List of superparticular intervals: Difference between revisions

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=<span style="color: #800080;">List of Superparticular Intervals</span>= 

[[Superparticular]] numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number other than 1. They appear frequently in [[Just Intonation]] and [[OverToneSeries|Harmonic Series]] music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio [[21_20|21/20]]. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[comma]]s are superparticular ratios.

The list below is ordered by [[harmonic limit]], or the largest prime involved in the prime factorization. [[36_35|36/35]], for instance, is an interval of the [[7-limit]], as it factors to (2<span style="vertical-align: super;">2</span>*3<span style="vertical-align: super;">2</span>)/(5*7), while 37/36 would belong to the 37-limit.

[[http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem|Størmer's theorem]] guarantees that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than 2/1, 3/2, 4/3, and 9/8.

See also: [[Gallery of Just Intervals]]. Many of the names below come from [[http://www.huygens-fokker.org/docs/intervals.html|here]].

||~ Ratio ||~ Cents Value ||~ Factorization ||~ Name(s) ||
||||||||~ 2-limit ||
|| [[2_1|2/1]] || 1200.000 || 2/1 || (perfect) unison, unity, perfect prime, tonic, duple ||
||||||||~ 3-limit ||
|| [[3_2|3/2]] || 701.995 || 3/2 || [[perfect fifth]], 3rd harmonic (octave reduced), diapente ||
|| [[4_3|4/3]] || 498.045 || 2<span style="vertical-align: super;">2</span>/3 || perfect fourth, 3rd subharmonic (octave reduced), diatessaron ||
|| [[9_8|9/8]] || 203.910 || 3<span style="vertical-align: super;">2</span>/2<span style="vertical-align: super;">3</span> || (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced) ||
||||||||~ 5-limit ||
|| [[5_4|5/4]] || 386.314 || 5/2<span style="vertical-align: super;">2</span> || (classic) (5-limit) major third, 5th harmonic (octave reduced) ||
|| [[6_5|6/5]] || 315.641 || (2*3)/5 || (classic) (5-limit) minor third ||
|| [[10_9|10/9]] || 182.404 || (2*5)/3<span style="vertical-align: super;">2</span> || classic (whole) tone, classic major second, minor whole tone ||
|| [[16_15|16/15]] || 111.713 || 2<span style="vertical-align: super;">4</span>/(3*5) || minor diatonic semitone, 15th subharmonic ||
|| [[25_24|25/24]] || 70.672 || 5<span style="vertical-align: super;">2</span>/(2<span style="vertical-align: super;">3</span>*3) || chroma, (classic) chromatic semitone, Zarlinian semitone ||
|| [[81_80|81/80]] || 21.506 || 3<span style="vertical-align: super;">4</span>/(2<span style="vertical-align: super;">4</span>*5) || syntonic comma, Didymus comma ||
||||||||~ 7-limit ||
|| [[7_6|7/6]] || 266.871 || 7/(2*3) || (septimal) subminor third, septimal minor third, augmented second ||
|| [[8_7|8/7]] || 231.174 || 2<span style="vertical-align: super;">3</span>/7 || (septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic ||
|| [[15_14|15/14]] || 119.443 || (3*5)/(2*7) || septimal diatonic semitone ||
|| [[21_20|21/20]] || 84.467 || (3*7)/(2<span style="vertical-align: super;">2</span>*5) || minor semitone, large septimal chromatic semitone ||
|| [[28_27|28/27]] || 62.961 || (2<span style="vertical-align: super;">2</span>*7)/3<span style="vertical-align: super;">3</span> || septimal chroma, small septimal chromatic semitone, Archytas' 1/3-tone ||
|| [[36_35|36/35]] || 48.770 || (2<span style="vertical-align: super;">2</span>*3<span style="vertical-align: super;">3</span>)/(5*7) || septimal quarter tone, septimal diesis ||
|| [[49_48|49/48]] || 35.697 || 7<span style="vertical-align: super;">2</span>/(2<span style="vertical-align: super;">4</span>*3) || large septimal diesis, slendro diesis, septimal 1/6-tone ||
|| [[50_49|50/49]] || 34.976 || (2*5<span style="vertical-align: super;">2</span>)/7<span style="vertical-align: super;">2</span> || septimal sixth-tone, jubilisma, small septimal diesis, tritonic diesis, Erlich's decatonic comma ||
|| [[64_63|64/63]] || 27.264 || 2<span style="vertical-align: super;">6</span>/(3<span style="vertical-align: super;">2</span>*7) || septimal comma, Archytas' comma ||
|| [[126_125|126/125]] || 13.795 || (2*3^2*7)/5^3 || starling comma, septimal semicomma ||
|| [[225_224|225/224]] || 7.7115 || (3^2*5^2)/(2^5*7) || marvel comma, septimal kleisma ||
|| [[2401_2400|2401/2400]] || 0.72120 || 7^4/(2^5*3*5^2) || breedsma ||
|| [[4375_4374|4375/4374]] || 0.39576 || (5^4*7)/(2*3^7) || ragisma ||
||||||||~ 11-limit ||
|| [[11_10|11/10]] || 165.004 || 11/(2*5) || (large) (undecimal) neutral second, 4/5-tone, Ptolemy's second ||
|| [[12_11|12/11]] || 150.637 || (2<span style="vertical-align: super;">2</span>*3)/11 || (small) (undecimal) neutral second, 3/4-tone ||
|| [[22_21|22/21]] || 80.537 || (2*11)/(3*7) || undecimal minor semitone ||
|| [[33_32|33/32]] || 53.273 || (3*11)/2<span style="vertical-align: super;">5</span> || unidecimal quarter tone, unidecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced) ||
|| [[45_44|45/44]] || 38.906 || (3<span style="vertical-align: super;">2</span>*5)/(2<span style="vertical-align: super;">2</span>*11) || 1/5-tone ||
|| [[55_54|55/54]] || 31.767 || (5*11)/(2*3<span style="vertical-align: super;">3</span>) ||   ||
|| [[56_55|56/55]] || 31.194 || (2<span style="vertical-align: super;">3</span>*7)/(5*11) ||   ||
|| [[99_98|99/98]] || 17.576 || (3<span style="vertical-align: super;">2</span>*11)/(2*7<span style="vertical-align: super;">2</span>) || small undecimal comma ||
|| [[100_99|100/99]] || 17.399 || (2<span style="vertical-align: super;">2</span>*5<span style="vertical-align: super;">2</span>)/(3<span style="vertical-align: super;">2</span>*11) || Ptolemy's comma ||
|| [[121_120|121/120]] || 14.376 || 11^2/(2^3*3*5) || undecimal seconds comma ||
|| [[176_175|176/175]] || 9.8646 || (2^4*11)/(5^2*7) ||   ||
|| [[243_242|243/242]] || 7.1391 || 2^5/(2*11^2) || neutral third comma ||
|| [[385_384|385/384]] || 4.5026 || (5*7*11)/(2^7*3) || keenanisma ||
|| [[441_440|441/440]] || 3.9302 || (3^2*7^2)/(2^3*5*11) || Werckmeister's undecimal septenarian schisma ||
|| [[540_539|540/539]] || 3.2090 || (2^2*3^3*5)/(7^2*11) || Swets' comma ||
