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</style><table class="infobox"><caption class="infobox-title">Neutral third</caption><tbody><tr><th scope="row" class="infobox-label"><a href="Inversion_(interval)" class="mw-redirect" title="Inversion (interval)">Inverse</a></th><td class="infobox-data"><a href="Neutral_sixth" title="Neutral sixth">neutral sixth</a></td></tr><tr><th colspan="2" class="infobox-header" style="background-color:palegoldenrod;">Name</th></tr><tr><th scope="row" class="infobox-label">Other names</th><td class="infobox-data">–</td></tr><tr><th scope="row" class="infobox-label">Abbreviation</th><td class="infobox-data">n3</td></tr><tr><th colspan="2" class="infobox-header" style="background-color:palegoldenrod;">Size</th></tr><tr><th scope="row" class="infobox-label"><a href="Semitone" title="Semitone">Semitones</a></th><td class="infobox-data">~3½</td></tr><tr><th scope="row" class="infobox-label"><a href="Interval_class" title="Interval class">Interval class</a></th><td class="infobox-data">~3½</td></tr><tr><th scope="row" class="infobox-label"><a href="Just_intonation" title="Just intonation">Just interval</a></th><td class="infobox-data">11:9,<sup id="cite_ref-Haluska_1-0" class="reference"><a href="#cite_note-Haluska-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 27:22,<sup id="cite_ref-Haluska_1-1" class="reference"><a href="#cite_note-Haluska-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> or 16:13<sup id="cite_ref-Haluska-xxiv_3-0" class="reference"><a href="#cite_note-Haluska-xxiv-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></td></tr><tr><th colspan="2" class="infobox-header" style="background-color:palegoldenrod;"><a href="Cent_(music)" title="Cent (music)">Cents</a></th></tr><tr><th scope="row" class="infobox-label"><a href="Quarter_tone" title="Quarter tone">24-Tone equal temperament</a></th><td class="infobox-data">350</td></tr><tr><th scope="row" class="infobox-label"><a href="Just_intonation" title="Just intonation">Just intonation</a></th><td class="infobox-data">347, 355, or 359</td></tr></tbody></table>
<p>A <b>neutral third</b>, (also known as a <b>submajor third</b>, or a <b>superminor third</b>), is a <a href="Interval_(music)" title="Interval (music)">musical interval</a> wider than a <a href="Minor_third" title="Minor third">minor third</a> <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/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":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="https://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"><span class="oo-ui-labelElement-label">play</span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable">ⓘ</sup></span> but narrower than a <a href="Major_third" title="Major third">major third</a> <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-2" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/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":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="https://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"><span class="oo-ui-labelElement-label">play</span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable">ⓘ</sup></span>. It is called the <b>neutral third</b> in 24-Tone Equal Temperament, and the <b>submajor third</b>, or the <b>superminor third</b>, in Just Intonation. It was named <i><b>neutral third</b></i> by Jan Pieter Land in 1880.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Land makes reference to the neutral third attributed to <a href="Mansour_Zalzal" title="Mansour Zalzal">Zalzal</a> (8th c.), described by <a href="Al-Farabi" title="Al-Farabi">Al-Farabi</a> (10th c.) as corresponding to a ratio of 27:22 (354.5 cents) and by <a href="Avicenna" title="Avicenna">Avicenna</a> (Ibn Sina, 11th c.) as 39:32 (342.5 cents).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The Zalzalian third may have been a mobile interval.
