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id="toc-Construction-sublist" class="vector-toc-list"> <li id="toc-12-tone_scale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#12-tone_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>12-tone scale</span> </div> </a> <ul id="toc-12-tone_scale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-C-based_construction_tables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#C-based_construction_tables"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>C-based construction tables</span> </div> </a> <ul id="toc-C-based_construction_tables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Justly_intonated_quarter-comma_meantone" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Justly_intonated_quarter-comma_meantone"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Justly intonated quarter-comma meantone</span> </div> </a> <ul id="toc-Justly_intonated_quarter-comma_meantone-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Greater_and_lesser_semitones" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Greater_and_lesser_semitones"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Greater and lesser semitones</span> </div> </a> <ul id="toc-Greater_and_lesser_semitones-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Size_of_intervals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Size_of_intervals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Size of intervals</span> </div> </a> <ul id="toc-Size_of_intervals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triads_in_the_chromatic_scale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Triads_in_the_chromatic_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Triads in the chromatic scale</span> </div> </a> <ul id="toc-Triads_in_the_chromatic_scale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_construction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alternative_construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Alternative construction</span> </div> </a> <button aria-controls="toc-Alternative_construction-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Alternative construction subsection</span> </button> <ul id="toc-Alternative_construction-sublist" class="vector-toc-list"> <li id="toc-Diatonic_scale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diatonic_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Diatonic scale</span> </div> </a> <ul id="toc-Diatonic_scale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chromatic_scale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chromatic_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Chromatic scale</span> </div> </a> <ul id="toc-Chromatic_scale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Comparison_with_31-tone_equal_temperament" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparison_with_31-tone_equal_temperament"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Comparison with 31-tone equal temperament</span> </div> </a> <ul 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codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5c/Just_perfect_fifth_on_C.mid/Just_perfect_fifth_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div> <div class="description"></div></div><hr /><div class="haudio"> <div class="listen-file-header"><a href="/wiki/File:Syntonic_comma_on_C.mid" title="File:Syntonic comma on C.mid">Syntonic comma on C</a></div> <div><span typeof="mw:File"><span><audio id="mwe_player_1" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="232" style="width:232px;" data-durationhint="7" data-mwtitle="Syntonic_comma_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/8/8b/Syntonic_comma_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8b/Syntonic_comma_on_C.mid/Syntonic_comma_on_C.mid.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8b/Syntonic_comma_on_C.mid/Syntonic_comma_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div> <div class="description"></div></div><hr /><div class="haudio"> <div class="listen-file-header"><a href="/wiki/File:Just_major_third_on_C.mid" title="File:Just major third on C.mid">Just major third on C</a></div> <div><span typeof="mw:File"><span><audio id="mwe_player_2" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="232" style="width:232px;" data-durationhint="7" data-mwtitle="Just_major_third_on_C.mid" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/2/2a/Just_major_third_on_C.mid" type="audio/midi" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/Just_major_third_on_C.mid/Just_major_third_on_C.mid.ogg" type="audio/ogg; 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codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/85/Quarter-comma_meantone_perfect_fifth_on_C.mid/Quarter-comma_meantone_perfect_fifth_on_C.mid.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div> <div class="description"></div></div></div></div> <div class="side-box-abovebelow"><hr /><i class="selfreference">Problems playing these files? See <a href="/wiki/Help:Media" title="Help:Media">media help</a>.</i></div> </div> <p><b>Quarter-comma meantone</b>, or <b><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 4 </span></span>⁠</span>-comma meantone</b>, was the most common <a href="/wiki/Meantone_temperament" title="Meantone temperament">meantone temperament</a> in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> is flattened by one quarter of a <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a> <span class="nowrap">( 81 : 80 ),</span> with respect to its <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a> used in <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a> (<a href="/wiki/Interval_ratio" title="Interval ratio">frequency ratio</a> <span class="nowrap">3 : 2 );</span> the result is <span class="nowrap"><span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 3 </span><span class="sr-only">/</span><span class="den"> 2 </span></span>⁠</span> × <span style="font-size:120%">[</span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 80 </span><span class="sr-only">/</span><span class="den"> 81 </span></span>⁠</span><span style="font-size:120%">]</span><sup><sup> 1 <i>/</i> 4</sup> </sup> = <span class="nowrap"><sup style="margin-right: -0.5em; vertical-align: 0.8em;">4</sup>√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span> ≈ 1.49535</span> ,</span> or a fifth of 696.578 <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>. (The 12th power of that value is 125, whereas 7 octaves is 128, and so falls 41.059 cents short.) This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned <a href="/wiki/Major_third" title="Major third">major thirds</a> (with a frequency ratio equal to <span class="nowrap"><a href="/wiki/Sesquiquartum" class="mw-redirect" title="Sesquiquartum">5:4</a>).</span> It was described by <a href="/wiki/Pietro_Aron" title="Pietro Aron">Pietro Aron</a> in his <i>Toscanello de la Musica</i> of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible".<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Later theorists <a href="/wiki/Gioseffo_Zarlino" title="Gioseffo Zarlino">Gioseffo Zarlino</a> and <a href="/wiki/Francisco_de_Salinas" title="Francisco de Salinas">Francisco de Salinas</a> described the tuning with mathematical exactitude. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=1" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a meantone tuning, we have different chromatic and diatonic <a href="/wiki/Semitone" title="Semitone">semitones</a>; the chromatic semitone is the difference between C and C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>, and the diatonic semitone the difference between C and D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>. In Pythagorean tuning, the diatonic semitone is often called the <a href="/wiki/Pythagorean_limma" class="mw-redirect" title="Pythagorean limma">Pythagorean limma</a> and the chromatic semitone <a href="/wiki/Pythagorean_apotome" class="mw-redirect" title="Pythagorean apotome">Pythagorean apotome</a>, but in Pythagorean tuning the apotome is larger, whereas in <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 4 </span></span>⁠</span> comma</span> meantone the limma is larger. Put another way, in Pythagorean tuning we have that C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> is higher than D<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>, whereas in <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 4 </span></span>⁠</span> comma</span> meantone we have C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> lower than D<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>. </p><p>In any meantone or Pythagorean tuning, where a <a href="/wiki/Whole_tone" class="mw-redirect" title="Whole tone">whole tone</a> is composed of one semitone of each kind, a <a href="/wiki/Major_third" title="Major third">major third</a> is two whole tones and therefore consists of two semitones of each kind, a <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> of meantone contains four diatonic and three chromatic semitones, and an <a href="/wiki/Octave" title="Octave">octave</a> seven diatonic and five chromatic semitones, it follows that: </p> <ul><li>Five fifths down and three octaves up make up a diatonic semitone, so that the <a href="/wiki/Pythagorean_limma" class="mw-redirect" title="Pythagorean limma">Pythagorean limma</a> is tempered to a diatonic semitone.</li> <li>Two fifths up and an octave down make up a whole tone consisting of one diatonic and one chromatic semitone.</li> <li>Four fifths up and two octaves down make up a major third, consisting of two diatonic and two chromatic semitones, or in other words two whole tones.</li></ul> <p>Thus, in Pythagorean tuning, where sequences of <a href="/wiki/Just_intonation" title="Just intonation">just</a> fifths (<a href="/wiki/Interval_ratio" title="Interval ratio">frequency ratio</a> <span class="nowrap"> 3 : 2 )</span> and octaves are used to produce the other intervals, a whole tone is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {~\left({\frac {3}{\ 2\ }}\right)^{2}}{2}}={\frac {\ \left({\frac {9}{\ 4\ }}\right)\ }{2}}={\frac {9}{\ 8\ }}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mrow> <mtext> </mtext> <mn>4</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mrow> <mtext> </mtext> <mn>8</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {~\left({\frac {3}{\ 2\ }}\right)^{2}}{2}}={\frac {\ \left({\frac {9}{\ 4\ }}\right)\ }{2}}={\frac {9}{\ 8\ }}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391de030ed5ec2ba9e9ef7f74f05b7aad015846e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.017ex; height:8.009ex;" alt="{\displaystyle {\frac {~\left({\frac {3}{\ 2\ }}\right)^{2}}{2}}={\frac {\ \left({\frac {9}{\ 4\ }}\right)\ }{2}}={\frac {9}{\ 8\ }}\ ,}"></span></dd></dl> <p>and a major third is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {~\left({\frac {3}{\ 2\ }}\right)^{4}}{2^{2}}}={\frac {\ \left({\frac {81}{\ 16\ }}\right)\ }{4}}={\frac {81}{\ 64\ }}\approx {\frac {80}{\ 64\ }}={\frac {5}{\ 4\ }}\ ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mrow> <mtext> </mtext> <mn>16</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mrow> <mtext> </mtext> <mn>64</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>80</mn> <mrow> <mtext> </mtext> <mn>64</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mrow> <mtext> </mtext> <mn>4</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {~\left({\frac {3}{\ 2\ }}\right)^{4}}{2^{2}}}={\frac {\ \left({\frac {81}{\ 16\ }}\right)\ }{4}}={\frac {81}{\ 64\ }}\approx {\frac {80}{\ 64\ }}={\frac {5}{\ 4\ }}\ ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f289f1d6cf28e9b8efd0bf7f60f4bbb5dca8e77c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.68ex; height:8.509ex;" alt="{\displaystyle {\frac {~\left({\frac {3}{\ 2\ }}\right)^{4}}{2^{2}}}={\frac {\ \left({\frac {81}{\ 16\ }}\right)\ }{4}}={\frac {81}{\ 64\ }}\approx {\frac {80}{\ 64\ }}={\frac {5}{\ 4\ }}\ ;}"></span></dd></dl> <p>the ratio of the different values is the <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a>, <span class="nowrap"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">80</span></span>⁠</span> .</span> </p><p>An interval of a seventeenth, consisting of sixteen diatonic and twelve chromatic semitones, such as the interval from D<sub>4</sub> to F<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>, can be equivalently obtained using either </p> <ul><li>a stack of four fifths (e.g. D<sub>4</sub> A<sub>4</sub> E<sub>5</sub> B<sub>5</sub> F<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>), or</li> <li>a stack of two octaves and one major third (e.g. D<sub>4</sub> D<sub>5</sub> D<sub>6</sub> F<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>).</li></ul> <p>This large interval of a seventeenth contains <span class="nowrap"> 5 + (5 − 1) + (5 − 1) + (5 − 1) = 20 − 3 = 17</span> <a href="/wiki/Staff_position" class="mw-redirect" title="Staff position">staff positions</a>. In Pythagorean tuning, the size of a seventeenth is defined using a stack of four justly tuned fifths (frequency ratio <span class="nowrap"> 3 : 2 ):</span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {3}{\ 2\ }}\right)^{4}={\frac {81}{\ 16\ }}={\frac {80}{\ 16\ }}\cdot {\frac {81}{\ 80\ }}=5\cdot {\frac {81}{\ 80\ }}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mrow> <mtext> </mtext> <mn>16</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>80</mn> <mrow> <mtext> </mtext> <mn>16</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mrow> <mtext> </mtext> <mn>80</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>5</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>81</mn> <mrow> <mtext> </mtext> <mn>80</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {3}{\ 2\ }}\right)^{4}={\frac {81}{\ 16\ }}={\frac {80}{\ 16\ }}\cdot {\frac {81}{\ 80\ }}=5\cdot {\frac {81}{\ 80\ }}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/963842eede818b58794f0b3386d79d920f131962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.968ex; height:6.676ex;" alt="{\displaystyle \left({\frac {3}{\ 2\ }}\right)^{4}={\frac {81}{\ 16\ }}={\frac {80}{\ 16\ }}\cdot {\frac {81}{\ 80\ }}=5\cdot {\frac {81}{\ 80\ }}~.