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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Equal temperament system in music</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Chromatische_toonladder.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Chromatische_toonladder.png/300px-Chromatische_toonladder.png" decoding="async" width="300" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Chromatische_toonladder.png/450px-Chromatische_toonladder.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Chromatische_toonladder.png/600px-Chromatische_toonladder.png 2x" data-file-width="1424" data-file-height="214" /></a><figcaption>12-tone equal temperament chromatic scale on C, one full octave ascending, notated only with sharps. <span data-nosnippet="" id="ooui-php-1" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f0\/ChromaticScaleUpDown.ogg\/ChromaticScaleUpDown.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play ascending and descending"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"ChromaticScaleUpDown.ogg"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f0/ChromaticScaleUpDown.ogg/ChromaticScaleUpDown.ogg.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">Play ascending and descending</a><a href="/wiki/File:ChromaticScaleUpDown.ogg" title="File:ChromaticScaleUpDown.ogg">ⓘ</a></figcaption></figure> 12 equal temperament (12-ET)<a href="#cite_note-12TET-1">[a]</a> is the musical system that divides the <a href="/wiki/Octave" title="Octave">octave</a> into 12 parts, all of which are <a href="/wiki/Equal_temperament" title="Equal temperament">equally tempered</a> (equally spaced) on a <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">logarithmic scale</a>, with a ratio equal to the <a href="/wiki/Twelfth_root_of_two" title="Twelfth root of two">12th root of 2</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt[{12}]{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt[{12}]{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f276a910a28604e3b8afd8ba62a4678a37193480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.107ex; height:3.009ex;" alt="{\textstyle {\sqrt[{12}]{2}}}"> ≈ 1.05946). That resulting smallest interval, <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>1⁄12 the width of an octave, is called a <a href="/wiki/Semitone" title="Semitone">semitone</a> or half step. Twelve-tone equal temperament is the most widespread system in music today. It has been the predominant tuning system of Western music, starting with <a href="/wiki/Classical_music" title="Classical music">classical music</a>, since the 18th century, and Europe almost exclusively used approximations of it for millennia before that.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] It has also been used in other cultures. In modern times, 12-ET is usually tuned relative to a <a href="/wiki/Standard_pitch" class="mw-redirect" title="Standard pitch">standard pitch</a> of 440 Hz, called <a href="/wiki/A440_(pitch_standard)" title="A440 (pitch standard)">A440</a>, meaning one note, <a href="/wiki/A_(musical_note)" title="A (musical note)">A</a>, is tuned to 440 <a href="/wiki/Hertz" title="Hertz">hertz</a> and all other notes are defined as some multiple of semitones apart from it, either higher or lower in <a href="/wiki/Frequency" title="Frequency">frequency</a>. The standard pitch has not always been 440 Hz. It has varied and generally risen over the past few hundred years.<a href="#cite_note-FOOTNOTEvon_HelmholtzEllis1885493–511-2">[1]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=1" title="Edit section: History">edit</a>]</div> The two figures frequently credited with the achievement of exact calculation of twelve-tone equal temperament are <a href="/wiki/Zhu_Zaiyu" title="Zhu Zaiyu">Zhu Zaiyu</a> (also romanized as Chu-Tsaiyu. Chinese: 朱載堉) in 1584 and <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a> in 1585. According to Fritz A. Kuttner, a critic of the theory,<a href="#cite_note-FOOTNOTEKuttner1975163-3">[2]</a> it is known that "Chu-Tsaiyu presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that "Simon Stevin offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later." The developments occurred independently.<a href="#cite_note-FOOTNOTEKuttner1975200-4">[3]</a> Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu<a href="#cite_note-FOOTNOTERobinson1980vii-5">[4]</a> and provides textual quotations as evidence.<a href="#cite_note-FOOTNOTENeedhamLingRobinson1962221-6">[5]</a> Zhu Zaiyu is quoted as saying that, in a text dating from 1584, "I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations."<a href="#cite_note-FOOTNOTENeedhamLingRobinson1962221-6">[5]</a> Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications."<a href="#cite_note-FOOTNOTEKuttner1975163-3">[2]</a> Kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament and that neither of the two should be treated as inventors.<a href="#cite_note-FOOTNOTEKuttner1975200-4">[3]</a> <div class="mw-heading mw-heading3"><h3 id="China">China</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=2" title="Edit section: China">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Early_history">Early history</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=3" title="Edit section: Early history">edit</a>]</div> A complete set of bronze chime bells, among many musical instruments found in the tomb of the Marquis Yi of Zeng (early Warring States, <abbr title="circa">c.</abbr> 5th century BCE in the Chinese Bronze Age), covers five full 7-note octaves in the key of C Major, including 12 note semi-tones in the middle of the range.<a href="#cite_note-FOOTNOTEKwang-chih_ChangPingfang_XuLiancheng_Lu2005140-7">[6]</a> An approximation for equal temperament was described by <a href="/w/index.php?title=He_Chengtian&action=edit&redlink=1" class="new" title="He Chengtian (page does not exist)">He Chengtian</a> [<a href="https://zh.wikipedia.org/wiki/%E4%BD%95%E6%89%BF%E5%A4%A9_(%E5%8D%97%E6%9C%9D)" class="extiw" title="zh:何承天 (南朝)">zh</a>], a mathematician of the <a href="/wiki/Southern_and_Northern_Dynasties" class="mw-redirect" title="Southern and Northern Dynasties">Southern and Northern Dynasties</a> who lived from 370 to 447.<a href="#cite_note-8">[7]</a> He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history: 900 849 802 758 715 677 638 601 570 536 509.5 479 450.<a href="#cite_note-FOOTNOTEBarbour200455–56-9">[8]</a> <div class="mw-heading mw-heading4"><h4 id="Zhu_Zaiyu">Zhu Zaiyu</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=4" title="Edit section: Zhu Zaiyu">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-2297.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-2297.jpg/310px-%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-2297.jpg" decoding="async" width="310" height="424" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-2297.jpg/465px-%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-2297.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-2297.jpg/620px-%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-2297.jpg 2x" data-file-width="731" data-file-height="1000" /></a><figcaption>Prince Zhu Zaiyu constructed 12 string equal temperament tuning instrument, front and back view</figcaption></figure> <a href="/wiki/Zhu_Zaiyu" title="Zhu Zaiyu">Zhu Zaiyu</a> (朱載堉), a prince of the <a href="/wiki/Ming_Dynasty" class="mw-redirect" title="Ming Dynasty">Ming</a> court, spent thirty years on research based on the equal temperament idea originally postulated by his father. He described his new pitch theory in his Fusion of Music and Calendar 律暦融通 published in 1580. This was followed by the publication of a detailed account of the new theory of the equal temperament with a precise numerical specification for 12-ET in his 5,000-page work Complete Compendium of Music and Pitch (Yuelü quan shu 樂律全書) in 1584.<a href="#cite_note-FOOTNOTEHart1998-10">[9]</a> An extended account is also given by Joseph Needham.<a href="#cite_note-FOOTNOTENeedhamLingRobinson1962221-6">[5]</a> Zhu obtained his result mathematically by dividing the length of string and pipe successively by 12√2 ≈ 1.059463, and for pipe length by 24√2,<a href="#cite_note-FOOTNOTENeedhamRonan1978385-11">[10]</a> such that after twelve divisions (an octave) the length was divided by a factor of 2: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\sqrt[{12}]{2}}\right)^{12}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\sqrt[{12}]{2}}\right)^{12}=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0e94dc870dfc2af0eb35a7b362eb3776054719" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.374ex; height:3.843ex;" alt="{\displaystyle \left({\sqrt[{12}]{2}}\right)^{12}=2}"> Similarly, after 84 divisions (7 octaves) the length was divided by a factor of 128: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\sqrt[{12}]{2}}\right)^{84}=2^{7}=128}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>84</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>=</mo> <mn>128</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\sqrt[{12}]{2}}\right)^{84}=2^{7}=128}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efc85da07cabc4f33cc3fc965e9ef6e74361e7c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.014ex; height:3.843ex;" alt="{\displaystyle \left({\sqrt[{12}]{2}}\right)^{84}=2^{7}=128}"> Zhu Zaiyu has been credited as the first person to solve the equal temperament problem mathematically.<a href="#cite_note-FOOTNOTECho2010-12">[11]</a> At least one researcher has proposed that <a href="/wiki/Matteo_Ricci" title="Matteo Ricci">Matteo Ricci</a>, a <a href="/wiki/Jesuit" class="mw-redirect" title="Jesuit">Jesuit</a> in China recorded this work in his personal journal<a href="#cite_note-FOOTNOTECho2010-12">[11]</a><a href="#cite_note-FOOTNOTELienhard1997-13">[12]</a> and may have transmitted the work back to Europe. (Standard resources on the topic make no mention of any such transfer.<a href="#cite_note-FOOTNOTEChristensen2002205-14">[13]</a>) In 1620, Zhu's work was referenced by a European mathematician.[<a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch">who?</a>]<a href="#cite_note-FOOTNOTELienhard1997-13">[12]</a> <a href="/wiki/J._Murray_Barbour" class="mw-redirect" title="J. Murray Barbour">Murray Barbour</a> said, "The first known appearance in print of the correct figures for equal temperament was in China, where Prince Tsaiyü's brilliant solution remains an enigma."<a href="#cite_note-FOOTNOTEBarbour20047-15">[14]</a> The 19th-century German physicist <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a> wrote in On the Sensations of Tone that a Chinese prince (see below) introduced a scale of seven notes, and that the division of the octave into twelve semitones was discovered in China.<a href="#cite_note-FOOTNOTEvon_HelmholtzEllis1885258-16">[15]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-1154.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-1154.jpg/220px-%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-1154.jpg" decoding="async" width="220" height="295" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-1154.jpg/330px-%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-1154.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-1154.jpg/440px-%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-1154.jpg 2x" data-file-width="747" data-file-height="1000" /></a><figcaption>Zhu Zaiyu's equal temperament pitch pipes</figcaption></figure> Zhu Zaiyu illustrated his equal temperament theory by the construction of a set of 36 bamboo tuning pipes ranging in 3 octaves, with instructions of the type of bamboo, color of paint, and detailed specification on their length and inner and outer diameters. He also constructed a 12-string tuning instrument, with a set of tuning pitch pipes hidden inside its bottom cavity. In 1890, <a href="/wiki/Victor-Charles_Mahillon" title="Victor-Charles Mahillon">Victor-Charles Mahillon</a>, curator of the Conservatoire museum in Brussels, duplicated a set of pitch pipes according to Zhu Zaiyu's specification. He said that the Chinese theory of tones knew more about the length of pitch pipes than its Western counterpart, and that the set of pipes duplicated according to the Zaiyu data proved the accuracy of this theory. <div class="mw-heading mw-heading3"><h3 id="Europe">Europe</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=5" title="Edit section: Europe">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:De_Spiegheling_der_signconst.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/De_Spiegheling_der_signconst.jpg/310px-De_Spiegheling_der_signconst.jpg" decoding="async" width="310" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/De_Spiegheling_der_signconst.jpg/465px-De_Spiegheling_der_signconst.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/De_Spiegheling_der_signconst.jpg/620px-De_Spiegheling_der_signconst.jpg 2x" data-file-width="653" data-file-height="491" /></a><figcaption>Simon Stevin's Van de Spiegheling der singconst <abbr title="circa">c.</abbr> 1605.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Early_history_2">Early history</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=6" title="Edit section: Early history">edit</a>]</div> One of the earliest discussions of equal temperament occurs in the writing of <a href="/wiki/Aristoxenus" title="Aristoxenus">Aristoxenus</a> in the 4th century BC.<a href="#cite_note-FOOTNOTETrue201861–74-17">[16]</a> <a href="/wiki/Vincenzo_Galilei" title="Vincenzo Galilei">Vincenzo Galilei</a> (father of <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a>) was one of the first practical advocates of twelve-tone equal temperament. He composed a set of dance suites on each of the 12 notes of the chromatic scale in all the "transposition keys", and published also, in his 1584 "<a href="/wiki/Fronimo_Dialogo" title="Fronimo Dialogo">Fronimo</a>", 24 + 1 <a href="/wiki/Ricercar" title="Ricercar">ricercars</a>.<a href="#cite_note-FOOTNOTEGalilei158480–89-18">[17]</a> He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).<a href="#cite_note-FOOTNOTEBarbour20048-19">[18]</a> Galilei's countryman and fellow <a href="/wiki/Lutenist" class="mw-redirect" title="Lutenist">lutenist</a> <a href="/w/index.php?title=Giacomo_Gorzanis&action=edit&redlink=1" class="new" title="Giacomo Gorzanis (page does not exist)">Giacomo Gorzanis</a> had written music based on equal temperament by 1567.<a href="#cite_note-FOOTNOTEde_Gorzanis1981-20">[19]</a> Gorzanis was not the only lutenist to explore all modes or keys: <a href="/wiki/Francesco_Spinacino" title="Francesco Spinacino">Francesco Spinacino</a> wrote a "Recercare de tutti li Toni" (<a href="/wiki/Ricercar" title="Ricercar">Ricercar</a> in all the Tones) as early as 1507.<a href="#cite_note-appstate.edu-21">[20]</a> In the 17th century lutenist-composer <a href="/wiki/John_Wilson_(composer)" title="John Wilson (composer)">John Wilson</a> wrote a set of 30 preludes including 24 in all the major/minor keys.<a href="#cite_note-FOOTNOTEWilson1997-22">[21]</a><a href="#cite_note-FOOTNOTEJorgens1986-23">[22]</a> <a href="/wiki/Henricus_Grammateus" title="Henricus Grammateus">Henricus Grammateus</a> drew a close approximation to equal temperament in 1518. The first tuning rules in equal temperament were given by <a href="/w/index.php?title=Giovani_Maria_Lanfranco&action=edit&redlink=1" class="new" title="Giovani Maria Lanfranco (page does not exist)">Giovani Maria Lanfranco</a> in his "Scintille de musica".<a href="#cite_note-Scintille-24">[23]</a> <a href="/wiki/Zarlino" class="mw-redirect" title="Zarlino">Zarlino</a> in his <a href="/wiki/Polemic" title="Polemic">polemic</a> with Galilei initially opposed equal temperament but eventually conceded to it in relation to the <a href="/wiki/Lute" title="Lute">lute</a> in his Sopplimenti musicali in 1588. <div class="mw-heading mw-heading4"><h4 id="Simon_Stevin">Simon Stevin</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=7" title="Edit section: Simon Stevin">edit</a>]</div> The first mention of equal temperament related to the <a href="/wiki/Twelfth_root_of_two" title="Twelfth root of two">twelfth root of two</a> in the West appeared in <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a>'s manuscript Van De Spiegheling der singconst (c. 1605), published posthumously nearly three centuries later in 1884.<a href="#cite_note-FOOTNOTECohen1987471–488-25">[24]</a> However, due to insufficient accuracy of his calculation, many of the chord length numbers he obtained were off by one or two units from the correct values.<a href="#cite_note-FOOTNOTEChristensen2002205-14">[13]</a> As a result, the frequency ratios of Simon Stevin's chords has no unified ratio, but one ratio per tone, which is claimed by Gene Cho as incorrect.