</p><p>Three distinct intervals may be termed neutral thirds:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p>
<ul><li>The <i>undecimal neutral third</i>, (also known as the <i>undecimal submajor third</i>, <i>undecimal superminor third</i>, <i>submajor third</i>, <i>superminor third</i>, <i>just undecimal neutral third</i>, <i>just undecimal submajor third</i>, or <i>just undecimal superminor third</i>), has a ratio of 11:9<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> between the <a href="Frequencies" class="mw-redirect" title="Frequencies">frequencies</a> of the two tones, or about 347.41 cents <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-3" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/ff\/Undecimal_neutral_third_on_C.mid\/Undecimal_neutral_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Undecimal neutral third on C.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="https://upload.wikimedia.org/wikipedia/commons/transcoded/f/ff/Undecimal_neutral_third_on_C.mid/Undecimal_neutral_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-labelElement-label">play</span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable">ⓘ</sup></span>. This ratio is the mathematical <a href="Mediant_(mathematics)" title="Mediant (mathematics)"> mediant</a> of the <a href="Major_third" title="Major third">major third</a> 5/4 and the <a href="Minor_third" title="Minor third">minor third</a> 6/5, and as such, has the property that if harmonic notes of frequency <i>f</i> and (11/9) <i>f</i> are played together, the <a href="Beat_frequency" class="mw-redirect" title="Beat frequency">beat frequency</a> of the 5th harmonic of the lower pitch against the 4th of the upper, i.e. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |5f-4(11/9)f|=(1/9)f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mn>5</mn>
<mi>f</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
<mo stretchy="false">(</mo>
<mn>11</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>9</mn>
<mo stretchy="false">)</mo>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mn>9</mn>
<mo stretchy="false">)</mo>
<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle |5f-4(11/9)f|=(1/9)f}</annotation>
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</math></span><img src="./4b9cfed9fd49e54a2a41949698b40f5d67e9269f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.149ex; height:2.843ex;" alt="{\displaystyle |5f-4(11/9)f|=(1/9)f}" loading="lazy"></span>, is the same as the beat frequency of the 6th harmonic of the lower pitch against the 5th of the upper, i.e. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |6f-5(11/9)f|=|-(1/9)f|=(1/9)f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mn>6</mn>
<mi>f</mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
<mo stretchy="false">(</mo>
<mn>11</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>9</mn>
<mo stretchy="false">)</mo>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>9</mn>
<mo stretchy="false">)</mo>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>9</mn>
<mo stretchy="false">)</mo>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |6f-5(11/9)f|=|-(1/9)f|=(1/9)f}</annotation>
</semantics>
</math></span><img src="./cd7d36c93b7568f67e26e0f4f2181ad458d82282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.957ex; height:2.843ex;" alt="{\displaystyle |6f-5(11/9)f|=|-(1/9)f|=(1/9)f}" loading="lazy"></span>. In this sense, it is the unique ratio which is equally well-tuned as a major and minor third.</li>
<li>A <i>tridecimal neutral third</i> <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-4" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/a\/aa\/Tridecimal_neutral_third_on_C.mid\/Tridecimal_neutral_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Tridecimal neutral third on C.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="https://upload.wikimedia.org/wikipedia/commons/transcoded/a/aa/Tridecimal_neutral_third_on_C.mid/Tridecimal_neutral_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-labelElement-label">play</span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable">ⓘ</sup></span> has a ratio of 16:13 between the <a href="Frequencies" class="mw-redirect" title="Frequencies">frequencies</a> of the two tones, or about 359.47 cents.<sup id="cite_ref-Haluska-xxiv_3-1" class="reference"><a href="#cite_note-Haluska-xxiv-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This is the largest neutral third, and occurs infrequently in music, as little music utilizes the 13th <a href="Harmonic" title="Harmonic">harmonic</a>. It is the <a href="Mediant_(mathematics)" title="Mediant (mathematics)">mediant</a> of the <a href="Septimal_major_third" title="Septimal major third">septimal major third</a> 9/7 and <a href="Septimal_minor_third" title="Septimal minor third">septimal minor third</a> 7/6, and as such, enjoys an analogous property with regard to the beating of the corresponding harmonics as above. That is, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |7f-6(16/13)f|=|9f-7(16/13)f|=(5/13)f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mn>7</mn>
<mi>f</mi>
<mo>−<!-- − --></mo>
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<mo stretchy="false">(</mo>
<mn>16</mn>
<mrow class="MJX-TeXAtom-ORD">
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<mo>=</mo>
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<mi>f</mi>
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<mn>16</mn>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">)</mo>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle |7f-6(16/13)f|=|9f-7(16/13)f|=(5/13)f}</annotation>