}"></span></dd></dl> <p>In quarter-comma meantone temperament, where a <a href="/wiki/Just_intonation" title="Just intonation">just</a> major third <span class="nowrap">(5:4)</span> is required, a slightly narrower seventeenth is obtained by stacking two octaves and a major third: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2}\cdot {\frac {5}{\ 4\ }}=5~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mrow> <mtext> </mtext> <mn>4</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>5</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2}\cdot {\frac {5}{\ 4\ }}=5~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9442a5d2ea31409373b9331fe560193abc5b3574" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.544ex; height:5.176ex;" alt="{\displaystyle 2^{2}\cdot {\frac {5}{\ 4\ }}=5~.}"></span></dd></dl> <p>By definition, however, a seventeenth of the same size <span class="nowrap">( 5 : 1 )</span> must be obtained, even in quarter-comma meantone, by stacking four fifths. Since justly tuned fifths, such as those used in Pythagorean tuning, produce a slightly wider seventeenth, in quarter-comma meantone the fifths must be slightly flattened to meet this requirement. Letting <span class="texhtml mvar" style="font-style:italic;">x</span> be the frequency ratio of the flattened fifth, it is desired that four fifths have a ratio of <span class="nowrap"> 5 : 1 ,</span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}=5\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>5</mn> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}=5\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8af973140012f887171e9e9d1d3bc6a74793fe29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.872ex; height:3.009ex;" alt="{\displaystyle x^{4}=5\ ,}"></span></dd></dl> <p>which implies that a fifth is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\sqrt[{4}]{5\ }}=5^{1/4}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>5</mn> <mtext> </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\sqrt[{4}]{5\ }}=5^{1/4}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ff524e4ada4ac682fa8b8e7f672eed513e9351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.294ex; height:3.176ex;" alt="{\displaystyle x={\sqrt[{4}]{5\ }}=5^{1/4}\ ,}"></span></dd></dl> <p>a whole tone, built by moving two fifths up and one octave down, is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {~x^{2}}{2}}={\frac {\ {\sqrt {5\ }}\ }{2}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {~x^{2}}{2}}={\frac {\ {\sqrt {5\ }}\ }{2}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95495c25cbccc976f371a1e43c23d4089f607fbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.803ex; height:5.843ex;" alt="{\displaystyle {\frac {~x^{2}}{2}}={\frac {\ {\sqrt {5\ }}\ }{2}}\ ,}"></span></dd></dl> <p>and a diatonic semitone, built by moving three octaves up and five fifths down, is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\;2^{3}\ }{\ x^{5}}}={\frac {8}{~5^{5/4}}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mrow> <mtext> </mtext> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\;2^{3}\ }{\ x^{5}}}={\frac {8}{~5^{5/4}}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac747d4543c281a8c9e2e8f157408563081ed83e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.882ex; height:6.343ex;" alt="{\displaystyle {\frac {\;2^{3}\ }{\ x^{5}}}={\frac {8}{~5^{5/4}}}~.}"></span></dd></dl> <p>Notice that, in quarter-comma meantone, the seventeenth is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">80</span></span>⁠</span> times narrower than in Pythagorean tuning. This difference in size, equal to about 21.506 <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>, is called the <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a>. This implies that the fifth is a quarter of a syntonic comma narrower than the justly tuned Pythagorean fifth. Namely, this system tunes the fifths in the ratio of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5^{1/4}\approx 1.495349\approx {\frac {\ 643\ }{430}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mo>≈</mo> <mn>1.495349</mn> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>643</mn> <mtext> </mtext> </mrow> <mn>430</mn> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{1/4}\approx 1.495349\approx {\frac {\ 643\ }{430}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf96638465e6f78e3c56647ac0a9caddc1a1e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.907ex; height:5.343ex;" alt="{\displaystyle 5^{1/4}\approx 1.495349\approx {\frac {\ 643\ }{430}}\ }"></span></dd></dl> <p>which is expressed in the logarithmic <a href="/wiki/Cents_(music)" class="mw-redirect" title="Cents (music)">cents</a> scale as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1200\ \log _{2}{5^{1/4}}\ {\mathsf {cents}}\approx 696.578\ {\mathsf {cents}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1200</mn> <mtext> </mtext> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">s</mi> </mrow> </mrow> <mo>≈</mo> <mn>696.578</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">s</mi> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1200\ \log _{2}{5^{1/4}}\ {\mathsf {cents}}\approx 696.578\ {\mathsf {cents}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0457a2a8a519da68eb6561ee71282433623be04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.994ex; height:3.343ex;" alt="{\displaystyle 1200\ \log _{2}{5^{1/4}}\ {\mathsf {cents}}\approx 696.578\ {\mathsf {cents}}\ ,}"></span></dd></dl> <p>which is slightly smaller (or flatter) than the ratio of a justly tuned fifth: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3}{2}}=1.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}=1.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc2d91e4e86736b164de701f33fe0c33d761efd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.069ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{2}}=1.5}"></span></dd></dl> <p>which is expressed in the logarithmic cents scale as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1200\log _{2}{\left({\frac {3}{2}}\right)}{\text{ cents}}\approx 701.955{\text{ cents}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1200</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> <mo>≈</mo> <mn>701.955</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1200\log _{2}{\left({\frac {3}{2}}\right)}{\text{ cents}}\approx 701.955{\text{ cents}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf926d7a3dce177448e568940275c5fa2a4bff57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.335ex; height:6.176ex;" alt="{\displaystyle 1200\log _{2}{\left({\frac {3}{2}}\right)}{\text{ cents}}\approx 701.955{\text{ cents}}~.}"></span></dd></dl> <p>The difference between these two sizes is a quarter of a syntonic comma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx 701.955-696.578\ {\mathsf {cents}}\approx 5.377\ {\mathsf {cents}}\approx {\frac {\ 21.506\ {\mathsf {cents}}\ }{4}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> <mn>701.955</mn> <mo>−</mo> <mn>696.578</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">s</mi> </mrow> </mrow> <mo>≈</mo> <mn>5.377</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">s</mi> </mrow> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>21.506</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">s</mi> </mrow> </mrow> <mtext> </mtext> </mrow> <mn>4</mn> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx 701.955-696.578\ {\mathsf {cents}}\approx 5.377\ {\mathsf {cents}}\approx {\frac {\ 21.506\ {\mathsf {cents}}\ }{4}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4522057f3797d124b434bf3e6dcd3b3687523cfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.448ex; height:5.176ex;" alt="{\displaystyle \approx 701.955-696.578\ {\mathsf {cents}}\approx 5.377\ {\mathsf {cents}}\approx {\frac {\ 21.506\ {\mathsf {cents}}\ }{4}}~.}"></span></dd></dl> <p>In sum, this system tunes the major thirds to the <a href="/wiki/Just_intonation" title="Just intonation">just</a> ratio of 5:4 (so, for instance, if A<sub>4</sub> is tuned to 440 <a href="/wiki/Hertz" title="Hertz">Hz</a>, C<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">5</sub></span></span> is tuned to 550 Hz), most of the whole tones (namely the <a href="/wiki/Major_second" title="Major second">major seconds</a>) in the ratio <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span>:2, and most of the semitones (namely the diatonic semitones or <a href="/wiki/Minor_second" class="mw-redirect" title="Minor second">minor seconds</a>) in the ratio <span class="nowrap"><span class="texhtml"> ( 8 : 5 )<sup><style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">5</span>⁄<span class="den">4</span></span></sup></span> .</span> This is achieved by tuning the seventeenth a syntonic comma flatter than the Pythagorean seventeenth, which implies tuning the fifth a quarter of a syntonic comma flatter than the <a href="/wiki/Just_intonation" title="Just intonation">just</a> ratio of 3:2. It is this that gives the system its name of <i>quarter-comma meantone</i>. </p> <div class="mw-heading mw-heading3"><h3 id="12-tone_scale">12-tone scale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=2" title="Edit section: 12-tone scale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The whole chromatic scale (a subset of which is the diatonic scale), can be constructed by starting from a given <i>base note</i>, and increasing or decreasing its frequency by one or more fifths. This method is identical to Pythagorean tuning, except for the size of the fifth, which is tempered as explained above. However, <a href="/wiki/Meantone_temperament" title="Meantone temperament">meantone temperaments</a> (except for <a href="/wiki/12_tone_equal_temperament" class="mw-redirect" title="12 tone equal temperament">12 <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">TET</span></span></a>) <i>cannot</i> fit into a 12-note keyboard; and like quarter-comma meantone, most require an infinite number of notes (although there is a very close approximation to quarter-comma that can fit into a keyboard with 31 keys per octave). When tuned to a 12-note keyboard many notes must be left out, and unless the tuning is <a href="/wiki/Well_temperament" title="Well temperament">"tempered"</a> to gloss over the missing notes, keyboard players who substitute the available nearest-pitch note (which is always the wrong pitch) for the <i>actual</i> appropriate quarter comma note (which <i>would</i> sound consonant, if it were available) create <a href="/wiki/Wolf_interval" title="Wolf interval">dissonant notes</a> in place of the <a href="/wiki/Dissonance_and_consonance" class="mw-redirect" title="Dissonance and consonance">consonant</a> quarter-comma note. </p><p>The construction table below illustrates how the pitches of the notes are obtained with respect to D (the <i>base note</i>), in a D-based scale (see <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a> for a more detailed explanation). For each note in the basic octave, the table provides the conventional name of the <a href="/wiki/Interval_(music)" title="Interval (music)">interval</a> from D (the base note), the formula to compute its frequency ratio, and the approximate values for its frequency ratio and size in cents. </p> <dl><dd><table class="wikitable" style="text-align: left"> <tbody><tr> <th>Note </th> <th>Interval from D </th> <th>Formula </th> <th>Freq.