List of superparticular intervals: Difference between revisions

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[[superparticular|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|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|comma]]s are superparticular ratios.
This '''list of superparticular intervals''' ordered by prime limit. It reaches to the [[101-limit]] and is complete up to the [[17-limit]].


The list below is ordered by [[Harmonic_Limit|harmonic limit]], or the largest prime involved in the prime factorization. [[36/35|36/35]], for instance, is an interval of the [[7-limit|7-limit]], as it factors to (2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)/(5*7), while 37/36 would belong to the 37-limit.
[[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]]. 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.


[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. OEIS [http://oeis.org/A145604 A145604] gives the number of superparticular ratios in each prime limit, and [http://oeis.org/A117581 A117581] 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).
The list below is ordered by [[harmonic limit]], or the largest prime involved in the prime factorization. [[36/35]], for instance, is an interval of the [[7-limit]], as it factors to (2<sup>2</sup>*3<sup>2</sup>)/(5*7), while 37/36 would belong to the 37-limit.


See also: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]. Many of the names below come from [http://www.huygens-fokker.org/docs/intervals.html here].
[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]]. [http://oeis.org/A002071 A002071 &#45;- OEIS] gives the number of superparticular ratios in each prime limit, [http://oeis.org/A145604 A145604 &#45; OEIS] shows the increment from limit to limit, and [http://oeis.org/A117581 A117581] 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).


remove factorization, uses the bad <i><nowiki><sup></nowiki></i> syntax and is redundant because monzo
See also [[gallery of just intervals]]. Many of the names below come from [http://www.huygens-fokker.org/docs/intervals.html here].


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Ratio
! Ratio
! | Cents
! Cents
! | Factorization
! Factorization
! | [[monzo|Monzo]]
! [[Monzo]]
! | Name(s)
! Name(s)
|-
|-
! colspan="5" | 2-limit (complete)
! colspan="5" | 2-limit (complete)
|-
|-
| | [[2/1|2/1]]
| [[2/1]]
| | 1200.000
| 1200.000
| | 2/1
| 2/1
| | | 1 &gt;
| {{Monzo|1}}
| | (perfect) unison, unity, perfect prime, tonic, duple
| octave, duple; ''after [[octave reduction]]:'' (perfect) unison, unity, perfect prime, tonic
|-
|-
! colspan="5" | 3-limit (complete)
! colspan="5" | 3-limit (complete)
|-
|-
| | [[3/2|3/2]]
| [[3/2]]
| | 701.995
| 701.995
| | 3/2
| 3/2
| | | -1 1 &gt;
| {{Monzo|-1 1}}
| | [[perfect_fifth|perfect fifth]], 3rd harmonic (octave reduced), diapente
| [[perfect fifth]], 3rd harmonic (octave reduced), diapente
|-
|-
| | [[4/3|4/3]]
| [[4/3]]
| | 498.045
| 498.045
| | 2<span style="font-size: 70%; vertical-align: super;">2</span>/3
| 2<span style="font-size: 70%; vertical-align: super;">2</span>/3
| | | 2 -1 &gt;
| {{Monzo|2 -1}}
| | perfect fourth, 3rd subharmonic (octave reduced), diatessaron
| perfect fourth, 3rd subharmonic (octave reduced), diatessaron
|-
|-
| | [[9/8|9/8]]
| [[9/8]]
| | 203.910
| 203.910
| | 3<span style="font-size: 70%; vertical-align: super;">2</span>/2<span style="font-size: 70%; vertical-align: super;">3</span>
| 3<span style="font-size: 70%; vertical-align: super;">2</span>/2<span style="font-size: 70%; vertical-align: super;">3</span>
| | | -3 2 &gt;
| {{Monzo|-3 2}}
| | (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced)
| (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced)
|-
|-
! colspan="5" | 5-limit (complete)
! colspan="5" | 5-limit (complete)
|-
|-
| | [[5/4|5/4]]
| [[5/4]]
| | 386.314
| 386.314
| | 5/2<span style="font-size: 70%; vertical-align: super;">2</span>
| 5/2<span style="font-size: 70%; vertical-align: super;">2</span>
| | | -2 0 1 &gt;
| {{Monzo|-2 0 1}}
| | (classic) (5-limit) major third, 5th harmonic (octave reduced)
| (classic) (5-limit) major third, 5th harmonic (octave reduced)
|-
|-
| | [[6/5|6/5]]
| [[6/5]]
| | 315.641
| 315.641
| | (2*3)/5
| (2*3)/5
| | | 1 1 -1 &gt;
| {{Monzo|1 1 -1}}
| | (classic) (5-limit) minor third
| (classic) (5-limit) minor third
|-
|-
| | [[10/9|10/9]]
| [[10/9]]
| | 182.404
| 182.404
| | (2*5)/3<span style="font-size: 70%; vertical-align: super;">2</span>
| (2*5)/3<span style="font-size: 70%; vertical-align: super;">2</span>
| | | 1 -2 1 &gt;
| {{Monzo|1 -2 1}}
| | classic (whole) tone, classic major second, minor whole tone
| classic (whole) tone, classic major second, minor whole tone
|-
|-
| | [[16/15|16/15]]
| [[16/15]]
| | 111.713
| 111.713
| | 2<span style="font-size: 70%; vertical-align: super;">4</span>/(3*5)
| 2<span style="font-size: 70%; vertical-align: super;">4</span>/(3*5)
| | | 4 -1 -1 &gt;
| {{Monzo|4 -1 -1}}
| | minor diatonic semitone, 15th subharmonic
| minor diatonic semitone, 15th subharmonic
|-
|-
| | [[25/24|25/24]]
| [[25/24]]
| | 70.672
| 70.672
| | 5<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3)
| 5<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3)
| | | -3 -1 2 &gt;
| {{Monzo|-3 -1 2}}
| | chroma, (classic) chromatic semitone, Zarlinian semitone
| chroma, (classic) chromatic semitone, Zarlinian semitone
|-
|-
| | [[81/80|81/80]]
| [[81/80]]
| | 21.506
| 21.506
| | (3/2)<span style="font-size: 70%; vertical-align: super;">4</span>/5
| (3/2)<span style="font-size: 70%; vertical-align: super;">4</span>/5
| | | -4 4 -1 &gt;
| {{Monzo|-4 4 -1}}
| | syntonic comma, Didymus comma
| syntonic comma, Didymus comma
|-
|-
! colspan="5" | 7-limit (complete)
! colspan="5" | 7-limit (complete)
|-
|-
| | [[7/6|7/6]]
| [[7/6]]
| | 266.871
| 266.871
| | 7/(2*3)
| 7/(2*3)
| | | -1 -1 0 1 &gt;
| {{Monzo|-1 -1 0 1 }}
| | (septimal) subminor third, septimal minor third, augmented second
| (septimal) subminor third, septimal minor third, augmented second
|-
|-
| | [[8/7|8/7]]
| [[8/7]]
| | 231.174
| 231.174
| | 2<span style="font-size: 70%; vertical-align: super;">3</span>/7
| 2<span style="font-size: 70%; vertical-align: super;">3</span>/7
| | | 3 0 0 -1 &gt;
| {{Monzo|3 0 0 -1}}
| | (septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic
| (septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic
|-
|-
| | [[15/14|15/14]]
| [[15/14]]
| | 119.443
| 119.443
| | (3*5)/(2*7)
| (3*5)/(2*7)
| | | -1 1 1 -1 &gt;
| {{Monzo|-1 1 1 -1}}
| | septimal diatonic semitone
| septimal diatonic semitone
|-
|-
| | [[21/20|21/20]]
| [[21/20]]
| | 84.467
| 84.467
| | (3*7)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*5)
| (3*7)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*5)
| | | -2 1 -1 1 &gt;
| {{Monzo|-2 1 -1 1}}
| | minor semitone, large septimal chromatic semitone
| minor semitone, large septimal chromatic semitone
|-
|-
| | [[28/27|28/27]]
| [[28/27]]
| | 62.961
| 62.961
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*7)/3<span style="font-size: 70%; vertical-align: super;">3</span>
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*7)/3<span style="font-size: 70%; vertical-align: super;">3</span>
| | | 2 -3 0 1 &gt;
| {{Monzo|2 -3 0 1}}
| | septimal chroma, small septimal chromatic semitone, Archytas' 1/3-tone
| septimal chroma, small septimal chromatic semitone, Archytas' 1/3-tone
|-
|-
| | [[36/35|36/35]]
| [[36/35]]
| | 48.770
