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List of superparticular intervals: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is a list of [[superparticular]] [[interval]]s ordered by [[prime limit]]. It reaches to the 127-limit and is complete up to the [[37-limit]].
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-09-17 10:54:38 UTC</tt>.<br>
: The original revision id was <tt>255045006</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>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="color: #800080;"&gt;List of Superparticular Intervals&lt;/span&gt;=


[[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 [[20_21|20/21]]. 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.
[[Wikipedia: Størmer's theorem|Størmer's theorem]] states 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]]. {{OEIS| A002071 }} gives the number of superparticular ratios in each prime limit, {{OEIS| A145604 }} shows the increment from limit to limit, and {{OEIS| A117581 }} gives the largest numerator for each prime limit (with some exceptions, such as the 23-limit, where the largest value is smaller than that of a smaller prime limit, in this case the 19-limit).


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.
== List of superparticular intervals ==
=== 2-limit ===
{| class="wikitable center-6" style="width:100%"
|-
! width="10%" | [[Ratio]]
! width="10%" | [[Cent]]s
! width="15%" | Factorization
! width="15%" | [[Monzo]]
! width="45%" | Name(s)
! width="5%" | Meta<ref name="ssp">Denoted by S-expressions, where s''k'' is defined as (''k''/(''k'' - 1))/((''k'' + 1)/''k''). See [[square superparticular]] for details.</ref>
|-
| [[2/1]]
| 1200.000
| 2/1
| {{Monzo| 1 }}
| Octave, duple, 2nd harmonic, diapason
|
|}


||~ Ratio ||~ Cents Value ||~ Factorization ||~ Prime Limit ||~ Name(s) ||
=== 3-limit ===
|| [[2_1|2/1]] || 1200.000 || 2/1 || 2 || (perfect) unison, unity, perfect prime, tonic, duple ||
{| class="wikitable center-6" style="width:100%"
|| [[3_2|3/2]] || 701.995 || 3/2 || 3 || [[perfect fifth]], 3rd harmonic (octave reduced), diapente ||
! width="10%" | [[Ratio]]
|| [[4_3|4/3]] || 498.045 || 2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;/3 || 3 || perfect fourth, 3rd subharmonic (octave reduced), diatessaron ||
! width="10%" | [[Cent]]s
|| [[5_4|5/4]] || 386.314 || 5/2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; || 5 || (classic) (5-limit) major third, 5th harmonic (octave reduced) ||
! width="15%" | Factorization
|| [[6_5|6/5]] || 315.641 || 3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;/5 || 5 || (classic) (5-limit) minor third ||
! width="15%" | [[Monzo]]
|| [[7_6|7/6]] || 266.871 || 7/3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; || 7 || (septimal) subminor third, septimal minor third, augmented second ||
! width="45%" | Name(s)
|| [[8_7|8/7]] || 231.174 || 2&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;/7 || 7 || (septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic ||
! width="5%" | Meta<ref name="ssp"/>
||  ||  ||  ||  ||  ||
|-
||  ||  ||  ||  ||  ||
| [[3/2]]
||  ||  ||  ||  ||  ||
| 701.955
||  ||  ||  ||  ||  ||
| 3/2
||  ||  ||  ||  ||  ||
| {{Monzo| -1 1 }}
||  ||  ||  ||  ||  ||
| Perfect fifth, octave-reduced 3rd harmonic, diapente
||  ||  ||  ||  ||  ||
|
||  ||  ||  ||  ||  ||
|-
||  ||  ||  ||  ||  ||
| [[4/3]]
||  ||  ||  ||  ||  ||
| 498.045
||  ||  ||  ||  ||  ||
| 2<sup>2</sup>/3
||  ||  ||  ||  ||  ||
| {{Monzo| 2 -1 }}
||  ||  ||  ||  ||  ||
| Perfect fourth, octave-reduced 3rd subharmonic, diatessaron
||  ||  ||  ||  ||  ||
| S2
||  ||  ||  ||  ||  ||
|-
||  ||  ||  ||  ||  ||
| [[9/8]]
||  ||  ||  ||  ||  ||
| 203.910
||  ||  ||  ||  ||  ||
| 3<sup>2</sup>/2<sup>3</sup>
||  ||  ||  ||  ||  ||
| {{monzo| -3 2 }}
||  ||  ||  ||  ||  ||
| Pythagorean whole tone, Pythagorean major second, <br>major whole tone, octave-reduced 9th harmonic, harmonic ninth
||  ||  ||  ||  ||  ||
| S3
||  ||  ||  ||  ||  ||
|}
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<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;List of Superparticular Intervals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="List of Superparticular Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #800080;"&gt;List of Superparticular Intervals&lt;/span&gt;&lt;/h1&gt;
&lt;br /&gt;
&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="/20_21"&gt;20/21&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;
&lt;br /&gt;


=== 5-limit ===
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Cent]]s
! width="15%" | Factorization
! width="15%" | [[Monzo]]
! width="45%" | Name(s)
! width="5%" | Meta<ref name="ssp"/>
|-
| [[5/4]]
| 386.314
| 5/2<sup>2</sup>
| {{Monzo| -2 0 1 }}
| Classic(al)/just major third, octave-reduced 5th harmonic
|
|-
| [[6/5]]
| 315.641
| (2×3)/5
| {{Monzo| 1 1 -1 }}
| Classic(al)/just minor third
|
|-
| [[10/9]]
| 182.404
| (2×5)/3<sup>2</sup>
| {{Monzo| 1 -2 1 }}
| Classic(al) (whole) tone, classic major second, minor whole tone