<br />ratio </th> <th>Size<br />(cents) </th> <th>Size (<a href="/wiki/31_equal_temperament" title="31 equal temperament">31-ET</a>) </th></tr> <tr> <td>A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td><a href="/wiki/Diminished_fifth" class="mw-redirect" title="Diminished fifth">diminished fifth</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-6}\cdot 2^{4}={\frac {\ 16{\sqrt {5\ }}\ }{25}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> <mn>25</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-6}\cdot 2^{4}={\frac {\ 16{\sqrt {5\ }}\ }{25}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d093b0983c30a07016290d973848e1ff5a720713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.658ex; height:5.843ex;" alt="{\displaystyle x^{-6}\cdot 2^{4}={\frac {\ 16{\sqrt {5\ }}\ }{25}}}"></span> </td> <td style="text-align: right">1.4311 </td> <td style="text-align: right">620.5 </td> <td style="text-align: right">16.03 </td></tr> <tr> <td>E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td><a href="/wiki/Minor_second" class="mw-redirect" title="Minor second">minor second</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-5}\cdot 2^{3}={\frac {\ 8{\sqrt {5\ }}\ x\ }{25}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> </mrow> <mn>25</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-5}\cdot 2^{3}={\frac {\ 8{\sqrt {5\ }}\ x\ }{25}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0227a059057f4abedce3c0ada86cb7c0861b38e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.406ex; height:5.843ex;" alt="{\displaystyle x^{-5}\cdot 2^{3}={\frac {\ 8{\sqrt {5\ }}\ x\ }{25}}}"></span> </td> <td style="text-align: right">1.0700 </td> <td style="text-align: right">117.1 </td> <td style="text-align: right">3.03 </td></tr> <tr> <td>B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td> <td><a href="/wiki/Minor_sixth" title="Minor sixth">minor sixth</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-4}\cdot 2^{3}={\frac {\ 8\ }{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>8</mn> <mtext> </mtext> </mrow> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-4}\cdot 2^{3}={\frac {\ 8\ }{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be3cdcbf5aba0492c49c3facf7800fb035a9dd89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.817ex; height:5.176ex;" alt="{\displaystyle x^{-4}\cdot 2^{3}={\frac {\ 8\ }{5}}}"></span> </td> <td style="text-align: right">1.6000 </td> <td style="text-align: right">813.7 </td> <td style="text-align: right">21.02 </td></tr> <tr> <td>F </td> <td><a href="/wiki/Minor_third" title="Minor third">minor third</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-3}\cdot 2^{2}={\frac {\ 4x\ }{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>4</mn> <mi>x</mi> <mtext> </mtext> </mrow> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-3}\cdot 2^{2}={\frac {\ 4x\ }{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4d5b17d750aa97b3be84bf0ff937d5fdbb3994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.146ex; height:5.176ex;" alt="{\displaystyle x^{-3}\cdot 2^{2}={\frac {\ 4x\ }{5}}}"></span> </td> <td style="text-align: right">1.1963 </td> <td style="text-align: right">310.3 </td> <td style="text-align: right">8.02 </td></tr> <tr> <td>C </td> <td><a href="/wiki/Minor_seventh" title="Minor seventh">minor seventh</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-2}\cdot 2^{2}={\frac {\ 4{\sqrt {5\ }}\ }{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-2}\cdot 2^{2}={\frac {\ 4{\sqrt {5\ }}\ }{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271ca01010f16a1e0cf43f53dc7ed5c9c4fbf0d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.496ex; height:5.843ex;" alt="{\displaystyle x^{-2}\cdot 2^{2}={\frac {\ 4{\sqrt {5\ }}\ }{5}}}"></span> </td> <td style="text-align: right">1.7889 </td> <td style="text-align: right">1006.8 </td> <td style="text-align: right">26.01 </td></tr> <tr> <td>G </td> <td><a href="/wiki/Perfect_fourth" title="Perfect fourth">perfect fourth</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}\cdot 2^{1}={\frac {\ 2{\sqrt {5\ }}\ x\ }{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> </mrow> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}\cdot 2^{1}={\frac {\ 2{\sqrt {5\ }}\ x\ }{5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/902fe0f252b740a39d947771d94fdc3a1873dbbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.406ex; height:5.843ex;" alt="{\displaystyle x^{-1}\cdot 2^{1}={\frac {\ 2{\sqrt {5\ }}\ x\ }{5}}}"></span> </td> <td style="text-align: right">1.3375 </td> <td style="text-align: right">503.4 </td> <td style="text-align: right">13.00 </td></tr> <tr> <td>D </td> <td><a href="/wiki/Unison" title="Unison">unison</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{0}\cdot 2^{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{0}\cdot 2^{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/850fb95bdca9b2b8b87f1850cc53d3dcf83143df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.541ex; height:2.676ex;" alt="{\displaystyle x^{0}\cdot 2^{0}=1}"></span> </td> <td style="text-align: right">1.0000 </td> <td style="text-align: right">0.0 </td> <td style="text-align: right">0.00 </td></tr> <tr> <td>A </td> <td><a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1}\cdot 2^{0}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1}\cdot 2^{0}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/825a659e60e9712d4a3ef185d0e43b271ca5ead2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.708ex; height:2.676ex;" alt="{\displaystyle x^{1}\cdot 2^{0}=x}"></span> </td> <td style="text-align: right">1.4953 </td> <td style="text-align: right">696.6 </td> <td style="text-align: right">18.00 </td></tr> <tr> <td>E </td> <td><a href="/wiki/Major_second" title="Major second">major second</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}\cdot 2^{-1}={\frac {\ {\sqrt {5\ }}\ }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}\cdot 2^{-1}={\frac {\ {\sqrt {5\ }}\ }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea3a197d6d393b1feddd323ae04364253512ed31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.333ex; height:5.843ex;" alt="{\displaystyle x^{2}\cdot 2^{-1}={\frac {\ {\sqrt {5\ }}\ }{2}}}"></span> </td> <td style="text-align: right">1.1180 </td> <td style="text-align: right">193.2 </td> <td style="text-align: right">4.99 </td></tr> <tr> <td>B </td> <td><a href="/wiki/Major_sixth" title="Major sixth">major sixth</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}\cdot 2^{-1}={\frac {\ {\sqrt {5\ }}\ x\ }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}\cdot 2^{-1}={\frac {\ {\sqrt {5\ }}\ x\ }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a18a1b6b3a3c03874370776b993494ac7e908d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.243ex; height:5.843ex;" alt="{\displaystyle x^{3}\cdot 2^{-1}={\frac {\ {\sqrt {5\ }}\ x\ }{2}}}"></span> </td> <td style="text-align: right">1.6719 </td> <td style="text-align: right">889.7 </td> <td style="text-align: right">22.98 </td></tr> <tr> <td>F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> </td> <td><a href="/wiki/Major_third" title="Major third">major third</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}\cdot 2^{-2}={\frac {\ 5\ }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>5</mn> <mtext> </mtext> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}\cdot 2^{-2}={\frac {\ 5\ }{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7261bb2013764ff79d587b4db8115cdd35f1036a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.817ex; height:5.176ex;" alt="{\displaystyle x^{4}\cdot 2^{-2}={\frac {\ 5\ }{4}}}"></span> </td> <td style="text-align: right">1.2500 </td> <td style="text-align: right">386.3 </td> <td style="text-align: right">9.98 </td></tr> <tr> <td>C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> </td> <td><a href="/wiki/Major_seventh" title="Major seventh">major seventh</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{5}\cdot 2^{-2}={\frac {\ 5\ x\ }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>5</mn> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> </mrow> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{5}\cdot 2^{-2}={\frac {\ 5\ x\ }{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e975e8267e4cb2a30f714db9ae97edf64898900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.727ex; height:5.176ex;" alt="{\displaystyle x^{5}\cdot 2^{-2}={\frac {\ 5\ x\ }{4}}}"></span> </td> <td style="text-align: right">1.8692 </td> <td style="text-align: right">1082.9 </td> <td style="text-align: right">27.97 </td></tr> <tr> <td>G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> </td> <td><a href="/wiki/Augmented_fourth" class="mw-redirect" title="Augmented fourth">augmented fourth</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{6}\cdot 2^{-3}={\frac {\ 5{\sqrt {5\ }}\ }{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{6}\cdot 2^{-3}={\frac {\ 5{\sqrt {5\ }}\ }{8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d20017fe3e3d90fbbcaf4544182cda34e672218a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.496ex; height:5.843ex;" alt="{\displaystyle x^{6}\cdot 2^{-3}={\frac {\ 5{\sqrt {5\ }}\ }{8}}}"></span> </td> <td style="text-align: right">1.3975 </td> <td style="text-align: right">579.5 </td> <td style="text-align: right">14.97 </td></tr></tbody></table></dd></dl> <p>In the formulas, <span class="nowrap"> <span class="texhtml"> <i>x</i> = <span class="nowrap"><sup style="margin-right: -0.5em; vertical-align: 0.8em;">4</sup>√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span> = 5<sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">4</span></span></sup></span> </span> is the size of the tempered perfect fifth, and the ratios <span class="nowrap"><span class="texhtml"> <i>x</i> : 1 </span> </span> or <span class="nowrap"><span class="texhtml"> 1 : <i>x</i> </span> </span> represent an ascending or descending tempered perfect fifth (i.e. an increase or decrease in frequency by <span class="texhtml mvar" style="font-style:italic;">x</span>), while <span class="nowrap"><span class="texhtml">2 : 1 </span> </span> or <span class="nowrap"> <span class="texhtml"> 1 : 2 </span> </span> represent an ascending or descending octave. </p><p>As in Pythagorean tuning, this method generates 13 pitches, with A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> and G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> nearly a quarter-tone apart. To build a 12-tone scale, typically A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> is arbitrarilly discarded. </p> <div class="mw-heading mw-heading3"><h3 id="C-based_construction_tables">C-based construction tables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=3" title="Edit section: C-based construction tables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The table above shows a D-based stack of fifths (i.e. a stack in which all ratios are expressed relative to D, and D has a ratio of <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 1 </span></span>⁠</span></span>). Since it is centered at D, the base note, this stack can be called <i>D-based symmetric</i>: </p> <dl><dd>A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–F–C–G–<b>D</b>–A–E–B–F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></dd></dl> <p>With the perfect fifth taken as <span class="nowrap"><sup style="margin-right: -0.5em; vertical-align: 0.8em;">4</sup>√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span>, the ends of this scale are 125 in frequency ratio apart, causing a gap of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">125</span><span class="sr-only">/</span><span class="den">128</span></span>⁠</span> (about two-fifths of a semitone) between its ends if they are normalized to the same octave. If the last step (here, G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>) is replaced by a copy of A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> but in the same octave as G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>, that will increase the interval C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> to a discord called a <a href="/wiki/Wolf_fifth" class="mw-redirect" title="Wolf fifth">wolf fifth</a>. (Note that in meantone systems there are no <a href="/wiki/Wolf_interval" title="Wolf interval">wolf intervals</a> when the actual, correct note is played: The wolf discord always is the result of naïvely trying to substitute the flat above for the required sharp below it, or vice-versa.) </p><p>Except for the size of the fifth, this is identical to the stack traditionally used in <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a>. Some authors prefer showing a C-based stack of fifths, ranging from A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> to G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>. Since C is not at its center, this stack is called <i>C-based asymmetric</i>: </p> <dl><dd>A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–F–<b>C</b>–G–D–A–E–B–F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></dd></dl> <p>Since the boundaries of this stack (A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> and G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>) are identical to those of the D-based symmetric stack, the note names of the 12-tone scale produced by this stack are also identical. The only difference is that the construction table shows intervals from C, rather than from D. Notice that 144 intervals can be formed from a 12-tone scale (see table below), which include intervals from C, D, and any other note. However, the construction table shows only 12 of them, in this case those starting from C. This is at the same time the main advantage and main disadvantage of the C-based asymmetric stack, as the intervals from C are commonly used, but since C is not at the center of this stack, they unfortunately include an <a href="/wiki/Augmented_fifth" title="Augmented fifth">augmented fifth</a> (i.e. the interval from C to G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>), instead of a <a href="/wiki/Minor_sixth" title="Minor sixth">minor sixth</a> (from C to A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>). This augmented fifth is an extremely dissonant <a href="/wiki/Wolf_interval" title="Wolf interval">wolf interval</a>, as it deviates by 41.1 cents (a <a href="/wiki/Diesis" title="Diesis">diesis</a> of ratio <span class="nowrap"> 128 : 125 ,</span> almost twice a <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a>) from the corresponding pure interval of <span class="nowrap"> 8 : 5 </span> or 813.7 cents. </p><p>On the contrary, the intervals from D shown in the table above, since D is at the center of the stack, do not include wolf intervals and include a pure minor sixth (from D to B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>), instead of an impure augmented fifth. Notice that in the above-mentioned set of 144 intervals pure minor sixths are more frequently observed than impure augmented fifths (see table below), and this is one of the