|| [[3025_3024|3025/3024]] || 0.57240 || (5^2*11^2)/(2^4*3^3*7) || Lehmerisma ||
|| [[9801_9800|9801/9800]] || 0.17665 || (3^4*11^2)/(2^3*5^2*7^2) || Gauss comma, kalisma ||
||||||||~ 13-limit ||
|| [[13_12|13/12]] || 138.573 || 13/(2<span style="vertical-align: super;">2</span>*3) || tridecimal 2/3-tone ||
|| [[14_13|14/13]] || 128.298 || (2*7)/13 || 2/3-tone, trienthird ||
|| [[26_25|26/25]] || 67.900 || (2*13)/5<span style="vertical-align: super;">2</span> || tridecimal 1/3-tone ||
|| [[27_26|27/26]] || 65.337 || 3<span style="vertical-align: super;">3</span>/(2*13) || tridecimal comma ||
|| [[40_39|40/39]] || 43.831 || (2<span style="vertical-align: super;">3</span>*5)/(3*13) || tridecimal minor diesis ||
|| [[65_64|65/64]] || 26.841 || (5*13)/2<span style="vertical-align: super;">6</span> || 13th-partial chroma ||
|| [[66_65|66/65]] || 26.432 || (2*3*11)/(5*13) ||   ||
|| [[78_77|78/77]] || 22.339 || (2*3*13)/(7*11) ||   ||
|| [[91_90|91/90]] || 19.130 || (7*13)/(2*3<span style="vertical-align: super;">2</span>*5) || [[The Biosphere|Biome]] comma ||
|| [[105_104|105/104]] || 16.567 ||   || small tridecimal comma ||
|| [[144_143|144/143]] || 12.064 ||   ||   ||
|| [[169_168|169/168]] || 10.274 ||   ||   ||
|| [[196_195|196/195]] || 8.8554 ||   ||   ||
|| [[325_324|325/324]] || 5.3351 ||   ||   ||
|| [[351_350|351/350]] || 4.9393 ||   ||   ||
|| [[352_351|352/351]] || 4.9253 ||   ||   ||
|| [[364_363|364/363]] || 4.7627 ||   ||   ||
|| [[625_624|625/624]] || 2.7722 ||   ||   ||
|| [[676_675|676/675]] || 2.5629 ||   ||   ||
|| [[729_728|729/728]] || 2.3764 ||   ||   ||
|| [[1001_1000|1001/1000]] || 1.7304 ||   ||   ||
|| [[1716_1715|1716/1715]] || 1.0092 ||   ||   ||
|| [[2080_2079|2080/2079]] || 0.83252 ||   ||   ||
|| [[4096_4095|4096/4095]] || 0.42272 || tridecimal schisma, Sagittal schismina ||   ||
|| [[4225_4224|4225/4224]] || 0.40981 ||   ||   ||
|| [[6656_6655|6656/6655]] || 0.26012 ||   ||   ||
|| [[10648_10647|10648/10647]] || 0.16260 || harmonisma ||   ||
|| [[123201_123200|123201/123200]] || 0.014052 ||   ||   ||
||||||||~ 17-limit (incomplete) ||
|| [[17_16|17/16]] || 104.955 || 17/2<span style="vertical-align: super;">4</span> || 17th harmonic (octave reduced) ||
|| [[18_17|18/17]] || 98.955 || (2*3<span style="vertical-align: super;">2</span>)/17 || Arabic lute index finger ||
|| [[34_33|34/33]] || 51.682 || (2*17)/(3*33) ||   ||
|| [[35_34|35/34]] || 50.184 || (5*7)/(2*17) || septendecimal 1/4-tone ||
|| [[51_50|51/50]] || 34.283 || (3*17)/(2*5<span style="vertical-align: super;">2</span>) || 17th-partial chroma ||
|| [[52_51|52/51]] || 33.617 || (2<span style="vertical-align: super;">2</span>*13)/(3*17) ||   ||
|| [[85_84|85/84]] || 20.488 || (5*17)/(2<span style="vertical-align: super;">2</span>*3*7) ||   ||
||||||||~ 19-limit (incomplete) ||
|| [[19_18|19/18]] || 93.603 || 19/(2*3<span style="vertical-align: super;">2</span>) || undevicesimal semitone ||
|| [[20_19|20/19]] || 88.801 || (2<span style="vertical-align: super;">2</span>*5)/19 || small undevicesimal semitone ||
|| [[39_38|39/38]] || 44.970 || (3*13)/(2*19) ||   ||
|| [[57_56|57/56]] || 30.642 || (3*19)/(2<span style="vertical-align: super;">3</span>*7) ||   ||
|| [[76_75|76/75]] || 22.931 || (2<span style="vertical-align: super;">2</span>*19)/(3*5<span style="vertical-align: super;">2</span>) ||   ||
|| [[77_76|77/76]] || 22.631 || (7*11)/(2<span style="vertical-align: super;">2</span>*19) ||   ||
|| [[96_95|96/95]] || 18.128 || (2<span style="vertical-align: super;">5</span>*3)/(5*19) ||   ||
||||||||~ 23-limit (incomplete) ||
|| [[23_22|23/22]] || 76.956 || 23/(2*11) ||   ||
|| [[24_23|24/23]] || 73.681 || (2<span style="vertical-align: super;">3</span>*3)/23 ||   ||
|| [[46_45|46/45]] || 38.051 || (2*23)/(3<span style="vertical-align: super;">2</span>*5) ||   ||
|| [[69_68|69/68]] || 25.274 || (3*23)/(2<span style="vertical-align: super;">2</span>*17) ||   ||
|| [[70_69|70/69]] || 24.910 || (2*5*7)/(3*23) ||   ||
|| [[92_91|92/91]] || 18.921 || (2<span style="vertical-align: super;">2</span>*23)/(7*13) ||   ||
||||||||~ 29-limit (incomplete) ||
|| [[29_28|29/28]] || 60.751 || 29/(2<span style="vertical-align: super;">2</span>*7) ||   ||
|| [[30_29|30/29]] || 58.692 || (2*3*5)/29 ||   ||
|| [[58_57|58/57]] || 30.109 || (2*29)/(3*19) ||   ||
|| [[88_87|88/87]] || 19.786 || (2<span style="vertical-align: super;">3</span>*11)/(3*29) ||   ||
||||||||~ 31-limit (incomplete) ||
|| [[31_30|31/30]] || 56.767 || 31/(2*3*5) ||   ||
|| [[32_31|32/31]] || 54.964 || 2<span style="vertical-align: super;">5</span>/31 || 31st subharmonic ||
|| [[63_62|63/62]] || 27.700 || (3<span style="vertical-align: super;">2</span>*7)/(2*31) ||   ||
|| [[93_92|93/92]] || 18.716 || (3*31)/(2<span style="vertical-align: super;">2</span>*23) ||   ||
||||||||~ 37-limit (incomplete) ||
|| [[37_36|37/36]] || 47.434 || 37/(2<span style="vertical-align: super;">2</span>*3<span style="vertical-align: super;">2</span>) ||   ||
|| [[38_37|38/37]] || 46.169 || (2*19)/37 ||   ||
|| [[75_74|75/74]] || 23.238 || (3*5<span style="vertical-align: super;">2</span>)/(2*37) ||   ||
||||||||~ 41-limit (incomplete) ||
|| [[41_40|41/40]] || 42.749 || 41/(2<span style="vertical-align: super;">3</span>*5) ||   ||
|| [[42_41|42/41]] || 41.719 || (2*3*7)/41 ||   ||
|| [[82_81|82/81]] || 21.242 || (2*41)/3<span style="vertical-align: super;">4</span> ||   ||
||||||||~ 43-limit (incomplete) ||
|| [[43_42|43/42]] || 40.737 || 43/(2*3*7) ||   ||
|| [[44_43|44/43]] || 39.800 || (2<span style="vertical-align: super;">2</span>*11)/43 ||   ||
|| [[86_85|86/85]] || 20.249 || (2*43)/(5*17) ||   ||
|| [[87_86|87/86]] || 20.014 || (3*29)/(2*43) ||   ||
||||||||~ 47-limit (incomplete) ||
|| [[47_46|47/46]] || 37.232 || 47/(2*23) ||   ||
|| [[48_47|48/47]] || 36.448 || (2<span style="vertical-align: super;">4</span>*3)/47 ||   ||
|| [[94_93|94/93]] || 18.516 || (2*47)/(3*31) ||   ||
|| [[95_94|95/94]] || 18.320 || (5*19)/(2*47) ||   ||
||||||||~ 53-limit (incomplete) ||
|| [[53_52|53/52]] || 32.977 || 53/(2<span style="vertical-align: super;">2</span>*13) ||   ||
|| [[54_53|54/53]] || 32.360 || (2*3<span style="vertical-align: super;">3</span>)/53 ||   ||
||||||||~ 59-limit (incomplete) ||
|| [[59_58|59/58]] || 29.594 || 59/(2*29) ||   ||