| 48.770
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">3</span>)/(5*7)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">3</span>)/(5*7)
| | | 2 2 -1 -1 &gt;
| {{Monzo|2 2 -1 -1}}
| | septimal quarter tone, septimal diesis
| septimal quarter tone, septimal diesis
|-
|-
| | [[49/48|49/48]]
| [[49/48]]
| | 35.697
| 35.697
| | 7<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">4</span>*3)
| 7<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">4</span>*3)
| | | -4 -1 0 2 &gt;
| {{Monzo|-4 -1 0 2}}
| | large septimal diesis, slendro diesis, septimal 1/6-tone
| large septimal diesis, slendro diesis, septimal 1/6-tone
|-
|-
| | [[50/49|50/49]]
| [[50/49]]
| | 34.976
| 34.976
| | 2*(5/7)<span style="font-size: 70%; vertical-align: super;">2</span>
| 2*(5/7)<span style="font-size: 70%; vertical-align: super;">2</span>
| | | 1 0 2 -2 &gt;
| {{Monzo|1 0 2 -2}}
| | septimal sixth-tone, jubilisma, small septimal diesis, tritonic diesis, Erlich's decatonic comma
| septimal sixth-tone, jubilisma, small septimal diesis, tritonic diesis, Erlich's decatonic comma
|-
|-
| | [[64/63|64/63]]
| [[64/63]]
| | 27.264
| 27.264
| | 2<span style="font-size: 70%; vertical-align: super;">6</span>/(3<span style="font-size: 70%; vertical-align: super;">2</span>*7)
| 2<span style="font-size: 70%; vertical-align: super;">6</span>/(3<span style="font-size: 70%; vertical-align: super;">2</span>*7)
| | | 6 -2 0 -1 &gt;
| {{Monzo|6 -2 0 -1}}
| | septimal comma, Archytas' comma
| septimal comma, Archytas' comma
|-
|-
| | [[126/125|126/125]]
| [[126/125]]
| | 13.795
| 13.795
| | (2*3<span style="font-size: 70%; vertical-align: super;">2</span>*7)/5<span style="font-size: 70%; vertical-align: super;">3</span>
| (2*3<span style="font-size: 70%; vertical-align: super;">2</span>*7)/5<span style="font-size: 70%; vertical-align: super;">3</span>
| | | 1 2 -3 1 &gt;
| {{Monzo|1 2 -3 1}}
| | starling comma, septimal semicomma
| starling comma, septimal semicomma
|-
|-
| | [[225/224|225/224]]
| [[225/224]]
| | 7.7115
| 7.7115
| | (3*5)<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">5</span>*7)
| (3*5)<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">5</span>*7)
| | | -5 2 2 -1 &gt;
| {{Monzo|-5 2 2 -1}}
| | marvel comma, septimal kleisma
| marvel comma, septimal kleisma
|-
|-
| | [[2401/2400|2401/2400]]
| [[2401/2400]]
| | 0.72120
| 0.72120
| | 7<span style="font-size: 70%; vertical-align: super;">4</span>/(2<span style="font-size: 70%; vertical-align: super;">5</span>*3*5<span style="font-size: 70%; vertical-align: super;">2</span>)
| 7<span style="font-size: 70%; vertical-align: super;">4</span>/(2<span style="font-size: 70%; vertical-align: super;">5</span>*3*5<span style="font-size: 70%; vertical-align: super;">2</span>)
| | | -5 -1 -2 4 &gt;
| {{Monzo|-5 -1 -2 4}}
| | breedsma
| breedsma
|-
|-
| | [[4375/4374|4375/4374]]
| [[4375/4374]]
| | 0.39576
| 0.39576
| | (5<span style="font-size: 70%; vertical-align: super;">4</span>*7)/(2*3<span style="font-size: 70%; vertical-align: super;">7</span>)
| (5<span style="font-size: 70%; vertical-align: super;">4</span>*7)/(2*3<span style="font-size: 70%; vertical-align: super;">7</span>)
| | | -1 -7 4 1 &gt;
| {{Monzo|-1 -7 4 1}}
| | ragisma
| ragisma
|-
|-
! colspan="5" | 11-limit (complete)
! colspan="5" | 11-limit (complete)
|-
|-
| | [[11/10|11/10]]
| [[11/10]]
| | 165.004
| 165.004
| | 11/(2*5)
| 11/(2*5)
| | | -1 0 -1 0 1 &gt;
| {{Monzo|-1 0 -1 0 1}}
| | (large) (undecimal) neutral second, 4/5-tone, Ptolemy's second
| (large) (undecimal) neutral second, 4/5-tone, Ptolemy's second
|-
|-
| | [[12/11|12/11]]
| [[12/11]]
| | 150.637
| 150.637
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*3)/11
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*3)/11
| | | 2 1 0 0 -1 &gt;
| {{Monzo|2 1 0 0 -1}}
| | (small) (undecimal) neutral second, 3/4-tone
| (small) (undecimal) neutral second, 3/4-tone
|-
|-
| | [[22/21|22/21]]
| [[22/21]]
| | 80.537
| 80.537
| | (2*11)/(3*7)
| (2*11)/(3*7)
| | | 1 -1 0 -1 1 &gt;
| {{Monzo|1 -1 0 -1 1}}
| | undecimal minor semitone
| undecimal minor semitone
|-
|-
| | [[33/32|33/32]]
| [[33/32]]
| | 53.273
| 53.273
| | (3*11)/2<span style="font-size: 70%; vertical-align: super;">5</span>
| (3*11)/2<span style="font-size: 70%; vertical-align: super;">5</span>
| | | -5 1 0 0 1 &gt;
| {{Monzo|-5 1 0 0 1}}
| | undecimal quarter tone, undecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced)
| undecimal quarter tone, undecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced)
|-
|-
| | [[45/44|45/44]]
| [[45/44]]
| | 38.906
| 38.906
| | (3/2)<span style="font-size: 70%; vertical-align: super;">2</span>*(5/11)
| (3/2)<span style="font-size: 70%; vertical-align: super;">2</span>*(5/11)
| | | -2 2 1 0 -1 &gt;
| {{Monzo|-2 2 1 0 -1}}
| | 1/5-tone
| 1/5-tone
|-
|-
| | [[55/54|55/54]]
| [[55/54]]
| | 31.767
| 31.767
| | (5*11)/(2*3<span style="font-size: 70%; vertical-align: super;">3</span>)
| (5*11)/(2*3<span style="font-size: 70%; vertical-align: super;">3</span>)
| | | -1 -3 1 0 1 &gt;
| {{Monzo|-1 -3 1 0 1}}
| | undecimal diasecundal comma, eleventyfive comma
| undecimal diasecundal comma, eleventyfive comma
|-
|-
| | [[56/55|56/55]]
| [[56/55]]
| | 31.194
| 31.194
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*7)/(5*11)
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*7)/(5*11)
| | | 3 0 -1 1 -1 &gt;
| {{Monzo|3 0 -1 1 -1}}
| | undecimal tritonic comma, konbini comma
| undecimal tritonic comma, konbini comma
|-
|-
| | [[99/98|99/98]]
| [[99/98]]
| | 17.576
| 17.576
| | (3/7)<span style="font-size: 70%; vertical-align: super;">2</span>*(11/2)
| (3/7)<span style="font-size: 70%; vertical-align: super;">2</span>*(11/2)
| | | -1 2 0 -2 1 &gt;
| {{Monzo|-1 2 0 -2 1}}
| | small undecimal comma, mothwellsma
| small undecimal comma, mothwellsma
|-
|-
| | [[100/99|100/99]]
| [[100/99]]
| | 17.399
| 17.399
| | (2*5/3)<span style="font-size: 70%; vertical-align: super;">2</span>/11)
| (2*5/3)<span style="font-size: 70%; vertical-align: super;">2</span>/11)
| | | 2 -2 2 0 -1 &gt;
| {{Monzo|2 -2 2 0 -1}}
| | Ptolemy's comma, ptolemisma
| Ptolemy's comma, ptolemisma
|-
|-
| | [[121/120|121/120]]
| [[121/120]]
| | 14.376
| 14.376
| | 11<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3*5)
| 11<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3*5)
| | | -3 -1 -1 0 2 &gt;
| {{Monzo|-3 -1 -1 0 2}}
| | undecimal seconds comma, biyatisma
| undecimal seconds comma, biyatisma
|-
|-
| | [[176/175|176/175]]
| [[176/175]]
| | 9.8646
| 9.8646
| | (2<span style="font-size: 70%; vertical-align: super;">4</span>*11)/(5<span style="font-size: 70%; vertical-align: super;">2</span>*7)
| (2<span style="font-size: 70%; vertical-align: super;">4</span>*11)/(5<span style="font-size: 70%; vertical-align: super;">2</span>*7)
| | | 4 0 -2 -1 1 &gt;
| {{Monzo|4 0 -2 -1 1}}
| | valinorsma
| valinorsma
|-
|-
| | [[243/242|243/242]]
| [[243/242]]
| | 7.1391
| 7.1391
| | 3<span style="font-size: 70%; vertical-align: super;">5</span>/(2*11<span style="font-size: 70%; vertical-align: super;">2</span>)
| 3<span style="font-size: 70%; vertical-align: super;">5</span>/(2*11<span style="font-size: 70%; vertical-align: super;">2</span>)
| | | -1 5 0 0 -2 &gt;
| {{Monzo|-1 5 0 0 -2}}
| | neutral third comma, rastma
| neutral third comma, rastma
|-
|-
| | [[385/384|385/384]]
| [[385/384]]
| | 4.5026
| 4.5026
| | (5*7*11)/(2<span style="font-size: 70%; vertical-align: super;">7</span>*3)