<a href="#cite_note-FOOTNOTECho2003223-26">[25]</a> The following were Simon Stevin's chord length from Van de Spiegheling der singconst:<a href="#cite_note-FOOTNOTECho2003222-27">[26]</a> <table class="wikitable" style="margin:auto;"> <tbody><tr> <th>Tone </th> <th>Chord 10000 from Simon Stevin </th> <th>Ratio </th> <th>Corrected chord </th></tr> <tr> <td>semitone </td> <td>9438 </td> <td>1.0595465 </td> <td>9438.7 </td></tr> <tr> <td>whole tone </td> <td>8909 </td> <td>1.0593781 </td> <td> </td></tr> <tr> <td>tone and a half </td> <td>8404 </td> <td>1.0600904 </td> <td>8409 </td></tr> <tr> <td>ditone </td> <td>7936 </td> <td>1.0594758 </td> <td>7937 </td></tr> <tr> <td>ditone and a half </td> <td>7491 </td> <td>1.0594046 </td> <td>7491.5 </td></tr> <tr> <td>tritone </td> <td>7071 </td> <td>1.0593975 </td> <td>7071.1 </td></tr> <tr> <td>tritone and a half </td> <td>6674 </td> <td>1.0594845 </td> <td>6674.2 </td></tr> <tr> <td>four-tone </td> <td>6298 </td> <td>1.0597014 </td> <td>6299 </td></tr> <tr> <td>four-tone and a half </td> <td>5944 </td> <td>1.0595558 </td> <td>5946 </td></tr> <tr> <td>five-tone </td> <td>5611 </td> <td>1.0593477 </td> <td>5612.3 </td></tr> <tr> <td>five-tone and a half </td> <td>5296 </td> <td>1.0594788 </td> <td>5297.2 </td></tr> <tr> <td>full tone </td> <td> </td> <td>1.0592000 </td> <td> </td></tr></tbody></table> A generation later, French mathematician <a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a> presented several equal tempered chord lengths obtained by Jean Beaugrand, Ismael Bouillaud, and Jean Galle.<a href="#cite_note-FOOTNOTEChristensen2002207-28">[27]</a> In 1630 <a href="/wiki/Johann_Faulhaber" title="Johann Faulhaber">Johann Faulhaber</a> published a 100-cent monochord table, which contained several errors due to his use of logarithmic tables. He did not explain how he obtained his results.<a href="#cite_note-FOOTNOTEChristensen200278-29">[28]</a> <div class="mw-heading mw-heading4"><h4 id="Baroque_era">Baroque era</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=8" title="Edit section: Baroque era">edit</a>]</div> From 1450 to about 1800, plucked instrument players (lutenists and guitarists) generally favored equal temperament,<a href="#cite_note-30">[29]</a> and the Brossard lute manuscript compiled in the last quarter of the 17th century contains a series of 18 preludes attributed to <a href="/wiki/Mlle_Bocquet" title="Mlle Bocquet">Bocquet</a> written in all keys, including the last prelude, entitled Prélude sur tous les tons, which enharmonically modulates through all keys.<a href="#cite_note-31">[30]</a>[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] <a href="/wiki/Angelo_Michele_Bartolotti" title="Angelo Michele Bartolotti">Angelo Michele Bartolotti</a> published a series of <a href="/wiki/Passacaglia" title="Passacaglia">passacaglias</a> in all keys, with connecting enharmonically modulating passages. Among the 17th-century keyboard composers <a href="/wiki/Girolamo_Frescobaldi" title="Girolamo Frescobaldi">Girolamo Frescobaldi</a> advocated equal temperament. Some theorists, such as <a href="/wiki/Giuseppe_Tartini" title="Giuseppe Tartini">Giuseppe Tartini</a>, were opposed to the adoption of equal temperament; they felt that degrading the purity of each chord degraded the aesthetic appeal of music, although <a href="/wiki/Andreas_Werckmeister" title="Andreas Werckmeister">Andreas Werckmeister</a> emphatically advocated equal temperament in his 1707 treatise published posthumously.<a href="#cite_note-32">[31]</a> Twelve-tone equal temperament took hold for a variety of reasons. It was a convenient fit for the existing keyboard design, and permitted total harmonic freedom with the burden of moderate impurity in every interval, particularly imperfect consonances. This allowed greater expression through <a href="/wiki/Modulation_(music)" title="Modulation (music)">enharmonic modulation</a>, which became extremely important in the 18th century in music of such composers as <a href="/wiki/Francesco_Geminiani" title="Francesco Geminiani">Francesco Geminiani</a>, <a href="/wiki/Wilhelm_Friedemann_Bach" title="Wilhelm Friedemann Bach">Wilhelm Friedemann Bach</a>, <a href="/wiki/Carl_Philipp_Emmanuel_Bach" class="mw-redirect" title="Carl Philipp Emmanuel Bach">Carl Philipp Emmanuel Bach</a>, and <a href="/wiki/Johann_Gottfried_M%C3%BCthel" title="Johann Gottfried Müthel">Johann Gottfried Müthel</a>.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Twelve-tone equal temperament did have some disadvantages, such as imperfect thirds, but as Europe switched to equal temperament, it changed the music that it wrote in order to accommodate the system and minimize dissonance.<a href="#cite_note-33">[b]</a> The progress of equal temperament from the mid-18th century on is described with detail in quite a few modern scholarly publications: It was already the temperament of choice during the Classical era (second half of the 18th century),[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] and it became standard during the Early Romantic era (first decade of the 19th century),[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] except for organs that switched to it more gradually, completing only in the second decade of the 19th century. (In England, some cathedral organists and choirmasters held out against it even after that date; <a href="/wiki/Samuel_Sebastian_Wesley" title="Samuel Sebastian Wesley">Samuel Sebastian Wesley</a>, for instance, opposed it all along. He died in 1876.)[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] A precise equal temperament is possible using the 17th century Sabbatini method of splitting the octave first into three tempered major thirds.<a href="#cite_note-FOOTNOTEDi_Veroli2009140,_142_and_256-34">[32]</a> This was also proposed by several writers during the Classical era. Tuning without beat rates but employing several checks, achieving virtually modern accuracy, was already done in the first decades of the 19th century.<a href="#cite_note-FOOTNOTEMoody2003-35">[33]</a> Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century.<a href="#cite_note-FOOTNOTEvon_HelmholtzEllis1885548-36">[34]</a> The ultimate precision was available with 2 decimal tables published by White in 1917.<a href="#cite_note-37">[35]</a> It is in the environment of equal temperament that the new styles of symmetrical tonality and <a href="/wiki/Polytonality" title="Polytonality">polytonality</a>, <a href="/wiki/Atonality" title="Atonality">atonal music</a> such as that written with the <a href="/wiki/Twelve_tone_technique" class="mw-redirect" title="Twelve tone technique">twelve tone technique</a> or <a href="/wiki/Serialism" title="Serialism">serialism</a>, and <a href="/wiki/Jazz" title="Jazz">jazz</a> (at least its piano component) developed and flourished. <div class="mw-heading mw-heading3"><h3 id="Comparison_of_historical_approximations_of_the_semitone">Comparison of historical approximations of the semitone</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=9" title="Edit section: Comparison of historical approximations of the semitone">edit</a>]</div> <table class="wikitable" style="margin:auto;"> <tbody><tr> <th>Year </th> <th>Name </th> <th>Ratio<a href="#cite_note-FOOTNOTEBarbour200455–78-38">[36]</a> </th> <th>Cents </th></tr> <tr> <td>400 </td> <td><a href="/w/index.php?title=He_Chengtian&action=edit&redlink=1" class="new" title="He Chengtian (page does not exist)">He Chengtian</a> </td> <td>1.060070671 </td> <td>101.0 </td></tr> <tr> <td>1580 </td> <td><a href="/wiki/Vincenzo_Galilei" title="Vincenzo Galilei">Vincenzo Galilei</a> </td> <td>18:17 [1.058823529] </td> <td>99.0 </td></tr> <tr> <td>1581 </td> <td><a href="/wiki/Zhu_Zaiyu" title="Zhu Zaiyu">Zhu Zaiyu</a> </td> <td>1.059463094 </td> <td>100.0 </td></tr> <tr> <td>1585 </td> <td><a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a> </td> <td>1.059546514 </td> <td>100.1 </td></tr> <tr> <td>1630 </td> <td><a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a> </td> <td>1.059322034 </td> <td>99.8 </td></tr> <tr> <td>1630 </td> <td><a href="/wiki/Johann_Faulhaber" title="Johann Faulhaber">Johann Faulhaber</a> </td> <td>1.059490385 </td> <td>100.0 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Mathematical_properties">Mathematical properties</h2>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=10" title="Edit section: Mathematical properties">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Monochord_ET.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Monochord_ET.png/220px-Monochord_ET.png" decoding="async" width="220" height="366" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Monochord_ET.png/330px-Monochord_ET.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Monochord_ET.png/440px-Monochord_ET.png 2x" data-file-width="456" data-file-height="758" /></a><figcaption>One octave of 12-ET on a monochord</figcaption></figure> In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a <a href="/wiki/Semitone" title="Semitone">semitone</a>, i.e. the <a href="/wiki/Interval_ratio" title="Interval ratio">frequency ratio</a> of the interval between two adjacent notes, is the <a href="/wiki/Twelfth_root_of_two" title="Twelfth root of two">twelfth root of two</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{12}]{2}}=2^{\frac {1}{12}}\approx 1.059463}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> </msup> <mo>≈</mo> <mn>1.059463</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{12}]{2}}=2^{\frac {1}{12}}\approx 1.059463}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232c2beab28b1c46c328080d982595d9ef196e08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.653ex; height:3.843ex;" alt="{\displaystyle {\sqrt[{12}]{2}}=2^{\frac {1}{12}}\approx 1.059463}"> This interval is divided into 100 <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>. <div class="mw-heading mw-heading3"><h3 id="Calculating_absolute_frequencies">Calculating absolute frequencies</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=11" title="Edit section: Calculating absolute frequencies">edit</a>]</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">See also: <a href="/wiki/Piano_key_frequencies" title="Piano key frequencies">Piano key frequencies</a></div> To find the frequency, <var style="padding-right: 1px;">P</var><var style="padding-right: 1px;">n</var>, of a note in 12-ET, the following definition may be used: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}=P_{a}\left({\sqrt[{12}]{2}}\right)^{(n-a)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}=P_{a}\left({\sqrt[{12}]{2}}\right)^{(n-a)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3381e111c79f7e16a073bbe05c6cabeaba2ff79a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.286ex; height:4.009ex;" alt="{\displaystyle P_{n}=P_{a}\left({\sqrt[{12}]{2}}\right)^{(n-a)}}"> In this formula <var style="padding-right: 1px;">P</var><var style="padding-right: 1px;">n</var> refers to the pitch, or frequency (usually in <a href="/wiki/Hertz" title="Hertz">hertz</a>), you are trying to find. <var style="padding-right: 1px;">P</var><var style="padding-right: 1px;">a</var> refers to the frequency of a reference pitch. n and a refer to numbers assigned to the desired pitch and the reference pitch, respectively. These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A4 (the reference pitch) is the 49th key from the left end of a piano (tuned to <a href="/wiki/A440_(pitch_standard)" title="A440 (pitch standard)">440 Hz</a>), and C4 (<a href="/wiki/Middle_C" class="mw-redirect" title="Middle C">middle C</a>), and F#4 are the 40th and 46th key respectively. These numbers can be used to find the frequency of C4 and F#4: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{3}P_{40}&=440\left({\sqrt[{12}]{2}}\right)^{(40-49)}&&\approx 261.626\ \mathrm {Hz} \\P_{46}&=440\left({\sqrt[{12}]{2}}\right)^{(46-49)}&&\approx 369.994\ \mathrm {Hz} \end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>40</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>440</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>40</mn> <mo>−</mo> <mn>49</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mn>261.626</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>46</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>440</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>46</mn> <mo>−</mo> <mn>49</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mn>369.994</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}P_{40}&=440\left({\sqrt[{12}]{2}}\right)^{(40-49)}&&\approx 261.626\ \mathrm {Hz} \\P_{46}&=440\left({\sqrt[{12}]{2}}\right)^{(46-49)}&&\approx 369.994\ \mathrm {Hz} \end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b2cc77efb874655e7633d236cfd2c9f7e5b8229" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:36.097ex; height:8.176ex;" alt="{\displaystyle {\begin{alignedat}{3}P_{40}&=440\left({\sqrt[{12}]{2}}\right)^{(40-49)}&&\approx 261.626\ \mathrm {Hz} \\P_{46}&=440\left({\sqrt[{12}]{2}}\right)^{(46-49)}&&\approx 369.994\ \mathrm {Hz} \end{alignedat}}}"> <div class="mw-heading mw-heading2"><h2 id="Just_intervals">Just intervals </h2>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=12" title="Edit section: Just intervals">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:12ed2-5Limit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/12ed2-5Limit.svg/250px-12ed2-5Limit.svg.png" decoding="async" width="250" height="360" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/12ed2-5Limit.svg/375px-12ed2-5Limit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/12ed2-5Limit.svg/500px-12ed2-5Limit.svg.png 2x" data-file-width="500" data-file-height="720" /></a><figcaption>5-Limit just intervals approximated in 12-ET</figcaption></figure> The intervals of 12-ET closely approximate some intervals in <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a>.<a href="#cite_note-FOOTNOTEPartch1979134-39">[37]</a> <div class="mw-heading mw-heading3"><h3 id="By_limit">By limit</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=13" title="Edit section: By limit">edit</a>]</div> 12 ET is very accurate in the 3 limit, but as one increases prime limits to 11, it gradually gets worse by about a sixth of a semitone each time. Its eleventh and thirteenth harmonics are extremely inaccurate. 12 ET's seventeenth and nineteenth harmonics are almost as accurate as its third harmonic, but by this point, the prime limit has gotten too high to sound consonant to most people.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading4"><h4 id="3_limit">3 limit</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=14" title="Edit section: 3 limit">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a></div> 12 ET has a very good approximation of the perfect fifth (<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>⁠ 3 /2⁠) and its <a href="/wiki/Inversion_(music)#Intervals" title="Inversion (music)">inversion</a>, the perfect fourth (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 4 /3⁠), especially for the division of the octave into a relatively small number of tones. Specifically, a just perfect fifth is only one fifty-first of a semitone sharper than the equally-tempered approximation. Because the major tone (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 9 /8⁠) is simply two perfect fifths minus an octave, and its inversion, the Pythagorean minor seventh (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 16 /9⁠), is simply two perfect fourths combined, they, for the most part, retain the accuracy of their predecessors; the error is doubled, but it remains small – so small, in fact, that humans cannot perceive it. One can continue to use fractions with higher powers of three, the next two being <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 27 /16⁠ and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 32 /27⁠, but as the terms of the fractions grow larger, they become less pleasing to the ear.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading4"><h4 id="5_limit">5 limit</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=15" title="Edit section: 5 limit">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Five-limit_tuning" title="Five-limit tuning">Five-limit tuning</a></div> 12 ET's approximation of the fifth harmonic (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 5 /4⁠) is approximately one-seventh of a semitone off. Because intervals that are less than a quarter of a scale step off still sound in tune, other five-limit intervals in 12 ET, such as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 5 /3⁠ and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 8 /5⁠, have similarly sized errors. The <a href="/wiki/Major_triad" class="mw-redirect" title="Major triad">major triad</a>, therefore, sounds in tune as its frequency ratio is approximately 4:5:6, further, merged with its first inversion, and two sub-octave tonics, it is 1:2:3:4:5:6, all six lowest natural harmonics of the bass tone.