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</math></span><img src="./56a60493ef6729508f8130d00bbaf16601ba5021.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.21ex; height:2.843ex;" alt="{\displaystyle |7f-6(16/13)f|=|9f-7(16/13)f|=(5/13)f}" loading="lazy"></span>.</li>
<li>An <i><a href="Equal_temperament" title="Equal temperament">equal-tempered</a> neutral third</i> <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-5" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/7\/77\/Neutral_third_on_C.mid\/Neutral_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Neutral third on C.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="https://upload.wikimedia.org/wikipedia/commons/transcoded/7/77/Neutral_third_on_C.mid/Neutral_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-labelElement-label">play</span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable">ⓘ</sup></span> is characterized by a difference of 350 <a href="Cent_(music)" title="Cent (music)">cents</a> between the two tones, slightly wider than the 11:9 ratio, and exactly half of an equal-tempered perfect fifth.</li></ul>
<p>These intervals are all within about 12 cents and are difficult for most people to distinguish by ear. Neutral thirds are roughly a <a href="Quarter_tone" title="Quarter tone">quarter tone</a> sharp from <a href="12_equal_temperament" title="12 equal temperament">12 equal temperament</a> minor thirds and a quarter tone flat from 12-ET major thirds. In just intonation, as well as in tunings such as <a href="31_equal_temperament" title="31 equal temperament">31-ET</a>, <a href="41_equal_temperament" title="41 equal temperament">41-ET</a>, or <a href="72_equal_temperament" title="72 equal temperament">72-ET</a>, which more closely approximate just intonation, the intervals are closer together.
</p><p>In addition to the above examples, a <i>square root neutral third</i> can be characterized by a ratio of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3/2}}}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {3/2}}}</annotation>
</semantics>
</math></span><img src="./9cf43edf8c5bb223b47f1e791713abfe88063cad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.811ex; height:4.843ex;" alt="{\displaystyle {\sqrt {3/2}}}" loading="lazy"></span> between two frequencies, being exactly half of a just <a href="Perfect_fifth" title="Perfect fifth">perfect fifth</a> of 3/2 and measuring about 350.98 cents. Such a definition stems from the two thirds traditionally making a fifth-based triad.
</p><p>A triad formed by two neutral thirds is neither major nor minor, thus the <b>neutral thirds triad</b> is ambiguous. While it is not found in twelve tone equal temperament it is found in others such as the quarter tone scale <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-6" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/c\/cb\/Neutral_thirds_triad_on_C.mid\/Neutral_thirds_triad_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Neutral thirds triad on C.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="https://upload.wikimedia.org/wikipedia/commons/transcoded/c/cb/Neutral_thirds_triad_on_C.mid/Neutral_thirds_triad_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-labelElement-label">Play</span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable">ⓘ</sup></span> and 31-tet <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-7" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/1\/12\/Neutral_thirds_triad_in_31-tet.mid\/Neutral_thirds_triad_in_31-tet.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Neutral thirds triad in 31-tet.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="https://upload.wikimedia.org/wikipedia/commons/transcoded/1/12/Neutral_thirds_triad_in_31-tet.mid/Neutral_thirds_triad_in_31-tet.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-labelElement-label">Play</span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable">ⓘ</sup></span>.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Occurrence_in_human_music">Occurrence in human music</h2></div>
<div class="mw-heading mw-heading3"><h3 id="In_infants'_song">In infants' song</h3></div>
<p><a href="Infants" class="mw-redirect" title="Infants">Infants</a> experiment with singing, and a few studies of individual infants' singing found that neutral thirds regularly arise in their improvisations. In two separate case studies of the progression and development of these improvisations, neutral thirds were found to arise in infants' songs after major and minor seconds and thirds, but before intervals smaller than a <a href="Semitone" title="Semitone">semitone</a> and also before intervals as large as a <a href="Perfect_fourth" title="Perfect fourth">perfect fourth</a> or larger.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="In_modern_classical_Western_music">In modern classical Western music</h3></div>
<p>The neutral third has been used by a number of modern composers, including <a href="Charles_Ives" title="Charles Ives">Charles Ives</a>, <a href="James_Tenney" title="James Tenney">James Tenney</a>, and <a href="Gayle_Young" title="Gayle Young">Gayle Young</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="In_traditional_music">In traditional music</h3></div>
<p><a href="Ptolemy" title="Ptolemy">Claudius Ptolemy</a> describes an "even diatonic" tuning which uses two justly tuned neutral thirds in <a href="https://archive.org/details/solomon-1999-ptolemy-harmonics" class="extiw external" title="iarchive:solomon-1999-ptolemy-harmonics"><i>Harmonikon</i></a> built off the 12:11 and 11:10 neutral seconds in compound intervals with 9:8 and 10:9 whole tones, forming the intervals: (12/11)*(9/8) = 27/22, (11/10)*(10/9) = 11/9. The latter of these is an interval found in the <a href="Harmonic_series_(music)" title="Harmonic series (music)">harmonic series</a> as the interval between partials 9 and 11.