| (5*7*11)/(2<span style="font-size: 70%; vertical-align: super;">7</span>*3)
| | | -7 -1 1 1 1 &gt;
| {{Monzo|-7 -1 1 1 1}}
| | keenanisma
| keenanisma
|-
|-
| | [[441/440|441/440]]
| [[441/440]]
| | 3.9302
| 3.9302
| | (3*7)<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*5*11)
| (3*7)<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*5*11)
| | | -3 2 -1 2 -1 &gt;
| {{Monzo|-3 2 -1 2 -1}}
| | Werckmeister's undecimal septenarian schisma, werckisma
| Werckmeister's undecimal septenarian schisma, werckisma
|-
|-
| | [[540/539|540/539]]
| [[540/539]]
| | 3.2090
| 3.2090
| | (2/7)<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">3</span>*5/11
| (2/7)<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">3</span>*5/11
| | | 2 3 1 -2 -1 &gt;
| {{Monzo|2 3 1 -2 -1}}
| | Swets' comma, swetisma
| Swets' comma, swetisma
|-
|-
| | [[3025/3024|3025/3024]]
| [[3025/3024]]
| | 0.57240
| 0.57240
| | (5*11)<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">4</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*7)
| (5*11)<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">4</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*7)
| | | -4 -3 2 -1 2 &gt;
| {{Monzo|-4 -3 2 -1 2}}
| | Lehmerisma
| Lehmerisma
|-
|-
| | [[9801/9800|9801/9800]]
| [[9801/9800]]
| | 0.17665
| 0.17665
| | [11/(5*7)]<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">4</span>/2<span style="font-size: 70%; vertical-align: super;">3</span>
| [11/(5*7)]<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">4</span>/2<span style="font-size: 70%; vertical-align: super;">3</span>
| | | -3 4 -2 -2 2 &gt;
| {{Monzo|-3 4 -2 -2 2}}
| | Gauss comma, kalisma
| Gauss comma, kalisma
|-
|-
! colspan="5" | 13-limit (complete)
! colspan="5" | 13-limit (complete)
|-
|-
| | [[13/12|13/12]]
| [[13/12]]
| | 138.573
| 138.573
| | 13/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3)
| 13/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3)
| | | -2 -1 0 0 0 1 &gt;
| {{Monzo|-2 -1 0 0 0 1}}
| | tridecimal 2/3-tone
| tridecimal 2/3-tone
|-
|-
| | [[14/13|14/13]]
| [[14/13]]
| | 128.298
| 128.298
| | (2*7)/13
| (2*7)/13
| | | 1 0 0 1 0 -1 &gt;
| {{Monzo|1 0 0 1 0 -1}}
| | 2/3-tone, trienthird
| 2/3-tone, trienthird
|-
|-
| | [[26/25|26/25]]
| [[26/25]]
| | 67.900
| 67.900
| | (2*13)/5<span style="font-size: 70%; vertical-align: super;">2</span>
| (2*13)/5<span style="font-size: 70%; vertical-align: super;">2</span>
| | | 1 0 -2 0 0 1 &gt;
| {{Monzo|1 0 -2 0 0 1}}
| | tridecimal 1/3-tone
| tridecimal 1/3-tone
|-
|-
| | [[27/26|27/26]]
| [[27/26]]
| | 65.337
| 65.337
| | 3<span style="font-size: 70%; vertical-align: super;">3</span>/(2*13)
| 3<span style="font-size: 70%; vertical-align: super;">3</span>/(2*13)
| | | -1 3 0 0 0 -1 &gt;
| {{Monzo|-1 3 0 0 0 -1}}
| | tridecimal comma
| tridecimal comma
|-
|-
| | [[40/39|40/39]]
| [[40/39]]
| | 43.831
| 43.831
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*5)/(3*13)
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*5)/(3*13)
| | | 3 -1 1 0 0 -1 &gt;
| {{Monzo|3 -1 1 0 0 -1}}
| | tridecimal minor diesis
| tridecimal minor diesis
|-
|-
| | [[65/64|65/64]]
| [[65/64]]
| | 26.841
| 26.841
| | (5*13)/2<span style="font-size: 70%; vertical-align: super;">6</span>
| (5*13)/2<span style="font-size: 70%; vertical-align: super;">6</span>
| | | -6 0 1 0 0 1 &gt;
| {{Monzo|-6 0 1 0 0 1}}
| | wilsorma, 13th-partial chroma
| wilsorma, 13th-partial chroma
|-
|-
| | [[66/65|66/65]]
| [[66/65]]
| | 26.432
| 26.432
| | (2*3*11)/(5*13)
| (2*3*11)/(5*13)
| | | 1 1 -1 0 1 -1 &gt;
| {{Monzo|1 1 -1 0 1 -1}}
| | winmeanma
| winmeanma
|-
|-
| | [[78/77|78/77]]
| [[78/77]]
| | 22.339
| 22.339
| | (2*3*13)/(7*11)
| (2*3*13)/(7*11)
| | | 1 1 0 -1 -1 1 &gt;
| {{Monzo|1 1 0 -1 -1 1}}
| | negustma
| negustma
|-
|-
| | [[91/90|91/90]]
| [[91/90]]
| | 19.130
| 19.130
| | (7*13)/(2*3<span style="font-size: 70%; vertical-align: super;">2</span>*5)
| (7*13)/(2*3<span style="font-size: 70%; vertical-align: super;">2</span>*5)
| | | -1 -2 -1 1 0 1 &gt;
| {{Monzo|-1 -2 -1 1 0 1}}
| | [[The_Biosphere|Biome]] comma, superleap comma
| [[The_Biosphere|Biome]] comma, superleap comma
|-
|-
| | [[105/104|105/104]]
| [[105/104]]
| | 16.567
| 16.567
| | (3*5*7)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*13)
| (3*5*7)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*13)
| | | -3 1 1 1 0 -1 &gt;
| {{Monzo|-3 1 1 1 0 -1}}
| | small tridecimal comma, animist comma
| small tridecimal comma, animist comma
|-
|-
| | [[144/143|144/143]]
| [[144/143]]
| | 12.064
| 12.064
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*3)<span style="font-size: 70%; vertical-align: super;">2</span>/(11*13)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*3)<span style="font-size: 70%; vertical-align: super;">2</span>/(11*13)
| | | 4 2 0 0 -1 -1 &gt;
| {{Monzo|4 2 0 0 -1 -1}}
| | grossma
| grossma
|-
|-
| | [[169/168|169/168]]
| [[169/168]]
| | 10.274
| 10.274
| | 13<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3*7)
| 13<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3*7)
| | | -3 -1 0 -1 0 2 &gt;
| {{Monzo|-3 -1 0 -1 0 2}}
| | buzurgisma, dhanvantarisma
| buzurgisma, dhanvantarisma
|-
|-
| | [[196/195|196/195]]
| [[196/195]]
| | 8.8554
| 8.8554
| | (2*7)<span style="font-size: 70%; vertical-align: super;">2</span>/(3*5*13)
| (2*7)<span style="font-size: 70%; vertical-align: super;">2</span>/(3*5*13)
| | | 2 -1 -1 2 0 -1 &gt;
| {{Monzo|2 -1 -1 2 0 -1}}
| | marveltwin comma
| marveltwin comma
|-
|-
| | [[325/324|325/324]]
| [[325/324]]
| | 5.3351
| 5.3351
| | (5<span style="font-size: 70%; vertical-align: super;">2</span>*13)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">4</span>)
| (5<span style="font-size: 70%; vertical-align: super;">2</span>*13)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">4</span>)
| | | -2 -4 2 0 0 1 &gt;
| {{Monzo|-2 -4 2 0 0 1}}
| |  
|  
|-
|-
| | [[351/350|351/350]]
| [[351/350]]
| | 4.9393
| 4.9393
| | (3/5)<span style="font-size: 70%; vertical-align: super;">2</span>*13/(2*7)
| (3/5)<span style="font-size: 70%; vertical-align: super;">2</span>*13/(2*7)
| | | -1 3 -2 -1 0 1 &gt;
| {{Monzo|-1 3 -2 -1 0 1}}
| | ratwolfsma
| ratwolfsma
|-
|-
| | [[352/351|352/351]]
| [[352/351]]
| | 4.9253
| 4.9253
| | (2<span style="font-size: 70%; vertical-align: super;">5</span>*11)/(3<span style="font-size: 70%; vertical-align: super;">2</span>*13)
| (2<span style="font-size: 70%; vertical-align: super;">5</span>*11)/(3<span style="font-size: 70%; vertical-align: super;">2</span>*13)
| | | 5 -3 0 0 1 -1 &gt;
| {{Monzo|5 -3 0 0 1 -1}}
| | minthma
| minthma
|-
|-
| | [[364/363|364/363]]
| [[364/363]]
| | 4.7627
| 4.7627
| | (2/11)<span style="font-size: 70%; vertical-align: super;">2</span>*7*13/3
| (2/11)<span style="font-size: 70%; vertical-align: super;">2</span>*7*13/3
| | | 2 -1 0 1 -2 1 &gt;
| {{Monzo|2 -1 0 1 -2 1}}
| | gentle comma
| gentle comma
|-
|-
| | [[625/624|625/624]]
| [[625/624]]
| | 2.7722
| 2.7722
| |  
|  
| | | -4 -1 4 0 0 -1 &gt;
| {{Monzo|-4 -1 4 0 0 -1}}
| | tunbarsma
| tunbarsma
|-
|-
| | [[676/675|676/675]]
| [[676/675]]
| | 2.5629
| 2.5629
| |  
|  
| | | 2 -3 -2 0 0 2 &gt;
| {{Monzo|2 -3 -2 0 0 2}}
| | island comma
| island comma
|-
|-
| | [[729/728|729/728]]
| [[729/728]]
| | 2.3764
| 2.3764
| |  
|  
| | | -3 6 0 -1 0 -1 &gt;
| {{Monzo|-3 6 0 -1 0 -1}}
| | squbema
| squbema
|-
|-
| | [[1001/1000|1001/1000]]
| [[1001/1000]]
| | 1.7304
| 1.7304
| |  
|  
| | | -3 0 -3 1 1 1 &gt;