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading4"><h4 id="7_limit">7 limit</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=16" title="Edit section: 7 limit">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/7-limit_tuning" title="7-limit tuning">7-limit tuning</a></div> 12 ET's approximation of the seventh harmonic (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 7 /4⁠) is about one-third of a semitone off. Because the error is greater than a quarter of a semitone, seven-limit intervals in 12 ET tend to sound out of tune. In the tritone fractions <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 7 /5⁠ and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 10 /7⁠, the errors of the fifth and seventh harmonics partially cancel each other out so that the just fractions are within a quarter of a semitone of their equally-tempered equivalents.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading4"><h4 id="11_and_13_limits">11 and 13 limits</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=17" title="Edit section: 11 and 13 limits">edit</a>]</div> The eleventh harmonic (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 11 /8⁠), at 551.32 cents, falls almost exactly halfway between the nearest two equally-tempered intervals in 12 ET and therefore is not approximated by either. In fact, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 11 /8⁠ is almost as far from any equally-tempered approximation as possible in 12 ET. The thirteenth harmonic (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 13 /8⁠), at two-fifths of a semitone sharper than a minor sixth, is almost as inaccurate. Although this means that the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 13 /11⁠ and also its inversion (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 22 /13⁠) are accurately approximated (specifically, by three semitones), since the errors of the eleventh and thirteenth harmonics mostly cancel out, most people who are not familiar with <a href="/wiki/Quarter_tone" title="Quarter tone">quarter tones</a> or microtonality will not be familiar with the eleventh and thirteenth harmonics. Similarly, while the error of the eleventh or thirteenth harmonic could be mostly canceled out by the error of the seventh harmonic, most Western musicians would not find the resulting fractions consonant since 12 ET does not approximate them accurately.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading4"><h4 id="17_and_19_limits">17 and 19 limits</h4>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=18" title="Edit section: 17 and 19 limits">edit</a>]</div> The seventeenth harmonic (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 17 /16⁠) is only about 5 cents sharper than one semitone in 12 ET. It can be combined with 12 ET's approximation of the third harmonic in order to yield <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 17 /12⁠, which is, as the next <a href="/wiki/Pell_number" title="Pell number">Pell approximation</a> after <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 7 /5⁠, only about three cents away from the equally-tempered tritone (the square root of two), and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 17 /9⁠, which is only one cent away from 12 ET's major seventh. The nineteenth harmonic is only about 2.5 cents flatter than three of 12 ET's semitones, so it can likewise be combined with the third harmonic to yield <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 19 /12⁠, which is about 4.5 cents flatter than an equally-tempered minor sixth, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 19 /18⁠, which is about 6.5 cents flatter than a semitone. However, because 17 and 19 are rather large for consonant ratios and most people are unfamiliar with 17 limit and 19 limit intervals, 17 limit and 19 limit intervals are not useful for most purposes, so they can likely not be judged as playing a part in any consonances of 12 ET.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Table">Table</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=19" title="Edit section: Table">edit</a>]</div> In the following table the sizes of various just intervals are compared against their equal-tempered counterparts, given as a ratio as well as <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>. Differences of less than six cents cannot be noticed by most people, and intervals that are more than a quarter of a step; which in this case is 25 cents, off sound out of tune.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <table class="wikitable collapsible" style="margin:auto;"> <tbody><tr> <th>Number of steps </th> <th>Note going up from C </th> <th>Exact value in 12-ET </th> <th>Decimal value in 12-ET </th> <th>Equally-tempered audio </th> <th>Cents </th> <th>Just intonation interval name </th> <th>Just intonation interval fraction </th> <th>Justly-intoned audio </th> <th>Cents in just intonation </th> <th>Difference </th></tr> <tr> <td>0 </td> <td><a href="/wiki/C_(musical_note)" title="C (musical note)">C</a> </td> <td>2<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">0⁄12 = 1 </td> <td>1 </td> <td><span data-nosnippet="" id="ooui-php-2" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/4c\/Unison_on_C.mid\/Unison_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Unison on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4c/Unison_on_C.mid/Unison_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Unison_on_C.mid" title="File:Unison on C.mid">ⓘ</a> </td> <td>0 </td> <td><a href="/wiki/Unison" title="Unison">Unison</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄1 = 1 </td> <td><span data-nosnippet="" id="ooui-php-3" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/4c\/Unison_on_C.mid\/Unison_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Unison on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4c/Unison_on_C.mid/Unison_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Unison_on_C.mid" title="File:Unison on C.mid">ⓘ</a> </td> <td>0 </td> <td>0 </td></tr> <tr> <td rowspan="15">1 </td> <td rowspan="15"><a href="/wiki/C%E2%99%AF_(musical_note)" title="C♯ (musical note)">C♯</a> or <a href="/wiki/D%E2%99%AD_(musical_note)" title="D♭ (musical note)">D♭</a> </td> <td rowspan="15">2<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{12}]{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{12}]{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc835f27425fb3140e1f75a5faa35b1e8b9efc35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.107ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{12}]{2}}}"> </td> <td rowspan="15">1.05946... </td> <td rowspan="15"><span data-nosnippet="" id="ooui-php-4" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/8\/8a\/Minor_second_on_C.mid\/Minor_second_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Minor second on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8a/Minor_second_on_C.mid/Minor_second_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Minor_second_on_C.mid" title="File:Minor second on C.mid">ⓘ</a> </td> <td rowspan="15">100 </td> <td><a href="/wiki/Septimal_third_tone" title="Septimal third tone">Septimal third tone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">28⁄27 = 1.03703... </td> <td><span data-nosnippet="" id="ooui-php-5" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/6\/60\/Septimal_minor_second_on_C.mid\/Septimal_minor_second_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal minor second on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/6/60/Septimal_minor_second_on_C.mid/Septimal_minor_second_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Septimal_minor_second_on_C.mid" title="File:Septimal minor second on C.mid">ⓘ</a> </td> <td>62.96 </td> <td>-37.04 </td></tr> <tr> <td><a href="/wiki/Just_chromatic_semitone" class="mw-redirect" title="Just chromatic semitone">Just chromatic semitone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">25⁄24 = 1.04166... </td> <td><span data-nosnippet="" id="ooui-php-6" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/1\/1d\/Just_chromatic_semitone_on_C.mid\/Just_chromatic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just chromatic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/1/1d/Just_chromatic_semitone_on_C.mid/Just_chromatic_semitone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">Play</a><a href="/wiki/File:Just_chromatic_semitone_on_C.mid" title="File:Just chromatic semitone on C.mid">ⓘ</a> </td> <td>70.67 </td> <td>-29.33 </td></tr> <tr> <td>Undecimal semitone </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">22⁄21 = 1.04761... </td> <td><span data-nosnippet="" id="ooui-php-7" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/6\/6d\/Undecimal_two-fifth_tone_on_C.mid\/Undecimal_two-fifth_tone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Undecimal two-fifth tone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6d/Undecimal_two-fifth_tone_on_C.mid/Undecimal_two-fifth_tone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Undecimal_two-fifth_tone_on_C.mid" title="File:Undecimal two-fifth tone on C.mid">ⓘ</a> </td> <td>80.54 </td> <td>-19.46 </td></tr> <tr> <td><a href="/wiki/Septimal_chromatic_semitone" title="Septimal chromatic semitone">Septimal chromatic semitone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 21⁄ 20 = 1.05 </td> <td><span data-nosnippet="" id="ooui-php-8" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/b\/ba\/Septimal_chromatic_semitone_on_C.mid\/Septimal_chromatic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal chromatic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/ba/Septimal_chromatic_semitone_on_C.mid/Septimal_chromatic_semitone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Septimal_chromatic_semitone_on_C.mid" title="File:Septimal chromatic semitone on C.mid">ⓘ</a> </td> <td>84.47 </td></tr> <tr> <td>Novendecimal chromatic semitone </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 20⁄ 19 = 1.05263... </td> <td><span data-nosnippet="" id="ooui-php-9" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/3\/39\/Novendecimal_augmented_unison_on_C.mid\/Novendecimal_augmented_unison_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Novendecimal augmented unison on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/3/39/Novendecimal_augmented_unison_on_C.mid/Novendecimal_augmented_unison_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Novendecimal_augmented_unison_on_C.mid" title="File:Novendecimal augmented unison on C.mid">ⓘ</a> </td> <td>88.80 </td></tr> <tr> <td><a href="/wiki/Pythagorean_diatonic_semitone" class="mw-redirect" title="Pythagorean diatonic semitone">Pythagorean diatonic semitone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 256⁄ 243 = 1.05349... </td> <td><span data-nosnippet="" id="ooui-php-10" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/6\/6e\/Pythagorean_minor_semitone_on_C.mid\/Pythagorean_minor_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean minor semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6e/Pythagorean_minor_semitone_on_C.mid/Pythagorean_minor_semitone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_minor_semitone_on_C.mid" title="File:Pythagorean minor semitone on C.mid">ⓘ</a> </td> <td>90.22 </td></tr> <tr> <td><a href="/wiki/Larger_chromatic_semitone" class="mw-redirect" title="Larger chromatic semitone">Larger chromatic semitone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 135⁄ 128 = 1.05468... </td> <td><span data-nosnippet="" id="ooui-php-11" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/1\/11\/Greater_chromatic_semitone_on_C.mid\/Greater_chromatic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Greater chromatic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/1/11/Greater_chromatic_semitone_on_C.mid/Greater_chromatic_semitone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Greater_chromatic_semitone_on_C.mid" title="File:Greater chromatic semitone on C.mid">ⓘ</a> </td> <td>92.18 </td></tr> <tr> <td>Novendecimal diatonic semitone </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 19⁄ 18 = 1.05555... </td> <td><span data-nosnippet="" id="ooui-php-12" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/0\/0d\/Novendecimal_minor_second_on_C.mid\/Novendecimal_minor_second_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Novendecimal minor second on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/0/0d/Novendecimal_minor_second_on_C.mid/Novendecimal_minor_second_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Novendecimal_minor_second_on_C.mid" title="File:Novendecimal minor second on C.mid">ⓘ</a> </td> <td>93.60 </td></tr> <tr> <td>Septadecimal chromatic semitone </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 18⁄ 17 = 1.05882... </td> <td><span data-nosnippet="" id="ooui-php-13" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/9\/94\/Just_minor_semitone_on_C.mid\/Just_minor_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just minor semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/94/Just_minor_semitone_on_C.mid/Just_minor_semitone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_minor_semitone_on_C.mid" title="File:Just minor semitone on C.mid">ⓘ</a> </td> <td>98.95 </td></tr> <tr> <td><a href="/wiki/Seventeenth_harmonic" class="mw-redirect" title="Seventeenth harmonic">Seventeenth harmonic</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 17⁄ 16 = 1.0625... </td> <td><span data-nosnippet="" id="ooui-php-14" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/48\/Just_major_semitone_on_C.mid\/Just_major_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/48/Just_major_semitone_on_C.mid/Just_major_semitone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_major_semitone_on_C.mid" title="File:Just major semitone on C.mid">ⓘ</a> </td> <td>104.96 </td> <td>+4.96 </td></tr> <tr> <td><a href="/wiki/Just_diatonic_semitone" class="mw-redirect" title="Just diatonic semitone">Just diatonic semitone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 16⁄ 15 = 1.06666... </td> <td><span data-nosnippet="" id="ooui-php-15" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/b\/be\/Just_diatonic_semitone_on_C.mid\/Just_diatonic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just diatonic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/be/Just_diatonic_semitone_on_C.mid/Just_diatonic_semitone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_diatonic_semitone_on_C.mid" title="File:Just diatonic semitone on C.mid">ⓘ</a> </td> <td>111.73 </td> <td>+11.73 </td></tr> <tr> <td><a href="/wiki/Pythagorean_chromatic_semitone" class="mw-redirect" title="Pythagorean chromatic semitone">Pythagorean chromatic semitone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 2187⁄ 2048 = 1.06787... </td> <td><span data-nosnippet="" id="ooui-php-16" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/3\/30\/Pythagorean_apotome_on_C.mid\/Pythagorean_apotome_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean apotome on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/3/30/Pythagorean_apotome_on_C.mid/Pythagorean_apotome_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_apotome_on_C.mid" title="File:Pythagorean apotome on C.mid">ⓘ</a> </td> <td>113.69 </td> <td>+13.69 </td></tr> <tr> <td><a href="/wiki/Septimal_diatonic_semitone" title="Septimal diatonic semitone">Septimal diatonic semitone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 15⁄ 14 = 1.07142... </td> <td><span data-nosnippet="" id="ooui-php-17" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/6\/69\/Septimal_diatonic_semitone_on_C.mid\/Septimal_diatonic_semitone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal diatonic semitone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/6/69/Septimal_diatonic_semitone_on_C.mid/Septimal_diatonic_semitone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Septimal_diatonic_semitone_on_C.mid" title="File:Septimal diatonic semitone on C.mid">ⓘ</a> </td> <td>119.44 </td> <td>+19.44 </td></tr> <tr> <td>Lesser tridecimal 2/3-tone </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 14⁄ 13 = 1.07692... </td> <td><span data-nosnippet="" id="ooui-php-18" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/5\/58\/Lesser_tridecimal_two-third_tone_on_C.mid\/Lesser_tridecimal_two-third_tone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Lesser tridecimal two-third tone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/58/Lesser_tridecimal_two-third_tone_on_C.mid/Lesser_tridecimal_two-third_tone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Lesser_tridecimal_two-third_tone_on_C.mid" title="File:Lesser tridecimal two-third tone on C.mid">ⓘ</a> </td> <td>128.30 </td> <td>+28.30 </td></tr> <tr> <td><a href="/wiki/Major_diatonic_semitone" class="mw-redirect" title="Major diatonic semitone">Major diatonic semitone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 27⁄ 25 = 1.08 </td> <td><span data-nosnippet="" id="ooui-php-19" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/4a\/Semitone_Maximus_on_C.mid\/Semitone_Maximus_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Semitone Maximus on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4a/Semitone_Maximus_on_C.mid/Semitone_Maximus_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Semitone_Maximus_on_C.mid" title="File:Semitone Maximus on C.mid">ⓘ</a> </td> <td>133.24 </td> <td>+33.24 </td></tr> <tr> <td rowspan="4">2 </td> <td rowspan="4"><a href="/wiki/D_(musical_note)" title="D (musical note)"> D</a> </td> <td rowspan="4">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 2⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{6}]{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{6}]{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd2e90711da1208f1bf08c8992ab44739cb9c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{6}]{2}}}"> </td> <td rowspan="4">1.12246... </td> <td rowspan="4"><span data-nosnippet="" id="ooui-php-20" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/b\/b9\/Major_second_on_C.mid\/Major_second_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major second on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/b9/Major_second_on_C.mid/Major_second_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Major_second_on_C.mid" title="File:Major second on C.mid">ⓘ</a> </td> <td rowspan="4">200 </td> <td><a href="/wiki/Pythagorean_diminished_third" class="mw-redirect" title="Pythagorean diminished third">Pythagorean diminished third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 65536⁄ 59049 = 1.10985... </td> <td><span data-nosnippet="" id="ooui-php-21" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/fd\/Pythagorean_diminished_third.mid\/Pythagorean_diminished_third.