|
|-
| [[16/15]]
| 111.731
| 2<sup>4</sup>/(3×5)
| {{Monzo| 4 -1 -1 }}
| Classic(al)/just diatonic semitone, 15th subharmonic
| S4
|-
| [[25/24]]
| 70.672
| 5<sup>2</sup>/(2<sup>3</sup>×3)
| {{Monzo| -3 -1 2 }}
| Classic(al)/just chromatic semitone, chroma, Zarlinian semitone
| S5
|-
| [[81/80]]
| 21.506
| (3/2)<sup>4</sup>/5
| {{Monzo| -4 4 -1 }}
| Syntonic comma, Didymus comma
| S9
|}


&lt;table class="wiki_table"&gt;
=== 7-limit ===
    &lt;tr&gt;
{| class="wikitable center-6" style="width:100%"
        &lt;th&gt;Ratio&lt;br /&gt;
! width="10%" | [[Ratio]]
&lt;/th&gt;
! width="10%" | [[Cent]]s
        &lt;th&gt;Cents Value&lt;br /&gt;
! width="15%" | Factorization
&lt;/th&gt;
! width="15%" | [[Monzo]]
        &lt;th&gt;Factorization&lt;br /&gt;
! width="45%" | Name(s)
&lt;/th&gt;
! width="5%" | Meta<ref name="ssp"/>
        &lt;th&gt;Prime Limit&lt;br /&gt;
|-
&lt;/th&gt;
| [[7/6]]
        &lt;th&gt;Name(s)&lt;br /&gt;
| 266.871
&lt;/th&gt;
| 7/(2×3)
    &lt;/tr&gt;
| {{Monzo| -1 -1 0 1 }}
    &lt;tr&gt;
| (Septimal) subminor third, septimal minor third
        &lt;td&gt;&lt;a class="wiki_link" href="/2_1"&gt;2/1&lt;/a&gt;&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;1200.000&lt;br /&gt;
| [[8/7]]
&lt;/td&gt;
| 231.174
        &lt;td&gt;2/1&lt;br /&gt;
| 2<sup>3</sup>/7
&lt;/td&gt;
| {{Monzo| 3 0 0 -1 }}
        &lt;td&gt;2&lt;br /&gt;
| (Septimal) supermajor second, septimal whole tone, <br>octave-reduced 7th subharmonic
&lt;/td&gt;
|
        &lt;td&gt;(perfect) unison, unity, perfect prime, tonic, duple&lt;br /&gt;
|-
&lt;/td&gt;
| [[15/14]]
    &lt;/tr&gt;
| 119.443
    &lt;tr&gt;
| (3×5)/(2×7)
        &lt;td&gt;&lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;&lt;br /&gt;
| {{Monzo| -1 1 1 -1 }}
&lt;/td&gt;
| Septimal major semitone, septimal diatonic semitone
        &lt;td&gt;701.995&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;3/2&lt;br /&gt;
| [[21/20]]
&lt;/td&gt;
| 84.467
        &lt;td&gt;3&lt;br /&gt;
| (3×7)/(2<sup>2</sup>×5)
&lt;/td&gt;
| {{Monzo| -2 1 -1 1 }}
        &lt;td&gt;&lt;a class="wiki_link" href="/perfect%20fifth"&gt;perfect fifth&lt;/a&gt;, 3rd harmonic (octave reduced), diapente&lt;br /&gt;
| Septimal minor semitone, large septimal chroma
&lt;/td&gt;
|
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| [[28/27]]
        &lt;td&gt;&lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;&lt;br /&gt;
| 62.961
&lt;/td&gt;
| (2<sup>2</sup>×7)/3<sup>3</sup>
        &lt;td&gt;498.045&lt;br /&gt;
| {{Monzo| 2 -3 0 1 }}
&lt;/td&gt;
| Septimal 1/3-tone, small septimal chroma, <br>(septimal) subminor second, septimal minor second, <br>trienstonic comma
        &lt;td&gt;2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;/3&lt;br /&gt;
|
&lt;/td&gt;
|-
        &lt;td&gt;3&lt;br /&gt;
| [[36/35]]
&lt;/td&gt;
| 48.770
        &lt;td&gt;perfect fourth, 3rd subharmonic (octave reduced), diatessaron&lt;br /&gt;
| (2×3)<sup>2</sup>/(5×7)
&lt;/td&gt;
| {{Monzo| 2 2 -1 -1 }}
    &lt;/tr&gt;
| Septimal 1/4-tone, mint comma
    &lt;tr&gt;
| S6
        &lt;td&gt;&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| [[49/48]]
        &lt;td&gt;386.314&lt;br /&gt;
| 35.697
&lt;/td&gt;
| 7<sup>2</sup>/(2<sup>4</sup>×3)
        &lt;td&gt;5/2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;&lt;br /&gt;
| {{Monzo| -4 -1 0 2 }}
&lt;/td&gt;
| Large septimal diesis, large septimal 1/6-tone, slendro diesis, semaphoresma
        &lt;td&gt;5&lt;br /&gt;
| S7
&lt;/td&gt;
|-
        &lt;td&gt;(classic) (5-limit) major third, 5th harmonic (octave reduced)&lt;br /&gt;
| [[50/49]]
&lt;/td&gt;
| 34.976
    &lt;/tr&gt;
| 2×(5/7)<sup>2</sup>
    &lt;tr&gt;
| {{Monzo| 1 0 2 -2 }}
        &lt;td&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;&lt;br /&gt;
| Small septimal diesis, small septimal 1/6-tone, septimal tritonic diesis, jubilisma
&lt;/td&gt;
|
        &lt;td&gt;315.641&lt;br /&gt;
|-
&lt;/td&gt;
| [[64/63]]
        &lt;td&gt;3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;/5&lt;br /&gt;
| 27.264
&lt;/td&gt;
| 2<sup>6</sup>/(3<sup>2</sup>×7)
        &lt;td&gt;5&lt;br /&gt;
| {{Monzo| 6 -2 0 -1 }}
&lt;/td&gt;
| Septimal comma, Archytas' comma
        &lt;td&gt;(classic) (5-limit) minor third&lt;br /&gt;
| S8
&lt;/td&gt;
|-
    &lt;/tr&gt;
| [[126/125]]
    &lt;tr&gt;
| 13.795
        &lt;td&gt;&lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;&lt;br /&gt;
| (2×3<sup>2</sup>×7)/5<sup>3</sup>
&lt;/td&gt;
| {{Monzo| 1 2 -3 1 }}
        &lt;td&gt;266.871&lt;br /&gt;
| Starling comma, septimal semicomma
&lt;/td&gt;
|
        &lt;td&gt;7/3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| [[225/224]]
        &lt;td&gt;7&lt;br /&gt;
| 7.7115
&lt;/td&gt;
| (3×5)<sup>2</sup>/(2<sup>5</sup>×7)
        &lt;td&gt;(septimal) subminor third, septimal minor third, augmented second&lt;br /&gt;
| {{Monzo| -5 2 2 -1 }}
&lt;/td&gt;
| Marvel comma, septimal kleisma
    &lt;/tr&gt;
| S15
    &lt;tr&gt;
|-
        &lt;td&gt;&lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;&lt;br /&gt;