| {{Monzo|-3 0 -3 1 1 1}}
| | sinbadma
| sinbadma
|-
|-
| | [[1716/1715|1716/1715]]
| [[1716/1715]]
| | 1.0092
| 1.0092
| |  
|  
| | | 2 1 -1 -3 1 1 &gt;
| {{Monzo|2 1 -1 -3 1 1}}
| | lummic comma
| lummic comma
|-
|-
| | [[2080/2079|2080/2079]]
| [[2080/2079]]
| | 0.83252
| 0.83252
| |  
|  
| | | 5 -3 1 -1 -1 1 &gt;
| {{Monzo|5 -3 1 -1 -1 1}}
| | ibnsinma
| ibnsinma
|-
|-
| | [[4096/4095|4096/4095]]
| [[4096/4095]]
| | 0.42272
| 0.42272
| |  
|  
| | | 12 -2 -1 -1 0 -1 &gt;
| {{Monzo|12 -2 -1 -1 0 -1}}
| | tridecimal schisma, Sagittal schismina
| tridecimal schisma, Sagittal schismina
|-
|-
| | [[4225/4224|4225/4224]]
| [[4225/4224]]
| | 0.40981
| 0.40981
| |  
|  
| | | -7 -1 2 0 -1 2 &gt;
| {{Monzo|-7 -1 2 0 -1 2}}
| | leprechaun comma
| leprechaun comma
|-
|-
| | [[6656/6655|6656/6655]]
| [[6656/6655]]
| | 0.26012
| 0.26012
| |  
|  
| | | 9 0 -1 0 -3 1 &gt;
| {{Monzo|9 0 -1 0 -3 1}}
| | jacobin comma
| jacobin comma
|-
|-
| | [[10648/10647|10648/10647]]
| [[10648/10647]]
| | 0.16260
| 0.16260
| |  
|  
| | | 3 -2 0 -1 3 -2 &gt;
| {{Monzo|3 -2 0 -1 3 -2}}
| | harmonisma
| harmonisma
|-
|-
| | [[123201/123200|123201/123200]]
| [[123201/123200]]
| | 0.014052
| 0.014052
| |  
|  
| | | -6 6 -2 -1 -1 2 &gt;
| {{Monzo|-6 6 -2 -1 -1 2}}
| | chalmersia
| chalmersia
|-
|-
! colspan="5" | 17-limit (complete)
! colspan="5" | 17-limit (complete)
|-
|-
| | [[17/16|17/16]]
| [[17/16]]
| | 104.955
| 104.955
| | 17/2<span style="font-size: 70%; vertical-align: super;">4</span>
| 17/2<span style="font-size: 70%; vertical-align: super;">4</span>
| | | -4 0 0 0 0 0 1 &gt;
| {{Monzo|-4 0 0 0 0 0 1}}
| | 17th harmonic (octave reduced)
| 17th harmonic (octave reduced)
|-
|-
| | [[18/17|18/17]]
| [[18/17]]
| | 98.955
| 98.955
| | (2*3<span style="font-size: 70%; vertical-align: super;">2</span>)/17
| (2*3<span style="font-size: 70%; vertical-align: super;">2</span>)/17
| | | 1 2 0 0 0 0 -1 &gt;
| {{Monzo|1 2 0 0 0 0 -1}}
| | Arabic lute index finger
| Arabic lute index finger
|-
|-
| | [[34/33|34/33]]
| [[34/33]]
| | 51.682
| 51.682
| | (2*17)/(3*11)
| (2*17)/(3*11)
| | | 1 -1 0 0 -1 0 1 &gt;
| {{Monzo|1 -1 0 0 -1 0 1}}
| |  
|  
|-
|-
| | [[35/34|35/34]]
| [[35/34]]
| | 50.184
| 50.184
| | (5*7)/(2*17)
| (5*7)/(2*17)
| | | -1 0 1 1 0 0 -1 &gt;
| {{Monzo|-1 0 1 1 0 0 -1}}
| | septendecimal 1/4-tone
| septendecimal 1/4-tone
|-
|-
| | [[51/50|51/50]]
| [[51/50]]
| | 34.283
| 34.283
| | (3*17)/(2*5<span style="font-size: 70%; vertical-align: super;">2</span>)
| (3*17)/(2*5<span style="font-size: 70%; vertical-align: super;">2</span>)
| | | -1 1 -2 0 0 0 1 &gt;
| {{Monzo|-1 1 -2 0 0 0 1}}
| | 17th-partial chroma
| 17th-partial chroma
|-
|-
| | [[52/51|52/51]]
| [[52/51]]
| | 33.617
| 33.617
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*13)/(3*17)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*13)/(3*17)
| | | 2 -1 0 0 0 1 -1 &gt;
| {{Monzo|2 -1 0 0 0 1 -1}}
| |  
|  
|-
|-
| | [[85/84|85/84]]
| [[85/84]]
| | 20.488
| 20.488
| | (5*17)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3*7)
| (5*17)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3*7)
| | | -2 -1 1 -1 0 0 1 &gt;
| {{Monzo|-2 -1 1 -1 0 0 1}}
| |  
|  
|-
|-
| | 120/119
| 120/119
| | 14.487
| 14.487
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*3*5)/(7*17)
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*3*5)/(7*17)
| | | 3 1 1 -1 0 0 -1 &gt;
| {{Monzo|3 1 1 -1 0 0 -1}}
| |  
|  
|-
|-
| | 136/135
| 136/135
| | 12.777
| 12.777
| | (2/3)<span style="font-size: 70%; vertical-align: super;">3</span>*17/5
| (2/3)<span style="font-size: 70%; vertical-align: super;">3</span>*17/5
| | | 3 -3 -1 0 0 0 1 &gt;
| {{Monzo|3 -3 -1 0 0 0 1}}
| |  
|  
|-
|-
| | 154/153
| 154/153
| | 11.278
| 11.278
| | (2*7*11)/(3<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| (2*7*11)/(3<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| | | 1 -2 0 1 1 0 -1 &gt;
| {{Monzo|1 -2 0 1 1 0 -1}}
| |  
|  
|-
|-
| | 170/169
| 170/169
| | 10.214
| 10.214
| | (2*5*17)/13<span style="font-size: 70%; vertical-align: super;">2</span>
| (2*5*17)/13<span style="font-size: 70%; vertical-align: super;">2</span>
| | | 1 0 1 0 0 -2 1 &gt;
| {{Monzo|1 0 1 0 0 -2 1}}
| |  
|  
|-
|-
| | 221/220
| 221/220
| | 7.8514
| 7.8514
| | (13*17)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*5*11)
| (13*17)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*5*11)
| | | -2 0 -1 0 -1 1 1 &gt;
| {{Monzo|-2 0 -1 0 -1 1 1}}
| |  
|  
|-
|-
| | 256/255
| 256/255
| | 6.7759
| 6.7759
| | (2<span style="font-size: 70%; vertical-align: super;">8</span>)/(3*5*17)
| (2<span style="font-size: 70%; vertical-align: super;">8</span>)/(3*5*17)
| | | 8 -1 -1 0 0 0 -1 &gt;
| {{Monzo|8 -1 -1 0 0 0 -1}}
| | 255th subharmonic
| 255th subharmonic
|-
|-
| | 273/272
| 273/272
| | 6.3532
| 6.3532
| | (3*7*13)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*17)
| (3*7*13)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*17)
| | | -4 1 0 1 0 1 -1 &gt;
| {{Monzo|-4 1 0 1 0 1 -1}}
| |  
|  
|-
|-
| | 289/288
| 289/288
| | 6.0008
| 6.0008
| | (17/3)<span style="font-size: 70%; vertical-align: super;">2</span>/2<span style="font-size: 70%; vertical-align: super;">5</span>
| (17/3)<span style="font-size: 70%; vertical-align: super;">2</span>/2<span style="font-size: 70%; vertical-align: super;">5</span>
| | | -5 -2 0 0 0 0 2 &gt;
| {{Monzo|-5 -2 0 0 0 0 2}}
| |  
|  
|-
|-
| | 375/374
| 375/374
| | 4.6228
| 4.6228
| | (3*5<span style="font-size: 70%; vertical-align: super;">3</span>)/(2*11*17)
| (3*5<span style="font-size: 70%; vertical-align: super;">3</span>)/(2*11*17)
| | | -1 1 3 0 -1 0 -1 &gt;
| {{Monzo|-1 1 3 0 -1 0 -1}}
| |  
|  
|-
|-
| | 442/441
| 442/441
| | 3.9213
| 3.9213
| | (2*13*17)/(3*7)<span style="font-size: 70%; vertical-align: super;">2</span>
| (2*13*17)/(3*7)<span style="font-size: 70%; vertical-align: super;">2</span>
| | | 1 -2 0 -2 0 1 1 &gt;
| {{Monzo|1 -2 0 -2 0 1 1}}
| |  
|  
|-
|-
| | 561/560
| 561/560
| | 3.0887
| 3.0887
| | (3*11*17)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*5*7)
| (3*11*17)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*5*7)
| | | -4 1 -1 -1 1 0 1 &gt;
| {{Monzo|-4 1 -1 -1 1 0 1}}
| |  
|  
|-
|-
| | 595/594
| 595/594
| | 2.9121
| 2.9121
| | (5*7*17)/(2*3<span style="font-size: 70%; vertical-align: super;">3</span>*11)
| (5*7*17)/(2*3<span style="font-size: 70%; vertical-align: super;">3</span>*11)
| | | -1 -3 1 1 -1 0 1 &gt;
| {{Monzo|-1 -3 1 1 -1 0 1}}
| |  
|  
|-
|-
| | 715/714
| 715/714
| | 2.4230
| 2.4230
| | (5*11*13)/(2*3*7*17)
| (5*11*13)/(2*3*7*17)
| | | -1 -1 1 -1 1 1 -1 &gt;
| {{Monzo|-1 -1 1 -1 1 1 -1}}
| |  
|  
|-
|-
| | 833/832
| 833/832
| | 2.0796
| 2.0796
| | (7<span style="font-size: 70%; vertical-align: super;">2</span>*17)/(2<span style="font-size: 70%; vertical-align: super;">6</span>*13)
| (7<span style="font-size: 70%; vertical-align: super;">2</span>*17)/(2<span style="font-size: 70%; vertical-align: super;">6</span>*13)
| | | -6 0 0 2 0 -1 1 &gt;
| {{Monzo|-6 0 0 2 0 -1 1}}
| |  
|  
|-
|-
| | 936/935
| 936/935
| | 1.8506
| 1.8506
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*13)/(5*11*17)
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*13)/(5*11*17)
| | | 3 2 -1 0 -1 1 -1 &gt;
| {{Monzo|3 2 -1 0 -1 1 -1}}
| |  
|  
|-
|-
| | 1089/1088
| 1089/1088
| | 1.5905
| 1.5905
| | (3<span style="font-size: 70%; vertical-align: super;">2</span>*11<span style="font-size: 70%; vertical-align: super;">2</span>)/(2<span style="font-size: 70%; vertical-align: super;">6</span>*17)