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean diminished third.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/fd/Pythagorean_diminished_third.mid/Pythagorean_diminished_third.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_diminished_third.mid" title="File:Pythagorean diminished third.mid">ⓘ</a> </td> <td>180.45 </td></tr> <tr> <td><a href="/wiki/Minor_tone" class="mw-redirect" title="Minor tone">Minor tone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 10⁄ 9 = 1.11111... </td> <td><span data-nosnippet="" id="ooui-php-22" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/8\/89\/Minor_tone_on_C.mid\/Minor_tone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Minor tone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/8/89/Minor_tone_on_C.mid/Minor_tone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Minor_tone_on_C.mid" title="File:Minor tone on C.mid">ⓘ</a> </td> <td>182.40 </td></tr> <tr> <td><a href="/wiki/Major_tone" class="mw-redirect" title="Major tone">Major tone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 9⁄ 8 = 1.125 </td> <td><span data-nosnippet="" id="ooui-php-23" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/7\/7b\/Major_tone_on_C.mid\/Major_tone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major tone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7b/Major_tone_on_C.mid/Major_tone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Major_tone_on_C.mid" title="File:Major tone on C.mid">ⓘ</a> </td> <td>203.91 </td> <td>+3.91 </td></tr> <tr> <td><a href="/wiki/Septimal_whole_tone" title="Septimal whole tone">Septimal whole tone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 8⁄ 7 = 1.14285... </td> <td><span data-nosnippet="" id="ooui-php-24" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/4f\/Septimal_major_second_on_C.mid\/Septimal_major_second_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal major second on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4f/Septimal_major_second_on_C.mid/Septimal_major_second_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Septimal_major_second_on_C.mid" title="File:Septimal major second on C.mid">ⓘ</a> </td> <td>231.17 </td> <td>+31.17 </td></tr> <tr> <td rowspan="6">3 </td> <td rowspan="6"><a href="/wiki/D%E2%99%AF_(musical_note)" title="D♯ (musical note)"> D♯</a> or <a href="/wiki/E%E2%99%AD_(musical_note)" title="E♭ (musical note)"> E♭</a> </td> <td rowspan="6">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 3⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{4}]{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{4}]{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa163183b2c3828db27e22253d454a643a4c936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{4}]{2}}}"> </td> <td rowspan="6">1.18920... </td> <td rowspan="6"><span data-nosnippet="" id="ooui-php-25" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/3\/31\/Minor_third_on_C.mid\/Minor_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Minor third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/3/31/Minor_third_on_C.mid/Minor_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Minor_third_on_C.mid" title="File:Minor third on C.mid">ⓘ</a> </td> <td rowspan="6">300 </td> <td><a href="/wiki/Septimal_minor_third" title="Septimal minor third">Septimal minor third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 7⁄ 6 = 1.16666... </td> <td><span data-nosnippet="" id="ooui-php-26" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/7\/73\/Septimal_minor_third_on_C.mid\/Septimal_minor_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal minor third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/7/73/Septimal_minor_third_on_C.mid/Septimal_minor_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Septimal_minor_third_on_C.mid" title="File:Septimal minor third on C.mid">ⓘ</a> </td> <td>266.87 </td></tr> <tr> <td><a href="/wiki/Tridecimal_minor_third" class="mw-redirect" title="Tridecimal minor third">Tridecimal minor third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 13⁄ 11 = 1.18181... </td> <td><span data-nosnippet="" id="ooui-php-27" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/d\/de\/Tridecimal_minor_third_on_C.mid\/Tridecimal_minor_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Tridecimal minor third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/d/de/Tridecimal_minor_third_on_C.mid/Tridecimal_minor_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Tridecimal_minor_third_on_C.mid" title="File:Tridecimal minor third on C.mid">ⓘ</a> </td> <td>289.21 </td></tr> <tr> <td><a href="/wiki/Pythagorean_minor_third" class="mw-redirect" title="Pythagorean minor third">Pythagorean minor third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 32⁄ 27 = 1.18518... </td> <td><span data-nosnippet="" id="ooui-php-28" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/9\/9f\/Pythagorean_minor_third_on_C.mid\/Pythagorean_minor_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean minor third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9f/Pythagorean_minor_third_on_C.mid/Pythagorean_minor_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_minor_third_on_C.mid" title="File:Pythagorean minor third on C.mid">ⓘ</a> </td> <td>294.13 </td></tr> <tr> <td><a href="/wiki/Nineteenth_harmonic" class="mw-redirect" title="Nineteenth harmonic">Nineteenth harmonic</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 19⁄ 16 = 1.1875 </td> <td><span data-nosnippet="" id="ooui-php-29" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/2\/29\/19th_harmonic_on_C.mid\/19th_harmonic_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"19th harmonic on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/29/19th_harmonic_on_C.mid/19th_harmonic_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:19th_harmonic_on_C.mid" title="File:19th harmonic on C.mid">ⓘ</a> </td> <td>297.51 </td></tr> <tr> <td><a href="/wiki/Just_minor_third" class="mw-redirect" title="Just minor third">Just minor third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 6⁄ 5 = 1.2 </td> <td><span data-nosnippet="" id="ooui-php-30" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/42\/Just_minor_third_on_C.mid\/Just_minor_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just minor third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/42/Just_minor_third_on_C.mid/Just_minor_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_minor_third_on_C.mid" title="File:Just minor third on C.mid">ⓘ</a> </td> <td>315.64 </td> <td>+15.64 </td></tr> <tr> <td><a href="/wiki/Pythagorean_augmented_second" class="mw-redirect" title="Pythagorean augmented second">Pythagorean augmented second</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 19683⁄ 16384 = 1.20135... </td> <td><span data-nosnippet="" id="ooui-php-31" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f3\/Pythagorean_augmented_second_on_C.mid\/Pythagorean_augmented_second_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean augmented second on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f3/Pythagorean_augmented_second_on_C.mid/Pythagorean_augmented_second_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_augmented_second_on_C.mid" title="File:Pythagorean augmented second on C.mid">ⓘ</a> </td> <td>317.60 </td> <td>+17.60 </td></tr> <tr> <td rowspan="5">4 </td> <td rowspan="5"><a href="/wiki/E_(musical_note)" title="E (musical note)"> E</a> </td> <td rowspan="5">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 4⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca071ab504481c2bb76081aacb03f5519930710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{2}}}"> </td> <td rowspan="5">1.25992... </td> <td rowspan="5"><span data-nosnippet="" id="ooui-php-32" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/9\/91\/Major_third_on_C.mid\/Major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/91/Major_third_on_C.mid/Major_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Major_third_on_C.mid" title="File:Major third on C.mid">ⓘ</a> </td> <td rowspan="5">400 </td> <td><a href="/wiki/Pythagorean_diminished_fourth" class="mw-redirect" title="Pythagorean diminished fourth">Pythagorean diminished fourth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 8192⁄ 6561 = 1.24859... </td> <td><span data-nosnippet="" id="ooui-php-33" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/2\/2c\/Pythagorean_diminished_fourth_on_C.mid\/Pythagorean_diminished_fourth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean diminished fourth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2c/Pythagorean_diminished_fourth_on_C.mid/Pythagorean_diminished_fourth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_diminished_fourth_on_C.mid" title="File:Pythagorean diminished fourth on C.mid">ⓘ</a> </td> <td>384.36 </td> <td>-15.64 </td></tr> <tr> <td><a href="/wiki/Just_major_third" class="mw-redirect" title="Just major third">Just major third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 5⁄ 4 = 1.25 </td> <td><span data-nosnippet="" id="ooui-php-34" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/2\/2a\/Just_major_third_on_C.mid\/Just_major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/Just_major_third_on_C.mid/Just_major_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_major_third_on_C.mid" title="File:Just major third on C.mid">ⓘ</a> </td> <td>386.31 </td> <td>-13.69 </td></tr> <tr> <td><a href="/wiki/Pythagorean_major_third" class="mw-redirect" title="Pythagorean major third">Pythagorean major third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 81⁄ 64 = 1.265625 </td> <td><span data-nosnippet="" id="ooui-php-35" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/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":["ext-phonos-PhonosButton","noexcerpt"]}"><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">play</a><a href="/wiki/File:Pythagorean_major_third_on_C.mid" title="File:Pythagorean major third on C.mid">ⓘ</a> </td> <td>407.82 </td> <td>+7.82 </td></tr> <tr> <td><a href="/wiki/Undecimal_major_third" class="mw-redirect" title="Undecimal major third">Undecimal major third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 14⁄ 11 = 1.27272... </td> <td><span data-nosnippet="" id="ooui-php-36" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/6\/67\/Undecimal_major_third_on_C.mid\/Undecimal_major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Undecimal major third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/6/67/Undecimal_major_third_on_C.mid/Undecimal_major_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">Play</a><a href="/wiki/File:Undecimal_major_third_on_C.mid" title="File:Undecimal major third on C.mid">ⓘ</a> </td> <td>417.51 </td> <td>+17.51 </td></tr> <tr> <td><a href="/wiki/Septimal_major_third" title="Septimal major third">Septimal major third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 9⁄ 7 = 1.28571... </td> <td><span data-nosnippet="" id="ooui-php-37" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/5\/55\/Septimal_major_third_on_C.mid\/Septimal_major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal major third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/55/Septimal_major_third_on_C.mid/Septimal_major_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Septimal_major_third_on_C.mid" title="File:Septimal major third on C.mid">ⓘ</a> </td> <td>435.08 </td> <td>+35.08 </td></tr> <tr> <td rowspan="2">5 </td> <td rowspan="2"><a href="/wiki/F_(musical_note)" title="F (musical note)"> F</a> </td> <td rowspan="2">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 5⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{12}]{32}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>32</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{12}]{32}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9744b1d14b93f31471a1ad0e7176cbd2e42f1a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.269ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{12}]{32}}}"> </td> <td rowspan="2">1.33484... </td> <td rowspan="2"><span data-nosnippet="" id="ooui-php-38" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/8\/87\/Perfect_fourth_on_C.mid\/Perfect_fourth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Perfect fourth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/8/87/Perfect_fourth_on_C.mid/Perfect_fourth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Perfect_fourth_on_C.mid" title="File:Perfect fourth on C.mid">ⓘ</a> </td> <td rowspan="2">500 </td> <td><a href="/wiki/Just_perfect_fourth" class="mw-redirect" title="Just perfect fourth">Just perfect fourth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 4⁄ 3 = 1.33333... </td> <td><span data-nosnippet="" id="ooui-php-39" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/b\/be\/Just_perfect_fourth_on_C.mid\/Just_perfect_fourth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just perfect fourth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/be/Just_perfect_fourth_on_C.mid/Just_perfect_fourth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_perfect_fourth_on_C.mid" title="File:Just perfect fourth on C.mid">ⓘ</a> </td> <td>498.04 </td> <td>-1.96 </td></tr> <tr> <td><a href="/wiki/Pythagorean_augmented_third" class="mw-redirect" title="Pythagorean augmented third">Pythagorean augmented third</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 177147⁄ 131072 = 1.35152... </td> <td><span data-nosnippet="" id="ooui-php-40" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/8\/86\/Pythagorean_augmented_third_on_C.mid\/Pythagorean_augmented_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean augmented third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/8/86/Pythagorean_augmented_third_on_C.mid/Pythagorean_augmented_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_augmented_third_on_C.mid" title="File:Pythagorean augmented third on C.mid">ⓘ</a> </td> <td>521.51 </td> <td>+21.51 </td></tr> <tr> <td rowspan="8">6 </td> <td rowspan="8"><a href="/wiki/F%E2%99%AF_(musical_note)" title="F♯ (musical note)"> F♯</a> or <a href="/wiki/G%E2%99%AD_(musical_note)" title="G♭ (musical note)"> G♭</a> </td> <td rowspan="8">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 6⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"> </td> <td rowspan="8">1.41421... </td> <td rowspan="8"><span data-nosnippet="" id="ooui-php-41" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/5\/58\/Tritone_on_C.mid\/Tritone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Tritone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/58/Tritone_on_C.mid/Tritone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Tritone_on_C.mid" title="File:Tritone on C.mid">ⓘ</a> </td> <td rowspan="8">600 </td> <td>Classic augmented fourth </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 25⁄ 18 = 1.38888... </td> <td><span data-nosnippet="" id="ooui-php-42" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/e\/ee\/Classic_augmented_fourth_on_C.mid\/Classic_augmented_fourth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Classic augmented fourth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/e/ee/Classic_augmented_fourth_on_C.mid/Classic_augmented_fourth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Classic_augmented_fourth_on_C.mid" title="File:Classic