special:BookSources/0-214-15689-3"><bdi>0-214-15689-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Ragas+of+Northern+Indian+Music&rft.place=London&rft.pub=Barrie+%26+Rockliff&rft.date=1968&rft.isbn=0-214-15689-3&rft.aulast=Dani%C3%A9lou&rft.aufirst=Alain&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" 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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBagchee1998" class="citation book cs1">Bagchee, Sandeep (1998). <i>Nad: Understanding Raga music</i>. India: BPI PVT Ltd. p. 23. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/81-86982-07-8" title="Special:BookSources/81-86982-07-8"><bdi>81-86982-07-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nad%3A+Understanding+Raga+music&rft.place=India&rft.pages=23&rft.pub=BPI+PVT+Ltd&rft.date=1998&rft.isbn=81-86982-07-8&rft.aulast=Bagchee&rft.aufirst=Sandeep&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-autogenerated1995-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-autogenerated1995_11-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaniélou1995" class="citation book cs1"><a href="/wiki/Alain_Dani%C3%A9lou" title="Alain Daniélou">Daniélou, Alain</a> (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/musicpowerofsoun0028dani"><i>Music and the Power of Sound: The influence of tuning and interval on consciousness</i></a></span> (Rep Sub ed.). Inner Traditions. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89281-336-9" title="Special:BookSources/0-89281-336-9"><bdi>0-89281-336-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Music+and+the+Power+of+Sound%3A+The+influence+of+tuning+and+interval+on+consciousness&rft.edition=Rep+Sub&rft.pub=Inner+Traditions&rft.date=1995&rft.isbn=0-89281-336-9&rft.aulast=Dani%C3%A9lou&rft.aufirst=Alain&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmusicpowerofsoun0028dani&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-autogenerated1999-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-autogenerated1999_12-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaniélou1999" class="citation book cs1"><a href="/wiki/Alain_Dani%C3%A9lou" title="Alain Daniélou">Daniélou, Alain</a> (1999). <i>Introduction to the Study of Musical Scales</i>. Oriental Book Reprint Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/81-7069-098-6" title="Special:BookSources/81-7069-098-6"><bdi>81-7069-098-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Study+of+Musical+Scales&rft.pub=Oriental+Book+Reprint+Corporation&rft.date=1999&rft.isbn=81-7069-098-6&rft.aulast=Dani%C3%A9lou&rft.aufirst=Alain&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-shruti-webst-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-shruti-webst_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-shruti-webst_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-shruti-webst_13-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-shruti-webst_13-3"><sup><i><b>d</b></i></sup></a></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://22shruti.com/">"22 shruti"</a>. <i>22shruti.com</i> (main page)<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-06-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=22shruti.com&rft.atitle=22+shruti&rft_id=https%3A%2F%2F22shruti.com%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStangeWickHinrichsen2018" class="citation journal cs1">Stange, Karolin; Wick, Christoph; Hinrichsen, Haye (October 2018) [June 2017]. <a rel="nofollow" class="external text" href="https://www.semanticscholar.org/paper/Playing-Music-in-Just-Intonation%3A-A-Dynamically-Stange-Wick/620b3a07a2d2f99b2286c41aaff076e6dd9074c1">"Playing music in just intonation – a dynamically-adapting tuning scheme"</a>. <i><a href="/wiki/Computer_Music_Journal" title="Computer Music Journal">Computer Music Journal</a></i>. <b>42</b> (3): <span class="nowrap">47–</span>62. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2Fcomj_a_00478">10.1162/comj_a_00478</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=Playing+music+in+just+intonation+%E2%80%93+a+dynamically-adapting+tuning+scheme&rft.volume=42&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E47-%3C%2Fspan%3E62&rft.date=2018-10&rft_id=info%3Adoi%2F10.1162%2Fcomj_a_00478&rft.aulast=Stange&rft.aufirst=Karolin&rft.au=Wick%2C+Christoph&rft.au=Hinrichsen%2C+Haye&rft_id=https%3A%2F%2Fwww.semanticscholar.org%2Fpaper%2FPlaying-Music-in-Just-Intonation%253A-A-Dynamically-Stange-Wick%2F620b3a07a2d2f99b2286c41aaff076e6dd9074c1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-:1-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-:1_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPartch1974" class="citation book cs1"><a href="/wiki/Harry_Partch" title="Harry Partch">Partch, Harry</a> (1974). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/genesisofmusicac00partc"><i>Genesis of a music : an account of a creative work, its roots and its fulfillments</i></a></span> (second, enlarged ed.). New York. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-306-71597-X" title="Special:BookSources/0-306-71597-X"><bdi>0-306-71597-X</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/624666">624666</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Genesis+of+a+music+%3A+an+account+of+a+creative+work%2C+its+roots+and+its+fulfillments&rft.place=New+York&rft.edition=second%2C+enlarged&rft.date=1974&rft_id=info%3Aoclcnum%2F624666&rft.isbn=0-306-71597-X&rft.aulast=Partch&rft.aufirst=Harry&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgenesisofmusicac00partc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></span> </li> <li id="cite_note-Helmholtz1885-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Helmholtz1885_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHelmholtz1885" class="citation book cs1"><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz, Hermann von</a> (1885). <i><a href="/wiki/On_the_Sensations_of_Tone_as_a_Physiological_Basis_for_the_Theory_of_Music" class="mw-redirect" title="On the Sensations of Tone as a Physiological Basis for the Theory of Music">On the Sensations of Tone as a Physiological Basis for the Theory of Music</a></i>. Longmans, Green. p. <a rel="nofollow" class="external text" href="https://archive.org/details/onsensationston00unkngoog/page/n301">276</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+the+Sensations+of+Tone+as+a+Physiological+Basis+for+the+Theory+of+Music&rft.pages=276&rft.pub=Longmans%2C+Green&rft.date=1885&rft.aulast=Helmholtz&rft.aufirst=Hermann+von&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span> Note the use of the "+" between just major thirds, "−" between just minor thirds, "|" between Pythagorean minor thirds, and "±" between perfect fifths.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenson,_David_J.2007" class="citation book cs1">Benson, David J. (2007). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/musicmathematica00bens_733"><i>Music: A Mathematical Offering</i></a></span>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/musicmathematica00bens_733/page/n185">172</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-85387-3" title="Special:BookSources/978-0-521-85387-3"><bdi>978-0-521-85387-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Music%3A+A+Mathematical+Offering&rft.pages=172&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-85387-3&rft.au=Benson%2C+David+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmusicmathematica00bens_733&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span><br />who cites <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEitz,_Carl_A.1891" class="citation book cs1"><a href="/wiki/Carl_Eitz" title="Carl Eitz">Eitz, Carl A.</a> (1891). <i>Das mathematisch-reine Tonsystem</i>. Leipzig.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Das+mathematisch-reine+Tonsystem&rft.place=Leipzig&rft.date=1891&rft.au=Eitz%2C+Carl+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></span> </li> <li id="cite_note-MonzoTonal-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-MonzoTonal_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MonzoTonal_18-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMonzo" class="citation encyclopaedia cs1">Monzo. "Helmholtz / Ellis / Wolf / Monzo system". <a rel="nofollow" class="external text" href="http://tonalsoft.com/enc/h/hewm.aspx"><i>Tonalsoft Encyclopaedia</i></a>. tonalsoft.com.