</p><p>The equal-tempered neutral third may be found in the <a href="Quarter_tone_scale" class="mw-redirect" title="Quarter tone scale">quarter tone scale</a> and in some traditional <a href="Arab_music" class="mw-redirect" title="Arab music">Arab music</a> (see also <a href="Arab_tone_system" title="Arab tone system">Arab tone system</a>). Undecimal neutral thirds appear in traditional <a href="Music_of_Georgia_(country)" title="Music of Georgia (country)">Georgian music</a>.<sup id="cite_ref-Caucasia1994_10-0" class="reference"><a href="#cite_note-Caucasia1994-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Neutral thirds are also found in American folk music.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="In_contemporary_popular_music">In contemporary popular music</h3></div>
<p><a href="Blue_note" title="Blue note">Blue notes</a> (a note found in <a href="Country_music" title="Country music">country music</a>, <a href="Blues" title="Blues">blues</a>, and some <a href="Rock_music" title="Rock music">rock music</a>) on the third note of a scale can be seen as a variant of a neutral third with the tonic, as they fall in between a major third and a minor third. Similarly the blue note on the seventh note of the scale can be seen as a neutral third with the dominant.
</p>
<div class="mw-heading mw-heading2"><h2 id="In_equal_temperaments">In equal temperaments</h2></div>
<p>Two steps of <a href="Equal_temperament#Five-,_seven-,_and_nine-tone_temperaments_in_ethnomusicology" title="Equal temperament">seven-tone equal temperament</a> form an interval of 342.8571 cents, which is within 5 cents of 347.4079 for the undecimal (11:9) neutral third.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This is an equal temperament in reasonably common use, at least in the form of "near seven equal", as it is a tuning used for Thai music as well as the Ugandan Chopi tradition of music.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>
</p><p>The neutral third also has good approximations in other commonly used equal temperaments including <a href="Quarter_tone" title="Quarter tone">24-ET</a> (7 steps, 350 cents<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>) and similarly by all multiples of 24 equal steps such as 48-ET<sup id="cite_ref-48edo_15-0" class="reference"><a href="#cite_note-48edo-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and 72-ET,<sup id="cite_ref-72edo_16-0" class="reference"><a href="#cite_note-72edo-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <a href="31_equal_temperament" title="31 equal temperament">31-ET</a> (9 steps, 348.39),<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <a href="34_equal_temperament" title="34 equal temperament">34-ET</a> (10 steps, 352.941 cents<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>), <a href="41_equal_temperament" title="41 equal temperament">41-ET</a> (12 steps, 351.22 cents<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>), and slightly less closely by <a href="53_equal_temperament" title="53 equal temperament">53-ET</a> (15 steps, 339.62 cents<sup id="cite_ref-53edo_20-0" class="reference"><a href="#cite_note-53edo-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>).