| [[2401/2400]]
&lt;/td&gt;
| 0.72120
        &lt;td&gt;231.174&lt;br /&gt;
| 7<sup>4</sup>/(2<sup>5</sup>×3×5<sup>2</sup>)
&lt;/td&gt;
| {{Monzo| -5 -1 -2 4 }}
        &lt;td&gt;2&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;/7&lt;br /&gt;
| Breedsma
&lt;/td&gt;
| S49
        &lt;td&gt;7&lt;br /&gt;
|-
&lt;/td&gt;
| [[4375/4374]]
        &lt;td&gt;(septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic&lt;br /&gt;
| 0.39576
&lt;/td&gt;
| (5<sup>4</sup>×7)/(2×3<sup>7</sup>)
    &lt;/tr&gt;
| {{Monzo| -1 -7 4 1 }}
    &lt;tr&gt;
| Ragisma
        &lt;td&gt;&lt;br /&gt;
|
&lt;/td&gt;
|}
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
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        &lt;td&gt;&lt;br /&gt;
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&lt;/td&gt;
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        &lt;td&gt;&lt;br /&gt;
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&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
=== 11-limit ===
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Cent]]s
! width="15%" | Factorization
! width="15%" | [[Monzo]]
! width="45%" | Name(s)
! width="5%" | Meta<ref name="ssp"/>
|-
| [[11/10]]
| 165.004
| 11/(2×5)
| {{Monzo| -1 0 -1 0 1 }}
| Large undecimal neutral second, <br>undecimal submajor second, Ptolemy's second
|
|-
| [[12/11]]
| 150.637
| (2<sup>2</sup>×3)/11
| {{Monzo| 2 1 0 0 -1 }}
| Small undecimal neutral second
|
|-
| [[22/21]]
| 80.537
| (2×11)/(3×7)
| {{Monzo| 1 -1 0 -1 1 }}
| Undecimal minor semitone
|
|-
| [[33/32]]
| 53.273
| (3×11)/2<sup>5</sup>
| {{Monzo| -5 1 0 0 1 }}
| Undecimal 1/4-tone, undecimal diesis, <br>al-Farabi's 1/4-tone, octave-reduced 33rd harmonic
|
|-
| [[45/44]]
| 38.906
| (3/2)<sup>2</sup>×(5/11)
| {{monzo| -2 2 1 0 -1 }}
| Undecimal 1/5-tone, cake comma
|
|-
| [[55/54]]
| 31.767
| (5×11)/(2×3<sup>3</sup>)
| {{Monzo| -1 -3 1 0 1 }}
| Telepathma, eleventyfive comma, <br>undecimal diasecundal comma
|
|-
| [[56/55]]
| 31.194
| (2<sup>3</sup>×7)/(5×11)
| {{Monzo| 3 0 -1 1 -1 }}
| Undecimal tritonic comma, konbini comma
|
|-
| [[99/98]]
| 17.576
| (3/7)<sup>2</sup>×(11/2)
| {{Monzo| -1 2 0 -2 1 }}
| Mothwellsma, small undecimal comma
|
|-
| [[100/99]]
| 17.399
| ((2×5)/3)<sup>2</sup>/11
| {{monzo| 2 -2 2 0 -1 }}
| Ptolemisma, Ptolemy's comma
| S10
|-
| [[121/120]]
| 14.376
| 11<sup>2</sup>/(2<sup>3</sup>×3×5)
| {{Monzo| -3 -1 -1 0 2 }}
| Biyatisma, undecimal seconds comma
| S11
|-
| [[176/175]]
| 9.8646
| (2<sup>4</sup>×11)/(5<sup>2</sup>×7)
| {{Monzo| 4 0 -2 -1 1 }}
| Valinorsma
|
|-
| [[243/242]]
| 7.1391
| 3<sup>5</sup>/(2×11<sup>2</sup>)
| {{Monzo| -1 5 0 0 -2 }}
| Rastma, neutral thirds comma
|
|-
| [[385/384]]
| 4.5026
| (5×7×11)/(2<sup>7</sup>×3)
| {{Monzo| -7 -1 1 1 1 }}
| Keenanisma
|
|-
| [[441/440]]
| 3.9302
| (3×7)<sup>2</sup>/(2<sup>3</sup>×5×11)
| {{Monzo| -3 2 -1 2 -1 }}
| Werckisma, Werckmeister's undecimal septenarian schisma
| S21
|-
| [[540/539]]
| 3.2090
| (2/7)<sup>2</sup>×((3<sup>3</sup>×5)/11)
| {{Monzo| 2 3 1 -2 -1 }}
| Swetisma, Swets' comma
|
|-
| [[3025/3024]]
| 0.57240
| (5×11)<sup>2</sup>/(2<sup>4</sup>×3<sup>3</sup>×7)
| {{Monzo| -4 -3 2 -1 2 }}
| Lehmerisma
| S55
|-
| [[9801/9800]]
| 0.17665
| ((3<sup>2</sup>×11)/(5×7))<sup>2</sup>/2<sup>3</sup>
| {{Monzo| -3 4 -2 -2 2 }}
| Kalisma, Gauss comma
| S99
|}
 
=== 13-limit ===
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Cent]]s
! width="15%" | Factorization
! width="15%" | [[Monzo]]
! width="45%" | Name(s)
! width="5%" | Meta<ref name="ssp"/>
|-
| [[13/12]]
| 138.573
| 13/(2<sup>2</sup>×3)
| {{Monzo| -2 -1 0 0 0 1 }}
| Large tridecimal 2/3-tone, <br>tridecimal neutral second
|
|-
| [[14/13]]
| 128.298
| (2×7)/13
| {{Monzo| 1 0 0 1 0 -1 }}
| Small tridecimal 2/3-tone, trienthird
|
|-
| [[26/25]]
| 67.900
| (2×13)/5<sup>2</sup>
| {{Monzo| 1 0 -2 0 0 1 }}
| Large tridecimal 1/3-tone
|
|-
| [[27/26]]
| 65.337
| 3<sup>3</sup>/(2×13)
| {{Monzo| -1 3 0 0 0 -1 }}
| Small tridecimal 1/3-tone
|
|-
| [[40/39]]
| 43.831
| (2<sup>3</sup>×5)/(3×13)
| {{Monzo| 3 -1 1 0 0 -1 }}
| Tridecimal minor diesis
|
|-
| [[65/64]]
| 26.841
| (5×13)/2<sup>6</sup>
| {{Monzo| -6 0 1 0 0 1 }}
| Wilsorma, 13th-partial chroma
|
|-
| [[66/65]]
| 26.432
| (2×3×11)/(5×13)
| {{Monzo| 1 1 -1 0 1 -1 }}
| Winmeanma
|
|-
| [[78/77]]
| 22.339
| (2×3×13)/(7×11)
| {{Monzo| 1 1 0 -1 -1 1 }}
| Negustma
|
|-
| [[91/90]]
| 19.130
| (7×13)/(2×3<sup>2</sup>×5)
| {{Monzo| -1 -2 -1 1 0 1 }}
| Biome comma, superleap comma
|
|-
| [[105/104]]
| 16.567
| (3×5×7)/(2<sup>3</sup>×13)
| {{Monzo| -3 1 1 1 0 -1 }}
| Animist comma, small tridecimal comma
|
|-
| [[144/143]]
| 12.064
| (2<sup>2</sup>×3)<sup>2</sup>/(11×13)
| {{Monzo| 4 2 0 0 -1 -1 }}
| Grossma
| S12
|-
| [[169/168]]
| 10.274
| 13<sup>2</sup>/(2<sup>3</sup>×3×7)
| {{Monzo| -3 -1 0 -1 0 2 }}
| Buzurgisma, dhanvantarisma
| S13
|-
| [[196/195]]
| 8.8554
| (2×7)<sup>2</sup>/(3×5×13)
| {{Monzo| 2 -1 -1 2 0 -1 }}
| Mynucuma
| S14
|-
| [[325/324]]
| 5.3351