augmented fourth on C.mid">ⓘ</a> </td> <td>568.72 </td> <td>-31.28 </td></tr> <tr> <td><a href="/wiki/Huygens%27_tritone" class="mw-redirect" title="Huygens' tritone">Huygens' tritone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 7⁄ 5 = 1.4 </td> <td><span data-nosnippet="" id="ooui-php-43" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/2\/24\/Lesser_septimal_tritone_on_C.mid\/Lesser_septimal_tritone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Lesser septimal tritone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/24/Lesser_septimal_tritone_on_C.mid/Lesser_septimal_tritone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Lesser_septimal_tritone_on_C.mid" title="File:Lesser septimal tritone on C.mid">ⓘ</a> </td> <td>582.51 </td> <td>-17.49 </td></tr> <tr> <td><a href="/wiki/Pythagorean_diminished_fifth" class="mw-redirect" title="Pythagorean diminished fifth">Pythagorean diminished fifth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 1024⁄ 729 = 1.40466... </td> <td><span data-nosnippet="" id="ooui-php-44" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/a\/a8\/Diminished_fifth_tritone_on_C.mid\/Diminished_fifth_tritone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Diminished fifth tritone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a8/Diminished_fifth_tritone_on_C.mid/Diminished_fifth_tritone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Diminished_fifth_tritone_on_C.mid" title="File:Diminished fifth tritone on C.mid">ⓘ</a> </td> <td>588.27 </td> <td>-11.73 </td></tr> <tr> <td><a href="/wiki/Just_augmented_fourth" class="mw-redirect" title="Just augmented fourth">Just augmented fourth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 45⁄ 32 = 1.40625 </td> <td><span data-nosnippet="" id="ooui-php-45" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/8\/80\/Just_augmented_fourth_on_C.mid\/Just_augmented_fourth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just augmented fourth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/8/80/Just_augmented_fourth_on_C.mid/Just_augmented_fourth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">Play</a><a href="/wiki/File:Just_augmented_fourth_on_C.mid" title="File:Just augmented fourth on C.mid">ⓘ</a> </td> <td>590.22 </td> <td>-9.78 </td></tr> <tr> <td><a href="/wiki/Just_diminished_fifth" class="mw-redirect" title="Just diminished fifth">Just diminished fifth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 64⁄ 45 = 1.42222... </td> <td><span data-nosnippet="" id="ooui-php-46" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f7\/Just_tritone_on_C.mid\/Just_tritone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just tritone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f7/Just_tritone_on_C.mid/Just_tritone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_tritone_on_C.mid" title="File:Just tritone on C.mid">ⓘ</a> </td> <td>609.78 </td> <td>+9.78 </td></tr> <tr> <td><a href="/wiki/Pythagorean_augmented_fourth" class="mw-redirect" title="Pythagorean augmented fourth">Pythagorean augmented fourth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 729⁄ 512 = 1.42382... </td> <td><span data-nosnippet="" id="ooui-php-47" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/c\/cf\/Pythagorean_augmented_fourth_on_C.mid\/Pythagorean_augmented_fourth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean augmented fourth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/c/cf/Pythagorean_augmented_fourth_on_C.mid/Pythagorean_augmented_fourth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_augmented_fourth_on_C.mid" title="File:Pythagorean augmented fourth on C.mid">ⓘ</a> </td> <td>611.73 </td> <td>+11.73 </td></tr> <tr> <td><a href="/wiki/Euler%27s_tritone" class="mw-redirect" title="Euler's tritone">Euler's tritone</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 10⁄ 7 = 1.42857... </td> <td><span data-nosnippet="" id="ooui-php-48" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/e\/e3\/Greater_septimal_tritone_on_C.mid\/Greater_septimal_tritone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Greater septimal tritone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e3/Greater_septimal_tritone_on_C.mid/Greater_septimal_tritone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">Play</a><a href="/wiki/File:Greater_septimal_tritone_on_C.mid" title="File:Greater septimal tritone on C.mid">ⓘ</a> </td> <td>617.49 </td> <td>+17.49 </td></tr> <tr> <td>Classic diminished fifth </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 36⁄ 25 = 1.44 </td> <td><span data-nosnippet="" id="ooui-php-49" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/b\/b8\/Just_diminished_fifth_on_C.mid\/Just_diminished_fifth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just diminished fifth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/b8/Just_diminished_fifth_on_C.mid/Just_diminished_fifth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_diminished_fifth_on_C.mid" title="File:Just diminished fifth on C.mid">ⓘ</a> </td> <td>631.28 </td> <td>+31.28 </td></tr> <tr> <td rowspan="2">7 </td> <td rowspan="2"><a href="/wiki/G_(musical_note)" title="G (musical note)"> G</a> </td> <td rowspan="2">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 7⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{12}]{128}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>128</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{12}]{128}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe1257282e06f592d5b60e9ce503586594b865c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.432ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{12}]{128}}}"> </td> <td rowspan="2">1.49830... </td> <td rowspan="2"><span data-nosnippet="" id="ooui-php-50" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/2\/20\/Perfect_fifth_on_C.mid\/Perfect_fifth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Perfect fifth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/20/Perfect_fifth_on_C.mid/Perfect_fifth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Perfect_fifth_on_C.mid" title="File:Perfect fifth on C.mid">ⓘ</a> </td> <td rowspan="2">700 </td> <td><a href="/wiki/Pythagorean_diminished_sixth" class="mw-redirect" title="Pythagorean diminished sixth">Pythagorean diminished sixth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 262144⁄ 177147 = 1.47981... </td> <td><span data-nosnippet="" id="ooui-php-51" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/d\/d2\/Pythagorean_diminished_sixth_on_C.mid\/Pythagorean_diminished_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean diminished sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d2/Pythagorean_diminished_sixth_on_C.mid/Pythagorean_diminished_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_diminished_sixth_on_C.mid" title="File:Pythagorean diminished sixth on C.mid">ⓘ</a> </td> <td>678.49 </td></tr> <tr> <td><a href="/wiki/Just_perfect_fifth" class="mw-redirect" title="Just perfect fifth">Just perfect fifth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 3⁄ 2 = 1.5 </td> <td><span data-nosnippet="" id="ooui-php-52" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/5\/5c\/Just_perfect_fifth_on_C.mid\/Just_perfect_fifth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just perfect fifth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5c/Just_perfect_fifth_on_C.mid/Just_perfect_fifth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_perfect_fifth_on_C.mid" title="File:Just perfect fifth on C.mid">ⓘ</a> </td> <td>701.96 </td> <td>+1.96 </td></tr> <tr> <td rowspan="5">8 </td> <td rowspan="5"><a href="/wiki/G%E2%99%AF_(musical_note)" title="G♯ (musical note)"> G♯</a> or <a href="/wiki/A%E2%99%AD_(musical_note)" title="A♭ (musical note)"> A♭</a> </td> <td rowspan="5">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 8⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb5da5142ec773ea1ba79813278a00c8d9ee202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{4}}}"> </td> <td rowspan="5">1.58740... </td> <td rowspan="5"><span data-nosnippet="" id="ooui-php-53" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/9\/9a\/Minor_sixth_on_C.mid\/Minor_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Minor sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Minor_sixth_on_C.mid/Minor_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Minor_sixth_on_C.mid" title="File:Minor sixth on C.mid">ⓘ</a> </td> <td rowspan="5">800 </td> <td><a href="/wiki/Septimal_minor_sixth" class="mw-redirect" title="Septimal minor sixth">Septimal minor sixth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 14⁄ 9 = 1.55555... </td> <td><span data-nosnippet="" id="ooui-php-54" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/3\/3a\/Septimal_minor_sixth_on_C.mid\/Septimal_minor_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal minor sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3a/Septimal_minor_sixth_on_C.mid/Septimal_minor_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Septimal_minor_sixth_on_C.mid" title="File:Septimal minor sixth on C.mid">ⓘ</a> </td> <td>764.92 </td> <td>-35.08 </td></tr> <tr> <td><a href="/wiki/Undecimal_minor_sixth" class="mw-redirect" title="Undecimal minor sixth">Undecimal minor sixth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 11⁄ 7 = 1.57142... </td> <td><span data-nosnippet="" id="ooui-php-55" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/7\/7c\/Undecimal_minor_sixth_on_C.mid\/Undecimal_minor_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Undecimal minor sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/7/7c/Undecimal_minor_sixth_on_C.mid/Undecimal_minor_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Undecimal_minor_sixth_on_C.mid" title="File:Undecimal minor sixth on C.mid">ⓘ</a> </td> <td>782.49 </td> <td>-17.51 </td></tr> <tr> <td><a href="/wiki/Pythagorean_minor_sixth" class="mw-redirect" title="Pythagorean minor sixth">Pythagorean minor sixth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 128⁄ 81 = 1.58024... </td> <td><span data-nosnippet="" id="ooui-php-56" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/6\/68\/Pythagorean_minor_sixth_on_C.mid\/Pythagorean_minor_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean minor sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/6/68/Pythagorean_minor_sixth_on_C.mid/Pythagorean_minor_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_minor_sixth_on_C.mid" title="File:Pythagorean minor sixth on C.mid">ⓘ</a> </td> <td>792.18 </td> <td>-7.82 </td></tr> <tr> <td><a href="/wiki/Just_minor_sixth" class="mw-redirect" title="Just minor sixth">Just minor sixth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 8⁄ 5 = 1.6 </td> <td><span data-nosnippet="" id="ooui-php-57" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/41\/Just_minor_sixth_on_C.mid\/Just_minor_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just minor sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/41/Just_minor_sixth_on_C.mid/Just_minor_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_minor_sixth_on_C.mid" title="File:Just minor sixth on C.mid">ⓘ</a> </td> <td>813.69 </td> <td>+13.69 </td></tr> <tr> <td><a href="/wiki/Pythagorean_augmented_fifth" class="mw-redirect" title="Pythagorean augmented fifth">Pythagorean augmented fifth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 6561⁄ 4096 = 1.60180... </td> <td><span data-nosnippet="" id="ooui-php-58" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/9\/90\/Pythagorean_augmented_fifth_on_C.mid\/Pythagorean_augmented_fifth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean augmented fifth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/90/Pythagorean_augmented_fifth_on_C.mid/Pythagorean_augmented_fifth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_augmented_fifth_on_C.mid" title="File:Pythagorean augmented fifth on C.mid">ⓘ</a> </td> <td>815.64 </td> <td>+15.64 </td></tr> <tr> <td rowspan="5">9 </td> <td rowspan="5"><a href="/wiki/A_(musical_note)" title="A (musical note)"> A</a> </td> <td rowspan="5">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 9⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{4}]{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{4}]{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675096532680297bc0b8e3ef793ed8b9271f628e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{4}]{8}}}"> </td> <td rowspan="5">1.68179... </td> <td rowspan="5"><span data-nosnippet="" id="ooui-php-59" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/d\/df\/Major_sixth_on_C.mid\/Major_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/d/df/Major_sixth_on_C.mid/Major_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Major_sixth_on_C.mid" title="File:Major sixth on C.mid">ⓘ</a> </td> <td rowspan="5">900 </td> <td><a href="/wiki/Pythagorean_diminished_seventh" class="mw-redirect" title="Pythagorean diminished seventh">Pythagorean diminished seventh</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 32768⁄ 19683 = 1.66478... </td> <td><span data-nosnippet="" id="ooui-php-60" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/b\/bb\/Pythagorean_diminished_seventh_on_C.mid\/Pythagorean_diminished_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean diminished seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bb/Pythagorean_diminished_seventh_on_C.mid/Pythagorean_diminished_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_diminished_seventh_on_C.mid" title="File:Pythagorean diminished seventh on C.mid">ⓘ</a> </td> <td>882.40 </td></tr> <tr> <td><a href="/wiki/Just_major_sixth" class="mw-redirect" title="Just major sixth">Just major sixth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 5⁄ 3 = 1.66666... </td> <td><span data-nosnippet="" id="ooui-php-61" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/9\/98\/Just_major_sixth_on_C.mid\/Just_major_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/98/Just_major_sixth_on_C.mid/Just_major_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_major_sixth_on_C.mid" title="File:Just major sixth on C.mid">ⓘ</a> </td> <td>884.36 </td></tr> <tr> <td><a href="/wiki/Nineteenth_subharmonic" class="mw-redirect" title="Nineteenth subharmonic">Nineteenth subharmonic</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 32⁄ 19 = 1.68421... </td> <td><span data-nosnippet="" id="ooui-php-62" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/5\/57\/19th_subharmonic_on_C.mid\/19th_subharmonic_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"19th subharmonic on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/57/19th_subharmonic_on_C.mid/19th_subharmonic_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:19th_subharmonic_on_C.mid" title="File:19th subharmonic on C.mid">ⓘ</a> </td> <td>902.49 </td> <td>+2.49 </td></tr> <tr> <td><a href="/wiki/Pythagorean_major_sixth" class="mw-redirect" title="Pythagorean major sixth">Pythagorean major sixth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 27⁄ 16 = 1.6875 </td> <td><span data-nosnippet="" id="ooui-php-63" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/5\/5e\/Pythagorean_major_sixth_on_C.mid\/Pythagorean_major_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean major sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5e/Pythagorean_major_sixth_on_C.mid/Pythagorean_major_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_major_sixth_on_C.mid" title="File:Pythagorean major sixth on C.mid">ⓘ</a> </td> <td>905.87 </td> <td>+5.87 </td></tr> <tr> <td><a href="/wiki/Septimal_major_sixth" class="mw-redirect" title="Septimal major sixth">Septimal major sixth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 12⁄ 7 = 1.71428... </td> <td><span data-nosnippet="" id="ooui-php-64" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/9\/94\/Septimal_major_sixth_on_C.mid\/Septimal_major_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Septimal major sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/94/Septimal_major_sixth_on_C.mid/Septimal_major_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">Play</a><a href="/wiki/File:Septimal_major_sixth_on_C.mid" title="File:Septimal major sixth on C.mid">ⓘ</a> </td> <td>933.13 </td> <td>+33.13 </td></tr> <tr> <td rowspan="4">10 </td> <td rowspan="4"><a href="/wiki/A%E2%99%AF_(musical_note)" title="A♯ (musical note)"> A♯</a> or <a href="/wiki/B%E2%99%AD_(musical_note)" title="B♭ (musical note)"> B♭</a> </td> <td rowspan="4">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 10⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{6}]{32}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>32</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{6}]{32}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75f5ed7285d717db47b68736ff2e37df9f71737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{6}]{32}}}"> </td> <td rowspan="4">1.78179... </td> <td