</p><p>Close approximations to the tridecimal neutral third (16:13) appear in <a href="53_equal_temperament" title="53 equal temperament">53-ET</a><sup id="cite_ref-53edo_20-1" class="reference"><a href="#cite_note-53edo-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and <a href="72_equal_temperament" title="72 equal temperament">72-ET</a>.<sup id="cite_ref-72edo_16-1" class="reference"><a href="#cite_note-72edo-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Both of these temperaments distinguish between the tridecimal (16:13) and undecimal (11:9) neutral thirds. All the other tuning systems mentioned above fail to distinguish between these intervals; they temper out the comma 144:143.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="List_of_pitch_intervals" title="List of pitch intervals">List of pitch intervals</a></li>
<li><a href="Microtonal_music" class="mw-redirect" title="Microtonal music">Microtonal music</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-Haluska-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Haluska_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Haluska_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Haluska, Jan (2003). <i>The Mathematical Theory of Tone Systems</i>, p. xxiii. <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8247-4714-3</bdi>. Undecimal neutral third and Zalzal's wosta.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/Neutral_third_scales">Neutral third scales</a>", on <i>Xenharmonic Wiki</i>.</span>
</li>
<li id="cite_note-Haluska-xxiv-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Haluska-xxiv_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Haluska-xxiv_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Haluska (2003), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=w7yZ1Nzb5awC&pg=PR24">p. xxiv</a>. Tridecimal neutral third.</span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">J. P. Land, <i>Over de toonladders der Arabische muziek</i>, 1880; <i>Recherches sur l'histoire de la gamme arabe</i>, 1884. See <a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a>, <i>On the Sensations of Tone,</i> (<a href="Alexander_John_Ellis" title="Alexander John Ellis">Alexander John Ellis</a>, trans.) (3rd ed., 1895), p. 281, note † (addition by Ellis).</span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Liberty Manik (1969). <i>Das Arabische Tonsystem im Mittelalter</i> (Leiden: Brill, 1969), 46–49.</span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Alois Hába, <i>Neue Harmonielehre des diatonischen, chromatischen, Viertel-, Drittel-, Sechstel- und Zwölftel-Tonsystems</i> (Leipzig: Fr. Kistner & C.F.W. Sigel, 1927), 143. [ISBN unspecified]. Cited in Skinner, Miles Leigh (2007). <i>Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky</i>, p. 25. ProQuest. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780542998478</bdi>.</span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Andrew Horner, Lydia Ayres (2002). <i>Cooking with <a href="Csound" title="Csound">Csound</a>: Woodwind and Brass Recipes</i>, p. 131. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-89579-507-8</bdi>. "Super-Major Second".</span>
</li>
<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="Bruno_Nettl" title="Bruno Nettl">Nettl, Bruno</a>. "Infant Musical Development and Primitive Music", <i><a href="Southwestern_Journal_of_Anthropology" class="mw-redirect" title="Southwestern Journal of Anthropology">Southwestern Journal of Anthropology</a></i>, vol. 12, no. 1, pp. 87–91. (Spring 1956) <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3628859">3628859</a></span>
</li>
<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Young, Gayle. "The Pitch Organization of Harmonium for James Tenney", <i><a href="Perspectives_of_New_Music" title="Perspectives of New Music">Perspectives of New Music</a></i>, vol. 26, no. 2. pp. 204–212 (Summer 1988) <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/833190">833190</a></span>
</li>
<li id="cite_note-Caucasia1994-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Caucasia1994_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFAronson,_Howard_I.1994" class="citation book cs1">Aronson, Howard I., ed. (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YbYWAQAAMAAJ"><i>The Annual of the Society for the Study of Caucasia</i></a>. Society for the Study of Caucasia. p. 93<span class="reference-accessdate">. Retrieved <span class="nowrap">14 April</span> 2011</span>.</cite></span>