|| [[60_59|60/59]] || 29.097 || (2<span style="vertical-align: super;">2</span>*3*5)/59 ||   ||
||||||||~ 61-limit (incomplete) ||
|| [[61_60|61/60]] || 28.616 || 61/(2<span style="vertical-align: super;">2</span>*3*5) ||   ||
|| [[62_61|62/61]] || 28.151 || (2*31)/61 ||   ||
||||||||~ 67-limit (incomplete) ||
|| [[67_66|67/66]] || 26.034 || 67/(2*3*11) ||   ||
|| [[68_67|68/67]] || 25.648 || (2<span style="vertical-align: super;">2</span>*17)/67 ||   ||
||||||||~ 71-limit (incomplete) ||
|| [[71_70|71/70]] || 24.557 || 71/(2*5*7) ||   ||
|| [[72_71|72/71]] || 24.213 || (2<span style="vertical-align: super;">3</span>*3<span style="vertical-align: super;">2</span>)/71 ||   ||
||||||||~ 73-limit (incomplete) ||
|| [[73_72|73/72]] || 23.879 || 73/(2<span style="vertical-align: super;">3</span>*3<span style="vertical-align: super;">2</span>) ||   ||
|| [[74_73|74/73]] || 23.555 || (2*37)/73 ||   ||
||||||||~ 79-limit (incomplete) ||
|| [[79_78|79/78]] || 22.054 || 79/(2*3*13) ||   ||
|| [[80_79|80/79]] || 21.777 || (2<span style="vertical-align: super;">4</span>*5)/79 ||   ||
||||||||~ 83-limit (incomplete) ||
|| [[83_82|83/82]] || 20.985 || 83/(2*41) ||   ||
|| [[84_83|84/83]] || 20.734 || (2<span style="vertical-align: super;">2</span>*3*7)/83 ||   ||
||||||||~ 89-limit (incomplete) ||
|| [[89_88|89/88]] || 19.562 || 89/(2<span style="vertical-align: super;">3</span>*11) ||   ||
|| [[90_89|90/89]] || 19.344 || (2*3<span style="vertical-align: super;">2</span>*5)/89 ||   ||
||||||||~ 97-limit (incomplete) ||
|| [[97_96|97/96]] || 17.940 || 97/(2<span style="vertical-align: super;">5</span>*3) ||   ||
|| [[98_97|98/97]] || 17.756 || (2*7<span style="vertical-align: super;">2</span>)/97 ||   ||
||||||||~ 101-limit (incomplete) ||
|| [[101_100|101/100]] || 17.226 || 101/(2<span style="vertical-align: super;">2</span>*5<span style="vertical-align: super;">2</span>) ||   ||

Original HTML content:

<html><head><title>List of Superparticular Intervals</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="List of Superparticular Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #800080;">List of Superparticular Intervals</span></h1>
 <br />
<a class="wiki_link" href="/Superparticular">Superparticular</a> numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number other than 1. They appear frequently in <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> and <a class="wiki_link" href="/OverToneSeries">Harmonic Series</a> music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio <a class="wiki_link" href="/21_20">21/20</a>. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common <a class="wiki_link" href="/comma">comma</a>s are superparticular ratios.<br />
<br />
The list below is ordered by <a class="wiki_link" href="/harmonic%20limit">harmonic limit</a>, or the largest prime involved in the prime factorization. <a class="wiki_link" href="/36_35">36/35</a>, for instance, is an interval of the <a class="wiki_link" href="/7-limit">7-limit</a>, as it factors to (2<span style="vertical-align: super;">2</span>*3<span style="vertical-align: super;">2</span>)/(5*7), while 37/36 would belong to the 37-limit.<br />
<br />
<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem" rel="nofollow">Størmer's theorem</a> guarantees that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than 2/1, 3/2, 4/3, and 9/8.<br />
<br />
See also: <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">Gallery of Just Intervals</a>. Many of the names below come from <a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/intervals.html" rel="nofollow">here</a>.<br />
<br />


<table class="wiki_table">
    <tr>
        <th>Ratio<br />
</th>
        <th>Cents Value<br />
</th>
        <th>Factorization<br />
</th>
        <th>Name(s)<br />
</th>
    </tr>
    <tr>
        <th colspan="4">2-limit<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/2_1">2/1</a><br />
</td>
        <td>1200.000<br />
</td>
        <td>2/1<br />
</td>
        <td>(perfect) unison, unity, perfect prime, tonic, duple<br />
</td>
    </tr>
    <tr>
        <th colspan="4">3-limit<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/3_2">3/2</a><br />
</td>
        <td>701.995<br />
</td>
        <td>3/2<br />
</td>
        <td><a class="wiki_link" href="/perfect%20fifth">perfect fifth</a>, 3rd harmonic (octave reduced), diapente<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/4_3">4/3</a><br />
</td>
        <td>498.045<br />
</td>
        <td>2<span style="vertical-align: super;">2</span>/3<br />
</td>
        <td>perfect fourth, 3rd subharmonic (octave reduced), diatessaron<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/9_8">9/8</a><br />
</td>
        <td>203.910<br />
</td>
        <td>3<span style="vertical-align: super;">2</span>/2<span style="vertical-align: super;">3</span><br />
</td>
        <td>(Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced)<br />
</td>
    </tr>
    <tr>
        <th colspan="4">5-limit<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/5_4">5/4</a><br />
</td>
        <td>386.314<br />
</td>
        <td>5/2<span style="vertical-align: super;">2</span><br />
</td>
        <td>(classic) (5-limit) major third, 5th harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/6_5">6/5</a><br />
</td>
        <td>315.641<br />
</td>
        <td>(2*3)/5<br />
</td>
        <td>(classic) (5-limit) minor third<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/10_9">10/9</a><br />
</td>
        <td>182.404<br />
</td>
        <td>(2*5)/3<span style="vertical-align: super;">2</span><br />
</td>
        <td>classic (whole) tone, classic major second, minor whole tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/16_15">16/15</a><br />
</td>
        <td>111.713<br />
</td>
        <td>2<span style="vertical-align: super;">4</span>/(3*5)<br />
</td>
        <td>minor diatonic semitone, 15th subharmonic<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/25_24">25/24</a><br />
</td>
        <td>70.672<br />
</td>
        <td>5<span style="vertical-align: super;">2</span>/(2<span style="vertical-align: super;">3</span>*3)<br />
</td>
        <td>chroma, (classic) chromatic semitone, Zarlinian semitone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/81_80">81/80</a><br />