rowspan="4"><span data-nosnippet="" id="ooui-php-65" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/a\/a5\/Minor_seventh_on_C.mid\/Minor_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Minor seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a5/Minor_seventh_on_C.mid/Minor_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Minor_seventh_on_C.mid" title="File:Minor seventh on C.mid">ⓘ</a> </td> <td rowspan="4">1000 </td> <td><a href="/wiki/Harmonic_seventh" title="Harmonic seventh">Harmonic seventh</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 7⁄ 4 = 1.75 </td> <td><span data-nosnippet="" id="ooui-php-66" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/d\/dd\/Harmonic_seventh_on_C.mid\/Harmonic_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Harmonic seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/d/dd/Harmonic_seventh_on_C.mid/Harmonic_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Harmonic_seventh_on_C.mid" title="File:Harmonic seventh on C.mid">ⓘ</a> </td> <td>968.83 </td> <td>-31.17 </td></tr> <tr> <td><a href="/wiki/Pythagorean_minor_seventh" class="mw-redirect" title="Pythagorean minor seventh">Pythagorean minor seventh</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 16⁄ 9 = 1.77777... </td> <td><span data-nosnippet="" id="ooui-php-67" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/e\/e9\/Lesser_just_minor_seventh_on_C.mid\/Lesser_just_minor_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Lesser just minor seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e9/Lesser_just_minor_seventh_on_C.mid/Lesser_just_minor_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Lesser_just_minor_seventh_on_C.mid" title="File:Lesser just minor seventh on C.mid">ⓘ</a> </td> <td>996.09 </td> <td>-3.91 </td></tr> <tr> <td><a href="/wiki/Large_minor_seventh" class="mw-redirect" title="Large minor seventh">Large minor seventh</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 9⁄ 5 = 1.8 </td> <td><span data-nosnippet="" id="ooui-php-68" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/e\/e2\/Greater_just_minor_seventh_on_C.mid\/Greater_just_minor_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Greater just minor seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e2/Greater_just_minor_seventh_on_C.mid/Greater_just_minor_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Greater_just_minor_seventh_on_C.mid" title="File:Greater just minor seventh on C.mid">ⓘ</a> </td> <td>1017.60 </td> <td>+17.60 </td></tr> <tr> <td><a href="/wiki/Pythagorean_augmented_sixth" class="mw-redirect" title="Pythagorean augmented sixth">Pythagorean augmented sixth</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 59049⁄ 32768 = 1.80203... </td> <td><span data-nosnippet="" id="ooui-php-69" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/2\/26\/Pythagorean_augmented_sixth_on_C.mid\/Pythagorean_augmented_sixth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean augmented sixth on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/26/Pythagorean_augmented_sixth_on_C.mid/Pythagorean_augmented_sixth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_augmented_sixth_on_C.mid" title="File:Pythagorean augmented sixth on C.mid">ⓘ</a> </td> <td>1019.55 </td> <td>+19.55 </td></tr> <tr> <td rowspan="6">11 </td> <td rowspan="6"><a href="/wiki/B_(musical_note)" title="B (musical note)"> B</a> </td> <td rowspan="6">2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 11⁄ 12 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{12}]{2048}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2048</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{12}]{2048}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/744decb84a19d221e7ce8df0ec3d315384fdd5ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.594ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{12}]{2048}}}"> </td> <td rowspan="6">1.88774... </td> <td rowspan="6"><span data-nosnippet="" id="ooui-php-70" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/c\/cb\/Major_seventh_on_C.mid\/Major_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/c/cb/Major_seventh_on_C.mid/Major_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Major_seventh_on_C.mid" title="File:Major seventh on C.mid">ⓘ</a> </td> <td rowspan="6">1100 </td> <td>Tridecimal neutral seventh </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 13⁄ 7 = 1.85714... </td> <td><span data-nosnippet="" id="ooui-php-71" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/44\/Tridecimal_neutral_seventh_on_C.mid\/Tridecimal_neutral_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Tridecimal neutral seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/44/Tridecimal_neutral_seventh_on_C.mid/Tridecimal_neutral_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Tridecimal_neutral_seventh_on_C.mid" title="File:Tridecimal neutral seventh on C.mid">ⓘ</a> </td> <td>1071.70 </td> <td>-28.30 </td></tr> <tr> <td><a href="/wiki/Pythagorean_diminished_octave" class="mw-redirect" title="Pythagorean diminished octave">Pythagorean diminished octave</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 4096⁄ 2187 = 1.87288... </td> <td><span data-nosnippet="" id="ooui-php-72" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/5\/59\/Pythagorean_diminished_octave_on_C.mid\/Pythagorean_diminished_octave_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean diminished octave on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/59/Pythagorean_diminished_octave_on_C.mid/Pythagorean_diminished_octave_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_diminished_octave_on_C.mid" title="File:Pythagorean diminished octave on C.mid">ⓘ</a> </td> <td>1086.31 </td> <td>-13.69 </td></tr> <tr> <td><a href="/wiki/Just_major_seventh" class="mw-redirect" title="Just major seventh">Just major seventh</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 15⁄ 8 = 1.875 </td> <td><span data-nosnippet="" id="ooui-php-73" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/a\/a4\/Just_major_seventh_on_C.mid\/Just_major_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/a/a4/Just_major_seventh_on_C.mid/Just_major_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_major_seventh_on_C.mid" title="File:Just major seventh on C.mid">ⓘ</a> </td> <td>1088.27 </td> <td>-11.73 </td></tr> <tr> <td><a href="/wiki/Seventeenth_subharmonic" class="mw-redirect" title="Seventeenth subharmonic">Seventeenth subharmonic</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 32⁄ 17 = 1.88235... </td> <td><span data-nosnippet="" id="ooui-php-74" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/4a\/17th_subharmonic_on_C.mid\/17th_subharmonic_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"17th subharmonic on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4a/17th_subharmonic_on_C.mid/17th_subharmonic_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:17th_subharmonic_on_C.mid" title="File:17th subharmonic on C.mid">ⓘ</a> </td> <td>1095.04 </td> <td>-4.96 </td></tr> <tr> <td><a href="/wiki/Pythagorean_major_seventh" class="mw-redirect" title="Pythagorean major seventh">Pythagorean major seventh</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 243⁄ 128 = 1.89843... </td> <td><span data-nosnippet="" id="ooui-php-75" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f4\/Pythagorean_major_seventh_on_C.mid\/Pythagorean_major_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean major seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f4/Pythagorean_major_seventh_on_C.mid/Pythagorean_major_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_major_seventh_on_C.mid" title="File:Pythagorean major seventh on C.mid">ⓘ</a> </td> <td>1109.78 </td> <td>+9.78 </td></tr> <tr> <td>Septimal major seventh </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 27⁄ 14 = 1.92857... </td> <td><span data-nosnippet="" id="ooui-php-76" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f4\/Pythagorean_major_seventh_on_C.mid\/Pythagorean_major_seventh_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Pythagorean major seventh on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f4/Pythagorean_major_seventh_on_C.mid/Pythagorean_major_seventh_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Pythagorean_major_seventh_on_C.mid" title="File:Pythagorean major seventh on C.mid">ⓘ</a> </td> <td>1137.04 </td> <td>+37.04 </td></tr> <tr> <td>12 </td> <td><a href="/wiki/C_(musical_note)" title="C (musical note)"> C</a> </td> <td>2 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 12⁄ 12 = 2 </td> <td>2 </td> <td><span data-nosnippet="" id="ooui-php-77" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f0\/Perfect_octave_on_C.mid\/Perfect_octave_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Perfect octave on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f0/Perfect_octave_on_C.mid/Perfect_octave_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Perfect_octave_on_C.mid" title="File:Perfect octave on C.mid">ⓘ</a> </td> <td>1200 </td> <td><a href="/wiki/Octave" title="Octave">Octave</a> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"> 2⁄ 1 = 2 </td> <td><span data-nosnippet="" id="ooui-php-78" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f0\/Perfect_octave_on_C.mid\/Perfect_octave_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Perfect octave on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f0/Perfect_octave_on_C.mid/Perfect_octave_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Perfect_octave_on_C.mid" title="File:Perfect octave on C.mid">ⓘ</a> </td> <td>1200.00 </td> <td>0 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Commas">Commas</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=20" title="Edit section: Commas">edit</a>]</div> 12-ET tempers out several <a href="/wiki/Comma_(music)" title="Comma (music)">commas</a>, meaning that there are several fractions close to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 1 /1⁠ that are treated as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 1 /1⁠ by 12-ET due to its mapping of different fractions to the same equally-tempered interval. For example, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠729/512⁠ (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 36 /29⁠) and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 1024 /729⁠ (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 210 /36⁠) are each mapped to the tritone, so they are treated as nominally the same interval; therefore, their quotient, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠531441/ 524288 ⁠ (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 312 /219⁠) is mapped to/treated as unison. This is the <a href="/wiki/Pythagorean_comma" title="Pythagorean comma">Pythagorean comma</a>, and it is 12-ET's only 3-limit comma. However, as one increases the prime limit and includes more intervals, the number of commas increases. 12-ET's most important five-limit comma is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/ 80 ⁠ (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠34/ 24 × 51 ⁠), which is known as the <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a> and is the factor between Pythagorean thirds and sixths and their just counterparts. 12-ET's other 5-limit commas include: <ul><li><a href="/wiki/Schisma" title="Schisma">Schisma</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠32805/ 32768 ⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 38 × 51 /215⁠ = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠531441/ 524288 ⁠)1 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/ 80 ⁠)−1</li> <li><a href="/wiki/Diaschisma" title="Diaschisma">Diaschisma</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2048/ 2025 ⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠211/ 34 × 52 ⁠ = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠531441/ 524288 ⁠)−1 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/ 80 ⁠)2</li> <li><a href="/wiki/Lesser_diesis" class="mw-redirect" title="Lesser diesis">Lesser diesis</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠128/ 125 ⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 27 /53⁠ = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠531441/ 524288 ⁠)−1 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/ 80 ⁠)3</li> <li><a href="/wiki/Greater_diesis" class="mw-redirect" title="Greater diesis">Greater diesis</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠648/ 625 ⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 23 × 34 /54⁠=(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠531441/ 524288 ⁠)−1 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/ 80 ⁠)4</li></ul> One of the 7-limit commas that 12-ET tempers out is the <a href="/wiki/Septimal_kleisma" title="Septimal kleisma">septimal kleisma</a>, which is equal to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠225/ 224 ⁠, or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 32×52 /25×71⁠. 12-ET's other 7-limit commas include: <ul><li><a href="/wiki/Septimal_semicomma" title="Septimal semicomma">Septimal semicomma</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠126/ 125 ⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 21 × 32 × 71 /53⁠ = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/ 80 ⁠)1 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠225/ 224 ⁠)−1</li> <li><a href="/wiki/Archytas%27_comma" class="mw-redirect" title="Archytas' comma">Archytas' comma</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠64/ 63 ⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠26/ 32 × 71 ⁠ = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠531441/ 524288 ⁠)−1 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/ 80 ⁠)2 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠225/ 224 ⁠)1</li> <li><a href="/wiki/Septimal_quarter_tone" title="Septimal quarter tone">Septimal quarter tone</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠36/ 35 ⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 22 × 32 /51 ×v71⁠ = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠531441/ 524288 ⁠)−1 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/80⁠)3 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠225/ 224 ⁠)1</li> <li><a href="/wiki/Jubilisma" class="mw-redirect" title="Jubilisma">Jubilisma</a>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠50/ 49 ⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠ 21 × 52 /72⁠ = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠531441/ 524288 ⁠)−1 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠81/ 80 ⁠)2 × (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠225/ 224 ⁠)2</li></ul> <div class="mw-heading mw-heading2"><h2 id="Similar_tuning_systems">Similar tuning systems</h2>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=21" title="Edit section: Similar tuning systems">edit</a>]</div> Historically, multiple tuning systems have been used that can be seen as slight variations of 12-TEDO, with twelve notes per octave but with some variation among interval sizes so that the notes are not quite equally-spaced. One example of this a three-limit scale where equally-tempered perfect fifths of 700 cents are replaced with justly-intoned perfect fifths of 701.955 cents. Because the two intervals differ by less than 2 cents, or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄600 of an octave, the two scales are very similar. In fact, the Chinese developed <a href="/wiki/Sh%C3%AD-%C3%A8r-l%C7%9C" class="mw-redirect" title="Shí-èr-lǜ">3-limit just intonation</a> at least a century before He Chengtian created the sequence of 12-TEDO.