</li>
<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Boswell, George W. "The Neutral Tone as a Function of Folk-Song Text", <i>Yearbook of the International Folk Music Council</i>, vol. 2, 1970, pp. 127–132 (1970) <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/767430">767430</a></span>
</li>
<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/7edo">7edo</a>", on <i>Xenharmonic Wiki</i>.</span>
</li>
<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Morton, David (1980). "The Music of Thailand", <i>Musics of Many Cultures</i>, p. 70. May, Elizabeth, ed. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-520-04778-8</bdi>.</span>
</li>
<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/24edo">24edo</a>", on <i>Xenharmonic Wiki</i>.</span>
</li>
<li id="cite_note-48edo-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-48edo_15-0">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/48edo">48edo</a>", on <i>Xenharmonic Wiki</i>.</span>
</li>
<li id="cite_note-72edo-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-72edo_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-72edo_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/72edo">72edo</a>", on <i>Xenharmonic Wiki</i>.</span>
</li>
<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/31edo">31edo</a>", on <i>Xenharmonic Wiki</i>.</span>
</li>
<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/34edo">34edo</a>", on <i>Xenharmonic Wiki</i>.</span>
</li>
<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/41edo">41edo</a>", on <i>Xenharmonic Wiki</i>.</span>
</li>
<li id="cite_note-53edo-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-53edo_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-53edo_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/53edo">53edo</a>", on <i>Xenharmonic Wiki</i>.</span>
</li>
</ol></div></div>
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</style><div id="Intervals258" style="font-size:114%;margin:0 4em"><a href="Interval_(music)" title="Interval (music)">Intervals</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Equal_temperament" title="Equal temperament">Twelve-<br>semitone</a><br>(post-Bach<br>Western)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="(Numbers_in_bracketsare_the_number_ofsemitones_in_theinterval.)86" scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><i>(Numbers in brackets<br>are the number of<br><a href="Semitone" title="Semitone">semitones</a> in the<br>interval.)</i></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;">Perfect</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Unison" title="Unison">unison</a> (0)</li>
<li><a href="Perfect_fourth" title="Perfect fourth">fourth</a> (5)</li>
<li><a href="Perfect_fifth" title="Perfect fifth">fifth</a> (7)</li>
<li><a href="Octave" title="Octave">octave</a> (12)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="Major_and_minor#Intervals_and_chords" title="Major and minor">Major</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Major_second" title="Major second">second</a> (2)</li>
<li><a href="Major_third" title="Major third">third</a> (4)</li>
<li><a href="Major_sixth" title="Major sixth">sixth</a> (9)</li>
<li><a href="Major_seventh" title="Major seventh">seventh</a> (11)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="Major_and_minor#Intervals_and_chords" title="Major and minor">Minor</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Minor_second" class="mw-redirect" title="Minor second">second</a> (1)</li>
<li><a href="Minor_third" title="Minor third">third</a> (3)</li>
<li><a href="Minor_sixth" title="Minor sixth">sixth</a> (8)</li>
<li><a href="Minor_seventh" title="Minor seventh">seventh</a> (10)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="Augmentation_(music)#Augmentation_of_intervals" title="Augmentation (music)">Augmented</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Augmented_unison" title="Augmented unison">unison</a> (1)</li>
<li><a href="Augmented_second" title="Augmented second">second</a> (3)</li>
<li><a href="Augmented_third" title="Augmented third">third</a> (5)</li>
<li><a href="Tritone" title="Tritone">fourth</a> (6)</li>
<li><a href="Augmented_fifth" title="Augmented fifth">fifth</a> (8)</li>
<li><a href="Augmented_sixth" title="Augmented sixth">sixth</a> (10)</li>