</td>
        <td>21.506<br />
</td>
        <td>3<span style="vertical-align: super;">4</span>/(2<span style="vertical-align: super;">4</span>*5)<br />
</td>
        <td>syntonic comma, Didymus comma<br />
</td>
    </tr>
    <tr>
        <th colspan="4">7-limit<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/7_6">7/6</a><br />
</td>
        <td>266.871<br />
</td>
        <td>7/(2*3)<br />
</td>
        <td>(septimal) subminor third, septimal minor third, augmented second<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/8_7">8/7</a><br />
</td>
        <td>231.174<br />
</td>
        <td>2<span style="vertical-align: super;">3</span>/7<br />
</td>
        <td>(septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/15_14">15/14</a><br />
</td>
        <td>119.443<br />
</td>
        <td>(3*5)/(2*7)<br />
</td>
        <td>septimal diatonic semitone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/21_20">21/20</a><br />
</td>
        <td>84.467<br />
</td>
        <td>(3*7)/(2<span style="vertical-align: super;">2</span>*5)<br />
</td>
        <td>minor semitone, large septimal chromatic semitone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/28_27">28/27</a><br />
</td>
        <td>62.961<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*7)/3<span style="vertical-align: super;">3</span><br />
</td>
        <td>septimal chroma, small septimal chromatic semitone, Archytas' 1/3-tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/36_35">36/35</a><br />
</td>
        <td>48.770<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*3<span style="vertical-align: super;">3</span>)/(5*7)<br />
</td>
        <td>septimal quarter tone, septimal diesis<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/49_48">49/48</a><br />
</td>
        <td>35.697<br />
</td>
        <td>7<span style="vertical-align: super;">2</span>/(2<span style="vertical-align: super;">4</span>*3)<br />
</td>
        <td>large septimal diesis, slendro diesis, septimal 1/6-tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/50_49">50/49</a><br />
</td>
        <td>34.976<br />
</td>
        <td>(2*5<span style="vertical-align: super;">2</span>)/7<span style="vertical-align: super;">2</span><br />
</td>
        <td>septimal sixth-tone, jubilisma, small septimal diesis, tritonic diesis, Erlich's decatonic comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/64_63">64/63</a><br />
</td>
        <td>27.264<br />
</td>
        <td>2<span style="vertical-align: super;">6</span>/(3<span style="vertical-align: super;">2</span>*7)<br />
</td>
        <td>septimal comma, Archytas' comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/126_125">126/125</a><br />
</td>
        <td>13.795<br />
</td>
        <td>(2*3^2*7)/5^3<br />
</td>
        <td>starling comma, septimal semicomma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/225_224">225/224</a><br />
</td>
        <td>7.7115<br />
</td>
        <td>(3^2*5^2)/(2^5*7)<br />
</td>
        <td>marvel comma, septimal kleisma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/2401_2400">2401/2400</a><br />
</td>
        <td>0.72120<br />
</td>
        <td>7^4/(2^5*3*5^2)<br />
</td>
        <td>breedsma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/4375_4374">4375/4374</a><br />
</td>
        <td>0.39576<br />
</td>
        <td>(5^4*7)/(2*3^7)<br />
</td>
        <td>ragisma<br />
</td>
    </tr>
    <tr>
        <th colspan="4">11-limit<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/11_10">11/10</a><br />
</td>
        <td>165.004<br />
</td>
        <td>11/(2*5)<br />
</td>
        <td>(large) (undecimal) neutral second, 4/5-tone, Ptolemy's second<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/12_11">12/11</a><br />
</td>
        <td>150.637<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*3)/11<br />
</td>
        <td>(small) (undecimal) neutral second, 3/4-tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/22_21">22/21</a><br />
</td>
        <td>80.537<br />
</td>
        <td>(2*11)/(3*7)<br />
</td>
        <td>undecimal minor semitone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/33_32">33/32</a><br />
</td>
        <td>53.273<br />
</td>
        <td>(3*11)/2<span style="vertical-align: super;">5</span><br />
</td>
        <td>unidecimal quarter tone, unidecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/45_44">45/44</a><br />
</td>
        <td>38.906<br />
</td>
        <td>(3<span style="vertical-align: super;">2</span>*5)/(2<span style="vertical-align: super;">2</span>*11)<br />
</td>
        <td>1/5-tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/55_54">55/54</a><br />
</td>
        <td>31.767<br />
</td>
        <td>(5*11)/(2*3<span style="vertical-align: super;">3</span>)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/56_55">56/55</a><br />
</td>
        <td>31.194<br />
</td>
        <td>(2<span style="vertical-align: super;">3</span>*7)/(5*11)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/99_98">99/98</a><br />
</td>
        <td>17.576<br />
</td>
        <td>(3<span style="vertical-align: super;">2</span>*11)/(2*7<span style="vertical-align: super;">2</span>)<br />
</td>
        <td>small undecimal comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/100_99">100/99</a><br />
</td>
        <td>17.399<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*5<span style="vertical-align: super;">2</span>)/(3<span style="vertical-align: super;">2</span>*11)<br />
</td>
        <td>Ptolemy's comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/121_120">121/120</a><br />
</td>
        <td>14.376<br />
</td>
        <td>11^2/(2^3*3*5)<br />
</td>
        <td>undecimal seconds comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/176_175">176/175</a><br />
</td>
        <td>9.8646<br />
</td>
        <td>(2^4*11)/(5^2*7)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/243_242">243/242</a><br />
</td>
        <td>7.1391<br />
</td>
        <td>2^5/(2*11^2)<br />
</td>
        <td>neutral third comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/385_384">385/384</a><br />
</td>
        <td>4.5026<br />
</td>
        <td>(5*7*11)/(2^7*3)<br />
</td>