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Helmholtz+%2F+Ellis+%2F+Wolf+%2F+Monzo+system&rft.btitle=Tonalsoft+Encyclopaedia&rft.pub=tonalsoft.com&rft.au=Monzo&rft_id=http%3A%2F%2Ftonalsoft.com%2Fenc%2Fh%2Fhewm.aspx&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarland1984" class="citation book cs1"><a href="/wiki/Peter_Garland_(composer)" title="Peter Garland (composer)">Garland, Peter</a>, ed. (1984). <i>The Music of James Tenney</i>. Soundings. Vol. 13. Santa Fe, New Mexico: Soundings Press. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/11371167">11371167</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Music+of+James+Tenney&rft.place=Santa+Fe%2C+New+Mexico&rft.series=Soundings&rft.pub=Soundings+Press&rft.date=1984&rft_id=info%3Aoclcnum%2F11371167&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Wannamaker, Robert, <i><a rel="nofollow" class="external text" href="https://www.press.uillinois.edu/books/?id=c043673">The Music of James Tenney, Volume 1: Contexts and Paradigms</a></i> (University of Illinois Press, 2021), 288-89.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKyle_Gann" class="citation web cs1"><a href="/wiki/Kyle_Gann" title="Kyle Gann">Kyle Gann</a>. <a rel="nofollow" class="external text" href="http://www.kylegann.com/tuning.html">"Just Intonation Explained"</a>. <i>KyleGann.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">28 February</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=KyleGann.com&rft.atitle=Just+Intonation+Explained&rft.au=Kyle+Gann&rft_id=http%3A%2F%2Fwww.kylegann.com%2Ftuning.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFonville1991" class="citation journal cs1"><a href="/wiki/John_Fonville" title="John Fonville">Fonville, John</a> (Summer 1991). "Ben Johnston's Extended Just Intonation: A guide for interpreters". <i><a href="/wiki/Perspectives_of_New_Music" title="Perspectives of New Music">Perspectives of New Music</a></i>. <b>29</b> (2): 121, <span class="nowrap">106–</span>137. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F833435">10.2307/833435</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/833435">833435</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Perspectives+of+New+Music&rft.atitle=Ben+Johnston%27s+Extended+Just+Intonation%3A+A+guide+for+interpreters&rft.ssn=summer&rft.volume=29&rft.issue=2&rft.pages=121%2C+%3Cspan+class%3D%22nowrap%22%3E106-%3C%2Fspan%3E137&rft.date=1991&rft_id=info%3Adoi%2F10.2307%2F833435&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F833435%23id-name%3DJSTOR&rft.aulast=Fonville&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStahnke,_Manfred2005" class="citation book cs1"><a href="/wiki/Manfred_Stahnke" title="Manfred Stahnke">Stahnke, Manfred</a>, ed. (2005). "The Extended Helmholtz-Ellis JI Pitch Notation: eine Notationsmethode für die natürlichen Intervalle". <i>Mikrotöne und Mehr – Auf György Ligetis Hamburger Pfaden</i>. Hamburg: von Bockel Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-932696-62-X" title="Special:BookSources/3-932696-62-X"><bdi>3-932696-62-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Extended+Helmholtz-Ellis+JI+Pitch+Notation%3A+eine+Notationsmethode+f%C3%BCr+die+nat%C3%BCrlichen+Intervalle&rft.btitle=Mikrot%C3%B6ne+und+Mehr+%E2%80%93+Auf+Gy%C3%B6rgy+Ligetis+Hamburger+Pfaden&rft.place=Hamburg&rft.pub=von+Bockel+Verlag&rft.date=2005&rft.isbn=3-932696-62-X&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSabat" class="citation web cs1">Sabat, Marc. <a rel="nofollow" class="external text" href="http://www.marcsabat.com/pdfs/notation.pdf">"The Extended Helmholtz Ellis JI Pitch Notation"</a> <span class="cs1-format">(PDF)</span>. Plainsound Music Edition<span class="reference-accessdate">. Retrieved <span class="nowrap">11 March</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Extended+Helmholtz+Ellis+JI+Pitch+Notation&rft.pub=Plainsound+Music+Edition&rft.aulast=Sabat&rft.aufirst=Marc&rft_id=http%3A%2F%2Fwww.marcsabat.com%2Fpdfs%2Fnotation.