| (3<span style="font-size: 70%; vertical-align: super;">2</span>*11<span style="font-size: 70%; vertical-align: super;">2</span>)/(2<span style="font-size: 70%; vertical-align: super;">6</span>*17)
| | | -6 2 0 0 2 0 -1 &gt;
| {{Monzo|-6 2 0 0 2 0 -1}}
| | twosquare comma
| twosquare comma
|-
|-
| | 1156/1155
| 1156/1155
| | 1.4983
| 1.4983
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*17<span style="font-size: 70%; vertical-align: super;">2</span>)/(3*5*7*11)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*17<span style="font-size: 70%; vertical-align: super;">2</span>)/(3*5*7*11)
| | | 2 -1 -1 -1 -1 0 2 &gt;
| {{Monzo|2 -1 -1 -1 -1 0 2}}
| |  
|  
|-
|-
| | 1225/1224
| 1225/1224
| | 1.4138
| 1.4138
| | (5<span style="font-size: 70%; vertical-align: super;">2</span>*7<span style="font-size: 70%; vertical-align: super;">2</span>)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| (5<span style="font-size: 70%; vertical-align: super;">2</span>*7<span style="font-size: 70%; vertical-align: super;">2</span>)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| | | -3 -2 2 2 0 0 -1 &gt;
| {{Monzo|-3 -2 2 2 0 0 -1}}
| |  
|  
|-
|-
| | 1275/1274
| 1275/1274
| | 1.3584
| 1.3584
| | (3*5<span style="font-size: 70%; vertical-align: super;">2</span>*17)/(2*7<span style="font-size: 70%; vertical-align: super;">2</span>*13)
| (3*5<span style="font-size: 70%; vertical-align: super;">2</span>*17)/(2*7<span style="font-size: 70%; vertical-align: super;">2</span>*13)
| | | -1 1 2 -2 0 -1 1 &gt;
| {{Monzo|-1 1 2 -2 0 -1 1}}
| |  
|  
|-
|-
| | 1701/1700
| 1701/1700
| | 1.0181
| 1.0181
| | (3<span style="font-size: 70%; vertical-align: super;">5</span>*7)/[(2*5)<span style="font-size: 70%; vertical-align: super;">2</span>*17]
| (3<span style="font-size: 70%; vertical-align: super;">5</span>*7)/[(2*5)<span style="font-size: 70%; vertical-align: super;">2</span>*17]
| | | -2 5 -2 1 0 0 -1 &gt;
| {{Monzo|-2 5 -2 1 0 0 -1}}
| |  
|  
|-
|-
| | 2058/2057
| 2058/2057
| | 0.8414
| 0.8414
| | (2*3*7<span style="font-size: 70%; vertical-align: super;">3</span>)/(11<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| (2*3*7<span style="font-size: 70%; vertical-align: super;">3</span>)/(11<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| | | 1 1 0 3 -2 0 -1 &gt;
| {{Monzo|1 1 0 3 -2 0 -1}}
| | xenisma
| xenisma
|-
|-
| | 2431/2430
| 2431/2430
| | 0.7123
| 0.7123
| | (11*13*17)/(2*3<span style="font-size: 70%; vertical-align: super;">5</span>*5)
| (11*13*17)/(2*3<span style="font-size: 70%; vertical-align: super;">5</span>*5)
| | | -1 -5 -1 0 1 1 1 &gt;
| {{Monzo|-1 -5 -1 0 1 1 1}}
| |  
|  
|-
|-
| | 2500/2499
| 2500/2499
| | 0.6926
| 0.6926
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*5<span style="font-size: 70%; vertical-align: super;">4</span>)/(3*7<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*5<span style="font-size: 70%; vertical-align: super;">4</span>)/(3*7<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| | | 2 -1 4 -2 0 0 -1 &gt;
| {{Monzo|2 -1 4 -2 0 0 -1}}
| |  
|  
|-
|-
| | 2601/2600
| 2601/2600
| | 0.6657
| 0.6657
| | (3<span style="font-size: 70%; vertical-align: super;">2</span>*17<span style="font-size: 70%; vertical-align: super;">2</span>)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*5<span style="font-size: 70%; vertical-align: super;">2</span>*13)
| (3<span style="font-size: 70%; vertical-align: super;">2</span>*17<span style="font-size: 70%; vertical-align: super;">2</span>)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*5<span style="font-size: 70%; vertical-align: super;">2</span>*13)
| | | -3 2 -2 0 0 -1 2 &gt;
| {{Monzo|-3 2 -2 0 0 -1 2}}
| |  
|  
|-
|-
| | 4914/4913
| 4914/4913
| | 0.3523
| 0.3523
| | (2*3<span style="font-size: 70%; vertical-align: super;">3</span>*7*13)/(17<span style="font-size: 70%; vertical-align: super;">3</span>)
| (2*3<span style="font-size: 70%; vertical-align: super;">3</span>*7*13)/(17<span style="font-size: 70%; vertical-align: super;">3</span>)
| | | 1 3 0 1 0 1 -3 &gt;
| {{Monzo|1 3 0 1 0 1 -3}}
| |  
|  
|-
|-
| | 5832/5831
| 5832/5831
| | 0.2969
| 0.2969
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">6</span>)/(7<span style="font-size: 70%; vertical-align: super;">3</span>*17)
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">6</span>)/(7<span style="font-size: 70%; vertical-align: super;">3</span>*17)
| | | 3 6 0 -3 0 0 -1 &gt;
| {{Monzo|3 6 0 -3 0 0 -1}}
| |  
|  
|-
|-
| | 12376/12375
| 12376/12375
| | 0.1399
| 0.1399
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*7*13*17)/(3<span style="font-size: 70%; vertical-align: super;">2</span>*5<span style="font-size: 70%; vertical-align: super;">3</span>*11)
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*7*13*17)/(3<span style="font-size: 70%; vertical-align: super;">2</span>*5<span style="font-size: 70%; vertical-align: super;">3</span>*11)
| | | 3 -2 -3 1 -1 1 1 &gt;
| {{Monzo|3 -2 -3 1 -1 1 1}}
| |  
|  
|-
|-
| | 14400/14399
| 14400/14399
| | 0.1202
| 0.1202
| | (2<span style="font-size: 70%; vertical-align: super;">6</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*5<span style="font-size: 70%; vertical-align: super;">2</span>)/(7*11<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| (2<span style="font-size: 70%; vertical-align: super;">6</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*5<span style="font-size: 70%; vertical-align: super;">2</span>)/(7*11<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| | | 6 2 2 -1 -2 0 -1 &gt;
| {{Monzo|6 2 2 -1 -2 0 -1}}
| |  
|  
|-
|-
| | 28561/28560
| 28561/28560
| | 0.0606
| 0.0606
| | (13<span style="font-size: 70%; vertical-align: super;">4</span>)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*3*5*7*17)
| (13<span style="font-size: 70%; vertical-align: super;">4</span>)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*3*5*7*17)
| | | -4 -1 -1 -1 0 4 -1 &gt;
| {{Monzo|-4 -1 -1 -1 0 4 -1}}
| |  
|  
|-
|-
| | 31213/31212
| 31213/31212
| | 0.0555
| 0.0555
| | (7<span style="font-size: 70%; vertical-align: super;">4</span>*13)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">3</span>*17<span style="font-size: 70%; vertical-align: super;">2</span>)
| (7<span style="font-size: 70%; vertical-align: super;">4</span>*13)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">3</span>*17<span style="font-size: 70%; vertical-align: super;">2</span>)
| | | -2 -3 0 4 0 1 -2 &gt;
| {{Monzo|-2 -3 0 4 0 1 -2}}
| |  
|  
|-
|-
| | 37180/37179
| 37180/37179
| | 0.0466
| 0.0466
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*5*11*13<span style="font-size: 70%; vertical-align: super;">2</span>)/(3<span style="font-size: 70%; vertical-align: super;">7</span>*17)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*5*11*13<span style="font-size: 70%; vertical-align: super;">2</span>)/(3<span style="font-size: 70%; vertical-align: super;">7</span>*17)
| | | 2 -7 1 0 1 2 -1 &gt;
| {{Monzo|2 -7 1 0 1 2 -1}}
| |  
|  
|-
|-
| | 194481/194480
| 194481/194480
| | 0.0089
| 0.0089
| | (3<span style="font-size: 70%; vertical-align: super;">4</span>*7<span style="font-size: 70%; vertical-align: super;">4</span>)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*5*11*13*17)
| (3<span style="font-size: 70%; vertical-align: super;">4</span>*7<span style="font-size: 70%; vertical-align: super;">4</span>)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*5*11*13*17)