reasons why it is not desirable to show an impure augmented fifth in the construction table. A <i>C-based symmetric</i> stack might be also used, to avoid the above-mentioned drawback: </p> <dl><dd>G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–D<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–F–<b>C</b>–G–D–A–E–B–F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></dd></dl> <p>In this stack, G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> and F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> have a similar frequency, and G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> is typically discarded. Also, the note between C and D is called D<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> rather than C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>, and the note between G and A is called A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> rather than G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>. The C-based symmetric stack is rarely used, possibly because it produces the <a href="/wiki/Wolf_fifth" class="mw-redirect" title="Wolf fifth">wolf fifth</a> in the unusual position of F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–D<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> instead of G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, where musicians accustomed to the previously used Pythagorean tuning might expect it). </p> <div class="mw-heading mw-heading2"><h2 id="Justly_intonated_quarter-comma_meantone">Justly intonated quarter-comma meantone</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=4" title="Edit section: Justly intonated quarter-comma meantone"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a> version of the quarter-comma meantone temperament may be constructed in the same way as <a href="/wiki/Johann_Kirnberger" title="Johann Kirnberger">Johann Kirnberger</a>'s <a href="/wiki/Schisma" title="Schisma">rational version</a> of <a href="/wiki/12-TET" class="mw-redirect" title="12-TET">12-TET</a>. The value of 5<sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">8</span></span></sup>·35<sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">3</span></span></sup> is very close to 4, which is why a 7-limit interval 6144:6125 (which is the difference between the 5-limit <a href="/wiki/Diesis" title="Diesis">diesis</a> 128:125 and the <a href="/wiki/Septimal_diesis" title="Septimal diesis">septimal diesis</a> 49:48), equal to 5.362 cents, appears very close to the quarter-comma (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">81</span><span class="sr-only">/</span><span class="den">80</span></span>⁠</span>)<sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">4</span></span></sup> of 5.377 cents. So the perfect fifth has the ratio of 6125:4096, which is the difference between three <a href="/wiki/Just_major_third" class="mw-redirect" title="Just major third">just major thirds</a> and two <a href="/wiki/Septimal_major_second" class="mw-redirect" title="Septimal major second">septimal major seconds</a>; four such fifths exceed the ratio of 5:1 by the tiny interval of 0.058 cents. The <a href="/wiki/Wolf_interval" title="Wolf interval">wolf fifth</a> there appears to be 49:32, the difference between the <a href="/wiki/Septimal_minor_seventh" class="mw-redirect" title="Septimal minor seventh">septimal minor seventh</a> and the <a href="/wiki/Septimal_major_second" class="mw-redirect" title="Septimal major second">septimal major second</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Greater_and_lesser_semitones">Greater and lesser semitones</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=5" title="Edit section: Greater and lesser semitones"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As discussed above, in the quarter-comma meantone temperament, </p> <ul><li>the ratio of a semitone is <i>S</i> = 8:5<sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">5</span>⁄<span class="den">4</span></span></sup>,</li> <li>the ratio of a tone is <i>T</i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span>:2.</li></ul> <p>The tones in the diatonic scale can be divided into pairs of semitones. However, since <i>S</i><sup>2</sup> is not equal to <i>T</i>, each tone must be composed of a pair of unequal semitones, <i>S</i>, and <i>X</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\cdot X=T.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⋅</mo> <mi>X</mi> <mo>=</mo> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\cdot X=T.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fa4f74ca405a7ee055dd71096b7d7f24f97505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.54ex; height:2.176ex;" alt="{\displaystyle S\cdot X=T.}"></span></dd></dl> <p>Hence, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\frac {T}{S}}={\frac {\sqrt {5}}{2}}{\Bigg /}{\frac {8}{5^{5/4}}}={\frac {5^{1/2}\cdot 5^{5/4}}{8\cdot 2}}={\frac {5^{7/4}}{16}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <mi>S</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="2.470em" minsize="2.470em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mn>16</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X={\frac {T}{S}}={\frac {\sqrt {5}}{2}}{\Bigg /}{\frac {8}{5^{5/4}}}={\frac {5^{1/2}\cdot 5^{5/4}}{8\cdot 2}}={\frac {5^{7/4}}{16}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc0728b64de9d5ad684147eb7267436983095d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.027ex; height:7.509ex;" alt="{\displaystyle X={\frac {T}{S}}={\frac {\sqrt {5}}{2}}{\Bigg /}{\frac {8}{5^{5/4}}}={\frac {5^{1/2}\cdot 5^{5/4}}{8\cdot 2}}={\frac {5^{7/4}}{16}}.}"></span></dd></dl> <p>Notice that <i>S</i> is 117.1 cents, and <i>X</i> is 76.0 cents. Thus, <i>S</i> is the greater semitone, and <i>X</i> is the lesser one. <i>S</i> is commonly called the <b>diatonic semitone</b> (or <a href="/wiki/Minor_second" class="mw-redirect" title="Minor second">minor second</a>), while <i>X</i> is called the <b>chromatic semitone</b> (or <a href="/wiki/Augmented_unison" title="Augmented unison">augmented unison</a>). </p><p>The sizes of <i>S</i> and <i>X</i> can be compared to the just intonated ratio 18:17 which is 99.0 cents. <i>S</i> deviates from it by +18.2 cents, and <i>X</i> by −22.9 cents. These two deviations are comparable to the syntonic comma (21.5 cents), which this system is designed to tune out from the Pythagorean major third. However, since even the just intonated ratio 18:17 sounds markedly dissonant, these deviations are considered acceptable in a semitone. </p><p>In quarter-comma meantone, the minor second is considered acceptable while the augmented unison sounds dissonant and should be avoided. </p> <div class="mw-heading mw-heading2"><h2 id="Size_of_intervals">Size of intervals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=6" title="Edit section: Size of intervals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The table above shows only intervals from D. However, intervals can be formed by starting from each of the above listed 12 notes. Thus, twelve intervals can be defined for each <i>interval type</i> (twelve unisons, twelve <a href="/wiki/Semitone" title="Semitone">semitones</a>, twelve intervals composed of 2 semitones, twelve intervals composed of 3 semitones, etc.). </p><p>As explained above, one of the twelve nominal "fifths" (the <a href="/wiki/Wolf_fifth" class="mw-redirect" title="Wolf fifth">wolf fifth</a>) has a different size with respect to the other eleven. For a similar reason, each of the other interval types (except for unisons and octaves) has two different sizes in quarter-comma meantone when truncated to fit into an octave that only permits 12 notes (whereas actual quarter-comma meantone requires approximately <a href="/wiki/31_equal_temperament" title="31 equal temperament">31 notes per octave</a>). This is the price paid for attempting to fit a many-note temperament onto a keyboard without enough distinct pitches per octave: The consequence is "fake" notes, for example, one of the so-called "fifths" is <i>not</i> a fifth, but really a quarter-comma <a href="/wiki/Diminished_sixth" title="Diminished sixth">diminished sixth</a>, whose pitch is a bad substitute for the needed fifth. </p><p>The table shows the approximate size of the notes in cents: The genuine notes are on a light grey background, the out-of-tune substitutes are on a red or orange background; the name for the genuine intervals are at the top or bottom of a column with plain grey background; the interval names of the bad substitutions are at opposite end, printed on a colored background. <a href="/wiki/Interval_(music)#Quality" title="Interval (music)">Interval names</a> are given in their standard shortened form.<sup id="cite_ref-short_form_intv_2-0" class="reference"><a href="#cite_note-short_form_intv-2"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> For instance, the size of the interval from D to A, which is a <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> (P5), can be found in the seventh column of the row labeled <b>D</b>. <a href="/wiki/Five-limit_tuning#The_justest_ratios" title="Five-limit tuning">strictly just</a> (or <i>pure</i>) intervals are shown in <b><a href="/wiki/Boldface" class="mw-redirect" title="Boldface">bold</a></b> font. <a href="/wiki/Wolf_interval" title="Wolf interval">Wolf intervals</a> are highlighted in red.<sup id="cite_ref-Wolf_3-0" class="reference"><a href="#cite_note-Wolf-3"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Size_of_intervals_in_D-based_symmetric_quarter-comma_meantone.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/Size_of_intervals_in_D-based_symmetric_quarter-comma_meantone.PNG" decoding="async" width="455" height="340" class="mw-file-element" data-file-width="455" data-file-height="340" /></a><figcaption>Approximate size in cents of the 144 intervals in D-based quarter-comma meantone tuning. <a href="/wiki/Interval_(music)#Quality" title="Interval (music)">Interval names</a> are given in their standard short-form.<sup id="cite_ref-short_form_intv_2-1" class="reference"><a href="#cite_note-short_form_intv-2"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> <a href="/wiki/5-limit_tuning#The_justest_ratios" class="mw-redirect" title="5-limit tuning">Purely just intervals</a> (which are only unisons, octaves, and some major thirds and minor sixths) are shown in <b><a href="/wiki/Boldface" class="mw-redirect" title="Boldface">bold</a></b> font. <a href="/wiki/Wolf_interval" title="Wolf interval">Wolf intervals</a> are highlighted in red.<sup id="cite_ref-Wolf_3-1" class="reference"><a href="#cite_note-Wolf-3"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> The red and gold colored intervals are out-of-tune substitutes for the missing correct meantone pitches, omitted because of the keyboard limit of 12 notes per octave.</figcaption></figure> <p>Surprisingly, although this tuning system was designed to produce purely consonant major thirds, only eight of the intervals that are thirds in <a href="/wiki/12_tone_equal_temperament" class="mw-redirect" title="12 tone equal temperament">12 <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">TET</span></span></a> are purely just <span class="nowrap">(5:4</span> or about 386.3 cents) in the truncated quarter comma shown on the table: The <i>actual</i> quarter-comma notes needed to start or end the interval of a third are missing from among the 12 available pitches, and substitution of nearby available-but-wrong notes leads to dissonant thirds. </p><p>The reason why the interval sizes vary throughout the scale is from using substitute notes, whose pitches are correctly tuned for a different use in the scale, instead of the genuine quarter comma notes for the in desired interval, creates out-of-tune intervals. The actual notes in a fully implemented quarter-comma scale (requiring about <a href="/wiki/31_equal_temperament" title="31 equal temperament">31 keys</a> per octave instead of only 12) would be consonant, like all of the uncolored intervals: The dissonance is the consequence of replacing the correct quarter-comma notes with wrong notes that happen to be assigned to the same key on the 12-tone keyboard. As mentioned above, the frequencies defined by construction for the twelve notes determine two different kinds of semitones (i.e. intervals between adjacent notes): </p> <ul><li>The minor second (m2), also called the diatonic semitone, with size</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\ ({\mathsf {\ or\ }}S_{1})={\frac {\ 8\ }{~5^{5/4}}}\approx \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif"> </mtext> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">r</mi> <mtext mathvariant="sans-serif"> </mtext> </mrow> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>8</mn> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≈</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\ ({\mathsf {\ or\ }}S_{1})={\frac {\ 8\ }{~5^{5/4}}}\approx \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b67a3b8bdef27add3d2bf28e87696ff88eda09aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.543ex; height:5.843ex;" alt="{\displaystyle S\ ({\mathsf {\ or\ }}S_{1})={\frac {\ 8\ }{~5^{5/4}}}\approx \ }"></span> 117.1 cents.