        <td>keenanisma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/441_440">441/440</a><br />
</td>
        <td>3.9302<br />
</td>
        <td>(3^2*7^2)/(2^3*5*11)<br />
</td>
        <td>Werckmeister's undecimal septenarian schisma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/540_539">540/539</a><br />
</td>
        <td>3.2090<br />
</td>
        <td>(2^2*3^3*5)/(7^2*11)<br />
</td>
        <td>Swets' comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/3025_3024">3025/3024</a><br />
</td>
        <td>0.57240<br />
</td>
        <td>(5^2*11^2)/(2^4*3^3*7)<br />
</td>
        <td>Lehmerisma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/9801_9800">9801/9800</a><br />
</td>
        <td>0.17665<br />
</td>
        <td>(3^4*11^2)/(2^3*5^2*7^2)<br />
</td>
        <td>Gauss comma, kalisma<br />
</td>
    </tr>
    <tr>
        <th colspan="4">13-limit<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/13_12">13/12</a><br />
</td>
        <td>138.573<br />
</td>
        <td>13/(2<span style="vertical-align: super;">2</span>*3)<br />
</td>
        <td>tridecimal 2/3-tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/14_13">14/13</a><br />
</td>
        <td>128.298<br />
</td>
        <td>(2*7)/13<br />
</td>
        <td>2/3-tone, trienthird<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/26_25">26/25</a><br />
</td>
        <td>67.900<br />
</td>
        <td>(2*13)/5<span style="vertical-align: super;">2</span><br />
</td>
        <td>tridecimal 1/3-tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/27_26">27/26</a><br />
</td>
        <td>65.337<br />
</td>
        <td>3<span style="vertical-align: super;">3</span>/(2*13)<br />
</td>
        <td>tridecimal comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/40_39">40/39</a><br />
</td>
        <td>43.831<br />
</td>
        <td>(2<span style="vertical-align: super;">3</span>*5)/(3*13)<br />
</td>
        <td>tridecimal minor diesis<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/65_64">65/64</a><br />
</td>
        <td>26.841<br />
</td>
        <td>(5*13)/2<span style="vertical-align: super;">6</span><br />
</td>
        <td>13th-partial chroma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/66_65">66/65</a><br />
</td>
        <td>26.432<br />
</td>
        <td>(2*3*11)/(5*13)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/78_77">78/77</a><br />
</td>
        <td>22.339<br />
</td>
        <td>(2*3*13)/(7*11)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/91_90">91/90</a><br />
</td>
        <td>19.130<br />
</td>
        <td>(7*13)/(2*3<span style="vertical-align: super;">2</span>*5)<br />
</td>
        <td><a class="wiki_link" href="/The%20Biosphere">Biome</a> comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/105_104">105/104</a><br />
</td>
        <td>16.567<br />
</td>
        <td><br />
</td>
        <td>small tridecimal comma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/144_143">144/143</a><br />
</td>
        <td>12.064<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/169_168">169/168</a><br />
</td>
        <td>10.274<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/196_195">196/195</a><br />
</td>
        <td>8.8554<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/325_324">325/324</a><br />
</td>
        <td>5.3351<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/351_350">351/350</a><br />
</td>
        <td>4.9393<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/352_351">352/351</a><br />
</td>
        <td>4.9253<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/364_363">364/363</a><br />
</td>
        <td>4.7627<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/625_624">625/624</a><br />
</td>
        <td>2.7722<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/676_675">676/675</a><br />
</td>
        <td>2.5629<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/729_728">729/728</a><br />
</td>
        <td>2.3764<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/1001_1000">1001/1000</a><br />
</td>
        <td>1.7304<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/1716_1715">1716/1715</a><br />
</td>
        <td>1.0092<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/2080_2079">2080/2079</a><br />
</td>
        <td>0.83252<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/4096_4095">4096/4095</a><br />
</td>
        <td>0.42272<br />
</td>
        <td>tridecimal schisma, Sagittal schismina<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/4225_4224">4225/4224</a><br />
</td>
        <td>0.40981<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/6656_6655">6656/6655</a><br />
</td>
        <td>0.26012<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/10648_10647">10648/10647</a><br />
</td>
        <td>0.16260<br />
</td>
        <td>harmonisma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/123201_123200">123201/123200</a><br />
</td>
        <td>0.014052<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">17-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/17_16">17/16</a><br />
</td>
        <td>104.955<br />
</td>
        <td>17/2<span style="vertical-align: super;">4</span><br />
</td>
        <td>17th harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/18_17">18/17</a><br />
</td>
        <td>98.955<br />
</td>
        <td>(2*3<span style="vertical-align: super;">2</span>)/17<br />
</td>
        <td>Arabic lute index finger<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/34_33">34/33</a><br />
</td>
        <td>51.682<br />
</td>
        <td>(2*17)/(3*33)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/35_34">35/34</a><br />
</td>
        <td>50.184<br />
</td>
        <td>(5*7)/(2*17)<br />
</td>
        <td>septendecimal 1/4-tone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/51_50">51/50</a><br />
</td>
        <td>34.283<br />
</td>
        <td>(3*17)/(2*5<span style="vertical-align: super;">2</span>)<br />
</td>
        <td>17th-partial chroma<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/52_51">52/51</a><br />
</td>
        <td>33.617<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*13)/(3*17)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/85_84">85/84</a><br />
</td>