<li><a href="Augmented_seventh" title="Augmented seventh">seventh</a> (12)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="Diminution#Diminution_of_intervals" title="Diminution">Diminished</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Diminished_second" title="Diminished second">second</a> (0)</li>
<li><a href="Diminished_third" title="Diminished third">third</a> (2)</li>
<li><a href="Diminished_fourth" title="Diminished fourth">fourth</a> (4)</li>
<li><a href="Tritone" title="Tritone">fifth</a> (6)</li>
<li><a href="Diminished_sixth" title="Diminished sixth">sixth</a> (7)</li>
<li><a href="Diminished_seventh" title="Diminished seventh">seventh</a> (9)</li>
<li><a href="Diminished_octave" title="Diminished octave">octave</a> (11)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><a href="Interval_(music)#Simple_and_compound" title="Interval (music)">Compound</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Ninth" title="Ninth">ninth</a> (13 or 14)</li>
<li><a href="Third_(chord)" title="Third (chord)">tenth</a> (15 or 16)</li>
<li><a href="Eleventh" title="Eleventh">eleventh</a> (17 or 18)</li>
<li><a href="Fifth_(chord)" title="Fifth (chord)">twelfth</a> (18 or 19)</li>
<li><a href="Thirteenth" title="Thirteenth">thirteenth</a> (20 or 21)</li>
<li><a href="Seventh_(chord)" title="Seventh (chord)">fourteenth</a> (22 or 23)</li>
<li><a href="Fifteenth" title="Fifteenth">fifteenth</a> (24)</li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other<br>tuning<br>systems</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><b><a href="24-tone_equal_temperament" class="mw-redirect" title="24-tone equal temperament">24-tone equal temperament</a></b><br><i>(Numbers in brackets refer<br>to fractional semitones.)</i></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Neutral57" scope="row" class="navbox-group" style="width:5em;font-weight:normal;"><div><br><a href="Neutral_interval" title="Neutral interval">Neutral</a><br><br></div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Quarter_tone" title="Quarter tone">quarter tone</a> (<span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>)</li>
<li><a href="Neutral_interval" title="Neutral interval">second</a> (<span class="frac">1<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li>
<li> (<span class="frac">3<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li>
<li><a href="Major_fourth_and_minor_fifth" title="Major fourth and minor fifth">major fourth</a> (<span class="frac">5<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li>
<li><a href="Major_fourth_and_minor_fifth" title="Major fourth and minor fifth">minor fifth</a> (<span class="frac">6<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li>
<li><a href="Neutral_sixth" title="Neutral sixth">sixth</a> (<span class="frac">8<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li>
<li><a href="Neutral_interval" title="Neutral interval">seventh</a> (<span class="frac">10<span class="sr-only">+</span><span class="num">1</span>⁄<span class="den">2</span></span>)</li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><b><a href="Just_intonation" title="Just intonation">Just intonations</a></b><br><i>(Numbers in brackets<br>refer to pitch ratios.)</i></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="7-limit_tuning" title="7-limit tuning">7-limit</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Septimal_quarter_tone" title="Septimal quarter tone">septimal quarter tone</a> (36:35)</li>
<li><a href="Septimal_third_tone" title="Septimal third tone">septimal third tone</a> (28:27)</li>
<li><a href="Septimal_chromatic_semitone" title="Septimal chromatic semitone">septimal chromatic semitone</a> (21:20)</li>
<li><a href="Septimal_diatonic_semitone" title="Septimal diatonic semitone">septimal diatonic semitone</a> (15:14)</li>
<li><a href="Septimal_whole_tone" title="Septimal whole tone">supermajor second</a> (8:7)</li>
<li><a href="Septimal_minor_third" title="Septimal minor third">subminor third</a> (7:6)</li>
<li><a href="Septimal_major_third" title="Septimal major third">supermajor third</a> (9:7)</li>
<li><a href="Septimal_tritone" title="Septimal tritone">subminor fifth</a> (7:5)</li>
<li><a href="Septimal_tritone" title="Septimal tritone">supermajor fourth</a> (10:7)</li>