</dd></dl></dd> <dd>(for instance, between D and E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>)</dd></dl> <ul><li>The augmented unison (A1), also called the <i>chromatic semitone</i>, with size</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\ ({\mathsf {\ or\ }}S_{2})={\frac {~5^{7/4}}{\ 16\ }}\approx \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif"> </mtext> <mi mathvariant="sans-serif">o</mi> <mi mathvariant="sans-serif">r</mi> <mtext mathvariant="sans-serif"> </mtext> </mrow> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mtext> </mtext> <mn>16</mn> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>≈</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\ ({\mathsf {\ or\ }}S_{2})={\frac {~5^{7/4}}{\ 16\ }}\approx \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c722e6ba573a71877e3362f1c4448803ef533db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.023ex; height:5.843ex;" alt="{\displaystyle X\ ({\mathsf {\ or\ }}S_{2})={\frac {~5^{7/4}}{\ 16\ }}\approx \ }"></span> 76.0 cents.</dd></dl></dd> <dd>(for instance, between C and C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>)</dd></dl> <p>Conversely, in an <a href="/wiki/Twelve_tone_equal_temperament" class="mw-redirect" title="Twelve tone equal temperament">equally tempered</a> chromatic scale, by definition the twelve pitches are equally spaced, all semitones having a size of exactly </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\mathsf {ET}}={\sqrt[{12}]{2\ }}=100{\mathsf {\ cents.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">E</mi> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mn>2</mn> <mtext> </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mn>100</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif"> </mtext> <mi mathvariant="sans-serif">c</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">t</mi> <mi mathvariant="sans-serif">s</mi> <mo mathvariant="sans-serif">.</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\mathsf {ET}}={\sqrt[{12}]{2\ }}=100{\mathsf {\ cents.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06411d85459778a799cbf69377e99e9a3cbff1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.354ex; height:3.009ex;" alt="{\displaystyle S_{\mathsf {ET}}={\sqrt[{12}]{2\ }}=100{\mathsf {\ cents.}}}"></span></dd></dl> <p>As a consequence all intervals of any given type have the same size (e.g., all major thirds have the same size, all fifths have the same size, etc.). The price paid, in this case, is that none of them is justly tuned and perfectly consonant, except, of course, for the unison and the octave. </p><p>For a comparison with other tuning systems, see also <a href="/wiki/Interval_(music)#Size_of_intervals_used_in_different_tuning_systems" title="Interval (music)">this table</a>. </p><p>By definition, in quarter-comma meantone, one so-called "perfect" fifth (P5 in the table) has a size of approximately 696.6 cents <span class="nowrap">( <span class="texhtml">700 − <i>ε</i></span> cents,</span> where <span class="nowrap"><span class="texhtml"><i>ε</i> ≈</span> 3.422 cents);</span> since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of <span class="nowrap"><span class="texhtml"> 700 + 11 <i>ε</i></span> cents,</span> which is about 737.6 cents (one of the <a href="/wiki/Wolf_fifth" class="mw-redirect" title="Wolf fifth">wolf fifths</a>). Notice that, as shown in the table, the latter interval, although used as a <a href="/wiki/Enharmonically_equivalent" class="mw-redirect" title="Enharmonically equivalent">substitute</a> for a fifth, the actual interval is really a <a href="/wiki/Diminished_sixth" title="Diminished sixth">diminished sixth</a> (<b>d6</b>), which is of course out of tune with the nearby but different fifth it replaces. Similarly, </p> <ul><li>10 <a href="/wiki/Major_second" title="Major second">major seconds</a> (<b>M2</b>) are ≈ 193.2 cents <span class="nowrap">( <span class="texhtml">200 − 2 <i>ε</i></span> ),</span> 2 <a href="/wiki/Diminished_third" title="Diminished third">diminished thirds</a> (<b>d3</b>) are ≈ 234.2 cents <span class="nowrap">( <span class="texhtml">200 + 10 <i>ε</i></span> ),</span> and their average is 200 cents;</li> <li>9 <a href="/wiki/Minor_third" title="Minor third">minor thirds</a> (<b>m3</b>) are ≈ 310.3 cents <span class="nowrap">( <span class="texhtml">300 + 3 <i>ε</i></span> ),</span> 3 <a href="/wiki/Augmented_second" title="Augmented second">augmented seconds</a> (<b>A2</b>) are ≈ 269.2 cents <span class="nowrap">( <span class="texhtml">300 − 9 <i>ε</i></span> ),</span> and their average is 300 cents;</li> <li>8 <a href="/wiki/Major_third" title="Major third">major thirds</a> (<b>M3</b>) are ≈ 386.3 cents <span class="nowrap">( <span class="texhtml">400 − 4 <i>ε</i></span> ),</span> 4 <a href="/wiki/Diminished_fourth" title="Diminished fourth">diminished fourths</a> (<b>d4</b>) are ≈ 427.4 cents <span class="nowrap">( <span class="texhtml">400 + 8 <i>ε</i></span> ),</span> and their average is 400 cents;</li> <li>7 diatonic <a href="/wiki/Semitone" title="Semitone">semitones</a> (<b>m2</b>) are ≈ 117.1 cents<span class="nowrap">( <span class="texhtml">100 + 5 <i>ε</i></span> ),</span> 5 chromatic semitones (<b>A1</b>) are ≈ 76.0 cents <span class="nowrap">( <span class="texhtml">100 − 7 <i>ε</i></span> ),</span> and their average is 100 cents.</li></ul> <p>In short, similar differences in width are observed for all interval types, except for unisons and octaves, and the excesses and deficits in width are all multiples of <span class="texhtml mvar" style="font-style:italic;">ε</span>, the difference between the quarter-comma meantone fifth and the average fifth required if one is to close the spiral of fifths into a circle. </p><p>Notice that, as an obvious consequence, each augmented or diminished interval is exactly <span class="nowrap"> <span class="texhtml">12 <i>ε</i></span> cents</span> (≈ 41.1 cents) wider or narrower than its enharmonic equivalent. For instance, the diminished sixth (or wolf fifth) is <span class="nowrap"> <span class="texhtml">12 <i>ε</i></span> cents</span> wider than each perfect fifth, and each augmented second is <span class="nowrap"> <span class="texhtml">12 <i>ε</i></span> cents</span> narrower than each minor third. This interval of size <span class="nowrap"> <span class="texhtml">12 <i>ε</i></span> cents</span> is known as a <a href="/wiki/Diesis" title="Diesis">diesis</a>, or <a href="/wiki/Diminished_second" title="Diminished second">diminished second</a>. This implies that <span class="texhtml mvar" style="font-style:italic;">ε</span> can be also defined as one twelfth of a diesis. </p> <div class="mw-heading mw-heading2"><h2 id="Triads_in_the_chromatic_scale">Triads in the chromatic scale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=7" title="Edit section: Triads in the chromatic scale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Major_triad" class="mw-redirect" title="Major triad">major triad</a> can be defined by a pair of intervals from the root note: a <a href="/wiki/Major_third" title="Major third">major third</a> (interval spanning 4 semitones) and a <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> (7 semitones). The <a href="/wiki/Minor_triad" class="mw-redirect" title="Minor triad">minor triad</a> can likewise be defined by a <a href="/wiki/Minor_third" title="Minor third">minor third</a> (3 semitones) and a perfect fifth (7 semitones). </p><p>As shown above, a chromatic scale has twelve intervals spanning seven semitones. Eleven of these are perfect fifths, while the twelfth is a diminished sixth. Since they span the same number of semitones, perfect fifths and diminished sixths are considered to be <a href="/wiki/Enharmonically_equivalent" class="mw-redirect" title="Enharmonically equivalent">enharmonically equivalent</a>. In an <a href="/wiki/Equal_temperament" title="Equal temperament">equally</a>-tuned chromatic scale, perfect fifths and diminished sixths have exactly the same size. The same is true for all the enharmonically equivalent intervals spanning 4 semitones (major thirds and diminished fourths), or 3 semitones (minor thirds and augmented seconds). However, in the meantone temperament this is not true. In this tuning system, enharmonically equivalent intervals may have different sizes, and some intervals may markedly deviate from their <a href="/wiki/Just_intonation" title="Just intonation">justly tuned</a> ideal ratios. As explained in the previous section, if the deviation is too large, then the given interval is not usable, either by itself or in a chord. </p><p>The following table focuses only on the above-mentioned three interval types, used to form major and minor triads. Each row shows three intervals of different types but which have the same root note. Each interval is specified by a pair of notes. To the right of each interval is listed the formula for the <a href="/wiki/Interval_ratio" title="Interval ratio">interval ratio</a>. The intervals diminished fourth, diminished sixth and augmented second may be regarded as <a href="/wiki/Wolf_interval" title="Wolf interval">wolf intervals</a>, and have their backgrounds set to pale red. <span class="texhtml mvar" style="font-style:italic;">S</span> and <span class="texhtml mvar" style="font-style:italic;">X</span> denote the ratio of the two abovementioned kinds of semitones (minor second and augmented unison). </p> <dl><dd><table class="wikitable"> <tbody><tr> <th colspan="2"><span style="font-size:120%">3 semitones</span><br />(m3 or A2) </th> <th colspan="2"><span style="font-size:120%">4 semitones</span><br />(M3 or d4) </th> <th colspan="2"><span style="font-size:120%">7 semitones</span><br />(P5 or d6) </th></tr> <tr> <th><span style="font-size:85%;">Interval</span> </th> <th><span class="nowrap"><span style="font-size:85%;">Ratio</span> </span> </th> <th><span style="font-size:85%;">Interval</span> </th> <th><span class="nowrap"><span style="font-size:85%;">Ratio</span> </span> </th> <th><span style="font-size:85%;">Interval</span> </th> <th><span class="nowrap"><span style="font-size:85%;">Ratio</span> </span> </th></tr> <tr> <td align="center">C–E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup></td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i></span> </td> <td align="center">C–E</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i><sup> 2</sup></span> </td> <td align="center">C–G</td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–E</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i></span> </td> <td align="center">C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–F</td> <td style="background:pink"><span class="texhtml"><i>S</i><sup> 3</sup> · <i>X</i></span> </td> <td align="center">C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">D–F</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i></span> </td> <td align="center">D–F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i><sup> 2</sup></span> </td> <td align="center">D–A</td> <td><span class="texhtml"><i>S</i><sup>4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td style="background:pink"><span class="texhtml"><i>S</i> · <i>X</i><sup> 2</sup></span> </td> <td align="center">E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–G</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i><sup> 2</sup></span> </td> <td align="center">E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup></td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">E–G</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i></span> </td> <td align="center">E–G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i><sup> 2</sup></span> </td> <td align="center">E–B</td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">F–G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td style="background:pink"><span class="texhtml"><i>S</i> · <i>X</i><sup> 2</sup></span> </td> <td align="center">F–A</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i><sup> 2</sup></span> </td> <td align="center">F–C</td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–A</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i></span> </td> <td align="center">F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–B<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></td> <td style="background:pink"><span class="texhtml"><i>S</i><sup> 3</sup> · <i>X</i></span> </td> <td align="center">F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">G–B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup></td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i></span> </td> <td align="center">G–B</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i><sup> 2</sup></span> </td> <td align="center">G–D</td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–B</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i></span> </td> <td align="center">G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–C</td> <td style="background:pink"><span class="texhtml"><i>S</i><sup> 3</sup> · <i>X</i></span> </td> <td align="center">G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>–E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></td> <td style="background:pink"><span class="texhtml"><i>S</i><sup> 5</sup> · <i>X</i><sup> 2</sup></span> </td></tr> <tr> <td align="center">A–C</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i></span> </td> <td align="center">A–C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i><sup> 2</sup></span> </td> <td align="center">A–E</td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></td> <td style="background:pink"><span class="texhtml"><i>S</i> · <i>X</i><sup> 2</sup></span> </td> <td align="center">B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–D</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i><sup> 2</sup></span> </td> <td align="center">B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–F</td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr> <tr> <td