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSecor,_George_D.Keenan,_David_C.2006" class="citation magazine cs1"><a href="/wiki/George_Secor" title="George Secor">Secor, George D.</a>; Keenan, David C. (2006). <a rel="nofollow" class="external text" href="http://sagittal.org/sagittal.pdf">"Sagittal: A Microtonal Notation System"</a> <span class="cs1-format">(PDF)</span>. <i>Xenharmonikôn: An Informal Journal of Experimental Music</i>. Vol. 18. pp. <span class="nowrap">1–</span>2 – via Sagittal.org.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Xenharmonik%C3%B4n%3A+An+Informal+Journal+of+Experimental+Music&rft.atitle=Sagittal%3A+A+Microtonal+Notation+System&rft.volume=18&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E2&rft.date=2006&rft.au=Secor%2C+George+D.&rft.au=Keenan%2C+David+C.&rft_id=http%3A%2F%2Fsagittal.org%2Fsagittal.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJust+intonation" class="Z3988"></span></span> </li> </ol></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=Just_intonation&action=edit&section=16" title="Edit section: External links"><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:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><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"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist"><a href="https://en.wikiversity.org/wiki/Sing_free/Prelude_and_Fugue_in_C_major_(ear_training)" class="extiw" title="v:Sing free/Prelude and Fugue in C major (ear training)">Wikiversity discusses a simplified just scale.</a></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110514002750/http://artofthestates.org/cgi-bin/genresearch.pl?genre=microtonal%2Fjust%20intonation">Art of the States: microtonal/just intonation</a> works using just intonation by American composers</li> <li><a rel="nofollow" class="external text" href="http://www.chrysalis-foundation.org/just_intonation.htm">The Chrysalis Foundation – Just Intonation: Two Definitions</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100630223629/http://users.rcn.com/dante.interport/justguitar.html">Dante Rosati's 21 Tone Just Intonation guitar</a></li> <li><a rel="nofollow" class="external text" href="http://alum.mit.edu/www/nowitzky/justint/">Just Intonation</a> by <a rel="nofollow" class="external text" href="http://alum.mit.edu/www/nowitzky/">Mark Nowitzky</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=d2I1zNw2w-c">Just intonation</a> compared with <a href="/wiki/Meantone_temperament" title="Meantone temperament">meantone</a> and <a href="/wiki/Equal_temperament" title="Equal temperament">12-equal</a> temperaments; a video featuring Pachelbel's canon.</li> <li><a rel="nofollow" class="external text" href="http://www.kylegann.com/tuning.html">Just Intonation Explained</a> by <a href="/wiki/Kyle_Gann" title="Kyle Gann">Kyle Gann</a></li> <li><a rel="nofollow" class="external text" href="http://www.ubu.com/sound/tellus_14.html">A selection of Just Intonation works edited by the Just Intonation Network</a> web published on the <a href="/wiki/Tellus_Audio_Cassette_Magazine" title="Tellus Audio Cassette Magazine">Tellus Audio Cassette Magazine</a> project archive at <a href="/wiki/UbuWeb" title="UbuWeb">UbuWeb</a></li> <li><a rel="nofollow" class="external text" href="http://www.medieval.org/emfaq/">Medieval Music and Arts Foundation</a></li> <li><a rel="nofollow" class="external text" href="http://www.MusicNovatory.com/justintonation.html">Music Novatory – Just Intonation</a></li> <li><a rel="nofollow" class="external text" href="http://www.patmissin.com/tunings/tun0.html">Why does Just Intonation sound so good?