| (5/(2×3<sup>2</sup>))<sup>2</sup>×13
| {{Monzo| -2 -4 2 0 0 1 }}
| Marveltwin comma
|
|-
| [[351/350]]
| 4.9393
| (3<sup>3</sup>×13)/(2×5<sup>2</sup>×7)
| {{Monzo| -1 3 -2 -1 0 1 }}
| Ratwolfsma
|
|-
| [[352/351]]
| 4.9253
| (2<sup>5</sup>×11)/(3<sup>3</sup>×13)
| {{Monzo| 5 -3 0 0 1 -1 }}
| Major minthma, major gentle comma
|
|-
| [[364/363]]
| 4.7627
| (2/11)<sup>2</sup>×((7×13)/3)
| {{Monzo| 2 -1 0 1 -2 1 }}
| Minor minthma, minor gentle comma
|
|-
| [[625/624]]
| 2.7722
| (5/2)<sup>4</sup>/(3×13)
| {{Monzo| -4 -1 4 0 0 -1 }}
| Tunbarsma
| S25
|-
| [[676/675]]
| 2.5629
| ((2×13)/5)<sup>2</sup>/3<sup>3</sup>
| {{Monzo| 2 -3 -2 0 0 2 }}
| Island comma
| S26
|-
| [[729/728]]
| 2.3764
| (3<sup>2</sup>/2)<sup>3</sup>/(7×13)
| {{Monzo| -3 6 0 -1 0 -1 }}
| Squbema
| S27
|-
| [[1001/1000]]
| 1.7304
| (7×11×13)/(2×5)<sup>3</sup>
| {{Monzo| -3 0 -3 1 1 1 }}
| Sinbadma
|
|-
| [[1716/1715]]
| 1.0092
| (2<sup>2</sup>×3×11×13)/(5×7<sup>3</sup>)
| {{Monzo| 2 1 -1 -3 1 1 }}
| Lummic comma
|
|-
| [[2080/2079]]
| 0.83252
| (2<sup>5</sup>×5×13)/(3<sup>3</sup>×7×11)
| {{Monzo| 5 -3 1 -1 -1 1 }}
| Ibnsinma, sinaisma
|
|-
| [[4096/4095]]
| 0.42272
| (2<sup>6</sup>/3)<sup>2</sup>/(5×7×13)
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| Minisma
| S64
|-
| [[4225/4224]]
| 0.40981
| (5×13)<sup>2</sup>/(2<sup>7</sup>×3×11)
| {{Monzo| -7 -1 2 0 -1 2 }}
| Leprechaun comma
| S65
|-
| [[6656/6655]]
| 0.26012
| (2<sup>3</sup>/11)<sup>3</sup>×(13/5)
| {{Monzo| 9 0 -1 0 -3 1 }}
| Jacobin comma
|
|-
| [[Harmonisma|10648/10647]]
| 0.16260
| (2×11)<sup>3</sup>/((3×13)<sup>2</sup>×7)
| {{Monzo| 3 -2 0 -1 3 -2 }}
| Harmonisma
|
|-
| [[Chalmersia|123201/123200]]
| 0.014052
| (3/2)<sup>6</sup>×(13/5)<sup>2</sup>/(7×11)
| {{Monzo| -6 6 -2 -1 -1 2 }}
| Chalmersia
| S351
|}
 
=== 17-limit ===
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Cent]]s
! width="15%" | Factorization
! width="15%" | [[Monzo]]
! width="45%" | Name(s)
! width="5%" | Meta<ref name="ssp"/>
|-
| [[17/16]]
| 104.955
| 17/2<sup>4</sup>
| {{Monzo| -4 0 0 0 0 0 1 }}
| Large septendecimal semitone, <br>octave-reduced 17th harmonic
|
|-
| [[18/17]]
| 98.955
| (2×3<sup>2</sup>)/17
| {{Monzo| 1 2 0 0 0 0 -1 }}
| Small septendecimal semitone, <br>Arabic lute index finger
|
|-
| [[34/33]]
| 51.682
| (2×17)/(3×11)
| {{Monzo| 1 -1 0 0 -1 0 1 }}
| Large septendecimal 1/4-tone
|
|-
| [[35/34]]
| 50.184
| (5×7)/(2×17)
| {{Monzo| -1 0 1 1 0 0 -1 }}
| Small septendecimal 1/4-tone
|
|-
| [[51/50]]
| 34.283
| (3×17)/(2×5<sup>2</sup>)
| {{Monzo| -1 1 -2 0 0 0 1 }}
| Large septendecimal 1/6-tone
|
|-
| [[52/51]]
| 33.617
| (2<sup>2</sup>×13)/(3×17)
| {{Monzo| 2 -1 0 0 0 1 -1 }}
| Small septendecimal 1/6-tone
|
|-
| [[85/84]]
| 20.488
| (5×17)/(2<sup>2</sup>×3×7)
| {{Monzo| -2 -1 1 -1 0 0 1 }}
| Monk comma
|
|-
| [[120/119]]
| 14.487
| (2<sup>3</sup>×3×5)/(7×17)
| {{Monzo| 3 1 1 -1 0 0 -1 }}
| Lynchisma
|
|-
| [[136/135]]
| 12.777
| (2/3)<sup>3</sup>×(17/5)
| {{Monzo| 3 -3 -1 0 0 0 1 }}
| Diatisma, septendecimal major second comma
|
|-
| [[154/153]]
| 11.278
| (2×7×11)/(3<sup>2</sup>×17)
| {{Monzo| 1 -2 0 1 1 0 -1 }}
| Augustma
|
|-
| [[170/169]]
| 10.214
| (2×5×17)/13<sup>2</sup>
| {{Monzo| 1 0 1 0 0 -2 1 }}
| Major naiadma
|
|-
| [[221/220]]
| 7.8514
| (13×17)/(2<sup>2</sup>×5×11)
| {{Monzo| -2 0 -1 0 -1 1 1 }}
| Minor naiadma
|
|-
| [[256/255]]
| 6.7759
| 2<sup>8</sup>/(3×5×17)
| {{Monzo| 8 -1 -1 0 0 0 -1 }}
| Charisma, charic comma, <br>septendecimal kleisma
| S16
|-
| [[273/272]]
| 6.3532
| (3×7×13)/(2<sup>4</sup>×17)
| {{Monzo| -4 1 0 1 0 1 -1 }}
| Tannisma, prototannisma
|
|-
| [[289/288]]
| 6.0008
| (17/3)<sup>2</sup>/2<sup>5</sup>
| {{Monzo| -5 -2 0 0 0 0 2 }}
| Semitonisma
| S17
|-
| [[375/374]]
| 4.6228
| (3×5<sup>3</sup>)/(2×11×17)
| {{Monzo| -1 1 3 0 -1 0 -1 }}
| Ursulisma
|
|-
| [[442/441]]
| 3.9213
| (2×13×17)/(3×7)<sup>2</sup>
| {{Monzo| 1 -2 0 -2 0 1 1 }}
| Seminaiadma
|
|-
| [[561/560]]
| 3.0887
| (3×11×17)/(2<sup>4</sup>×5×7)
| {{Monzo| -4 1 -1 -1 1 0 1 }}
| Monardisma, tsaharuk comma
|
|-
| [[595/594]]
| 2.9121
| (5×7×17)/(2×3<sup>3</sup>×11)
| {{Monzo| -1 -3 1 1 -1 0 1 }}
| Dakotisma
|
|-
| [[715/714]]
| 2.4230
| (5×11×13)/(2×3×7×17)
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| September comma, septembrisma
|
|-
| [[833/832]]
| 2.0796
| (7<sup>2</sup>×17)/(2<sup>6</sup>×13)
| {{Monzo| -6 0 0 2 0 -1 1 }}
| Horizma, horizon comma
|
|-
| [[936/935]]
| 1.8506
| (2<sup>3</sup>×3<sup>2</sup>×13)/(5×11×17)
| {{Monzo| 3 2 -1 0 -1 1 -1 }}
| Ainisma, ainic comma
|
|-
| [[1089/1088]]
| 1.5905
| (3×11)<sup>2</sup>/(2<sup>6</sup>×17)
| {{Monzo| -6 2 0 0 2 0 -1 }}
| Twosquare comma
| S33
|-
| [[1156/1155]]
| 1.4983
| (2×17)<sup>2</sup>/(3×5×7×11)
| {{Monzo| 2 -1 -1 -1 -1 0 2 }}
| Quadrantonisma
| S34
|-
| [[1225/1224]]
| 1.4138
| (5×7)<sup>2</sup>/(2<sup>3</sup>×3<sup>2</sup>×17)
| {{Monzo| -3 -2 2 2 0 0 -1 }}
| Noellisma
| S35
|-
| [[1275/1274]]
| 1.3584
| (3×5<sup>2</sup>×17)/(2×7<sup>2</sup>×13)
| {{Monzo| -1 1 2 -2 0 -1 1 }}
| Cimbrisma
|
|-
| [[1701/1700]]
| 1.0181