        <td>20.488<br />
</td>
        <td>(5*17)/(2<span style="vertical-align: super;">2</span>*3*7)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">19-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/19_18">19/18</a><br />
</td>
        <td>93.603<br />
</td>
        <td>19/(2*3<span style="vertical-align: super;">2</span>)<br />
</td>
        <td>undevicesimal semitone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/20_19">20/19</a><br />
</td>
        <td>88.801<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*5)/19<br />
</td>
        <td>small undevicesimal semitone<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/39_38">39/38</a><br />
</td>
        <td>44.970<br />
</td>
        <td>(3*13)/(2*19)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/57_56">57/56</a><br />
</td>
        <td>30.642<br />
</td>
        <td>(3*19)/(2<span style="vertical-align: super;">3</span>*7)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/76_75">76/75</a><br />
</td>
        <td>22.931<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*19)/(3*5<span style="vertical-align: super;">2</span>)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/77_76">77/76</a><br />
</td>
        <td>22.631<br />
</td>
        <td>(7*11)/(2<span style="vertical-align: super;">2</span>*19)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/96_95">96/95</a><br />
</td>
        <td>18.128<br />
</td>
        <td>(2<span style="vertical-align: super;">5</span>*3)/(5*19)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">23-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/23_22">23/22</a><br />
</td>
        <td>76.956<br />
</td>
        <td>23/(2*11)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/24_23">24/23</a><br />
</td>
        <td>73.681<br />
</td>
        <td>(2<span style="vertical-align: super;">3</span>*3)/23<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/46_45">46/45</a><br />
</td>
        <td>38.051<br />
</td>
        <td>(2*23)/(3<span style="vertical-align: super;">2</span>*5)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/69_68">69/68</a><br />
</td>
        <td>25.274<br />
</td>
        <td>(3*23)/(2<span style="vertical-align: super;">2</span>*17)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/70_69">70/69</a><br />
</td>
        <td>24.910<br />
</td>
        <td>(2*5*7)/(3*23)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/92_91">92/91</a><br />
</td>
        <td>18.921<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*23)/(7*13)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">29-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/29_28">29/28</a><br />
</td>
        <td>60.751<br />
</td>
        <td>29/(2<span style="vertical-align: super;">2</span>*7)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/30_29">30/29</a><br />
</td>
        <td>58.692<br />
</td>
        <td>(2*3*5)/29<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/58_57">58/57</a><br />
</td>
        <td>30.109<br />
</td>
        <td>(2*29)/(3*19)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/88_87">88/87</a><br />
</td>
        <td>19.786<br />
</td>
        <td>(2<span style="vertical-align: super;">3</span>*11)/(3*29)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">31-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/31_30">31/30</a><br />
</td>
        <td>56.767<br />
</td>
        <td>31/(2*3*5)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/32_31">32/31</a><br />
</td>
        <td>54.964<br />
</td>
        <td>2<span style="vertical-align: super;">5</span>/31<br />
</td>
        <td>31st subharmonic<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/63_62">63/62</a><br />
</td>
        <td>27.700<br />
</td>
        <td>(3<span style="vertical-align: super;">2</span>*7)/(2*31)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/93_92">93/92</a><br />
</td>
        <td>18.716<br />
</td>
        <td>(3*31)/(2<span style="vertical-align: super;">2</span>*23)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">37-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/37_36">37/36</a><br />
</td>
        <td>47.434<br />
</td>
        <td>37/(2<span style="vertical-align: super;">2</span>*3<span style="vertical-align: super;">2</span>)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/38_37">38/37</a><br />
</td>
        <td>46.169<br />
</td>
        <td>(2*19)/37<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/75_74">75/74</a><br />
</td>
        <td>23.238<br />
</td>
        <td>(3*5<span style="vertical-align: super;">2</span>)/(2*37)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">41-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/41_40">41/40</a><br />
</td>
        <td>42.749<br />
</td>
        <td>41/(2<span style="vertical-align: super;">3</span>*5)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/42_41">42/41</a><br />
</td>
        <td>41.719<br />
</td>
        <td>(2*3*7)/41<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/82_81">82/81</a><br />
</td>
        <td>21.242<br />
</td>
        <td>(2*41)/3<span style="vertical-align: super;">4</span><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">43-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/43_42">43/42</a><br />
</td>
        <td>40.737<br />
</td>
        <td>43/(2*3*7)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/44_43">44/43</a><br />
</td>
        <td>39.800<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*11)/43<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/86_85">86/85</a><br />
</td>
        <td>20.249<br />
</td>
        <td>(2*43)/(5*17)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/87_86">87/86</a><br />
</td>
        <td>20.014<br />
</td>
        <td>(3*29)/(2*43)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">47-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/47_46">47/46</a><br />
</td>
        <td>37.232<br />
</td>
        <td>47/(2*23)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/48_47">48/47</a><br />
</td>
        <td>36.448<br />
</td>
        <td>(2<span style="vertical-align: super;">4</span>*3)/47<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/94_93">94/93</a><br />