<a href="#cite_note-FOOTNOTENeedhamLingRobinson1962170–171-40">[38]</a> Likewise, Pythagorean tuning, which was developed by ancient Greeks, was the predominant system in Europe until during the Renaissance, when Europeans realized that dissonant intervals such as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">81⁄64<a href="#cite_note-FOOTNOTEBenwardSaker200356-41">[39]</a> could be made more consonant by tempering them to simpler ratios like <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">5⁄4, resulting in Europe developing a series of <a href="/wiki/Meantone_temperament" title="Meantone temperament">meantone temperaments</a> that slightly modified the interval sizes but could still be viewed as an approximate of 12-TEDO. Due to meantone temperaments' tendency to concentrate error onto one enharmonic perfect fifth, making it <a href="/wiki/Wolf_fifth" class="mw-redirect" title="Wolf fifth">very dissonant</a>, European music theorists, such as Andreas Werckmeister, Johann Philipp Kirnberger, Francesco Antonio Vallotti, and Thomas Young, created various <a href="/wiki/Well_temperament" title="Well temperament">well temperaments</a> with the goal of dividing up the commas in order to reduce the dissonance of the worst-affected intervals. Werckmeister and Kirnberger were each dissatisfied with his first temperament and therefore created multiple temperaments, the latter temperaments more closely approximating equal temperament than the former temperaments. Likewise, Europe as a whole gradually transitioned from meantone and well temperaments to 12-TEDO, the system that it still uses today. <div class="mw-heading mw-heading2"><h2 id="Subsets">Subsets</h2>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=22" title="Edit section: Subsets">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Scale_(music)" title="Scale (music)">Scale (music)</a></div> While some types of music, such as <a href="/wiki/Serialism" title="Serialism">serialism</a>, use all twelve notes of 12-TEDO, most music only uses notes from a particular subset of 12-TEDO known as a scale. Many different types of scales exist. The most popular type of scale in 12-TEDO is meantone. Meantone refers to any scale where all of its notes are consecutive on the circle of fifths. Meantone scales of different sizes exist, and some meantone scales used include <a href="/wiki/Pentatonic_scale#Pentatonic_scales_found_by_running_up_the_keys_C,_D,_E,_G_and_A" title="Pentatonic scale">five-note meantone</a>, <a href="/wiki/Diatonic_scale" title="Diatonic scale">seven-note meantone</a>, and <a href="/wiki/Blues_scale#Nonatonic" title="Blues scale">nine-note meantone</a>. Meantone is present in the design of Western instruments. For example, the keys of a piano and its predecessors are structured so that the white keys form a seven-note meantone scale and the black keys form a five-note meantone scale. Another example is that guitars and other string instruments with at least five strings are typically tuned so that their open strings form a five-note meantone scale. Other scales used in 12-TEDO include the <a href="/wiki/Ascending_melodic_minor_scale" class="mw-redirect" title="Ascending melodic minor scale">ascending melodic minor scale</a>, the <a href="/wiki/Harmonic_minor" class="mw-redirect" title="Harmonic minor">harmonic minor</a>, the <a href="/wiki/Harmonic_major" class="mw-redirect" title="Harmonic major">harmonic major</a>, the <a href="/wiki/Diminished_scale" class="mw-redirect" title="Diminished scale">diminished scale</a>, and the <a href="/wiki/In_scale" title="In scale">in scale</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=23" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Equal_temperament" title="Equal temperament">Equal temperament</a></li> <li><a href="/wiki/Just_intonation" title="Just intonation">Just intonation</a></li> <li><a href="/wiki/Musical_acoustics" title="Musical acoustics">Musical acoustics</a> (the physics of music)</li> <li><a href="/wiki/Music_and_mathematics" title="Music and mathematics">Music and mathematics</a></li> <li><a href="/wiki/Microtonal_music" class="mw-redirect" title="Microtonal music">Microtonal music</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/Diatonic_and_chromatic" title="Diatonic and chromatic">Diatonic and chromatic</a></li> <li><a href="/wiki/Electronic_tuner" title="Electronic tuner">Electronic tuner</a></li> <li><a href="/wiki/Musical_tuning" title="Musical tuning">Musical tuning</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=24" title="Edit section: References">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Footnotes">Footnotes</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=25" title="Edit section: Footnotes">edit</a>]</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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-12TET-1"><a href="#cite_ref-12TET_1-0">^</a> Also known as twelve-tone equal temperament (12-TET), 12-tone equal division of the octave (12-TEDO), 12 equal division of 2/1 (12-ED2), 12 equal division of the octave (12-EDO); informally abbreviated to twelve equal or referred to as equal temperament without qualification in <a href="/wiki/Western_world" title="Western world">Western countries</a>. </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=26" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEvon_HelmholtzEllis1885493–511-2"><a href="#cite_ref-FOOTNOTEvon_HelmholtzEllis1885493–511_2-0">^</a> <a href="#CITEREFvon_HelmholtzEllis1885">von Helmholtz & Ellis 1885</a>, pp. 493–511. </li> <li id="cite_note-FOOTNOTEKuttner1975163-3">^ <a href="#cite_ref-FOOTNOTEKuttner1975163_3-0">a</a> <a href="#cite_ref-FOOTNOTEKuttner1975163_3-1">b</a> <a href="#CITEREFKuttner1975">Kuttner 1975</a>, p. 163. </li> <li id="cite_note-FOOTNOTEKuttner1975200-4">^ <a href="#cite_ref-FOOTNOTEKuttner1975200_4-0">a</a> <a href="#cite_ref-FOOTNOTEKuttner1975200_4-1">b</a> <a href="#CITEREFKuttner1975">Kuttner 1975</a>, p. 200. </li> <li id="cite_note-FOOTNOTERobinson1980vii-5"><a href="#cite_ref-FOOTNOTERobinson1980vii_5-0">^</a> <a href="#CITEREFRobinson1980">Robinson 1980</a>, p. vii: Chu-Tsaiyu the first formulator of the mathematics of "equal temperament" anywhere in the world </li> <li id="cite_note-FOOTNOTENeedhamLingRobinson1962221-6">^ <a href="#cite_ref-FOOTNOTENeedhamLingRobinson1962221_6-0">a</a> <a href="#cite_ref-FOOTNOTENeedhamLingRobinson1962221_6-1">b</a> <a href="#cite_ref-FOOTNOTENeedhamLingRobinson1962221_6-2">c</a> <a href="#CITEREFNeedhamLingRobinson1962">Needham, Ling & Robinson 1962</a>, p. 221. </li> <li id="cite_note-FOOTNOTEKwang-chih_ChangPingfang_XuLiancheng_Lu2005140-7"><a href="#cite_ref-FOOTNOTEKwang-chih_ChangPingfang_XuLiancheng_Lu2005140_7-0">^</a> <a href="#CITEREFKwang-chih_ChangPingfang_XuLiancheng_Lu2005">Kwang-chih Chang, Pingfang Xu & Liancheng Lu 2005</a>, p. 140. </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <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="CITEREFGoodmanLien2009" class="citation journal cs1">Goodman, Howard L.; Lien, Y. Edmund (April 2009). "A Third Century AD Chinese System of Di-Flute Temperament: Matching Ancient Pitch-Standards and Confronting Modal Practice". <a href="/wiki/The_Galpin_Society_Journal" class="mw-redirect" title="The Galpin Society Journal">The Galpin Society Journal</a>. 62. Galpin Society: 7. <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/20753625">20753625</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Galpin+Society+Journal&rft.atitle=A+Third+Century+AD+Chinese+System+of+Di-Flute+Temperament%3A+Matching+Ancient+Pitch-Standards+and+Confronting+Modal+Practice&rft.volume=62&rft.pages=7&rft.date=2009-04&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F20753625%23id-name%3DJSTOR&rft.aulast=Goodman&rft.aufirst=Howard+L.&rft.au=Lien%2C+Y.+Edmund&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"> </li> <li id="cite_note-FOOTNOTEBarbour200455–56-9"><a href="#cite_ref-FOOTNOTEBarbour200455–56_9-0">^</a> <a href="#CITEREFBarbour2004">Barbour 2004</a>, pp. 55–56. </li> <li id="cite_note-FOOTNOTEHart1998-10"><a href="#cite_ref-FOOTNOTEHart1998_10-0">^</a> <a href="#CITEREFHart1998">Hart 1998</a>. </li> <li id="cite_note-FOOTNOTENeedhamRonan1978385-11"><a href="#cite_ref-FOOTNOTENeedhamRonan1978385_11-0">^</a> <a href="#CITEREFNeedhamRonan1978">Needham & Ronan 1978</a>, p. 385. </li> <li id="cite_note-FOOTNOTECho2010-12">^ <a href="#cite_ref-FOOTNOTECho2010_12-0">a</a> <a href="#cite_ref-FOOTNOTECho2010_12-1">b</a> <a href="#CITEREFCho2010">Cho 2010</a>. </li> <li id="cite_note-FOOTNOTELienhard1997-13">^ <a href="#cite_ref-FOOTNOTELienhard1997_13-0">a</a> <a href="#cite_ref-FOOTNOTELienhard1997_13-1">b</a> <a href="#CITEREFLienhard1997">Lienhard 1997</a>. </li> <li id="cite_note-FOOTNOTEChristensen2002205-14">^ <a href="#cite_ref-FOOTNOTEChristensen2002205_14-0">a</a> <a href="#cite_ref-FOOTNOTEChristensen2002205_14-1">b</a> <a href="#CITEREFChristensen2002">Christensen 2002</a>, p. 205. </li> <li id="cite_note-FOOTNOTEBarbour20047-15"><a href="#cite_ref-FOOTNOTEBarbour20047_15-0">^</a> <a href="#CITEREFBarbour2004">Barbour 2004</a>, p. 7. </li> <li id="cite_note-FOOTNOTEvon_HelmholtzEllis1885258-16"><a href="#cite_ref-FOOTNOTEvon_HelmholtzEllis1885258_16-0">^</a> <a href="#CITEREFvon_HelmholtzEllis1885">von Helmholtz & Ellis 1885</a>, p. 258. </li> <li id="cite_note-FOOTNOTETrue201861–74-17"><a href="#cite_ref-FOOTNOTETrue201861–74_17-0">^</a> <a href="#CITEREFTrue2018">True 2018</a>, pp. 61–74. </li> <li id="cite_note-FOOTNOTEGalilei158480–89-18"><a href="#cite_ref-FOOTNOTEGalilei158480–89_18-0">^</a> <a href="#CITEREFGalilei1584">Galilei 1584</a>, pp. 80–89. </li> <li id="cite_note-FOOTNOTEBarbour20048-19"><a href="#cite_ref-FOOTNOTEBarbour20048_19-0">^</a> <a href="#CITEREFBarbour2004">Barbour 2004</a>, p. 8. </li> <li id="cite_note-FOOTNOTEde_Gorzanis1981-20"><a href="#cite_ref-FOOTNOTEde_Gorzanis1981_20-0">^</a> <a href="#CITEREFde_Gorzanis1981">de Gorzanis 1981</a>. </li> <li id="cite_note-appstate.edu-21"><a href="#cite_ref-appstate.edu_21-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110725182053/http://www.library.appstate.edu/music/lute/16index/tspi07a.html">"Spinacino 1507a: Thematic Index"</a>. Appalachian State University. Archived from <a rel="nofollow" class="external text" href="http://www.library.appstate.edu/music/lute/16index/tspi07a.html">the original</a> on 2011-07-25. Retrieved 2012-06-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Spinacino+1507a%3A+Thematic+Index&rft.pub=Appalachian+State+University&rft_id=http%3A%2F%2Fwww.library.appstate.edu%2Fmusic%2Flute%2F16index%2Ftspi07a.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"> </li> <li id="cite_note-FOOTNOTEWilson1997-22"><a href="#cite_ref-FOOTNOTEWilson1997_22-0">^</a> <a href="#CITEREFWilson1997">Wilson 1997</a>. </li> <li id="cite_note-FOOTNOTEJorgens1986-23"><a href="#cite_ref-FOOTNOTEJorgens1986_23-0">^</a> <a href="#CITEREFJorgens1986">Jorgens 1986</a>. </li> <li id="cite_note-Scintille-24"><a href="#cite_ref-Scintille_24-0">^</a> "Scintille de musica", (Brescia, 1533), p. 132 </li> <li id="cite_note-FOOTNOTECohen1987471–488-25"><a href="#cite_ref-FOOTNOTECohen1987471–488_25-0">^</a> <a href="#CITEREFCohen1987">Cohen 1987</a>, pp. 471–488. </li> <li id="cite_note-FOOTNOTECho2003223-26"><a href="#cite_ref-FOOTNOTECho2003223_26-0">^</a> <a href="#CITEREFCho2003">Cho 2003</a>, p. 223. </li> <li id="cite_note-FOOTNOTECho2003222-27"><a href="#cite_ref-FOOTNOTECho2003222_27-0">^</a> <a href="#CITEREFCho2003">Cho 2003</a>, p. 222. </li> <li id="cite_note-FOOTNOTEChristensen2002207-28"><a href="#cite_ref-FOOTNOTEChristensen2002207_28-0">^</a> <a href="#CITEREFChristensen2002">Christensen 2002</a>, p. 207. </li> <li id="cite_note-FOOTNOTEChristensen200278-29"><a href="#cite_ref-FOOTNOTEChristensen200278_29-0">^</a> <a href="#CITEREFChristensen2002">Christensen 2002</a>, p. 78. </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> Lindley, Mark. Lutes, Viols, Temperaments. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-28883-5" title="Special:BookSources/978-0-521-28883-5">978-0-521-28883-5</a> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> Vm7 6214 </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> Andreas Werckmeister (1707), Musicalische Paradoxal-Discourse </li> <li id="cite_note-FOOTNOTEDi_Veroli2009140,_142_and_256-34"><a href="#cite_ref-FOOTNOTEDi_Veroli2009140,_142_and_256_34-0">^</a> <a href="#CITEREFDi_Veroli2009">Di Veroli 2009</a>, pp. 140, 142 and 256. </li> <li id="cite_note-FOOTNOTEMoody2003-35"><a href="#cite_ref-FOOTNOTEMoody2003_35-0">^</a> <a href="#CITEREFMoody2003">Moody 2003</a>. </li> <li id="cite_note-FOOTNOTEvon_HelmholtzEllis1885548-36"><a href="#cite_ref-FOOTNOTEvon_HelmholtzEllis1885548_36-0">^</a> <a href="#CITEREFvon_HelmholtzEllis1885">von Helmholtz & Ellis 1885</a>, p. 548. </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhite1946" class="citation book cs1">White, William Braid (1946) [1917]. Piano Tuning and Allied Arts (5th enlarged ed.). Boston, Massachusetts: Tuners Supply Co. p. 68.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Piano+Tuning+and+Allied+Arts&rft.place=Boston%2C+Massachusetts&rft.pages=68&rft.edition=5th+enlarged&rft.pub=Tuners+Supply+Co.&rft.date=1946&rft.aulast=White&rft.aufirst=William+Braid&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"> </li> <li id="cite_note-FOOTNOTEBarbour200455–78-38"><a href="#cite_ref-FOOTNOTEBarbour200455–78_38-0">^</a> <a href="#CITEREFBarbour2004">Barbour 2004</a>, pp. 55–78. </li> <li id="cite_note-FOOTNOTEPartch1979134-39"><a href="#cite_ref-FOOTNOTEPartch1979134_39-0">^</a> <a href="#CITEREFPartch1979">Partch 1979</a>, p. 134. </li> <li id="cite_note-FOOTNOTENeedhamLingRobinson1962170–171-40"><a href="#cite_ref-FOOTNOTENeedhamLingRobinson1962170–171_40-0">^</a> <a href="#CITEREFNeedhamLingRobinson1962">Needham, Ling & Robinson 1962</a>, pp. 170–171. </li> <li id="cite_note-FOOTNOTEBenwardSaker200356-41"><a href="#cite_ref-FOOTNOTEBenwardSaker200356_41-0">^</a> <a href="#CITEREFBenwardSaker2003">Benward & Saker 2003</a>, p. 56. </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=27" title="Edit section: Sources">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-hanging-indents refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarbour2004" class="citation book cs1">Barbour, James Murray (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=G-pG77pmlp4C&pg=PA55">Tuning and Temperament: A Historical Survey</a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43406-3" title="Special:BookSources/978-0-486-43406-3"><bdi>978-0-486-43406-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tuning+and+Temperament%3A+A+Historical+Survey&rft.pub=Courier+Corporation&rft.date=2004&rft.isbn=978-0-486-43406-3&rft.aulast=Barbour&rft.aufirst=James+Murray&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DG-pG77pmlp4C%26pg%3DPA55&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenwardSaker2003" class="citation book cs1">Benward, Bruce; Saker, Marilyn (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qkYJAQAAMAAJ&pg=PA56">Music in Theory and Practice</a>. Vol. 1. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-294261-3" title="Special:BookSources/978-0-07-294261-3"><bdi>978-0-07-294261-3</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+in+Theory+and+Practice&rft.pub=McGraw-Hill&rft.date=2003&rft.isbn=978-0-07-294261-3&rft.aulast=Benward&rft.aufirst=Bruce&rft.au=Saker%2C+Marilyn&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqkYJAQAAMAAJ%26pg%3DPA56&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCho2003" class="citation book cs1">Cho, Gene J. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lXoYAQAAIAAJ">The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century</a>. E. Mellen Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7734-6941-9" title="Special:BookSources/978-0-7734-6941-9"><bdi>978-0-7734-6941-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Discovery+of+Musical+Equal+Temperament+in+China+and+Europe+in+the+Sixteenth+Century&rft.pub=E.+Mellen+Press&rft.date=2003&rft.isbn=978-0-7734-6941-9&rft.aulast=Cho&rft.aufirst=Gene+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlXoYAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCho2010" class="citation journal cs1">Cho, Gene J. (2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120315013436/http://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htm">"The Significance of the Discovery of the Musical Equal Temperament in the Cultural History"</a>. Journal of Xinghai Conservatory of Music. Archived from <a rel="nofollow" class="external text" href="http://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htm">the original</a> on 2012-03-15. Retrieved 2020-04-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Xinghai+Conservatory+of+Music&rft.atitle=The+Significance+of+the+Discovery+of+the+Musical+Equal+Temperament+in+the+Cultural+History&rft.date=2010&rft.aulast=Cho&rft.aufirst=Gene+J.&rft_id=http%3A%2F%2Fen.cnki.com.cn%2FArticle_en%2FCJFDTOTAL-XHYY201002002.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristensen2002" class="citation book cs1">Christensen, Thomas (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ioa9uW2t7AQC">The Cambridge History of Western Music Theory</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-62371-1" title="Special:BookSources/978-0-521-62371-1"><bdi>978-0-521-62371-1</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+Cambridge+History+of+Western+Music+Theory&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-521-62371-1&rft.aulast=Christensen&rft.aufirst=Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dioa9uW2t7AQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1987" class="citation journal cs1">Cohen, H. Floris (1987). "Simon Stevin's equal division of the octave". Annals of Science. 44 (5). Informa UK Limited: 471–488. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00033798700200311">10.1080/00033798700200311</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-3790">0003-3790</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Science&rft.atitle=Simon+Stevin%27s+equal+division+of+the+octave&rft.volume=44&rft.issue=5&rft.pages=471-488&rft.date=1987&rft_id=info%3Adoi%2F10.1080%2F00033798700200311&rft.issn=0003-3790&rft.aulast=Cohen&rft.aufirst=H.