| | | -4 4 -1 4 -1 -1 -1&gt;
| {{Monzo|-4 4 -1 4 -1 -1 -1}}
| | scintillisma
| scintillisma
|-
|-
| | 336141/336140
| 336141/336140
| | 0.0052
| 0.0052
| | (3<span style="font-size: 70%; vertical-align: super;">2</span>*13<span style="font-size: 70%; vertical-align: super;">3</span>*17)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*5*7<span style="font-size: 70%; vertical-align: super;">5</span>)
| (3<span style="font-size: 70%; vertical-align: super;">2</span>*13<span style="font-size: 70%; vertical-align: super;">3</span>*17)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*5*7<span style="font-size: 70%; vertical-align: super;">5</span>)
| | | -2 2 -1 -5 0 3 1 &gt;
| {{Monzo|-2 2 -1 -5 0 3 1}}
| |  
|  
|-
|-
! colspan="5" | 19-limit (incomplete)
! colspan="5" | 19-limit (incomplete)
|-
|-
| | [[19/18|19/18]]
| [[19/18]]
| | 93.603
| 93.603
| | 19/(2*3<span style="font-size: 70%; vertical-align: super;">2</span>)
| 19/(2*3<span style="font-size: 70%; vertical-align: super;">2</span>)
| | | -1 -2 0 0 0 0 0 1 &gt;
| {{Monzo|-1 -2 0 0 0 0 0 1}}
| | undevicesimal semitone
| undevicesimal semitone
|-
|-
| | [[20/19|20/19]]
| [[20/19]]
| | 88.801
| 88.801
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*5)/19
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*5)/19
| | | 2 0 1 0 0 0 0 -1 &gt;
| {{Monzo|2 0 1 0 0 0 0 -1}}
| | small undevicesimal semitone
| small undevicesimal semitone
|-
|-
| | [[39/38|39/38]]
| [[39/38]]
| | 44.970
| 44.970
| | (3*13)/(2*19)
| (3*13)/(2*19)
| | | -1 1 0 0 0 1 0 -1 &gt;
| {{Monzo|-1 1 0 0 0 1 0 -1}}
| |  
|  
|-
|-
| | [[57/56|57/56]]
| [[57/56]]
| | 30.642
| 30.642
| | (3*19)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*7)
| (3*19)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*7)
| | | -3 1 0 -1 0 0 0 1 &gt;
| {{Monzo|-3 1 0 -1 0 0 0 1}}
| |  
|  
|-
|-
| | [[76/75|76/75]]
| [[76/75]]
| | 22.931
| 22.931
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*19)/(3*5<span style="font-size: 70%; vertical-align: super;">2</span>)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*19)/(3*5<span style="font-size: 70%; vertical-align: super;">2</span>)
| | | 2 -1 -2 0 0 0 0 1 &gt;
| {{Monzo|2 -1 -2 0 0 0 0 1}}
| |  
|  
|-
|-
| | [[77/76|77/76]]
| [[77/76]]
| | 22.631
| 22.631
| | (7*11)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*19)
| (7*11)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*19)
| | | -2 0 0 1 1 0 0 -1 &gt;
| {{Monzo|-2 0 0 1 1 0 0 -1}}
| |  
|  
|-
|-
| | [[96/95|96/95]]
| [[96/95]]
| | 18.128
| 18.128
| | (2<span style="font-size: 70%; vertical-align: super;">5</span>*3)/(5*19)
| (2<span style="font-size: 70%; vertical-align: super;">5</span>*3)/(5*19)
| | | 5 1 -1 0 0 0 0 -1 &gt;
| {{Monzo|5 1 -1 0 0 0 0 -1}}
| |  
|  
|-
|-
| | 133/132
| 133/132
| | 13.066
| 13.066
| | (19*7)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3*11)
| (19*7)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3*11)
| | | -2 -1 0 1 -1 0 0 1 &gt;
| {{Monzo|-2 -1 0 1 -1 0 0 1}}
| |  
|  
|-
|-
| | 153/152
| 153/152
| | 11.352
| 11.352
| | (3<span style="font-size: 70%; vertical-align: super;">2</span>*17)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*19)
| (3<span style="font-size: 70%; vertical-align: super;">2</span>*17)/(2<span style="font-size: 70%; vertical-align: super;">3</span>*19)
| | | -3 2 0 0 0 0 1 -1 &gt;
| {{Monzo|-3 2 0 0 0 0 1 -1}}
| |  
|  
|-
|-
| | 171/170
| 171/170
| | 10.154
| 10.154
| | (3<span style="font-size: 70%; vertical-align: super;">2</span>*19)/(2*5*17)
| (3<span style="font-size: 70%; vertical-align: super;">2</span>*19)/(2*5*17)
| | | -1 2 -1 0 0 0 -1 1 &gt;
| {{Monzo|-1 2 -1 0 0 0 -1 1}}
| |  
|  
|-
|-
| | 190/189
| 190/189
| | 9.1358
| 9.1358
| | (2*5*19)/(3<span style="font-size: 70%; vertical-align: super;">3</span>*7)
| (2*5*19)/(3<span style="font-size: 70%; vertical-align: super;">3</span>*7)
| | | 1 -3 1 -1 0 0 0 1 &gt;
| {{Monzo|1 -3 1 -1 0 0 0 1}}
| |  
|  
|-
|-
| | 209/208
| 209/208
| | 8.3033
| 8.3033
| | (11*19)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*13)
| (11*19)/(2<span style="font-size: 70%; vertical-align: super;">4</span>*13)
| | | -4 0 0 0 1 -1 0 1 &gt;
| {{Monzo|-4 0 0 0 1 -1 0 1}}
| |  
|  
|-
|-
| | 210/209
| 210/209
| | 8.2637
| 8.2637
| | (2*3*5*7)/(11*19)
| (2*3*5*7)/(11*19)
| | | 1 1 1 1 -1 0 0 -1 &gt;
| {{Monzo|1 1 1 1 -1 0 0 -1}}
| |  
|  
|-
|-
| | 286/285
| 286/285
| | 6.0639
| 6.0639
| | (2*11*13)/(3*5*19)
| (2*11*13)/(3*5*19)
| | | 1 -1 -1 0 1 1 0 -1 &gt;
| {{Monzo|1 -1 -1 0 1 1 0 -1}}
| |  
|  
|-
|-
| | 324/323
| 324/323
| | 5.3516
| 5.3516
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">4</span>)/(17*19)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">4</span>)/(17*19)
| | | 2 4 0 0 0 0 -1 -1 &gt;
| {{Monzo|2 4 0 0 0 0 -1 -1}}
| |  
|  
|-
|-
| | 343/342
| 343/342
| | 5.0547
| 5.0547
| | 7<span style="font-size: 70%; vertical-align: super;">4</span>/(2*3<span style="font-size: 70%; vertical-align: super;">3</span>*19)
| 7<span style="font-size: 70%; vertical-align: super;">4</span>/(2*3<span style="font-size: 70%; vertical-align: super;">3</span>*19)
| | | -1 -2 0 3 0 0 0 -1 &gt;
| {{Monzo|-1 -2 0 3 0 0 0 -1}}
| |  
|  
|-
|-
| | 361/360
| 361/360
| | 4.8023
| 4.8023
| | 19<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*5)
| 19<span style="font-size: 70%; vertical-align: super;">2</span>/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>*5)
| | | -3 -2 -1 0 0 0 0 2 &gt;
| {{Monzo|-3 -2 -1 0 0 0 0 2}}
| |  
|  
|-
|-
| | 400/399
| 400/399
| | 4.3335
| 4.3335
| | (2<span style="font-size: 70%; vertical-align: super;">4</span>*5<span style="font-size: 70%; vertical-align: super;">2</span>)/(3*7*19)
| (2<span style="font-size: 70%; vertical-align: super;">4</span>*5<span style="font-size: 70%; vertical-align: super;">2</span>)/(3*7*19)
| | | 4 -1 2 -1 0 0 0 -1 &gt;
| {{Monzo|4 -1 2 -1 0 0 0 -1}}
| |  
|  
|-
|-
| | 456/455
| 456/455
| | 3.8007
| 3.8007
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*3*19)/(5*7*13)
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*3*19)/(5*7*13)
| | | 3 1 -1 -1 0 -1 0 1 &gt;
| {{Monzo|3 1 -1 -1 0 -1 0 1}}
| |  
|  
|-
|-
| | 476/475
| 476/475
| | 3.6409
| 3.6409
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*7*17)/(5<span style="font-size: 70%; vertical-align: super;">2</span>*19)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*7*17)/(5<span style="font-size: 70%; vertical-align: super;">2</span>*19)
| | | 2 0 -2 1 0 0 1 -1 &gt;
| {{Monzo|2 0 -2 1 0 0 1 -1}}
| |  
|  
|-
|-
| | 495/494
| 495/494
| | 3.501
| 3.501
| | (3<span style="font-size: 70%; vertical-align: super;">2</span>*5*11)/(2*13*19)
| (3<span style="font-size: 70%; vertical-align: super;">2</span>*5*11)/(2*13*19)
| | | -1 2 1 0 1 -1 0 -1 &gt;
| {{Monzo|-1 2 1 0 1 -1 0 -1}}
| |  
|  
|-
|-
| | 513/512
| 513/512
| | 3.378
| 3.378
| | (3<span style="font-size: 70%; vertical-align: super;">3</span>*19)/2<span style="font-size: 70%; vertical-align: super;">9</span>
| (3<span style="font-size: 70%; vertical-align: super;">3</span>*19)/2<span style="font-size: 70%; vertical-align: super;">9</span>
| | | -9 3 0 0 0 0 0 1 &gt;
| {{Monzo|-9 3 0 0 0 0 0 1}}
| | 513th harmonic
| 513th harmonic
|-
|-
! colspan="5" | 23-limit (incomplete)
! colspan="5" | 23-limit (incomplete)
|-
|-
| | [[23/22|23/22]]
| [[23/22]]
| | 76.956
| 76.956
| | 23/(2*11)
| 23/(2*11)
| |  
|  
| |  
|  
|-
|-
| | [[24/23|24/23]]
| [[24/23]]
| | 73.681
| 73.681