</a></li> <li><a rel="nofollow" class="external text" href="http://www.anaphoria.com/wilson.html">The Wilson Archives</a></li> <li>Barbieri, Patrizio. <a rel="nofollow" class="external text" href="http://www.patriziobarbieri.it/1.htm">Enharmonic instruments and music, 1470–1900</a>. (2008) Latina, Il Levante</li> <li><a rel="nofollow" class="external text" href="http://www.mediafire.com/download.php?ljr44lwzoyj">22 Note Just Intonation Keyboard Software with 12 Indian Instrument Sounds</a> Libreria Editrice</li> <li><a rel="nofollow" class="external text" href="http://www.plainsound.org">Plainsound Music Edition</a> – JI music and research, information about the Helmholtz-Ellis JI Pitch Notation</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol 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class">Pitch class</a></li> <li><a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">Consonance and dissonance</a></li> <li><a href="/wiki/List_of_pitch_intervals" title="List of pitch intervals">List of musical intervals</a></li> <li><a href="/wiki/List_of_intervals_in_5-limit_just_intonation" title="List of intervals in 5-limit just intonation">List of intervals in 5-limit just intonation</a></li> <li><a href="/wiki/List_of_meantone_intervals" title="List of meantone intervals">List of meantone intervals</a></li> <li><a href="/wiki/Microtonal_music" class="mw-redirect" title="Microtonal music">Microtone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Just intonation</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/Euler%E2%80%93Fokker_genus" title="Euler–Fokker genus">Euler–Fokker genus</a></li> <li><a href="/wiki/Harmonic_scale" title="Harmonic scale">Harmonic scale</a></li> <li><a href="/wiki/Harry_Partch%27s_43-tone_scale" title="Harry Partch's 43-tone scale">Harry Partch's 43-tone scale</a></li> <li><a href="/wiki/Hexany" title="Hexany">Hexany</a></li> <li><a href="/wiki/Limit_(music)" title="Limit (music)">Limit</a> <ul><li><a href="/wiki/Five-limit_tuning" title="Five-limit tuning">5-limit</a></li> <li><a href="/wiki/7-limit_tuning" title="7-limit tuning">7-limit</a></li></ul></li> <li><a href="/wiki/List_of_compositions_in_just_intonation" title="List of compositions in just intonation">List of compositions</a></li> <li><a href="/wiki/Otonality_and_Utonality" class="mw-redirect" title="Otonality and Utonality">Otonality</a></li> <li><a href="/wiki/Ptolemy%27s_intense_diatonic_scale" title="Ptolemy's intense diatonic scale">Ptolemy's intense diatonic scale</a></li> <li><a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a></li> <li><a href="/wiki/Scale_of_harmonics" title="Scale of harmonics">Scale of harmonics</a></li> <li><a href="/wiki/Tonality_diamond" title="Tonality diamond">Tonality diamond</a></li> <li><a href="/wiki/Tonality_flux" title="Tonality flux">Tonality flux</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Musical_temperament" title="Musical temperament">Temperaments</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Equal_temperament" title="Equal temperament">Equal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Whole-tone_scale" title="Whole-tone scale">6-tone</a></li> <li><a href="/wiki/12_equal_temperament" title="12 equal temperament">12-tone</a></li> <li><a href="/wiki/15_equal_temperament" title="15 equal temperament">15-tone</a></li> <li><a href="/wiki/17_equal_temperament" title="17 equal temperament">17-tone</a></li> <li><a href="/wiki/19_equal_temperament" title="19 equal temperament">19-tone</a></li> <li><a href="/wiki/22_equal_temperament" title="22 equal temperament">22-tone</a></li> <li><a href="/wiki/23_equal_temperament" title="23 equal temperament">23-tone</a></li> <li><a href="/wiki/Quarter_tone" title="Quarter tone">24-tone</a> (<a href="/wiki/List_of_quarter_tone_pieces" title="List of quarter tone pieces">pieces</a>)</li> <li><a href="/wiki/31_equal_temperament" title="31 equal temperament">31-tone</a></li> <li><a href="/wiki/34_equal_temperament" title="34 equal temperament">34-tone</a></li> <li><a href="/wiki/41_equal_temperament" title="41 equal temperament">41-tone</a></li> <li><a href="/wiki/53_equal_temperament" title="53 equal temperament">53-tone</a></li> <li><a href="/wiki/58_equal_temperament" title="58 equal temperament">58-tone</a></li> <li><a href="/wiki/72_equal_temperament" title="72 equal temperament">72-tone</a></li> <li><a href="/wiki/96_equal_temperament" title="96 equal temperament">96-tone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Regular_temperament" title="Regular temperament">Linear</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Meantone_temperament" title="Meantone