+Floris&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Gorzanis1981" class="citation book cs1 cs1-prop-foreign-lang-source">de Gorzanis, G. (1981). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=W7w5AQAAIAAJ">Intabolatura di liuto: I-III</a>. Intabolatura di liuto: I-III (in Italian). Minkoff. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-8266-0721-2" title="Special:BookSources/978-2-8266-0721-2"><bdi>978-2-8266-0721-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Intabolatura+di+liuto%3A+I-III&rft.series=Intabolatura+di+liuto%3A+I-III&rft.pub=Minkoff&rft.date=1981&rft.isbn=978-2-8266-0721-2&rft.aulast=de+Gorzanis&rft.aufirst=G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DW7w5AQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDi_Veroli2009" class="citation book cs1">Di Veroli, Claudio (2009). Unequal Temperaments: Theory, History and Practice (2nd ed.). Bray, Ireland: Bray Baroque.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Unequal+Temperaments%3A+Theory%2C+History+and+Practice&rft.place=Bray%2C+Ireland&rft.edition=2nd&rft.pub=Bray+Baroque&rft.date=2009&rft.aulast=Di+Veroli&rft.aufirst=Claudio&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalilei1584" class="citation book cs1"><a href="/wiki/Vincenzo_Galilei" title="Vincenzo Galilei">Galilei, Vincenzo</a> (1584). <a rel="nofollow" class="external text" href="https://www.scribd.com/document/44080020/Fronimo-Dialogo-di-Vincentio-Galilei">Il Fronimo</a>. Venice: <a href="/wiki/Girolamo_Scotto" title="Girolamo Scotto">Girolamo Scotto</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Il+Fronimo&rft.place=Venice&rft.pub=Girolamo+Scotto&rft.date=1584&rft.aulast=Galilei&rft.aufirst=Vincenzo&rft_id=https%3A%2F%2Fwww.scribd.com%2Fdocument%2F44080020%2FFronimo-Dialogo-di-Vincentio-Galilei&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHart1998" class="citation cs2">Hart, Roger (1998), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120305174554/http://uts.cc.utexas.edu/~rhart/papers/quantifying.html">Quantifying Ritual: Political Cosmology, Courtly Music, and Precision Mathematics in Seventeenth-Century China</a>, Departments of History and Asian Studies, University of Texas, Austin, archived from <a rel="nofollow" class="external text" href="http://uts.cc.utexas.edu/~rhart/papers/quantifying.html">the original</a> on 2012-03-05, retrieved 2012-03-20</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantifying+Ritual%3A+Political+Cosmology%2C+Courtly+Music%2C+and+Precision+Mathematics+in+Seventeenth-Century+China&rft.pub=Departments+of+History+and+Asian+Studies%2C+University+of+Texas%2C+Austin&rft.date=1998&rft.aulast=Hart&rft.aufirst=Roger&rft_id=http%3A%2F%2Futs.cc.utexas.edu%2F~rhart%2Fpapers%2Fquantifying.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJorgens1986" class="citation book cs1">Jorgens, Elise Bickford (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yTh5zQEACAAJ">English Song, 1600-1675: Facsimiles of Twenty-six Manuscripts and an Edition of the Texts</a>. Garland. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780824082314" title="Special:BookSources/9780824082314"><bdi>9780824082314</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=English+Song%2C+1600-1675%3A+Facsimiles+of+Twenty-six+Manuscripts+and+an+Edition+of+the+Texts&rft.pub=Garland&rft.date=1986&rft.isbn=9780824082314&rft.aulast=Jorgens&rft.aufirst=Elise+Bickford&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyTh5zQEACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuttner1975" class="citation journal cs1">Kuttner, Fritz A. (May 1975). <a rel="nofollow" class="external text" href="https://sites.evergreen.edu/thewordintheear-fall/wp-content/uploads/sites/316/2014/09/ChineseEqualTemperament.pdf">"Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory"</a> (PDF). Ethnomusicology. 19 (2): 163–206. <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%2F850355">10.2307/850355</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/850355">850355</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ethnomusicology&rft.atitle=Prince+Chu+Tsai-Y%C3%BC%27s+Life+and+Work%3A+A+Re-Evaluation+of+His+Contribution+to+Equal+Temperament+Theory&rft.volume=19&rft.issue=2&rft.pages=163-206&rft.date=1975-05&rft_id=info%3Adoi%2F10.2307%2F850355&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F850355%23id-name%3DJSTOR&rft.aulast=Kuttner&rft.aufirst=Fritz+A.&rft_id=https%3A%2F%2Fsites.evergreen.edu%2Fthewordintheear-fall%2Fwp-content%2Fuploads%2Fsites%2F316%2F2014%2F09%2FChineseEqualTemperament.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKwang-chih_ChangPingfang_XuLiancheng_Lu2005" class="citation book cs1">Kwang-chih Chang; Pingfang Xu; Liancheng Lu (2005). "The eastern Zhou and the growth of regionalism". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sP-PN2StH2cC">The Formation of Chinese Civilization: An Archaeological Perspective</a>. Xu Pingfang, Shao Wangping, Zhang Zhongpei, Wang Renxiang. Yale University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-300-09382-7" title="Special:BookSources/978-0-300-09382-7"><bdi>978-0-300-09382-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+eastern+Zhou+and+the+growth+of+regionalism&rft.btitle=The+Formation+of+Chinese+Civilization%3A+An+Archaeological+Perspective&rft.pub=Yale+University+Press&rft.date=2005&rft.isbn=978-0-300-09382-7&rft.au=Kwang-chih+Chang&rft.au=Pingfang+Xu&rft.au=Liancheng+Lu&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsP-PN2StH2cC&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLienhard1997" class="citation web cs1">Lienhard, John H. (1997). <a rel="nofollow" class="external text" href="http://www.uh.edu/engines/epi380.htm">"Equal Temperament"</a>. The Engines of Our Ingenuity. University of Houston. Retrieved 2014-10-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Engines+of+Our+Ingenuity&rft.atitle=Equal+Temperament&rft.date=1997&rft.aulast=Lienhard&rft.aufirst=John+H.&rft_id=http%3A%2F%2Fwww.uh.edu%2Fengines%2Fepi380.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoody2003" class="citation journal cs1">Moody, Richard (February 2003). "Early Equal Temperament, An Aural Perspective: Claude Montal 1836". Piano Technicians Journal. Kansas City.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Piano+Technicians+Journal&rft.atitle=Early+Equal+Temperament%2C+An+Aural+Perspective%3A+Claude+Montal+1836&rft.date=2003-02&rft.aulast=Moody&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeedhamLingRobinson1962" class="citation book cs1"><a href="/wiki/Joseph_Needham" title="Joseph Needham">Needham, Joseph</a>; Ling, Wang; Robinson, Kenneth G. (1962). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oJ9nayZZ2oEC&pg=PA221">Science and Civilisation in China</a>. Volume 4 - Part 1. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-05802-5" title="Special:BookSources/978-0-521-05802-5"><bdi>978-0-521-05802-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+and+Civilisation+in+China&rft.series=Volume+4+-+Part+1&rft.pub=Cambridge+University+Press&rft.date=1962&rft.isbn=978-0-521-05802-5&rft.aulast=Needham&rft.aufirst=Joseph&rft.au=Ling%2C+Wang&rft.au=Robinson%2C+Kenneth+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoJ9nayZZ2oEC%26pg%3DPA221&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeedhamRonan1978" class="citation book cs1">Needham, Joseph; Ronan, Colin A. (1978). The Shorter Science and Civilisation in China. Volume 4 - Part 1. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Shorter+Science+and+Civilisation+in+China&rft.series=Volume+4+-+Part+1&rft.pub=Cambridge+University+Press&rft.date=1978&rft.aulast=Needham&rft.aufirst=Joseph&rft.au=Ronan%2C+Colin+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPartch1979" class="citation book cs1">Partch, Harry (1979). <a rel="nofollow" class="external text" href="https://archive.org/details/genesismusicacco00part">Genesis of a Music</a> (2nd ed.). Da Capo Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-306-80106-X" title="Special:BookSources/0-306-80106-X"><bdi>0-306-80106-X</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.edition=2nd&rft.pub=Da+Capo+Press&rft.date=1979&rft.isbn=0-306-80106-X&rft.aulast=Partch&rft.aufirst=Harry&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgenesismusicacco00part&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobinson1980" class="citation book cs1">Robinson, Kenneth (1980). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lYyfAAAAMAAJ">A Critical Study of Chu Tsai-yü's Contribution to the Theory of Equal Temperament in Chinese Music</a>. Volume 9 of Sinologica Coloniensia. Wiesbaden: Steiner. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-515-02732-8" title="Special:BookSources/978-3-515-02732-8"><bdi>978-3-515-02732-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Critical+Study+of+Chu+Tsai-y%C3%BC%27s+Contribution+to+the+Theory+of+Equal+Temperament+in+Chinese+Music&rft.place=Wiesbaden&rft.series=Volume+9+of+Sinologica+Coloniensia&rft.pub=Steiner&rft.date=1980&rft.isbn=978-3-515-02732-8&rft.aulast=Robinson&rft.aufirst=Kenneth&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlYyfAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSethares2005" class="citation book cs1">Sethares, William A. (2005). Tuning, Timbre, Spectrum, Scale (2nd ed.). London: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-85233-797-4" title="Special:BookSources/1-85233-797-4"><bdi>1-85233-797-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tuning%2C+Timbre%2C+Spectrum%2C+Scale&rft.place=London&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=2005&rft.isbn=1-85233-797-4&rft.aulast=Sethares&rft.aufirst=William+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrue2018" class="citation journal cs1">True, Timothy (2018). <a rel="nofollow" class="external text" href="https://doi.org/10.15385%2Fjmo.2018.9.2.2">"The Battle Between Impeccable Intonation and Maximized Modulation"</a>. Musical Offerings. 9 (2): 61–74. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.15385%2Fjmo.2018.9.2.2">10.15385/jmo.2018.9.2.2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Musical+Offerings&rft.atitle=The+Battle+Between+Impeccable+Intonation+and+Maximized+Modulation&rft.volume=9&rft.issue=2&rft.pages=61-74&rft.date=2018&rft_id=info%3Adoi%2F10.15385%2Fjmo.2018.9.2.2&rft.aulast=True&rft.aufirst=Timothy&rft_id=https%3A%2F%2Fdoi.org%2F10.15385%252Fjmo.2018.9.2.2&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_HelmholtzEllis1885" class="citation book cs1"><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz, Hermann</a>; <a href="/wiki/Alexander_J._Ellis" class="mw-redirect" title="Alexander J. Ellis">Ellis, Alexander J.</a> (1885). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GwE6AAAAIAAJ&pg=PA493">On the Sensations of Tone as a Physiological Basis for the Theory of Music</a> (2nd ed.). London: Longmans, Green.</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.place=London&rft.edition=2nd&rft.pub=Longmans%2C+Green&rft.date=1885&rft.aulast=von+Helmholtz&rft.aufirst=Hermann&rft.au=Ellis%2C+Alexander+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGwE6AAAAIAAJ%26pg%3DPA493&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilson1997" class="citation web cs1">Wilson, John (1997). <a rel="nofollow" class="external text" href="https://diapason.xentonic.org/dp/dp049.html">"Thirty preludes in all (24) keys for lute [DP 49]"</a>. The Diapason Press. Retrieved 27 October 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Thirty+preludes+in+all+%2824%29+keys+for+lute+%5BDP+49%5D&rft.pub=The+Diapason+Press&rft.date=1997&rft.aulast=Wilson&rft.aufirst=John&rft_id=https%3A%2F%2Fdiapason.xentonic.org%2Fdp%2Fdp049.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A12+equal+temperament" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Further_reading">Further reading</h3>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=28" title="Edit section: Further reading">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li>Duffin, Ross W. How Equal Temperament Ruined Harmony (and Why You Should Care). W.W. Norton & Company, 2007.</li> <li>Jorgensen, Owen. Tuning. Michigan State University Press, 1991. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-87013-290-3" title="Special:BookSources/0-87013-290-3">0-87013-290-3</a></li> <li>Khramov, Mykhaylo. "Approximation of 5-limit just intonation. Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave", <a rel="nofollow" class="external text" href="http://www.sigmap.org/Abstracts/2008/abstracts.htm">Proceedings of the International Conference SIGMAP-2008</a>[<a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot">permanent dead link</a>‍], 26–29 July 2008, <a href="/wiki/Porto" title="Porto">Porto</a>, pp. 181–184, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-989-8111-60-9" title="Special:BookSources/978-989-8111-60-9">978-989-8111-60-9</a></li> <li>Surjodiningrat, W., Sudarjana, P.J., and Susanto, A. (1972) Tone measurements of outstanding Javanese gamelans in Jogjakarta and Surakarta, Gadjah Mada University Press, Jogjakarta 1972. Cited on <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050127000731/http://web.telia.com/~u57011259/pelog_main.htm">https://web.archive.org/web/20050127000731/http://web.telia.com/~u57011259/pelog_main.htm</a>. Retrieved May 19, 2006.</li> <li>Stewart, P. J. (2006) "From Galaxy to Galaxy: Music of the Spheres" <a rel="nofollow" class="external autonumber" href="https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxncmFucGhpNjE4fGd4OjRhN2VjNTNhOGY1ZmRkNDA">[1]</a></li> <li><a rel="nofollow" class="external text" href="https://archive.org/stream/onsensationston01helmgoog#page/n0/mode/2up">Sensations of Tone</a> a foundational work on acoustics and the perception of sound by Hermann von Helmholtz. Especially Appendix XX: Additions by the Translator, pages 430–556, (pdf pages 451–577)]</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=12_equal_temperament&action=edit&section=29" title="Edit section: External links">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://en.xen.wiki/w/EDO_vs_ET">Xenharmonic wiki on EDOs vs. Equal Temperaments</a></li> <li><a rel="nofollow" class="external text" href="http://www.huygens-fokker.org/index_en.html">Huygens-Fokker Foundation Centre for Microtonal Music</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20041011224640/http://www.fortunecity.com/tinpan/lennon/362/english/acoustics.htm">A.Orlandini: Music Acoustics</a></li> <li><a rel="nofollow" class="external text" href="http://digicoll.library.wisc.edu/cgi-bin/HistSciTech/HistSciTech-idx?type=turn&entity=HistSciTech001000270617&isize=M&q1=temperament">"Temperament" from A supplement to Mr. Chambers's cyclopædia (1753)</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> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090215045859/http://www.patriziobarbieri.it/1.htm">Archived</a> 2009-02-15 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. (2008) Latina, Il Levante Libreria Editrice</li> <li><a rel="nofollow" class="external text" href="http://www.interdependentscience.com/music/calliopist.html">Fractal Microtonal Music</a>, Jim Kukula.</li> <li><a rel="nofollow" class="external text" href="https://www.academia.edu/5210832/18th_Century_Quotes_on_J.S._Bachs_Temperament">All existing 18th century quotes on J.S. Bach and temperament</a></li> <li>Dominic Eckersley: "<a rel="nofollow" class="external text" href="https://www.academia.edu/3368760/Rosetta_Revisited_Bachs_Very_Ordinary_Temperament">Rosetta Revisited: Bach's Very Ordinary Temperament</a>"</li> <li><a rel="nofollow" class="external text" href="http://home.deds.nl/~broekaert/Well%20Tempering.html">Well Temperaments, based on the Werckmeister Definition</a></li> <li><a rel="nofollow" class="external text" href="https://en.xen.wiki/images/b/b3/Zetamusic5.pdf">FAVORED CARDINALITIES OF SCALES</a> by PETER BUCH</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 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(music)">Interval</a></li> <li><a href="/wiki/Interval_ratio" title="Interval ratio">Interval ratio</a></li> <li><a href="/wiki/Pitch_class" title="Pitch 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 href="/wiki/Just_intonation" title="Just intonation">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 class="mw-selflink selflink">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 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><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=12_equal_temperament&oldid=1258631704">https://en.wikipedia.org/w/index.php?title=12_equal_temperament&oldid=1258631704</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Equal_temperaments" title="Category:Equal temperaments">Equal temperaments</a></li><li><a href="/wiki/Category:Chinese_discoveries" 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