| (3<sup>5</sup>×7)/((2×5)<sup>2</sup>×17)
| {{Monzo| -2 5 -2 1 0 0 -1 }}
| Palingenetic comma, palingenesis
|
|-
| [[2058/2057]]
| 0.84143
| (2×3×7<sup>3</sup>)/(11<sup>2</sup>×17)
| {{Monzo| 1 1 0 3 -2 0 -1 }}
| Xenisma
|
|-
| [[2431/2430]]
| 0.71230
| (11×13×17)/(2×3<sup>5</sup>×5)
| {{Monzo| -1 -5 -1 0 1 1 1 }}
| Heptacircle comma
|
|-
| [[2500/2499]]
| 0.69263
| (2×5<sup>2</sup>)<sup>2</sup>/(3×7<sup>2</sup>×17)
| {{Monzo| 2 -1 4 -2 0 0 -1 }}
| Sperasma
| S50
|-
| [[2601/2600]]
| 0.66573
| (3×17)<sup>2</sup>/(2<sup>3</sup>×5<sup>2</sup>×13)
| {{Monzo| -3 2 -2 0 0 -1 2 }}
| Sextantonisma
| S51
|-
| [[4914/4913]]
| 0.35234
| (2×3<sup>3</sup>×7×13)/17<sup>3</sup>
| {{Monzo| 1 3 0 1 0 1 -3 }}
| Baladisma
|
|-
| [[5832/5831]]
| 0.29688
| (2×3<sup>2</sup>)<sup>3</sup>/(7<sup>3</sup>×17)
| {{Monzo| 3 6 0 -3 0 0 -1 }}
| Chlorisma
|
|-
| [[Flashma|12376/12375]]
| 0.13989
| (2<sup>3</sup>×7×13×17)/(3<sup>2</sup>×5<sup>3</sup>×11)
| {{Monzo| 3 -2 -3 1 -1 1 1 }}
| Flashma
|
|-
| [[Sparkisma|14400/14399]]
| 0.12023
| (2<sup>3</sup>×3×5)<sup>2</sup>/(7×11<sup>2</sup>×17)
| {{monzo| 6 2 2 -1 -2 0 -1 }}
| Sparkisma
| S120
|-
| [[28561/28560]]
| 0.060616
| (13/2)<sup>4</sup>/(3×5×7×17)
| {{Monzo| -4 -1 -1 -1 0 4 -1 }}
| Pisanoisma
| S169
|-
| [[E-shaped comma|31213/31212]]
| 0.055466
| (7<sup>4</sup>×13)/(2<sup>2</sup>×3<sup>3</sup>×17<sup>2</sup>)
| {{Monzo| -2 -3 0 4 0 1 -2 }}
| E-shaped comma
|
|-
| [[Lateral comma|37180/37179]]
| 0.046564
| (2<sup>2</sup>×5×11×13<sup>2</sup>)/(3<sup>7</sup>×17)
| {{Monzo| 2 -7 1 0 1 2 -1 }}
| Lateral comma
|
|-
| [[Scintillisma|194481/194480]]
| 0.0089018
| (3×7)<sup>4</sup>/(2<sup>4</sup>×5×11×13×17)
| {{Monzo| -4 4 -1 4 -1 -1 -1 }}
| Scintillisma
| S441
|-
| [[Aksial comma|336141/336140]]
| 0.0051503
| (3<sup>2</sup>×13<sup>3</sup>×17)/(2<sup>2</sup>×5×7<sup>5</sup>)
| {{Monzo| -2 2 -1 -5 0 3 1 }}
| Aksial comma
|
|}
 
=== 19-limit ===
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Cent]]s
! width="15%" | Factorization
! width="15%" | [[Monzo]]
! width="45%" | Name(s)
! width="5%" | Meta<ref name="ssp"/>
|-
| [[19/18]]
| 93.603
| 19/(2×3<sup>2</sup>)
| {{Monzo| -1 -2 0 0 0 0 0 1 }}
| Large undevicesimal semitone
|
|-
| [[20/19]]
| 88.801
| (2<sup>2</sup>×5)/19
| {{Monzo| 2 0 1 0 0 0 0 -1 }}
| Small undevicesimal semitone
|
|-
| [[39/38]]
| 44.970
| (3×13)/(2×19)
| {{Monzo| -1 1 0 0 0 1 0 -1 }}
| Undevicesimal diesis, <br>undevicesimal 2/9-tone
|
|-
| [[57/56]]
| 30.642
| (3×19)/(2<sup>3</sup>×7)
| {{Monzo| -3 1 0 -1 0 0 0 1 }}
| Hendrix comma
|
|-
| [[76/75]]
| 22.931
| (2<sup>2</sup>×19)/(3×5<sup>2</sup>)
| {{Monzo| 2 -1 -2 0 0 0 0 1 }}
| Large undevicesimal 1/9-tone
|
|-
| [[77/76]]
| 22.631
| (7×11)/(2<sup>2</sup>×19)
| {{Monzo| -2 0 0 1 1 0 0 -1 }}
| Small undevicesimal 1/9-tone
|
|-
| [[96/95]]
| 18.128
| (2<sup>5</sup>×3)/(5×19)
| {{Monzo| 5 1 -1 0 0 0 0 -1 }}
| 19th-partial chroma
|
|-
| [[133/132]]
| 13.066
| (7×19)/(2<sup>2</sup>×3×11)
| {{Monzo| -2 -1 0 1 -1 0 0 1 }}
| Minithirdma
|
|-
| [[153/152]]
| 11.352
| (3<sup>2</sup>×17)/(2<sup>3</sup>×19)
| {{Monzo| -3 2 0 0 0 0 1 -1 }}
| Ganassisma, Ganassi's comma
|
|-
| [[171/170]]
| 10.154
| (3<sup>2</sup>×19)/(2×5×17)
| {{Monzo| -1 2 -1 0 0 0 -1 1 }}
| Malcolmisma
|
|-
| [[190/189]]
| 9.1358
| (2×5×19)/(3<sup>3</sup>×7)
| {{Monzo| 1 -3 1 -1 0 0 0 1 }}
| Cotylisma
|
|-
| [[209/208]]
| 8.3033
| (11×19)/(2<sup>4</sup>×13)
| {{Monzo| -4 0 0 0 1 -1 0 1 }}
| Yama comma
|
|-
| [[210/209]]
| 8.2637
| (2×3×5×7)/(11×19)
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
| Spleen comma
|
|-
| [[286/285]]
| 6.0639
| (2×11×13)/(3×5×19)
| {{Monzo| 1 -1 -1 0 1 1 0 -1 }}
| Chthonisma
|
|-
| [[324/323]]
| 5.3516
| (2×3<sup>2</sup>)<sup>2</sup>/(17×19)
| {{Monzo| 2 4 0 0 0 0 -1 -1 }}
| Photisma
| S18
|-
| [[343/342]]
| 5.0547
| 7<sup>3</sup>/(2×3<sup>2</sup>×19)
| {{Monzo| -1 -2 0 3 0 0 0 -1 }}
| Nutrisma
|
|-
| [[361/360]]
| 4.8023
| 19<sup>2</sup>/(2<sup>3</sup>×3<sup>2</sup>×5)
| {{Monzo| -3 -2 -1 0 0 0 0 2 }}
| Go comma, Dudon comma
| S19
|-
| [[400/399]]
| 4.3335
| (2<sup>2</sup>×5)<sup>2</sup>/(3×7×19)
| {{Monzo| 4 -1 2 -1 0 0 0 -1 }}
| Devichroma
| S20
|-
| [[456/455]]
| 3.8007
| (2<sup>3</sup>×3×19)/(5×7×13)
| {{Monzo| 3 1 -1 -1 0 -1 0 1 }}
| Abnobisma
|
|-
| [[476/475]]
| 3.6409
| (2<sup>2</sup>×7×17)/(5<sup>2</sup>×19)
| {{Monzo| 2 0 -2 1 0 0 1 -1 }}
| Hedwigma
|
|-
| [[495/494]]
| 3.5010
| (3<sup>2</sup>×5×11)/(2×13×19)
| {{Monzo| -1 2 1 0 1 -1 0 -1 }}
| Eulalisma
|
|-
| [[513/512]]
| 3.3780
| (3<sup>3</sup>×19)/2<sup>9</sup>
| {{Monzo| -9 3 0 0 0 0 0 1 }}
| Undevicesimal comma, undevicesimal schisma, <br>Boethius' comma, 513th harmonic
|
|-
| [[969/968]]
| 1.7875
| (3×17×19)/(2<sup>3</sup>×11<sup>2</sup>)
| {{Monzo| -3 1 0 0 -2 0 1 1 }}
| Kingfisher comma
|
|-
| [[1216/1215]]
| 1.4243
| (2<sup>6</sup>×19)/(3<sup>5</sup>×5)
| {{Monzo| 6 -5 -1 0 0 0 0 1 }}
| Password comma, Eratosthenes' comma
|
|-
| [[1331/1330]]
| 1.3012
| 11<sup>3</sup>/(2×5×7×19)
| {{Monzo| -1 0 -1 -1 3 0 0 -1 }}
| Solvejgsma
|
|-
| [[1445/1444]]
| 1.1985
| 5×(17/(2×19))<sup>2</sup>