<li><a href="Harmonic_seventh" title="Harmonic seventh">subminor seventh</a> (7:4)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Limit_(music)" title="Limit (music)">Higher-limit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Minor_diatonic_semitone" title="Minor diatonic semitone">minor diatonic semitone</a> (17-limit)</li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other<br>intervals</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><b>Groups</b></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Microtonal_music" class="mw-redirect" title="Microtonal music">Microtone</a></li>
<li><a href="List_of_intervals_in_5-limit_just_intonation" title="List of intervals in 5-limit just intonation">5-limit</a></li>
<li><a href="Comma_(music)" title="Comma (music)">Comma</a></li>
<li><a href="Pseudo-octave" title="Pseudo-octave">Pseudo-octave</a></li>
<li><a href="Pythagorean_interval" title="Pythagorean interval">Pythagorean interval</a></li>
<li><a href="Subminor_and_supermajor" title="Subminor and supermajor">Subminor and supermajor</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><i><b>Semitones</b></i></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Pythagorean_limma" class="mw-redirect" title="Pythagorean limma">Pythagorean limma</a></li>
<li><a href="Pythagorean_apotome" class="mw-redirect" title="Pythagorean apotome">Pythagorean apotome</a></li>
<li><a href="Major_limma" title="Major limma">Major limma</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><i><b>Quarter tones</b></i></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Quarter_tone" title="Quarter tone">Quarter tone</a></li>
<li><a href="Septimal_quarter_tone" title="Septimal quarter tone">Septimal quarter tone</a></li>
<li><a href="Undecimal_quarter_tone" class="mw-redirect" title="Undecimal quarter tone">Undecimal quarter tone</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><b><a href="Comma_(music)" title="Comma (music)">Commas</a></b></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Pythagorean_comma" title="Pythagorean comma">Pythagorean comma</a> (23.5 cents)</li>
<li><a href="Syntonic_comma" title="Syntonic comma">Syntonic comma</a> (21.5 cents)</li>
<li><a href="Holdrian_comma" class="mw-redirect" title="Holdrian comma">Holdrian comma</a> (22.6 cents)</li>
<li><a href="Septimal_comma" title="Septimal comma">Septimal comma</a> (27.3 cents)</li></ul>
<ul><li><a href="Diesis" title="Diesis">Lesser diesis</a> (41.1 cents)</li>
<li><a href="Diesis" title="Diesis">Greater diesis</a> (62.6 cents)</li>
<li><a href="Septimal_diesis" title="Septimal diesis">Septimal diesis</a> (35.7 cents)</li></ul>
<ul><li><a href="Diaschisma" title="Diaschisma">Diaschisma</a> (19.5 cents)</li>
<li><a href="Semicomma" title="Semicomma">Semicomma</a> (10.1 cents)</li>
<li><a href="Septimal_semicomma" title="Septimal semicomma">Septimal semicomma</a> (13.8 cents)</li>
<li><a href="Kleisma" title="Kleisma">Kleisma</a> (8.1 cents)</li>
<li><a href="Septimal_kleisma" title="Septimal kleisma">Septimal kleisma</a> (7.7 cents)</li></ul>
<ul><li><a href="Schisma" title="Schisma">Schisma</a> (1.95 cents)</li>
<li><a href="Breedsma" title="Breedsma">Breedsma</a> (0.72 cents)</li>
<li><a href="Ragisma" title="Ragisma">Ragisma</a> (0.4 cents)</li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><b>Measurement</b></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Cent_(music)" title="Cent (music)">Cent</a></li>
<li><a href="Cent_(music)#Centitones" title="Cent (music)">Centitone</a></li>
<li><a href="Millioctave" title="Millioctave">Millioctave</a></li>
<li><a href="Savart" title="Savart">Savart</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:6em;font-weight:normal;"><b>Others</b></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Wolf_interval" title="Wolf interval">Wolf</a></li>
<li><a href="Ditone" title="Ditone">Ditone</a></li>
<li><a href="Semiditone" class="mw-redirect" title="Semiditone">Semiditone</a></li>
<li><a href="George_Secor#Secor" title="George Secor">Secor</a></li>
<li><a href="Incomposite_interval" title="Incomposite interval">Incomposite interval</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><span class="noviewer" typeof="mw:File"><span title="List-Class article"></span></span> <a href="List_of_pitch_intervals" title="List of pitch intervals">List of pitch intervals</a></div></td></tr></tbody></table></div></div><!--htdig_noindex--><div><div class="zim-footer">
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