temperament">Meantone</a> (<a href="/wiki/Quarter-comma_meantone" title="Quarter-comma meantone">quarter-comma</a>, <a href="/wiki/Septimal_meantone_temperament" title="Septimal meantone temperament">septimal</a>)</li> <li><a href="/wiki/Schismatic_temperament" title="Schismatic temperament">Schismatic</a></li> <li><a href="/wiki/Miracle_temperament" class="mw-redirect" title="Miracle temperament">Miracle</a></li> <li><a href="/wiki/Magic_temperament" title="Magic temperament">Magic</a></li> <li><a href="/wiki/Regular_diatonic_tuning" title="Regular diatonic tuning">Regular diatonic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Irregular</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Well_temperament" title="Well temperament">Well temperament</a>/<a href="/wiki/Temperament_ordinaire" title="Temperament ordinaire">Temperament ordinaire</a> (<a href="/wiki/Kirnberger_temperament" title="Kirnberger temperament">Kirnberger</a>, <a href="/wiki/Vallotti_temperament" title="Vallotti temperament">Vallotti</a>, <a href="/wiki/Werckmeister_temperament" title="Werckmeister temperament">Werckmeister</a>, <a href="/wiki/Young_temperament" title="Young temperament">Young</a>)</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Traditional<br /> non-Western</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/Chinese_musicology" title="Chinese musicology">Chinese musicology</a></li> <li><a href="/wiki/Sh%C3%AD-%C3%A8r-l%C7%9C" class="mw-redirect" title="Shí-èr-lǜ">Shí-èr-lǜ</a></li> <li><a href="/wiki/Dastgah" class="mw-redirect" title="Dastgah">Dastgah</a></li> <li>Maqam <ul><li><a href="/wiki/Arabic_maqam" title="Arabic maqam">Arabic maqam</a></li> <li><a href="/wiki/Turkish_makam" title="Turkish makam">Turkish makam</a></li> <li><a href="/wiki/Mugham" title="Mugham">Mugham</a></li> <li><a href="/wiki/Muqam" title="Muqam">Muqam</a></li></ul></li> <li><a href="/wiki/Octoechos" title="Octoechos">Octoechos</a></li> <li><a href="/wiki/Pelog" title="Pelog">Pelog</a></li> <li><a href="/wiki/Raga" title="Raga">Raga</a> (<a href="/wiki/Carnatic_raga" title="Carnatic raga">Carnatic raga</a>)</li> <li><a href="/wiki/Slendro" title="Slendro">Slendro</a></li> <li><a href="/wiki/Tetrachord" title="Tetrachord">Tetrachord</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Non-octave</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/833_cents_scale" title="833 cents scale">833 cents scale</a></li> <li><a href="/wiki/A12_scale" title="A12 scale">A12 scale</a></li> <li><a href="/wiki/Alpha_scale" title="Alpha scale">Alpha scale</a></li> <li><a href="/wiki/Beta_scale" title="Beta scale">Beta scale</a></li> <li><a href="/wiki/Gamma_scale" title="Gamma scale">Gamma scale</a></li> <li><a href="/wiki/Delta_scale" title="Delta scale">Delta scale</a></li> <li><a href="/wiki/Bohlen%E2%80%93Pierce_scale" title="Bohlen–Pierce scale">Lambda scale</a> (Bohlen–Pierce)</li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Instrument_intonation328" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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:Instrument_tunings" title="Template:Instrument tunings"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Instrument_tunings" title="Template talk:Instrument tunings"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Instrument_tunings" title="Special:EditPage/Template:Instrument tunings"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Instrument_intonation328" style="font-size:114%;margin:0 4em">Instrument <a href="/wiki/Intonation_(music)" title="Intonation (music)">intonation</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">Just intonation</a> in any key</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/String_instrument#Bowing" title="String instrument">Fretless string instrument</a></li> <li><a href="/wiki/Pedal_steel_guitar" title="Pedal steel guitar">Pedal steel guitar</a></li> <li><a href="/wiki/Trombone" title="Trombone">Trombone</a></li> <li><a href="/wiki/Timpani" title="Timpani">Timpani</a></li> <li><a href="/wiki/Human_voice" title="Human voice">Human