| {{Monzo| -2 0 1 0 0 0 2 -2 }}
| Aureusma
|
|-
| [[1521/1520]]
| 1.1386
| (3×13)<sup>2</sup>/(2<sup>4</sup>×5×19)
| {{Monzo| -4 2 -1 0 0 2 0 -1 }}
| Pinkanberry
| S39
|-
| [[1540/1539]]
| 1.1245
| (2<sup>2</sup>×5×7×11)/(3<sup>4</sup>×19)
| {{Monzo| 2 -4 1 1 1 0 0 -1 }}
| Kevolisma
|
|-
| [[1729/1728]]
| 1.0016
| (7×13×19)/(2<sup>2</sup>×3)<sup>3</sup>
| {{Monzo| -6 -3 0 1 0 1 0 1 }}
| Ramanujanisma
|
|-
| [[2376/2375]]
| 0.72879
| ((2×3)/5)<sup>3</sup>×(11/19)
| {{Monzo| 3 3 -3 0 1 0 0 -1 }}
| Trichthonisma
|
|-
| [[2432/2431]]
| 0.71200
| (2<sup>7</sup>×19)/(11×13×17)
| {{Monzo| 7 0 0 0 -1 -1 -1 1 }}
| Blumeyer comma
|
|-
| [[2926/2925]]
| 0.59177
| (2×7×11×19)/((3×5)<sup>2</sup>×13)
| {{Monzo| 1 -2 -2 1 1 -1 0 1 }}
| Neovulture comma, neovulturisma
|
|-
| [[3136/3135]]
| 0.55214
| (2<sup>3</sup>×7)<sup>2</sup>/(3×5×11×19)
| {{Monzo| 6 -1 -1 2 -1 0 0 -1 }}
| Neomirkwai comma, neomirkwaisma
| S56
|-
| [[3250/3249]]
| 0.53277
| (2×5<sup>3</sup>×13)/(3×19)<sup>2</sup>
| {{Monzo| 1 -2 3 0 0 1 0 -2 }}
| Martebisma
|
|-
| [[4200/4199]]
| 0.41225
| (2<sup>3</sup>×3×5<sup>2</sup>×7)/(13×17×19)
| {{Monzo| 3 1 2 1 0 -1 -1 -1 }}
| Neosatanisma
|
|-
| [[5776/5775]]
| 0.29975
| (2<sup>2</sup>×19)<sup>2</sup>/(3×5<sup>2</sup>×7×11)
| {{Monzo| 4 -1 -2 -1 -1 0 0 2 }}
| Neovish comma, neovishma
| S76
|-
| [[5929/5928]]
| 0.29202
| (7×11)<sup>2</sup>/(2<sup>3</sup>×3×13×19)
| {{Monzo| -3 -1 0 2 2 -1 0 -1 }}
| Manzanisma
| S77
|-
| [[5985/5984]]
| 0.28929
| (3<sup>2</sup>×5×7×19)/(2<sup>5</sup>×11×17)
| {{Monzo| -5 2 1 1 -1 0 -1 1 }}
| Neogrendel comma, neogrendelisma
|
|-
| [[6175/6174]]
| 0.28038
| (5<sup>2</sup>×13×19)/(2×3<sup>2</sup>×7<sup>3</sup>)
| {{Monzo| -1 -2 2 -3 0 1 0 1 }}
| Neonewtisma
|
|-
| [[6860/6859]]
| 0.25238
| (2<sup>2</sup>×5×7<sup>3</sup>)/19<sup>3</sup>
| {{Monzo| 2 0 1 3 0 0 0 -3 }}
| Devicubisma
|
|-
| 10241/10240
| 0.16906
| (7<sup>2</sup>×11×19)/(2<sup>11</sup>×5)
| {{Monzo| -11 0 -1 2 1 0 0 1 }}
|
|
|-
| 10830/10829
| 0.15986
| (2×3×5×19<sup>2</sup>)/(7<sup>2</sup>×13×17)
| {{Monzo| 1 1 1 -2 0 -1 -1 2 }}
|
|
|-
| [[12636/12635]]
| 0.13701
| (2<sup>2</sup>×3<sup>5</sup>×13)/(5×7×19<sup>2</sup>)
| {{Monzo| 2 5 -1 -1 0 1 0 -2 }}
| Padriellisma
|
|-
| 13377/13376
| 0.12942
| (3×7<sup>3</sup>×13)/(2<sup>6</sup>×11×19)
| {{Monzo| -6 1 0 3 -1 1 0 -1 }}
|
|
|-
| 14080/14079
| 0.12296
| (2<sup>8</sup>×5×11)/(3×13×19<sup>2</sup>)
| {{Monzo| 8 -1 1 0 1 -1 0 -2 }}
|
|
|-
| 14365/14364
| 0.12052
| (5×13<sup>2</sup>×17)/(2<sup>2</sup>×3<sup>3</sup>×7×19)
| {{Monzo| -2 -3 1 -1 0 2 1 -1 }}
|
|
|-
| 23409/23408
| 0.073957
| ((3/2)<sup>2</sup>×17)<sup>2</sup>/(7×11×19)
| {{Monzo| -4 4 0 -1 -1 0 2 -1 }}
|
| S153
|-
| 27456/27455
| 0.063056
| (2<sup>6</sup>×3×11×13)/(5×17<sup>2</sup>×19)
| {{Monzo| 6 1 -1 0 1 1 -2 -1 }}
|
|
|-
| 28900/28899
| 0.059905
| ((2×5×17)/(3×13))<sup>2</sup>/19
| {{Monzo| 2 -2 2 0 0 -2 2 -1 }}
|
| S170
|-
| 43681/43680
| 0.039634
| (11×19)<sup>2</sup>/(2<sup>5</sup>×3×5×7×13)
| {{Monzo| -5 -1 -1 -1 2 -1 0 2 }}
|
| S209
|-
| 89376/89375
| 0.019370
| (2<sup>5</sup>×3×7<sup>2</sup>×19)/(5<sup>4</sup>×11×13)
| {{Monzo| 5 1 -4 2 -1 -1 0 1 }}
|
|
|-
| 104976/104975
| 0.016492
| (2×3<sup>2</sup>)<sup>4</sup>/(5<sup>2</sup>×13×17×19)
| {{Monzo| 4 8 -2 0 0 -1 -1 -1 }}
|
| S324
|-
| [[Decimillisma|165376/165375]]
| 0.010469
| (2<sup>9</sup>×17×19)/((3×5)<sup>3</sup>×7<sup>2</sup>)
| {{Monzo| 9 -3 -3 -2 0 0 1 1 }}
| Decimillisma
|
|-
| 228096/228095
| 0.0075900
| ((2<sup>2</sup>×3)/7)<sup>4</sup>×(11/(5×19))
| {{Monzo| 8 4 -1 -4 1 0 0 -1 }}
|
|
|-
| 601426/601425
| 0.0028786
| (2×7<sup>2</sup>×17×19<sup>2</sup>)/(3<sup>7</sup>×5<sup>2</sup>×11)
| {{Monzo| 1 -7 -2 2 -1 0 1 2 }}
|
|
|-
| [[Devicisma|633556/633555]]
| 0.0027326
| (2<sup>2</sup>×7×11<sup>3</sup>×17)/(3<sup>3</sup>×5×13×19<sup>2</sup>)
| {{Monzo| 2 -3 -1 1 3 -1 1 -2 }}
| Devicisma
|
|-
| 709632/709631
| 0.0024396
| (2<sup>10</sup>×3<sup>2</sup>×7×11)/(13<sup>3</sup>×17×19)
| {{Monzo| 10 2 0 1 1 -3 -1 -1 }}
|
|
|-
| 5909761/5909760
| 0.00029294
| (11×13×17)<sup>2</sup>/(2<sup>8</sup>×3<sup>5</sup>×5×19)
| {{Monzo| -8 -5 -1 0 2 2 2 -1 }}
|
| S2431
|-
| <font style="font-size:0.88em">[[11859211/11859210]]</font>
| 0.00014598
| (19/(3×11))<sup>4</sup>×((7×13)/(2×5))
| {{Monzo| -1 -4 -1 1 -4 1 0 4 }}
| Tredekisma
|
|}
 
=== 23-limit ===
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Cent]]s
! width="15%" | Factorization
! width="15%" | [[Monzo]]
! width="45%" | Name(s)
! width="5%" | Meta<ref name="ssp"/>
|-
| [[23/22]]
| 76.956
| 23/(2×11)
| {{Monzo| -1 0 0 0 -1 0 0 0 1 }}
| Large vicesimotertial semitone
|
|-
| [[24/23]]
| 73.681
| (2<sup>3</sup>×3)/23
| {{Monzo| 3 1 0 0 0 0 0 0 -1 }}
| Small vicesimotertial semitone
|
|-
| [[46/45]]
| 38.051
| (2×23)/(3<sup>2</sup>×5)
| {{Monzo| 1 -2 -1 0 0 0 0 0 1 }}
| Vicesimotertial 1/5-tone
|
|-
| [[69/68]]
| 25.274
| (3×23)/(2<sup>2</sup>×17)
| {{Monzo| -2 1 0 0 0 0 -1 0 1 }}
| Large vicesimotertial 1/8-tone
|
|-
| [[70/69]]
| 24.910
| (2×5×7)/(3×23)
| {{Monzo| 1 -1 1 1 0 0 0 0 -1 }}