</td>
        <td>18.516<br />
</td>
        <td>(2*47)/(3*31)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/95_94">95/94</a><br />
</td>
        <td>18.320<br />
</td>
        <td>(5*19)/(2*47)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">53-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/53_52">53/52</a><br />
</td>
        <td>32.977<br />
</td>
        <td>53/(2<span style="vertical-align: super;">2</span>*13)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/54_53">54/53</a><br />
</td>
        <td>32.360<br />
</td>
        <td>(2*3<span style="vertical-align: super;">3</span>)/53<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">59-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/59_58">59/58</a><br />
</td>
        <td>29.594<br />
</td>
        <td>59/(2*29)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/60_59">60/59</a><br />
</td>
        <td>29.097<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*3*5)/59<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">61-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/61_60">61/60</a><br />
</td>
        <td>28.616<br />
</td>
        <td>61/(2<span style="vertical-align: super;">2</span>*3*5)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/62_61">62/61</a><br />
</td>
        <td>28.151<br />
</td>
        <td>(2*31)/61<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">67-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/67_66">67/66</a><br />
</td>
        <td>26.034<br />
</td>
        <td>67/(2*3*11)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/68_67">68/67</a><br />
</td>
        <td>25.648<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*17)/67<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">71-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/71_70">71/70</a><br />
</td>
        <td>24.557<br />
</td>
        <td>71/(2*5*7)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/72_71">72/71</a><br />
</td>
        <td>24.213<br />
</td>
        <td>(2<span style="vertical-align: super;">3</span>*3<span style="vertical-align: super;">2</span>)/71<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">73-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/73_72">73/72</a><br />
</td>
        <td>23.879<br />
</td>
        <td>73/(2<span style="vertical-align: super;">3</span>*3<span style="vertical-align: super;">2</span>)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/74_73">74/73</a><br />
</td>
        <td>23.555<br />
</td>
        <td>(2*37)/73<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">79-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/79_78">79/78</a><br />
</td>
        <td>22.054<br />
</td>
        <td>79/(2*3*13)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/80_79">80/79</a><br />
</td>
        <td>21.777<br />
</td>
        <td>(2<span style="vertical-align: super;">4</span>*5)/79<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">83-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/83_82">83/82</a><br />
</td>
        <td>20.985<br />
</td>
        <td>83/(2*41)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/84_83">84/83</a><br />
</td>
        <td>20.734<br />
</td>
        <td>(2<span style="vertical-align: super;">2</span>*3*7)/83<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">89-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/89_88">89/88</a><br />
</td>
        <td>19.562<br />
</td>
        <td>89/(2<span style="vertical-align: super;">3</span>*11)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/90_89">90/89</a><br />
</td>
        <td>19.344<br />
</td>
        <td>(2*3<span style="vertical-align: super;">2</span>*5)/89<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">97-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/97_96">97/96</a><br />
</td>
        <td>17.940<br />
</td>
        <td>97/(2<span style="vertical-align: super;">5</span>*3)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/98_97">98/97</a><br />
</td>
        <td>17.756<br />
</td>
        <td>(2*7<span style="vertical-align: super;">2</span>)/97<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <th colspan="4">101-limit (incomplete)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/101_100">101/100</a><br />
</td>
        <td>17.226<br />
</td>
        <td>101/(2<span style="vertical-align: super;">2</span>*5<span style="vertical-align: super;">2</span>)<br />
</td>
        <td><br />
</td>
    </tr>
</table>

</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-10-28 04:28:07 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-10-28 04:40:39 UTC</tt>.<br>
-: The original revision id was <tt>269489636</tt>.<br>
+: The original revision id was <tt>269491218</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 10: / Line 10: @@
 [[Superparticular]] numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number other than 1. They appear frequently in [[Just Intonation]] and [[OverToneSeries|Harmonic Series]] music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio [[21_20|21/20]]. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[comma]]s are superparticular ratios.
-In addition to names and cents values, the list below includes the factorization of each superparticular ratio as well as the largest prime involved. This is relevant when considering which intervals are characteristic of which [[harmonic limit]]s. [[36_35|36/35]], for instance, is an interval of the [[7-limit]], as it factors to (2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(5*7), while 37/36 would belong to the 37-limit.
+The list below is ordered by [[harmonic limit]], or the largest prime involved in the prime factorization. [[36_35|36/35]], for instance, is an interval of the [[7-limit]], as it factors to (2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(5*7), while 37/36 would belong to the 37-limit.
+[[http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem|Størmer's theorem]] guarantees that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than 2/1, 3/2, 4/3, and 9/8.
 See also: [[Gallery of Just Intervals]]. Many of the names below come from [[http://www.huygens-fokker.org/docs/intervals.html|here]].