voice</a></li> <li><a href="/wiki/Synthesizer" title="Synthesizer">Synthesizer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dynamic intonation adjustment</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/Crook_(music)" title="Crook (music)">Crook</a></li> <li><a href="/wiki/String_instrument#Bowing" title="String instrument">Fretted string instrument</a></li> <li><a href="/wiki/Wind_instrument" title="Wind instrument">Wind instruments</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Just intonation in one key</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/Bagpipes" title="Bagpipes">Bagpipes</a></li> <li><a href="/wiki/Natural_horn" title="Natural horn">Natural horn</a></li> <li><a href="/wiki/Tromba_marina" title="Tromba marina">Tromba marina</a></li> <li><a href="/wiki/Long-string_instrument" title="Long-string instrument">Long-string instrument</a></li> <li><a href="/wiki/Harmonica" title="Harmonica">Harmonica</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Retunable to a just key</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/Keyboard_instrument" title="Keyboard instrument">Keyboard instruments</a> <ul><li><a href="/wiki/Split_sharp" title="Split sharp">Split sharp</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Harmonic" title="Harmonic">Flageolet tones</a> (harmonics) or<br />natural <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">overtone series</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/Guqin" title="Guqin">Guqin</a></li> <li><a href="/wiki/%C4%90%C3%A0n_b%E1%BA%A7u" title="Đàn bầu">Đàn bầu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical just-intoned string<br />part relation with additional <a href="/wiki/3rd_bridge" title="3rd bridge">3rd bridge</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/Monochord" title="Monochord">Monochord</a></li> <li><a href="/wiki/Bradford_Reed" title="Bradford Reed">Pencilina</a></li> <li><a href="/wiki/Moodswinger" title="Moodswinger">Moodswinger</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Tonality12" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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:Tonality" title="Template:Tonality"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tonality" title="Template talk:Tonality"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tonality" title="Special:EditPage/Template:Tonality"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tonality12" style="font-size:114%;margin:0 4em"><a href="/wiki/Tonality" title="Tonality">Tonality</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cadence" title="Cadence">Cadence</a></li> <li><a href="/wiki/Circle_of_fifths" title="Circle of fifths">Circle of fifths</a></li> <li><a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">Consonance and dissonance</a></li> <li><a href="/wiki/Diatonic_scale" title="Diatonic scale">Diatonic scale</a></li> <li><a href="/wiki/Function_(music)" title="Function (music)">Diatonic function</a> <ul><li><a href="/wiki/Secondary_chord" title="Secondary chord">Secondary function</a></li></ul></li> <li><a href="/wiki/Figured_bass" title="Figured bass">Figured bass</a></li> <li><a class="mw-selflink selflink">Just intonation</a></li> <li><a href="/wiki/Key_(music)" title="Key (music)">Key</a></li> <li><a href="/wiki/Major_and_minor" title="Major and minor">Major and minor</a></li> <li><a href="/wiki/Modulation_(music)" title="Modulation (music)">Modulation</a></li> <li><a href="/wiki/Neotonality" title="Neotonality">Neotonality</a></li> <li><a href="/wiki/Ostinato" title="Ostinato">Ostinato</a></li> <li><a href="/wiki/Otonality_and_utonality" title="Otonality and utonality">Otonality and utonality</a></li> <li><a href="/wiki/Parallel_key" title="Parallel key">Parallel key</a></li> <li><a href="/wiki/Polytonality" title="Polytonality">Polytonality</a></li> <li><a href="/wiki/Progressive_tonality" title="Progressive tonality">Progressive tonality</a></li> <li><a href="/wiki/Schenkerian_analysis" title="Schenkerian analysis">Schenkerian analysis</a></li> <li><a href="/wiki/Sonata_form" title="Sonata form">Sonata form</a></li> <li><a