align="center">B–D</td> <td><span class="texhtml"><i>S</i><sup> 2</sup> · <i>X</i></span> </td> <td align="center">B–E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup></td> <td style="background:pink"><span class="texhtml"><i>S</i><sup> 3</sup> · <i>X</i></span> </td> <td align="center">B–F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml"><i>S</i><sup> 4</sup> · <i>X</i><sup> 3</sup></span> </td></tr></tbody></table></dd></dl> <p>First, look at the last two columns on the right. All the 7 semitone intervals except one have a ratio of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{4}\cdot X^{3}\approx 1.4953\approx 696.6{\text{ cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>≈</mo> <mn>1.4953</mn> <mo>≈</mo> <mn>696.6</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{4}\cdot X^{3}\approx 1.4953\approx 696.6{\text{ cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6034ce4a54b59e01b0bcaae005bbf805cd5100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:31.018ex; height:2.676ex;" alt="{\displaystyle S^{4}\cdot X^{3}\approx 1.4953\approx 696.6{\text{ cents}}}"></span></dd></dl> <p>which deviates by −5.4 cents from the just 3:2 of 702.0 cents. Five cents is small and acceptable. On the other hand, the diminished sixth from G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> to E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> has a ratio of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{5}\cdot X^{2}\approx 1.5312\approx 737.6{\text{ cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <mn>1.5312</mn> <mo>≈</mo> <mn>737.6</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{5}\cdot X^{2}\approx 1.5312\approx 737.6{\text{ cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21944671b202082f4740440030a57d8320689779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:31.018ex; height:2.676ex;" alt="{\displaystyle S^{5}\cdot X^{2}\approx 1.5312\approx 737.6{\text{ cents}}}"></span></dd></dl> <p>which deviates by +35.7 cents from the just perfect fifth. Thirty-five cents is beyond the acceptable range. </p><p>Now look at the two columns in the middle. Eight of the twelve 4-semitone intervals have a ratio of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}\cdot X^{2}=1.25\approx 386.3{\text{ cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1.25</mn> <mo>≈</mo> <mn>386.3</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}\cdot X^{2}=1.25\approx 386.3{\text{ cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd679e72a6e2db88be7bfe02b263f83b09e1aaf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:28.693ex; height:2.676ex;" alt="{\displaystyle S^{2}\cdot X^{2}=1.25\approx 386.3{\text{ cents}}}"></span></dd></dl> <p>which is exactly a just 5:4. On the other hand, the four diminished fourths with roots at C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>, F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>, G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> and B have a ratio of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}\cdot X=1.28\approx 427.4{\text{ cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mi>X</mi> <mo>=</mo> <mn>1.28</mn> <mo>≈</mo> <mn>427.4</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{3}\cdot X=1.28\approx 427.4{\text{ cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c0862059ad59e08dd630a3f8abb736c26d697d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.622ex; height:2.676ex;" alt="{\displaystyle S^{3}\cdot X=1.28\approx 427.4{\text{ cents}}}"></span></dd></dl> <p>which deviates by +41.1 cents from the just major third. Again, this sounds badly out of tune. </p><p>Major triads are formed out of both major thirds and perfect fifths. If either of the two intervals is substituted by a wolf interval (d6 instead of P5, or d4 instead of M3), then the triad is not acceptable. Therefore, major triads with root notes of C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>, F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>, G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> and B are not used in meantone scales whose fundamental note is C. </p><p>Now look at the first two columns on the left. Nine of the twelve 3-semitone intervals have a ratio of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}\cdot X\approx 1.1963\approx 310.3{\text{ cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>X</mi> <mo>≈</mo> <mn>1.1963</mn> <mo>≈</mo> <mn>310.3</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}\cdot X\approx 1.1963\approx 310.3{\text{ cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c881d1d85a86a8a7d10a569e9146d703baf59c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:29.947ex; height:2.676ex;" alt="{\displaystyle S^{2}\cdot X\approx 1.1963\approx 310.3{\text{ cents}}}"></span></dd></dl> <p>which deviates by −5.4 cents from the just 6:5 of 315.6 cents. Five cents is acceptable. On the other hand, the three augmented seconds whose roots are E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>, F and B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> have a ratio of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\cdot X^{2}\approx 1.1682\approx 269.2{\text{ cents}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⋅</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <mn>1.1682</mn> <mo>≈</mo> <mn>269.2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> cents</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\cdot X^{2}\approx 1.1682\approx 269.2{\text{ cents}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25fa3ffe6ba8f97ced436a68becf373ad4b5db44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:29.941ex; height:2.676ex;" alt="{\displaystyle S\cdot X^{2}\approx 1.1682\approx 269.2{\text{ cents}}}"></span></dd></dl> <p>which deviates by −46.4 cents from the just minor third. It is a close match, however, for the 7:6 <a href="/wiki/Septimal_minor_third" title="Septimal minor third">septimal minor third</a> of 266.9 cents, deviating by +2.3 cents. These augmented seconds, though sufficiently consonant by themselves, will sound "exotic" or atypical when played together with a perfect fifth. </p><p>Minor triads are formed out of both minor thirds and fifths. If either of the two intervals are substituted by an enharmonically equivalent interval (d6 instead of P5, or A2 instead of m3), then the triad will not sound good. Therefore, minor triads with root notes of E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>, F, G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> and B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> are not used in the meantone scale defined above. </p> <dl><dd><dl><dd><table> <caption><span style="font-size:120%"> <b>Summary of utility of notes for triads</b> </span> </caption> <tbody><tr style="vertical-align:bottom;"> <td align="right">Usable major triad tonics</td> <td></td> <td>C, D, E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>, E, F, G, A, B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td></tr> <tr style="vertical-align:bottom;"> <td align="right">Usable minor triad tonics</td> <td></td> <td>C, C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>, D, E, F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>, G, A, B </td></tr> <tr style="vertical-align:bottom;"> <td align="right"><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> Usable for tonics of both<br />major and minor triads</td> <td></td> <td>C, D, E, G, A </td></tr> <tr style="vertical-align:bottom;"> <td align="right">Only usable for<br />major triad tonics</td> <td></td> <td>E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>, F, B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup> </td></tr> <tr style="vertical-align:bottom;"> <td align="right">Only usable for<br />minor triad tonics</td> <td></td> <td>C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>, F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>, B </td></tr> <tr style="vertical-align:bottom;"> <td align="right">Not usable as a major<br />nor minor triad tonic</td> <td></td> <td>G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> </td></tr></tbody></table></dd></dl></dd></dl> <p>Note carefully that the limitations of what triads are feasible is determined by the choice to only allow 12 notes per octave, to conform with a standard piano keyboard. It is not a limitation of meantone tuning, <i>per se</i>, but rather the fact that sharps are different from the flats of the notes above them, and standard 12 note keyboards are built on the false assumption that they should be the same. As discussed above, G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> is a different pitch that A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>, as are all other "enharmonic" pairs of sharps and flats in quarter comma meantone: Each requires a separate key on the keyboard and neither can substitute for the other. This is, in fact a property of <i>all</i> other tuning systems, with the exception of <a href="/wiki/12_tone_equal_temperament" class="mw-redirect" title="12 tone equal temperament">12 tone equal temperament</a> (alone among all equal temperaments) and <a href="/wiki/Well_temperament" title="Well temperament">well temperaments</a> of all types. The limited chordal options is not a fault in meantone tunings; it is the consequence of needing more notes in the octave than is available on some modern equal tempered instruments.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_construction">Alternative construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=8" title="Edit section: Alternative construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As discussed above, in the quarter-comma meantone temperament truncated to only 12 notes, </p> <ul><li>the ratio of a greater (diatonic) semitone is <i>S</i> = 8:5<sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">5</span>⁄<span class="den">4</span></span></sup>,</li> <li>the ratio of a lesser (chromatic) semitone is <i>X</i> = 5<sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">7</span>⁄<span class="den">4</span></span></sup>:16,</li> <li>the ratio of most whole tones is <i>T</i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span>:2,</li> <li>the ratio of most fifths is <i>P</i> = <span class="nowrap"><sup style="margin-right: -0.5em; vertical-align: 0.8em;">4</sup>√<span style="border-top:1px solid; padding:0 0.1em;">5</span></span>.</li></ul> <p>It can be verified through calculation that most whole tones (namely, the major seconds) are composed of one greater and one lesser semitone: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=S\cdot X={\frac {8}{5^{5/4}}}\cdot {\frac {5^{7/4}}{16}}={\frac {\sqrt {5}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mi>S</mi> <mo>⋅</mo> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mn>16</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=S\cdot X={\frac {8}{5^{5/4}}}\cdot {\frac {5^{7/4}}{16}}={\frac {\sqrt {5}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0085bdd92c00ab263a6a64364b0cd47dcc6aff43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.744ex; height:6.509ex;" alt="{\displaystyle T=S\cdot X={\frac {8}{5^{5/4}}}\cdot {\frac {5^{7/4}}{16}}={\frac {\sqrt {5}}{2}}.}"></span></dd></dl> <p>Similarly, a fifth is typically composed of three tones and one greater semitone: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=T^{3}\cdot S={\frac {5^{3/2}}{2^{3}}}\cdot {\frac {8}{5^{5/4}}}={\sqrt[{4}]{5}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=T^{3}\cdot S={\frac {5^{3/2}}{2^{3}}}\cdot {\frac {8}{5^{5/4}}}={\sqrt[{4}]{5}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d030302ae19405bca7c7b3408da3994a43edaca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.811ex; height:6.509ex;" alt="{\displaystyle P=T^{3}\cdot S={\frac {5^{3/2}}{2^{3}}}\cdot {\frac {8}{5^{5/4}}}={\sqrt[{4}]{5}},}"></span></dd></dl> <p>which is equivalent to four greater and three lesser semitones: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=T^{3}\cdot S=S^{4}\cdot X^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mi>S</mi> <mo>=</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=T^{3}\cdot S=S^{4}\cdot X^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1abfa03bb94a7ea8e7d5389d95e72c36a567bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.848ex; height:2.676ex;" alt="{\displaystyle P=T^{3}\cdot S=S^{4}\cdot X^{3}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Diatonic_scale">Diatonic scale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=9" title="Edit section: Diatonic scale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Diatonic_scale" title="Diatonic scale">diatonic scale</a> can be constructed by starting from the fundamental note and multiplying it either by a meantone <span class="texhtml mvar" style="font-style:italic;">T</span> to move up by one large step or by a semitone <span class="texhtml mvar" style="font-style:italic;">S</span> to move up by a small step. </p> <pre>C D E F G A B C<span class="nowrap" style="padding-left:0.05em;">′</span> D<span class="nowrap" style="padding-left:0.05em;">′</span> ‖----|----|----|----‖----|----|----‖----| <span class="texhtml mvar" style="font-style:italic;">T</span> <span class="texhtml mvar" style="font-style:italic;">T</span> <span class="texhtml mvar" style="font-style:italic;">S</span> <span class="texhtml mvar" style="font-style:italic;">T</span> <span class="texhtml mvar" style="font-style:italic;">T</span> <span class="texhtml mvar" style="font-style:italic;">T</span> <span class="texhtml