@@ Line 50: / Line 52: @@
 || [[55_54|55/54]] || 31.767 || (5*11)/(2*3&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;) ||   ||
 || [[56_55|56/55]] || 31.194 || (2&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;*7)/(5*11) ||   ||
-|| [[99_98|99/98]] || 17.576 || (3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*11)/(2*7&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;) ||   ||
+|| [[99_98|99/98]] || 17.576 || (3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*11)/(2*7&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;) || small undecimal comma ||
 || [[100_99|100/99]] || 17.399 || (2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*5&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*11) || Ptolemy's comma ||
-|| [[121_120|121/120]] || 14.376 || 11^2/(2^3*3*5) ||   ||
+|| [[121_120|121/120]] || 14.376 || 11^2/(2^3*3*5) || undecimal seconds comma ||
 || [[176_175|176/175]] || 9.8646 || (2^4*11)/(5^2*7) ||   ||
-|| [[243_242|243/242]] || 7.1391 || 2^5/(2*11^2) ||   ||
+|| [[243_242|243/242]] || 7.1391 || 2^5/(2*11^2) || neutral third comma ||
 || [[385_384|385/384]] || 4.5026 || (5*7*11)/(2^7*3) || keenanisma ||
-|| [[441_440|441/440]] || 3.9302 || (3^2*7^2)/(2^3*5*11) ||   ||
+|| [[441_440|441/440]] || 3.9302 || (3^2*7^2)/(2^3*5*11) || Werckmeister's undecimal septenarian schisma ||
-|| [[540_539|540/539]] || 3.2090 || (2^2*3^3*5)/(7^2*11) ||   ||
+|| [[540_539|540/539]] || 3.2090 || (2^2*3^3*5)/(7^2*11) || Swets' comma ||
-|| [[3025_3024|3025/3024]] || 0.57240 || (5^2*11^2)/(2^4*3^3*7) ||   ||
+|| [[3025_3024|3025/3024]] || 0.57240 || (5^2*11^2)/(2^4*3^3*7) || Lehmerisma ||
-|| [[9801_9800|9801/9800]] || 0.17665 || (3^4*11^2)/(2^3*5^2*7^2) || Gauss comma ||
+|| [[9801_9800|9801/9800]] || 0.17665 || (3^4*11^2)/(2^3*5^2*7^2) || Gauss comma, kalisma ||
 ||||||||~ 13-limit ||
 || [[13_12|13/12]] || 138.573 || 13/(2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3) || tridecimal 2/3-tone ||
@@ Line 69: / Line 71: @@
 || [[66_65|66/65]] || 26.432 || (2*3*11)/(5*13) ||   ||
 || [[78_77|78/77]] || 22.339 || (2*3*13)/(7*11) ||   ||
-|| [[91_90|91/90]] || 19.130 || (7*13)/(2*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*5) ||   ||
+|| [[91_90|91/90]] || 19.130 || (7*13)/(2*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*5) || [[The Biosphere|Biome]] comma ||
-|| [[105_104|105/104]] || 16.567 ||   ||   ||
+|| [[105_104|105/104]] || 16.567 ||   || small tridecimal comma ||
 || [[144_143|144/143]] || 12.064 ||   ||   ||
 || [[169_168|169/168]] || 10.274 ||   ||   ||
@@ Line 84: / Line 86: @@
 || [[1716_1715|1716/1715]] || 1.0092 ||   ||   ||
 || [[2080_2079|2080/2079]] || 0.83252 ||   ||   ||
-|| [[4096_4095|4096/4095]] || 0.42272 ||   ||   ||
+|| [[4096_4095|4096/4095]] || 0.42272 || tridecimal schisma, Sagittal schismina ||   ||
 || [[4225_4224|4225/4224]] || 0.40981 ||   ||   ||
 || [[6656_6655|6656/6655]] || 0.26012 ||   ||   ||
-|| [[10648_10647|10648/10647]] || 0.16260 ||   ||   ||
+|| [[10648_10647|10648/10647]] || 0.16260 || harmonisma ||   ||
 || [[123201_123200|123201/123200]] || 0.014052 ||   ||   ||
 ||||||||~ 17-limit (incomplete) ||
@@ Line 177: / Line 179: @@
 &lt;a class="wiki_link" href="/Superparticular"&gt;Superparticular&lt;/a&gt; numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number other than 1. They appear frequently in &lt;a class="wiki_link" href="/Just%20Intonation"&gt;Just Intonation&lt;/a&gt; and &lt;a class="wiki_link" href="/OverToneSeries"&gt;Harmonic Series&lt;/a&gt; music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio &lt;a class="wiki_link" href="/21_20"&gt;21/20&lt;/a&gt;. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s are superparticular ratios.&lt;br /&gt;
 &lt;br /&gt;
-In addition to names and cents values, the list below includes the factorization of each superparticular ratio as well as the largest prime involved. This is relevant when considering which intervals are characteristic of which &lt;a class="wiki_link" href="/harmonic%20limit"&gt;harmonic limit&lt;/a&gt;s. &lt;a class="wiki_link" href="/36_35"&gt;36/35&lt;/a&gt;, for instance, is an interval of the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, as it factors to (2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(5*7), while 37/36 would belong to the 37-limit.&lt;br /&gt;
+The list below is ordered by &lt;a class="wiki_link" href="/harmonic%20limit"&gt;harmonic limit&lt;/a&gt;, or the largest prime involved in the prime factorization. &lt;a class="wiki_link" href="/36_35"&gt;36/35&lt;/a&gt;, for instance, is an interval of the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, as it factors to (2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(5*7), while 37/36 would belong to the 37-limit.&lt;br /&gt;
+&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem" rel="nofollow"&gt;Størmer's theorem&lt;/a&gt; guarantees that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than 2/1, 3/2, 4/3, and 9/8.&lt;br /&gt;
 &lt;br /&gt;
 See also: &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Gallery of Just Intervals&lt;/a&gt;. Many of the names below come from &lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/intervals.html" rel="nofollow"&gt;here&lt;/a&gt;.&lt;br /&gt;
@@ Line 521: / Line 525: @@
          &lt;td&gt;(3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*11)/(2*7&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;small undecimal comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 541: / Line 545: @@
          &lt;td&gt;11^2/(2^3*3*5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;undecimal seconds comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 561: / Line 565: @@
          &lt;td&gt;2^5/(2*11^2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;neutral third comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 581: / Line 585: @@
          &lt;td&gt;(3^2*7^2)/(2^3*5*11)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Werckmeister's undecimal septenarian schisma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 591: / Line 595: @@
          &lt;td&gt;(2^2*3^3*5)/(7^2*11)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Swets' comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 601: / Line 605: @@
          &lt;td&gt;(5^2*11^2)/(2^4*3^3*7)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Lehmerisma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 611: / Line 615: @@
          &lt;td&gt;(3^4*11^2)/(2^3*5^2*7^2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Gauss comma&lt;br /&gt;
+         &lt;td&gt;Gauss comma, kalisma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 705: / Line 709: @@
          &lt;td&gt;(7*13)/(2*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/The%20Biosphere"&gt;Biome&lt;/a&gt; comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 715: / Line 719: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;small tridecimal comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 853: / Line 857: @@
          &lt;td&gt;0.42272&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;tridecimal schisma, Sagittal schismina&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 883: / Line 887: @@
          &lt;td&gt;0.16260&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;harmonisma&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;

List of superparticular intervals: Difference between revisions

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