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*3)/23
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*3)/23
| |  
|  
| |  
|  
|-
|-
| | [[46/45|46/45]]
| [[46/45]]
| | 38.051
| 38.051
| | (2*23)/(3<span style="font-size: 70%; vertical-align: super;">2</span>*5)
| (2*23)/(3<span style="font-size: 70%; vertical-align: super;">2</span>*5)
| |  
|  
| |  
|  
|-
|-
| | [[69/68|69/68]]
| [[69/68]]
| | 25.274
| 25.274
| | (3*23)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| (3*23)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*17)
| |  
|  
| |  
|  
|-
|-
| | [[70/69|70/69]]
| [[70/69]]
| | 24.910
| 24.910
| | (2*5*7)/(3*23)
| (2*5*7)/(3*23)
| |  
|  
| |  
|  
|-
|-
| | [[92/91|92/91]]
| [[92/91]]
| | 18.921
| 18.921
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*23)/(7*13)
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*23)/(7*13)
| |  
|  
| |  
|  
|-
|-
| | 115/114
| 115/114
| | 15.120
| 15.120
| | (5*23)/(2*3*19)
| (5*23)/(2*3*19)
| |  
|  
| |  
|  
|-
|-
| | 161/160
| 161/160
| | 10.7865
| 10.7865
| | (7*23)/(2<span style="font-size: 70%; vertical-align: super;">5</span>*5)
| (7*23)/(2<span style="font-size: 70%; vertical-align: super;">5</span>*5)
| |  
|  
| |  
|  
|-
|-
| | 162/161
| 162/161
| | 10.720
| 10.720
| | (2*3<span style="font-size: 70%; vertical-align: super;">4</span>)/(7*23)
| (2*3<span style="font-size: 70%; vertical-align: super;">4</span>)/(7*23)
| |  
|  
| |  
|  
|-
|-
| | 208/207
| 208/207
| | 8.343
| 8.343
| | (2<span style="font-size: 70%; vertical-align: super;">4</span>*13)/(23*9)
| (2<span style="font-size: 70%; vertical-align: super;">4</span>*13)/(23*9)
| |  
|  
| |  
|  
|-
|-
| | 576/575
| 576/575
| | 3.008
| 3.008
| | (2<span style="font-size: 70%; vertical-align: super;">6</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)/(23*25)
| (2<span style="font-size: 70%; vertical-align: super;">6</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)/(23*25)
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 29-limit (incomplete)
! colspan="5" | 29-limit (incomplete)
|-
|-
| | [[29/28|29/28]]
| [[29/28]]
| | 60.751
| 60.751
| | 29/(2<span style="font-size: 70%; vertical-align: super;">2</span>*7)
| 29/(2<span style="font-size: 70%; vertical-align: super;">2</span>*7)
| |  
|  
| |  
|  
|-
|-
| | [[30/29|30/29]]
| [[30/29]]
| | 58.692
| 58.692
| | (2*3*5)/29
| (2*3*5)/29
| |  
|  
| |  
|  
|-
|-
| | [[58/57|58/57]]
| [[58/57]]
| | 30.109
| 30.109
| | (2*29)/(3*19)
| (2*29)/(3*19)
| |  
|  
| |  
|  
|-
|-
| | [[88/87|88/87]]
| [[88/87]]
| | 19.786
| 19.786
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*11)/(3*29)
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*11)/(3*29)
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 31-limit (incomplete)
! colspan="5" | 31-limit (incomplete)
|-
|-
| | [[31/30|31/30]]
| [[31/30]]
| | 56.767
| 56.767
| | 31/(2*3*5)
| 31/(2*3*5)
| |  
|  
| |  
|  
|-
|-
| | [[32/31|32/31]]
| [[32/31]]
| | 54.964
| 54.964
| | 2<span style="font-size: 70%; vertical-align: super;">5</span>/31
| 2<span style="font-size: 70%; vertical-align: super;">5</span>/31
| |  
|  
| | 31st subharmonic
| 31st subharmonic
|-
|-
| | [[63/62|63/62]]
| [[63/62]]
| | 27.700
| 27.700
| | (3<span style="font-size: 70%; vertical-align: super;">2</span>*7)/(2*31)
| (3<span style="font-size: 70%; vertical-align: super;">2</span>*7)/(2*31)
| |  
|  
| |  
|  
|-
|-
| | [[93/92|93/92]]
| [[93/92]]
| | 18.716
| 18.716
| | (3*31)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*23)
| (3*31)/(2<span style="font-size: 70%; vertical-align: super;">2</span>*23)
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 37-limit (incomplete)
! colspan="5" | 37-limit (incomplete)
|-
|-
| | [[37/36|37/36]]
| [[37/36]]
| | 47.434
| 47.434
| | 37/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)
| 37/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)
| |  
|  
| |  
|  
|-
|-
| | [[38/37|38/37]]
| [[38/37]]
| | 46.169
| 46.169
| | (2*19)/37
| (2*19)/37
| |  
|  
| |  
|  
|-
|-
| | [[75/74|75/74]]
| [[75/74]]
| | 23.238
| 23.238
| | (3*5<span style="font-size: 70%; vertical-align: super;">2</span>)/(2*37)
| (3*5<span style="font-size: 70%; vertical-align: super;">2</span>)/(2*37)
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 41-limit (incomplete)
! colspan="5" | 41-limit (incomplete)
|-
|-
| | [[41/40|41/40]]
| [[41/40]]
| | 42.749
| 42.749
| | 41/(2<span style="font-size: 70%; vertical-align: super;">3</span>*5)
| 41/(2<span style="font-size: 70%; vertical-align: super;">3</span>*5)
| |  
|  
| |  
|  
|-
|-
| | [[42/41|42/41]]
| [[42/41]]
| | 41.719
| 41.719
| | (2*3*7)/41
| (2*3*7)/41
| |  
|  
| |  
|  
|-
|-
| | [[82/81|82/81]]
| [[82/81]]
| | 21.242
| 21.242
| | (2*41)/3<span style="font-size: 70%; vertical-align: super;">4</span>
| (2*41)/3<span style="font-size: 70%; vertical-align: super;">4</span>
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 43-limit (incomplete)
! colspan="5" | 43-limit (incomplete)
|-
|-
| | [[43/42|43/42]]
| [[43/42]]
| | 40.737
| 40.737
| | 43/(2*3*7)
| 43/(2*3*7)
| |  
|  
| |  
|  
|-
|-
| | [[44/43|44/43]]
| [[44/43]]
| | 39.800
| 39.800
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*11)/43
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*11)/43
| |  
|  
| |  
|  
|-
|-
| | [[86/85|86/85]]
| [[86/85]]
| | 20.249
| 20.249
| | (2*43)/(5*17)
| (2*43)/(5*17)
| |  
|  
| |  
|  
|-
|-
| | [[87/86|87/86]]
| [[87/86]]
| | 20.014
| 20.014
| | (3*29)/(2*43)
| (3*29)/(2*43)
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 47-limit (incomplete)
! colspan="5" | 47-limit (incomplete)
|-
|-
| | [[47/46|47/46]]
| [[47/46]]
| | 37.232
| 37.232
| | 47/(2*23)
| 47/(2*23)
| |  
|  
| |  
|  
|-
|-
| | [[48/47|48/47]]
| [[48/47]]
| | 36.448
| 36.448
| | (2<span style="font-size: 70%; vertical-align: super;">4</span>*3)/47
| (2<span style="font-size: 70%; vertical-align: super;">4</span>*3)/47
| |  
|  
| |  
|  
|-
|-
| | [[94/93|94/93]]
| [[94/93]]
| | 18.516
| 18.516
| | (2*47)/(3*31)
| (2*47)/(3*31)
| |  
|  
| |  
|  
|-
|-
| | [[95/94|95/94]]
| [[95/94]]
| | 18.320
| 18.320
| | (5*19)/(2*47)
| (5*19)/(2*47)
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 53-limit (incomplete)
! colspan="5" | 53-limit (incomplete)
|-
|-
| | [[53/52|53/52]]
| [[53/52]]
| | 32.977
| 32.977
| | 53/(2<span style="font-size: 70%; vertical-align: super;">2</span>*13)
| 53/(2<span style="font-size: 70%; vertical-align: super;">2</span>*13)
| |  
|  
| |  
|  
|-
|-
| | [[54/53|54/53]]
| [[54/53]]
| | 32.360
| 32.360
| | (2*3<span style="font-size: 70%; vertical-align: super;">3</span>)/53
| (2*3<span style="font-size: 70%; vertical-align: super;">3</span>)/53
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 59-limit (incomplete)
! colspan="5" | 59-limit (incomplete)
|-
|-
| | [[59/58|59/58]]
| [[59/58]]
| | 29.594
| 29.594
| | 59/(2*29)
| 59/(2*29)
| |  
|  
| |  
|  
|-
|-
| | [[60/59|60/59]]
| [[60/59]]
| | 29.097
| 29.097
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*3*5)/59
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*3*5)/59
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 61-limit (incomplete)
! colspan="5" | 61-limit (incomplete)
|-
|-
| | [[61/60|61/60]]
| [[61/60]]
| | 28.616
| 28.616
| | 61/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3*5)
| 61/(2<span style="font-size: 70%; vertical-align: super;">2</span>*3*5)
| |  
|  
| |  
|  
|-
|-
| | [[62/61|62/61]]
| [[62/61]]
| | 28.151