mvar" style="font-style:italic;">S</span> <span class="texhtml mvar" style="font-style:italic;">T</span> </pre> <p>The resulting interval sizes with respect to the base note C are shown in the following table. To emphasize the repeating pattern, the formulas use the symbol <span class="texhtml"><i>P</i> ≡ <i>T</i><sup> 3</sup> <i>S</i> </span> to represent a <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> (<a href="/wiki/Penta-" class="mw-redirect" title="Penta-"><i>penta</i></a>): </p> <dl><dd><table class="wikitable"> <tbody><tr style="vertical-align:bottom;"> <th>Note<br />name </th> <th>Formula </th> <th>Ratio </th> <th>Quarter<br />comma<br />(<a href="/wiki/Musical_cents" class="mw-redirect" title="Musical cents">cents</a>) </th> <th><a href="/wiki/Pythagorean_temperament" class="mw-redirect" title="Pythagorean temperament">Pythagorean</a><br />(cents) </th> <th><a href="/wiki/12_tone_equal_temperament" class="mw-redirect" title="12 tone equal temperament"><span class="nowrap">12 <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">TET</span></span></span></a><br />(cents) </th></tr> <tr> <td>C</td> <td><span class="texhtml">1</span></td> <td>1.0000</td> <td align="center">0.0</td> <td align="center">0.0</td> <td align="center">0 </td></tr> <tr> <td>D</td> <td><span class="texhtml mvar" style="font-style:italic;">T</span></td> <td>1.1180</td> <td align="center">193.2</td> <td align="center">203.9</td> <td align="center">200 </td></tr> <tr> <td>E</td> <td><span class="texhtml mvar" style="font-style:italic;">T</span><sup> 2</sup></td> <td>1.2500</td> <td align="center">386.3</td> <td align="center">407.8</td> <td align="center">400 </td></tr> <tr> <td>F</td> <td><span class="texhtml mvar" style="font-style:italic;">T</span><sup> 2</sup> <span class="texhtml mvar" style="font-style:italic;">S</span></td> <td>1.3375</td> <td align="center">503.4</td> <td align="center">498.0</td> <td align="center">500 </td></tr> <tr> <td>G</td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml">1</span></td> <td>1.4953</td> <td align="center">696.6</td> <td align="center">702.0</td> <td align="center">700 </td></tr> <tr> <td>A</td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml mvar" style="font-style:italic;">T</span></td> <td>1.6719</td> <td align="center">889.7</td> <td align="center">905.9</td> <td align="center">900 </td></tr> <tr> <td>B</td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml mvar" style="font-style:italic;">T</span><sup> 2</sup></td> <td>1.8692</td> <td align="center">1082.9</td> <td align="center">1109.8</td> <td align="center">1100 </td></tr> <tr> <td>C<span class="nowrap" style="padding-left:0.05em;">′</span></td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml mvar" style="font-style:italic;">T</span><sup> 2</sup> <span class="texhtml mvar" style="font-style:italic;">S</span></td> <td>2.0000</td> <td align="center">1200.0</td> <td align="center">1200.0</td> <td align="center">1200 </td></tr></tbody></table></dd></dl> <p><span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/4\/47\/Quarter-comma_meantone_major_chord_on_C.mid\/Quarter-comma_meantone_major_chord_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play tonic major chord"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Quarter-comma meantone major chord on C.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/4/47/Quarter-comma_meantone_major_chord_on_C.mid/Quarter-comma_meantone_major_chord_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Play tonic major chord</span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:Quarter-comma_meantone_major_chord_on_C.mid" title="File:Quarter-comma meantone major chord on C.mid">ⓘ</a></sup></span> <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-2" class="noexcerpt ext-phonos-PhonosButton oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/2\/2a\/Just_major_third_on_C.mid\/Just_major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play major third"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major third on C.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/Just_major_third_on_C.mid/Just_major_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Play major third</span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:Just_major_third_on_C.mid" title="File:Just major third on C.mid">ⓘ</a></sup></span> <span 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\/8\/85\/Quarter-comma_meantone_perfect_fifth_on_C.mid\/Quarter-comma_meantone_perfect_fifth_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"Play perfect fifth"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Quarter-comma meantone perfect fifth on C.mid"},"classes":["noexcerpt","ext-phonos-PhonosButton"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/8/85/Quarter-comma_meantone_perfect_fifth_on_C.mid/Quarter-comma_meantone_perfect_fifth_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label">Play perfect fifth</span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:Quarter-comma_meantone_perfect_fifth_on_C.mid" title="File:Quarter-comma meantone perfect fifth on C.mid">ⓘ</a></sup></span> </p> <div class="mw-heading mw-heading3"><h3 id="Chromatic_scale">Chromatic scale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=10" title="Edit section: Chromatic scale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Construction of a quarter-comma meantone <a href="/wiki/Chromatic_scale" title="Chromatic scale">chromatic scale</a> can proceed by stacking a sequence of 12 semitones, each of which may be either the longer diatonic <span class="nowrap">( <span class="texhtml mvar" style="font-style:italic;">S</span> )</span> or the shorter chromatic <span class="nowrap">( <span class="texhtml mvar" style="font-style:italic;">Χ</span> ).</span> </p> <pre>C C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> D E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> E F F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> G G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> A B<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> B C<span class="nowrap" style="padding-left:0.05em;">′</span> C<span class="nowrap" style="padding-left:0.05em;">′</span><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> ‖----|----|----|----|----|----|----‖----|----|----|----|----‖----| <span class="texhtml mvar" style="font-style:italic;">Χ</span> <span class="texhtml mvar" style="font-style:italic;">S</span> <span class="texhtml mvar" style="font-style:italic;">S</span> <span class="texhtml mvar" style="font-style:italic;">Χ</span> <span class="texhtml mvar" style="font-style:italic;">S</span> <span class="texhtml mvar" style="font-style:italic;">Χ</span> <span class="texhtml mvar" style="font-style:italic;">S</span> <span class="texhtml mvar" style="font-style:italic;">Χ</span> <span class="texhtml mvar" style="font-style:italic;">S</span> <span class="texhtml mvar" style="font-style:italic;">S</span> <span class="texhtml mvar" style="font-style:italic;">Χ</span> <span class="texhtml mvar" style="font-style:italic;">S</span> <span class="texhtml mvar" style="font-style:italic;">Χ</span> </pre> <p>Notice that this scale is an extension of the diatonic scale shown in the previous table. Only five notes have been added: C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>, E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>, G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> and B<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> (a <a href="/wiki/Pentatonic_scale" title="Pentatonic scale">pentatonic scale</a>). </p><p>As explained above, an identical scale was originally defined and produced by using a sequence of tempered fifths, ranging from E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> (five fifths below D) to G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span> (six fifths above D), rather than a sequence of semitones. This more conventional approach, similar to the <i>D-based</i> <a href="/wiki/Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a> system, explains the reason why the <span class="texhtml mvar" style="font-style:italic;">Χ</span> and <span class="texhtml mvar" style="font-style:italic;">S</span> semitones are arranged in the particular and apparently arbitrary sequence shown above. </p><p>The interval sizes with respect to the base note C are presented in the following table. The frequency ratios are computed as shown by the formulas. Delta is the difference in cents between meantone and <a href="/wiki/12_tone_equal_temperament" class="mw-redirect" title="12 tone equal temperament"><span class="nowrap">12 <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">TET</span></span></span></a>; the column titled "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 4 </span></span>⁠</span>-c" is the difference in quarter-<a href="/wiki/Syntonic_comma" title="Syntonic comma">commas</a> between meantone and Pythagorean tuning. Note that <span class="nowrap"><span class="texhtml"> <i>Χ</i> ≡ <i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">T</span><span class="sr-only">/</span><span class="den"> S </span></span>⁠</span></i> </span> ,</span> so that <span class="nowrap"><span class="texhtml"> <i>Χ S</i> = <i>T</i> </span> ;</span> most of the <span class="texhtml mvar" style="font-style:italic;">Χ</span> steps appearing in the chart above disappear in the table below, because they combine with a preceding <span class="texhtml mvar" style="font-style:italic;">S</span> and become a <span class="texhtml mvar" style="font-style:italic;">T</span>. </p> <dl><dd><table class="wikitable"> <tbody><tr style="vertical-align:bottom;"> <th>Note<br />name </th> <th>Formula </th> <th>Frequency<br />ratio </th> <th>Quarter<br />comma<br />(<a href="/wiki/Musical_cents" class="mw-redirect" title="Musical cents">cents</a>) </th> <th><a href="/wiki/12_tone_equal_temperament" class="mw-redirect" title="12 tone equal temperament"><span class="nowrap">12 <span class="smallcaps"><span style="font-variant: small-caps; text-transform: lowercase;">TET</span></span></span></a><br />(cents) </th> <th>Delta<br />(cents) </th> <th><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 4 </span></span>⁠</span>-c </th></tr> <tr> <td>C</td> <td><span class="texhtml">1</span></td> <td>1.0000</td> <td align="right">0.0</td> <td align="right">0</td> <td align="right">0.0</td> <td align="right">0 </td></tr> <tr> <td>C<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml mvar" style="font-style:italic;">Χ</span></td> <td>1.0449</td> <td align="right">76.0</td> <td align="right">100</td> <td align="right">−24.0</td> <td align="right">−7 </td></tr> <tr> <td>D</td> <td><span class="texhtml mvar" style="font-style:italic;">T</span></td> <td>1.1180</td> <td align="right">193.2</td> <td align="right">200</td> <td align="right">−6.8</td> <td align="right">−2 </td></tr> <tr> <td>E<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup></td> <td><span class="texhtml mvar" style="font-style:italic;">T</span> <span class="texhtml mvar" style="font-style:italic;">S</span></td> <td>1.1963</td> <td align="right">310.3</td> <td align="right">300</td> <td align="right">+10.3</td> <td align="right">+3 </td></tr> <tr> <td>E</td> <td><span class="texhtml mvar" style="font-style:italic;">T</span><sup> 2</sup></td> <td>1.2500</td> <td align="right">386.3</td> <td align="right">400</td> <td align="right">−13.7</td> <td align="right">−4 </td></tr> <tr> <td>F</td> <td><span class="texhtml mvar" style="font-style:italic;">T</span><sup> 2</sup> <span class="texhtml mvar" style="font-style:italic;">S</span></td> <td>1.3375</td> <td align="right">503.4</td> <td align="right">500</td> <td align="right">+3.4</td> <td align="right">+1 </td></tr> <tr> <td>F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml mvar" style="font-style:italic;">T</span><sup> 3</sup></td> <td>1.3975</td> <td align="right">579.5</td> <td align="right">600</td> <td align="right">−20.5</td> <td align="right">−6 </td></tr> <tr> <td>G</td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml">1</span></td> <td>1.4953</td> <td align="right">696.6</td> <td align="right">700</td> <td align="right">−3.4</td> <td align="right">−1 </td></tr> <tr> <td>G<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml mvar" style="font-style:italic;">Χ</span></td> <td>1.5625</td> <td align="right">772.6</td> <td align="right">800</td> <td align="right">−27.4</td> <td align="right">−8 </td></tr> <tr> <td>A</td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml mvar" style="font-style:italic;">T</span></td> <td>1.6719</td> <td align="right">889.7</td> <td align="right">900</td> <td align="right">−10.3</td> <td align="right">−3 </td></tr> <tr> <td>B<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup></td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml mvar" style="font-style:italic;">T</span> <span class="texhtml mvar" style="font-style:italic;">S</span></td> <td>1.7889</td> <td align="right">1006.8</td> <td align="right">1000</td> <td align="right">+6.8</td> <td align="right">2 </td></tr> <tr> <td>B</td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml mvar" style="font-style:italic;">T</span><sup> 2</sup></td> <td>1.8692</td> <td align="right">1082.9</td> <td align="right">1100</td> <td align="right">−17.1</td> <td align="right">−5 </td></tr> <tr> <td>C<span class="nowrap" style="padding-left:0.05em;">′</span></td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml mvar" style="font-style:italic;">T</span><sup> 2</sup> <span class="texhtml