href="/wiki/Tonality_diamond" title="Tonality diamond">Tonality diamond</a></li> <li><a href="/wiki/Tonicization" title="Tonicization">Tonicization</a></li> <li><a href="/wiki/Voice_leading" title="Voice leading">Voice leading</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"></div><div role="navigation" class="navbox" aria-labelledby="Fractions_and_ratios54" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="4"><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:Fractions_and_ratios" title="Template:Fractions and ratios"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Fractions_and_ratios" title="Template talk:Fractions and ratios"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Fractions_and_ratios" title="Special:EditPage/Template:Fractions and ratios"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Fractions_and_ratios54" style="font-size:114%;margin:0 4em"><a href="/wiki/Fraction" title="Fraction">Fractions</a> and <a href="/wiki/Ratio" title="Ratio">ratios</a></div></th></tr><tr><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 2px 0 0;min-width: 60px"><div><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Unicode_0x0025.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/50px-Unicode_0x0025.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/75px-Unicode_0x0025.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Unicode_0x0025.svg/100px-Unicode_0x0025.svg.png 2x" data-file-width="16" data-file-height="16" /></a></span></div></td><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Division and ratio</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/Division_(mathematics)" title="Division (mathematics)">Dividend</a> ÷ <a href="/wiki/Divisor" title="Divisor">Divisor</a> = <a href="/wiki/Quotient" title="Quotient">Quotient</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div><span class="mw-default-size" typeof="mw:File"><span><img alt="The ratio of width to height of standard-definition television." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/66px-Aspect-ratio-4x3.svg.png" decoding="async" width="66" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/99px-Aspect-ratio-4x3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Aspect-ratio-4x3.svg/131px-Aspect-ratio-4x3.svg.png 2x" data-file-width="139" data-file-height="106" /></span></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Fraction</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><span style="font-size:120%"><span class="sfrac">⁠<span class="tion"><span class="num">Numerator</span><span class="sr-only">/</span><span class="den">Denominator</span></span>⁠</span></span> = Quotient</li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_fraction" title="Algebraic fraction">Algebraic</a></li> <li><a href="/wiki/Aspect_ratio" title="Aspect ratio">Aspect</a></li> <li><a href="/wiki/Binary_number" title="Binary number">Binary</a></li> <li><a href="/wiki/Continued_fraction" title="Continued fraction">Continued</a></li> <li><a href="/wiki/Decimal#Decimal_fractions" title="Decimal">Decimal</a></li> <li><a href="/wiki/Dyadic_rational" title="Dyadic rational">Dyadic</a></li> <li><a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden</a> <ul><li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver</a></li></ul></li> <li><a href="/wiki/Integer" title="Integer">Integer</a></li> <li><a href="/wiki/Irreducible_fraction" title="Irreducible fraction">Irreducible</a> <ul><li><a href="/wiki/Reduction_(mathematics)" title="Reduction (mathematics)">Reduction</a></li></ul></li> <li><a class="mw-selflink selflink">Just intonation</a></li> <li><a href="/wiki/Lowest_common_denominator" title="Lowest common denominator">LCD</a></li> <li><a href="/wiki/Interval_(music)" title="Interval (music)">Musical interval</a></li> <li><a href="/wiki/Paper_size" title="Paper size">Paper size</a></li> <li><a href="/wiki/Percentage" title="Percentage">Percentage</a></li> <li><a href="/wiki/Unit_fraction" title="Unit fraction">Unit</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox472" style="padding:3px"><table 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