| Small vicesimotertial 1/8-tone
|
|-
| [[92/91]]
| 18.921
| (2<sup>2</sup>×23)/(7×13)
| {{Monzo| 2 0 0 -1 0 -1 0 0 1 }}
| Undinisma
|
|-
| [[115/114]]
| 15.120
| (5×23)/(2×3×19)
| {{Monzo| -1 -1 1 0 0 0 0 -1 1 }}
| Yarmanisma
|
|-
| [[161/160]]
| 10.787
| (7×23)/(2<sup>5</sup>×5)
| {{Monzo| -5 0 -1 1 0 0 0 0 1 }}
| Major kirnbergerisma
|
|-
| [[162/161]]
| 10.720
| (2×3<sup>4</sup>)/(7×23)
| {{Monzo| 1 4 0 -1 0 0 0 0 -1 }}
| Minor kirnbergerisma
|
|-
| [[208/207]]
| 8.3433
| (2<sup>4</sup>×13)/(3<sup>2</sup>×23)
| {{Monzo| 4 -2 0 0 0 1 0 0 -1 }}
| Vicetone comma
|
|-
| [[231/230]]
| 7.5108
| (3×7×11)/(2×5×23)
| {{Monzo| -1 1 -1 1 1 0 0 0 -1 }}
| Major neutravicema
|
|-
| [[253/252]]
| 6.8564
| (11×23)/((2×3)<sup>2</sup>×7)
| {{Monzo| -2 -2 0 -1 1 0 0 0 1 }}
| Middle neutravicema
|
|-
| [[276/275]]
| 6.2840
| (2<sup>2</sup>×3×23)/(5<sup>2</sup>×11)
| {{Monzo| 2 1 -2 0 -1 0 0 0 1 }}
| Minor neutravicema
|
|-
| [[300/299]]
| 5.7804
| ((2×5)<sup>2</sup>×3)/(13×23)
| {{Monzo| 2 1 2 0 0 -1 0 0 -1 }}
| Major naiadvicema
|
|-
| [[323/322]]
| 5.3682
| (17×19)/(2×7×23)
| {{Monzo| -1 0 0 -1 0 0 1 1 -1 }}
| Major semivicema
|
|-
| [[391/390]]
| 4.4334
| (17×23)/(2×3×5×13)
| {{Monzo| -1 -1 -1 0 0 -1 1 0 1 }}
| Minor naiadvicema
|
|-
| [[392/391]]
| 4.4221
| (2<sup>3</sup>×7<sup>2</sup>)/(17×23)
| {{Monzo| 3 0 0 2 0 0 -1 0 -1 }}
| Minor semivicema
|
|-
| [[460/459]]
| 3.7676
| (2<sup>2</sup>×5×23)/(3<sup>3</sup>×17)
| {{Monzo| 2 -3 1 0 0 0 -1 0 1 }}
| Scanisma, vicewolf comma
|
|-
| [[484/483]]
| 3.5806
| (2×11)<sup>2</sup>/(3×7×23)
| {{Monzo| 2 -1 0 -1 2 0 0 0 -1 }}
| Pittsburghisma
| S22
|-
| [[507/506]]
| 3.4180
| (3×13<sup>2</sup>)/(2×11×23)
| {{Monzo| -1 1 0 0 -1 2 0 0 -1 }}
| Laodicisma
|
|-
| [[529/528]]
| 3.2758
| 23<sup>2</sup>/(2<sup>4</sup>×3×11)
| {{Monzo| -4 -1 0 0 -1 0 0 0 2 }}
| Preziosisma
| S23
|-
| [[576/575]]
| 3.0082
| ((2<sup>3</sup>×3)/5)<sup>2</sup>/23
| {{Monzo| 6 2 -2 0 0 0 0 0 -1 }}
| Worcester comma
| S24
|-
| [[736/735]]
| 2.3538
| (2<sup>5</sup>×23)/(3×5×7<sup>2</sup>)
| {{Monzo| 5 -1 -1 -2 0 0 0 0 1 }}
| Harvardisma
|
|-
| [[760/759]]
| 2.2794
| (2<sup>3</sup>×5×19)/(3×11×23)
| {{Monzo| 3 -1 1 0 -1 0 0 1 -1 }}
| Squadronisma
|
|-
| [[875/874]]
| 1.9797
| (5<sup>3</sup>×7)/(2×19×23)
| {{Monzo| -1 0 3 1 0 0 0 -1 -1 }}
| Nymphisma
|
|-
| [[897/896]]
| 1.9311
| (3×13×23)/(2<sup>7</sup>×7)
| {{Monzo| -7 1 0 -1 0 1 0 0 1 }}
| Lysistratisma
|
|-
| [[1105/1104]]
| 1.5674
| (5×13×17)/(2<sup>4</sup>×3×23)
| {{Monzo| -4 -1 1 0 0 1 1 0 -1 }}
| Fragarisma
|
|-
| [[1197/1196]]
| 1.4469
| (3<sup>2</sup>×7×19)/(2<sup>2</sup>×13×23)
| {{Monzo| -2 2 0 1 0 -1 0 1 -1 }}
| Rodessisma
|
|-
| [[1288/1287]]
| 1.3446
| (2<sup>3</sup>×7×23)/(3<sup>2</sup>×11×13)
| {{Monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| Santisma, triaphonisma
|
|-
| [[1496/1495]]
| 1.1576
| (2<sup>3</sup>×11×17)/(5×13×23)
| {{Monzo| 3 0 -1 0 1 -1 1 0 -1 }}
| Turkisma
|
|-
| [[1863/1862]]
| 0.92952
| (3<sup>4</sup>×23)/(2×7<sup>2</sup>×19)
| {{Monzo| -1 4 0 -2 0 0 0 -1 1 }}
| Antinousisma
|
|-
| [[2024/2023]]
| 0.85556
| (2<sup>3</sup>×11×23)/(7×17<sup>2</sup>)
| {{Monzo| 3 0 0 -1 1 0 -2 0 1 }}
| Artifisma, insincere comma
|
|-
| [[2025/2024]]
| 0.85514
| (3<sup>2</sup>×5)<sup>2</sup>/(2<sup>3</sup>×11×23)
| {{Monzo| -3 4 2 0 -1 0 0 0 -1 }}
| Cupcake comma, cupcakesma
| S45
|-
| [[2185/2184]]
| 0.79251
| (5×19×23)/(2<sup>3</sup>×3×7×13)
| {{Monzo| -3 -1 1 -1 0 -1 0 1 1 }}
| Guangdongisma
|
|-
| [[2300/2299]]
| 0.75287
| ((2×5)/11)<sup>2</sup>×(23/19)
| {{Monzo| 2 0 2 0 -2 0 0 -1 1 }}
| Travellisma
|
|-
| [[2646/2645]]
| 0.65441
| (2×3<sup>3</sup>×7<sup>2</sup>)/(5×23<sup>2</sup>)
| {{Monzo| 1 3 -1 2 0 0 0 0 -2 }}
| Biyativice comma, biyativicema
|
|-
| [[2737/2736]]
| 0.63265
| (7×17×23)/(2<sup>4</sup>×3<sup>2</sup>×19)
| {{Monzo| -4 -2 0 1 0 0 1 -1 1 }}
| Kotkisma
|
|-
| [[3060/3059]]
| 0.56586
| ((2×3)<sup>2</sup>×5×17)/(7×19×23)
| {{Monzo| 2 2 1 -1 0 0 1 -1 -1 }}
| Vicious comma, viciousma
|
|-
| [[3381/3380]]
| 0.51212
| (3×7<sup>2</sup>×23)/(2<sup>2</sup>×5×13<sup>2</sup>)
| {{Monzo| -2 1 -1 2 0 -2 0 0 1 }}
| Mikkolisma
|
|-
| [[3520/3519]]
| 0.49190
| (2<sup>6</sup>×5×11)/(3<sup>2</sup>×17×23)
| {{Monzo| 6 -2 1 0 1 0 -1 0 -1 }}
| Vicedim comma, vicedimma
|
|-
| [[3888/3887]]
| 0.44533
| (2<sup>4</sup>×3<sup>5</sup>)/(13<sup>2</sup>×23)
| {{Monzo| 4 5 0 0 0 -2 0 0 -1 }}
| Shoalma, vicetride comma
|
|-
| [[4693/4692]]
| 0.36893
| (13×19<sup>2</sup>)/(2<sup>2</sup>×3×17×23)
| {{Monzo| -2 -1 0 0 0 1 -1 2 -1 }}
| Viceaug comma, viceaugma
|
|-
| [[4761/4760]]
| 0.36367
| (3×23)<sup>2</sup>/(2<sup>3</sup>×5×7×17)
| {{Monzo| -3 2 -1 -1 0 0 -1 0 2 }}
| Demiquartervice comma
| S69
|-
| [[5083/5082]]
| 0.34063
| (13×17×23)/(2×3×7×11<sup>2</sup>)
| {{Monzo| -1 -1 0 -1 -2 1 1 0 1 }}
| Broadviewsma
|
|-
| [[7866/7865]]
| 0.22010
| (2×3<sup>2</sup>×19×23)/(5×11<sup>2</sup>×13)
| {{Monzo| 1 2 -1 0 -2 -1 0 1 1 }}
|
|
|-
| [[8281/8280]]
| 0.20907
| (7×13)<sup>2</sup>/(2<sup>3</sup>×3<sup>2</sup>×5×23)
| {{Monzo| -3 -2 -1 2 0 2 0 0 -1 }}
|
| S91
|-