mvar" style="font-style:italic;">S</span></td> <td>2.0000</td> <td align="right">1200.0</td> <td align="right">1200</td> <td align="right">0.0</td> <td align="right">0 </td></tr> <tr> <td>C<span class="nowrap" style="padding-left:0.05em;">′</span><sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup></td> <td><span class="texhtml mvar" style="font-style:italic;">P</span> <span class="texhtml mvar" style="font-style:italic;">T</span><sup> 3</sup></td> <td>2.0898</td> <td align="right">1276.0</td> <td align="right">1300</td> <td align="right">−24.0</td> <td align="right">−7 </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Comparison_with_31-tone_equal_temperament">Comparison with 31-tone equal temperament</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=11" title="Edit section: Comparison with 31-tone equal temperament"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The perfect fifth of quarter-comma meantone, expressed as a fraction of an octave, is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> log<sub>2</sub> 5. Since log<sub>2</sub> 5 is an <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>, a chain of meantone fifths never closes (i.e. never equals a chain of octaves). However, the <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">continued fraction approximations</a> to this irrational fraction number allow us to find equal divisions of the octave which do close; the denominators of these are 1, 2, 5, 7, 12, 19, 31, 174, 205, 789, ... From this we find that 31 quarter-comma meantone fifths come close to closing, and conversely <a href="/wiki/31_equal_temperament" title="31 equal temperament">31 equal temperament</a> represents a good approximation to quarter-comma meantone. </p> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=12" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-short_form_intv-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-short_form_intv_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-short_form_intv_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><b><a href="/wiki/Augmentation_(music)#augmented_interval_anchor" title="Augmentation (music)">A</a></b> for <i><a href="/wiki/Augmentation_(music)#augmented_interval_anchor" title="Augmentation (music)">augmented</a></i> (sharpened); <b><a href="/wiki/Diminution#diminished_interval_anchor" title="Diminution">d</a></b> for <i><a href="/wiki/Diminution#diminished_interval_anchor" title="Diminution">diminished</a></i> (flattened perfect, or flattened minor); <b><a href="/wiki/Interval_(music)#major_minor_perfect_anchor" title="Interval (music)">M</a></b> for <i>major</i>; <b><a href="/wiki/Interval_(music)#major_minor_perfect_anchor" title="Interval (music)">m</a></b> for <i><a href="/wiki/Interval_(music)#major_minor_perfect_anchor" title="Interval (music)">minor</a></i>; <b><a href="/wiki/Interval_(music)#major_minor_perfect_anchor" title="Interval (music)">P</a></b> for <i><a href="/wiki/Interval_(music)#major_minor_perfect_anchor" title="Interval (music)">perfect</a></i> (even though <a href="/wiki/Perfect_fourth" title="Perfect fourth">P4</a> and <a href="/wiki/Perfect_fifth" title="Perfect fifth">P5</a> are not <i>actually</i> "perfect" fourths or fifths in meantone).</span> </li> <li id="cite_note-Wolf-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Wolf_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Wolf_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"Wolf" intervals are defined here in practice as any presumably consonant interval (which are all composed of 3, 4, 5, 7, 8, or 9 diatonic or chromatic semitones) whose size differs from <a href="/wiki/Just_intonation" title="Just intonation">just intonation</a> by a <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a>, or more. Those are: Major and minor thirds or sixths, perfect fourths or fifths, and all their <a href="/wiki/Enharmonic_equivalent" class="mw-redirect" title="Enharmonic equivalent">enharmonic equivalents</a>. On the other hand, presumed dissonant intervals (made up of 1, 2, 6, 10, or 11 semitones) are major and minor seconds or sevenths, tritones, and their <a href="/wiki/Enharmonic_equivalent" class="mw-redirect" title="Enharmonic equivalent">enharmonic equivalents</a>. Since they are already expected to be <a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">dissonant</a>, whether they are justly tuned or not, they are not marked as "wolf" intervals even when they do deviate from just intonation by more than one <a href="/wiki/Syntonic_comma" title="Syntonic comma">syntonic comma</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"> It is interesting that the five notes (C, D, E, G, A) that are usable as tonics for both a major and minor triads, themselves constitute a major <a href="/wiki/Pentatonic_scale" title="Pentatonic scale">pentatonic scale</a>. </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"> Many instruments are able to accommodate more distinct notes in an octave than pianos. For example, an <a href="/wiki/Orchestral_harp" class="mw-redirect" title="Orchestral harp">orchestral harp</a> has distinct pitches for every diatonic note, and all of its sharps, and all of its flats, giving 21 notes for each octave, and each octave can be tuned independently from every other. Fretless stringed instruments like the <a href="/wiki/Violin_family" title="Violin family">violin family</a>, <a href="/wiki/Fretless_guitar" title="Fretless guitar">fretless guitars</a>, and many <a href="/wiki/Plucked_string_instrument" title="Plucked string instrument">lute-family</a> are continuously tunable, as is the <a href="/wiki/Singing" title="Singing">human voice</a>. <dl><dd></dd></dl> There are some rare piano keyboards made with split black keys that have either 17 or 19 notes per octave. Less rare are double-manual harpsichords, multi-manual <a href="/wiki/Organ_(instrument)" class="mw-redirect" title="Organ (instrument)">acoustic organs</a>, and other instruments incorporating multiple-keyboards (such as stacked <a href="/wiki/MIDI" title="MIDI">MIDI</a> keyboards) which conveniently provide 24 notes per octave, or 36, or 48, and so on. <dl><dd></dd></dl> Having instruments with a larger number of possible notes allows using more of the individual pitches on the meantone scale, extending into all sharps and flats and even some double sharps and double flats, at a minimum (i.e., a 17 note per octave keyboard). The feasible triads for instruments with more notes per octaves is wider, and for quarter comma meantone and tunings close to it, can even add <a href="/wiki/Harmonic_seventh" title="Harmonic seventh">harmonic sevenths</a> (a "lost chord" that comes gratis in quarter-comma as an <a href="/wiki/Augmented_sixth" title="Augmented sixth">augmented sixth</a>, e.g. C-E-G-A<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup>) to a few triads – and of course, with more notes per octave come more extended harmonies (see <a href="/wiki/31_tone_equal_temperament" class="mw-redirect" title="31 tone equal temperament">31 tone equal temperament</a> for a circulating tuning system close to quarter-comma that has equally well-tuned extended harmonic notes addable to all triads, e.g. all 7th harmonics and limited 11th and 13th harmonics).</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLindley1974" class="citation journal cs1">Lindley, Mark (1974). "Early 16th century keyboard temperaments". <i><a href="/wiki/Musica_Disciplina" class="mw-redirect" title="Musica Disciplina">Musica Disciplina</a></i>. <b>28</b>: <span class="nowrap">129–</span>151. <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/20532169">20532169</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Musica+Disciplina&rft.atitle=Early+16th+century+keyboard+temperaments&rft.volume=28&rft.pages=%3Cspan+class%3D%22nowrap%22%3E129-%3C%2Fspan%3E151&rft.date=1974&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F20532169%23id-name%3DJSTOR&rft.aulast=Lindley&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuarter-comma+meantone" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quarter-comma_meantone&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>"<a rel="nofollow" class="external text" href="https://en.xen.wiki/w/Quarter-comma_meantone">Quarter-comma meantone</a>", on <i>Xenharmonic Wiki</i>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output 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class">Pitch class</a></li> <li><a href="/wiki/Consonance_and_dissonance" title="Consonance and dissonance">Consonance and dissonance</a></li> <li><a href="/wiki/List_of_pitch_intervals" title="List of pitch intervals">List of musical intervals</a></li> <li><a href="/wiki/List_of_intervals_in_5-limit_just_intonation" title="List of intervals in 5-limit just intonation">List of intervals in 5-limit just intonation</a></li> <li><a href="/wiki/List_of_meantone_intervals" title="List of meantone intervals">List of meantone intervals</a></li> <li><a href="/wiki/Microtonal_music" class="mw-redirect" title="Microtonal music">Microtone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a 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 href="/wiki/12_equal_temperament" title="12 equal temperament">12-tone</a></li> <li><a href="/wiki/15_equal_temperament" title="15 equal temperament">15-tone</a></li> <li><a href="/wiki/17_equal_temperament" title="17 equal temperament">17-tone</a></li> <li><a href="/wiki/19_equal_temperament" title="19 equal temperament">19-tone</a></li> <li><a href="/wiki/22_equal_temperament" title="22 equal temperament">22-tone</a></li> <li><a href="/wiki/23_equal_temperament" title="23 equal temperament">23-tone</a></li> <li><a href="/wiki/Quarter_tone" title="Quarter tone">24-tone</a> (<a href="/wiki/List_of_quarter_tone_pieces" title="List of quarter tone pieces">pieces</a>)</li> <li><a href="/wiki/31_equal_temperament" title="31 equal temperament">31-tone</a></li> <li><a href="/wiki/34_equal_temperament" title="34 equal temperament">34-tone</a></li> <li><a href="/wiki/41_equal_temperament" title="41 equal temperament">41-tone</a></li> <li><a href="/wiki/53_equal_temperament" title="53 equal temperament">53-tone</a></li> <li><a href="/wiki/58_equal_temperament" title="58 equal temperament">58-tone</a></li> <li><a href="/wiki/72_equal_temperament" title="72 equal temperament">72-tone</a></li> <li><a href="/wiki/96_equal_temperament" title="96 equal temperament">96-tone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Regular_temperament" title="Regular temperament">Linear</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Meantone_temperament" title="Meantone temperament">Meantone</a> (<a class="mw-selflink selflink">quarter-comma</a>, <a href="/wiki/Septimal_meantone_temperament" title="Septimal meantone temperament">septimal</a>)</li> <li><a href="/wiki/Schismatic_temperament" title="Schismatic temperament">Schismatic</a></li> <li><a href="/wiki/Miracle_temperament" class="mw-redirect" title="Miracle temperament">Miracle</a></li> <li><a href="/wiki/Magic_temperament" title="Magic temperament">Magic</a></li> <li><a href="/wiki/Regular_diatonic_tuning" title="Regular diatonic tuning">Regular diatonic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Irregular</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="/wiki/Well_temperament" title="Well temperament">Well temperament</a>/<a href="/wiki/Temperament_ordinaire" title="Temperament ordinaire">Temperament ordinaire</a> (<a href="/wiki/Kirnberger_temperament" title="Kirnberger temperament">Kirnberger</a>, <a href="/wiki/Vallotti_temperament" title="Vallotti temperament">Vallotti</a>, <a href="/wiki/Werckmeister_temperament" title="Werckmeister temperament">Werckmeister</a>, <a href="/wiki/Young_temperament" title="Young temperament">Young</a>)</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Traditional<br /> non-Western</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chinese_musicology" title="Chinese musicology">Chinese musicology</a></li> <li><a href="/wiki/Sh%C3%AD-%C3%A8r-l%C7%9C" class="mw-redirect" title="Shí-èr-lǜ">Shí-èr-lǜ</a></li> <li><a href="/wiki/Dastgah" class="mw-redirect" title="Dastgah">Dastgah</a></li> <li>Maqam <ul><li><a href="/wiki/Arabic_maqam" title="Arabic maqam">Arabic maqam</a></li> <li><a href="/wiki/Turkish_makam" title="Turkish makam">Turkish makam</a></li> <li><a href="/wiki/Mugham" title="Mugham">Mugham</a></li> <li><a href="/wiki/Muqam" title="Muqam">Muqam</a></li></ul></li> <li><a href="/wiki/Octoechos" title="Octoechos">Octoechos</a></li> <li><a href="/wiki/Pelog" title="Pelog">Pelog</a></li> <li><a href="/wiki/Raga" title="Raga">Raga</a> (<a href="/wiki/Carnatic_raga" title="Carnatic raga">Carnatic raga</a>)</li> <li><a href="/wiki/Slendro" title="Slendro">Slendro</a></li> <li><a href="/wiki/Tetrachord" title="Tetrachord">Tetrachord</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Non-octave</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/833_cents_scale" title="833 cents scale">833 cents scale</a></li> <li><a href="/wiki/A12_scale" title="A12 scale">A12 scale</a></li> <li><a href="/wiki/Alpha_scale" title="Alpha scale">Alpha scale</a></li> <li><a href="/wiki/Beta_scale" title="Beta scale">Beta scale</a></li> <li><a href="/wiki/Gamma_scale" title="Gamma scale">Gamma scale</a></li> <li><a href="/wiki/Delta_scale" title="Delta scale">Delta scale</a></li> <li><a href="/wiki/Bohlen%E2%80%93Pierce_scale" title="Bohlen–Pierce scale">Lambda scale</a> (Bohlen–Pierce)</li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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