| [[8625/8624]]
| 0.20073
| (3×5<sup>3</sup>×23)/(2<sup>4</sup>×7<sup>2</sup>×11)
| {{Monzo| -4 1 3 -2 -1 0 0 0 1 }}
| Beerglass comma
|
|-
| [[10626/10625]]
| 0.16293
| (2×3×7×11×23)/(5<sup>4</sup>×17)
| {{Monzo| 1 1 -4 1 1 0 -1 0 1 }}
| Demiglace comma
|
|-
| 11271/11270
| 0.15361
| (3×13×17<sup>2</sup>)/(2×5×7<sup>2</sup>×23)
| {{Monzo| -1 1 -1 -2 0 1 2 0 -1 }}
|
|
|-
| 11662/11661
| 0.14846
| (2×7<sup>3</sup>×17)/(3×13<sup>2</sup>×23)
| {{Monzo| 1 -1 0 3 0 -2 1 0 -1 }}
|
|
|-
| [[Vicetertisma|12168/12167]]
| 0.14228
| (2/23)<sup>3</sup>×(3×13)<sup>2</sup>
| {{Monzo| 3 2 0 0 0 2 0 0 -3 }}
| Vicetertisma
|
|-
| 16929/16928
| 0.10227
| (3<sup>4</sup>×11×19)/(2<sup>5</sup>×23<sup>2</sup>)
| {{Monzo| -5 4 0 0 1 0 0 1 -2 }}
|
|
|-
| 19551/19550
| 0.088552
| (3×7<sup>3</sup>×19)/(2×5<sup>2</sup>×17×23)
| {{Monzo| -1 1 -2 3 0 0 -1 1 -1 }}
|
|
|-
| 21505/21504
| 0.080506
| (5×11×17×23)/(2<sup>10</sup>×3×7)
| {{Monzo| -10 -1 1 -1 1 0 1 0 1 }}
|
|
|-
| 21736/21735
| 0.079650
| (2<sup>3</sup>×11×13×19)/(3<sup>3</sup>×5×7×23)
| {{Monzo| 3 -3 -1 -1 1 1 0 1 -1 }}
|
|
|-
| 23276/23275
| 0.074380
| ((2×23)/(5×7))<sup>2</sup>×(11/19)
| {{Monzo| 2 0 -2 -2 1 0 0 -1 2 }}
|
|
|-
| [[Joshuavoisma|25025/25024]]
| 0.069182
| (5<sup>2</sup>×7×11×13)/(2<sup>6</sup>×17×23)
| {{Monzo| -6 0 2 1 1 1 -1 0 -1 }}
| Joshuavoisma
|
|-
| [[Diarithmedia|25921/25920]]
| 0.066790
| (7×23)<sup>2</sup>/(2<sup>6</sup>×3<sup>4</sup>×5)
| {{Monzo| -6 -4 -1 2 0 0 0 0 2 }}
| Diarithmedia
| S161
|-
| 43264/43263
| 0.040016
| (2<sup>4</sup>×13)<sup>2</sup>/(3<sup>2</sup>×11×19×23)
| {{Monzo| 8 -2 0 0 -1 2 0 -1 -1 }}
|
| S208
|-
| 52326/52325
| 0.033086
| (2×3<sup>4</sup>×17×19)/(5<sup>2</sup>×7×13×23)
| {{Monzo| 1 4 -2 -1 0 -1 1 1 -1 }}
|
|
|-
| 71875/71874
| 0.024087
| (5<sup>5</sup>×23)/(2×(3×11)<sup>3</sup>)
| {{Monzo| -1 -3 5 0 -3 0 0 0 1 }}
|
|
|-
| 75141/75140
| 0.023040
| (3<sup>3</sup>×11<sup>2</sup>×23)/(2<sup>2</sup>×5×13×17<sup>2</sup>)
| {{Monzo| -2 3 -1 0 2 -1 -2 0 1 }}
|
|
|-
| 76545/76544
| 0.022617
| (3<sup>7</sup>×5×7)/(2<sup>8</sup>×13×23)
| {{Monzo| -8 7 1 1 0 -1 0 0 -1 }}
|
|
|-
| 104329/104328
| 0.016594
| (17×19)<sup>2</sup>/(2<sup>3</sup>×3<sup>4</sup>×7×23)
| {{Monzo| -3 -4 0 -1 0 0 2 2 -1 }}
|
| S323
|-
| 122452/122451
| 0.014138
| (2<sup>2</sup>×11<sup>3</sup>×23)/(3×7<sup>4</sup>×17)
| {{Monzo| 2 -1 0 -4 3 0 -1 0 1 }}
|
|
|-
| 126225/126224
| 0.013716
| (3<sup>3</sup>×5<sup>2</sup>×11×17)/(2<sup>4</sup>×7<sup>3</sup>×23)
| {{Monzo| -4 3 2 -3 1 0 1 0 -1 }}
|
|
|-
| 152881/152880
| 0.011324
| (17×23)<sup>2</sup>/(2<sup>4</sup>×3×5×7<sup>2</sup>×13)
| {{Monzo| -4 -1 -1 -2 0 -1 2 0 2 }}
|
| S391
|-
| 202125/202124
| 0.0085652
| (3×5<sup>3</sup>×7<sup>2</sup>×11)/(2<sup>2</sup>×13<sup>3</sup>×23)
| {{Monzo| -2 1 3 2 1 -3 0 0 -1 }}
|
|
|-
| 264385/264384
| 0.0065482
| (5×11<sup>2</sup>×19×23)/(2<sup>6</sup>×3<sup>5</sup>×17)
| {{Monzo| -6 -5 1 0 2 0 -1 1 1 }}
|
|
|-
| 282625/282624
| 0.0061256
| (5<sup>3</sup>×7×17×19)/(2<sup>12</sup>×3×23)
| {{Monzo| -12 -1 3 1 0 0 1 1 -1 }}
|
|
|-
| 328510/328509
| 0.0052700
| (2×5×7×13×19<sup>2</sup>)/(3×23)<sup>3</sup>
| {{Monzo| 1 -3 1 1 0 1 0 2 -3 }}
|
|
|-
| 2023425/2023424
| 0.00085560
| ((3×5×23)<sup>2</sup>×17)/(2<sup>13</sup>×13×19)
| {{Monzo| -13 2 2 0 0 -1 1 -1 2 }}
|
|
|-
| 4096576/4096575
| 0.00042261
| ((2<sup>3</sup>×11×23)/(3<sup>2</sup>×5×17))<sup>2</sup>/7
| {{Monzo| 6 -4 -2 -1 2 0 -2 0 2 }}
|
| S2024
|-
| 5142501/5142500
| 0.00033665
| 3<sup>3</sup>×((7×13)/(2×5<sup>2</sup>×11))<sup>2</sup>×(23/17)
| {{Monzo| -2 3 -4 2 -2 2 -1 0 1 }}
|
|
|}
 
=== Higher-limit ===
See:
* [[List of superparticular intervals/29-limit]]
* [[List of superparticular intervals/31-limit]]
* [[List of superparticular intervals/37-limit]]
* [[List of superparticular intervals/41-limit]]
* [[List of superparticular intervals/43-limit]]
* [[List of superparticular intervals/47-limit]]
* [[List of superparticular intervals/53-limit]]
* [[List of superparticular intervals/59-limit]]
* [[List of superparticular intervals/61-limit]]
* [[List of superparticular intervals/67-limit]]
* [[List of superparticular intervals/71-limit]]
* [[List of superparticular intervals/73-limit]]
* [[List of superparticular intervals/79-limit]]
* [[List of superparticular intervals/83-limit]]
* [[List of superparticular intervals/89-limit]]
* [[List of superparticular intervals/97-limit]]
* [[List of superparticular intervals/101-limit]]
* [[List of superparticular intervals/103-limit]]
* [[List of superparticular intervals/107-limit]]
* [[List of superparticular intervals/109-limit]]
* [[List of superparticular intervals/113-limit]]
* [[List of superparticular intervals/127-limit]]
 
== See also ==
* [[Gallery of just intervals]]
 
== Notes ==
<references/>
 
== External links ==
* [http://www.huygens-fokker.org/docs/intervals.html ''List of intervals''] on the Huygens-Fokker Foundation website
 
[[Category:Lists of intervals]]
[[Category:Superparticular ratios|*]]
Retrieved from "https://en.xen.wiki/w/List_of_superparticular_intervals"