| 28.151
| | (2*31)/61
| (2*31)/61
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 67-limit (incomplete)
! colspan="5" | 67-limit (incomplete)
|-
|-
| | [[67/66|67/66]]
| [[67/66]]
| | 26.034
| 26.034
| | 67/(2*3*11)
| 67/(2*3*11)
| |  
|  
| |  
|  
|-
|-
| | [[68/67|68/67]]
| [[68/67]]
| | 25.648
| 25.648
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*17)/67
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*17)/67
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 71-limit (incomplete)
! colspan="5" | 71-limit (incomplete)
|-
|-
| | [[71/70|71/70]]
| [[71/70]]
| | 24.557
| 24.557
| | 71/(2*5*7)
| 71/(2*5*7)
| |  
|  
| |  
|  
|-
|-
| | [[72/71|72/71]]
| [[72/71]]
| | 24.213
| 24.213
| | (2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)/71
| (2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)/71
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 73-limit (incomplete)
! colspan="5" | 73-limit (incomplete)
|-
|-
| | [[73/72|73/72]]
| [[73/72]]
| | 23.879
| 23.879
| | 73/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)
| 73/(2<span style="font-size: 70%; vertical-align: super;">3</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)
| |  
|  
| |  
|  
|-
|-
| | [[74/73|74/73]]
| [[74/73]]
| | 23.555
| 23.555
| | (2*37)/73
| (2*37)/73
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 79-limit (incomplete)
! colspan="5" | 79-limit (incomplete)
|-
|-
| | [[79/78|79/78]]
| [[79/78]]
| | 22.054
| 22.054
| | 79/(2*3*13)
| 79/(2*3*13)
| |  
|  
| |  
|  
|-
|-
| | [[80/79|80/79]]
| [[80/79]]
| | 21.777
| 21.777
| | (2<span style="font-size: 70%; vertical-align: super;">4</span>*5)/79
| (2<span style="font-size: 70%; vertical-align: super;">4</span>*5)/79
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 83-limit (incomplete)
! colspan="5" | 83-limit (incomplete)
|-
|-
| | [[83/82|83/82]]
| [[83/82]]
| | 20.985
| 20.985
| | 83/(2*41)
| 83/(2*41)
| |  
|  
| |  
|  
|-
|-
| | [[84/83|84/83]]
| [[84/83]]
| | 20.734
| 20.734
| | (2<span style="font-size: 70%; vertical-align: super;">2</span>*3*7)/83
| (2<span style="font-size: 70%; vertical-align: super;">2</span>*3*7)/83
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 89-limit (incomplete)
! colspan="5" | 89-limit (incomplete)
|-
|-
| | [[89/88|89/88]]
| [[89/88]]
| | 19.562
| 19.562
| | 89/(2<span style="font-size: 70%; vertical-align: super;">3</span>*11)
| 89/(2<span style="font-size: 70%; vertical-align: super;">3</span>*11)
| |  
|  
| |  
|  
|-
|-
| | [[90/89|90/89]]
| [[90/89]]
| | 19.344
| 19.344
| | (2*3<span style="font-size: 70%; vertical-align: super;">2</span>*5)/89
| (2*3<span style="font-size: 70%; vertical-align: super;">2</span>*5)/89
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 97-limit (incomplete)
! colspan="5" | 97-limit (incomplete)
|-
|-
| | [[97/96|97/96]]
| [[97/96]]
| | 17.940
| 17.940
| | 97/(2<span style="font-size: 70%; vertical-align: super;">5</span>*3)
| 97/(2<span style="font-size: 70%; vertical-align: super;">5</span>*3)
| |  
|  
| |  
|  
|-
|-
| | [[98/97|98/97]]
| [[98/97]]
| | 17.756
| 17.756
| | (2*7<span style="font-size: 70%; vertical-align: super;">2</span>)/97
| (2*7<span style="font-size: 70%; vertical-align: super;">2</span>)/97
| |  
|  
| |  
|  
|-
|-
! colspan="5" | 101-limit (incomplete)
! colspan="5" | 101-limit (incomplete)
|-
|-
| | [[101/100|101/100]]
| [[101/100]]
| | 17.226
| 17.226
| | 101/(2<span style="font-size: 70%; vertical-align: super;">2</span>*5<span style="font-size: 70%; vertical-align: super;">2</span>)
| 101/(2<span style="font-size: 70%; vertical-align: super;">2</span>*5<span style="font-size: 70%; vertical-align: super;">2</span>)
| |  
|  
| |  
|  
|-
|-
| | [[102/101|102/101]]
| [[102/101]]
| | 17.057
| 17.057
| | (2*3*17)/101
| (2*3*17)/101
| |  
|  
| |  
|  
|}
|}
[[Category:interval_list]]
[[Category:interval_list]]
[[Category:superparticular]]
[[Category:superparticular]]

Revision as of 14:56, 25 October 2018

This list of superparticular intervals ordered by prime limit. It reaches to the 101-limit and is complete up to the 17-limit.

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 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. 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 commas are superparticular ratios.

The list below is ordered by harmonic limit, or the largest prime involved in the prime factorization. 36/35, for instance, is an interval of the 7-limit, as it factors to (22*32)/(5*7), while 37/36 would belong to the 37-limit.

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. A002071 -- OEIS gives the number of superparticular ratios in each prime limit, A145604 - OEIS shows the increment from limit to limit, and A117581 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).

See also gallery of just intervals. Many of the names below come from here.

Ratio Cents Factorization Monzo Name(s)
2-limit (complete)
2/1 1200.000 2/1 [1⟩ octave, duple; after octave reduction: (perfect) unison, unity, perfect prime, tonic
3-limit (complete)
3/2 701.995 3/2 [-1 1⟩ perfect fifth, 3rd harmonic (octave reduced), diapente
4/3 498.045 22/3 [2 -1⟩ perfect fourth, 3rd subharmonic (octave reduced), diatessaron
9/8 203.910 32/23 [-3 2⟩ (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced)
5-limit (complete)
5/4 386.314 5/22 [-2 0 1⟩ (classic) (5-limit) major third, 5th harmonic (octave reduced)
6/5 315.641 (2*3)/5 [1 1 -1⟩ (classic) (5-limit) minor third
10/9 182.404 (2*5)/32 [1 -2 1⟩ classic (whole) tone, classic major second, minor whole tone
16/15 111.713 24/(3*5) [4 -1 -1⟩ minor diatonic semitone, 15th subharmonic
25/24 70.672 52/(23*3) [-3 -1 2⟩ chroma, (classic) chromatic semitone, Zarlinian semitone
81/80 21.506 (3/2)4/5 [-4 4 -1⟩ syntonic comma, Didymus comma
7-limit (complete)
7/6 266.871 7/(2*3) [-1 -1 0 1⟩ (septimal) subminor third, septimal minor third, augmented second
8/7 231.174 23/7 [3 0 0 -1⟩ (septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic
15/14 119.443 (3*5)/(2*7) [-1 1 1 -1⟩ septimal diatonic semitone
21/20 84.467 (3*7)/(22*5) [-2 1 -1 1⟩ minor semitone, large septimal chromatic semitone
28/27 62.961 (22*7)/33 [2 -3 0 1⟩ septimal chroma, small septimal chromatic semitone, Archytas' 1/3-tone
36/35 48.770 (22*33)/(5*7) [2 2 -1 -1⟩ septimal quarter tone, septimal diesis
49/48 35.697 72/(24*3) [-4 -1 0 2⟩ large septimal diesis, slendro diesis, septimal 1/6-tone
50/49 34.976 2*(5/7)2 [1 0 2 -2⟩ septimal sixth-tone, jubilisma, small septimal diesis, tritonic diesis, Erlich's decatonic comma
64/63 27.264 26/(32*7) [6 -2 0 -1⟩ septimal comma, Archytas' comma
126/125 13.795 (2*32*7)/53 [1 2 -3 1⟩ starling comma, septimal semicomma
225/224 7.7115 (3*5)2/(25*7) [-5 2 2 -1⟩ marvel comma, septimal kleisma
2401/2400 0.72120 74/(25*3*52) [-5 -1 -2 4⟩ breedsma
4375/4374 0.39576 (54*7)/(2*37) [-1 -7 4 1⟩ ragisma
11-limit (complete)
11/10 165.004 11/(2*5) [-1 0 -1 0 1⟩ (large) (undecimal) neutral second, 4/5-tone, Ptolemy's second
12/11 150.637 (22*3)/11 [2 1 0 0 -1⟩ (small) (undecimal) neutral second, 3/4-tone
22/21 80.537 (2*11)/(3*7) [1 -1 0 -1 1⟩ undecimal minor semitone
33/32 53.273 (3*11)/25 [-5 1 0 0 1⟩ undecimal quarter tone, undecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced)
45/44 38.906 (3/2)2*(5/11) [-2 2 1 0 -1⟩ 1/5-tone
55/54 31.767 (5*11)/(2*33) [-1 -3 1 0 1⟩ undecimal diasecundal comma, eleventyfive comma
56/55 31.194 (23*7)/(5*11) [3 0 -1 1 -1⟩ undecimal tritonic comma, konbini comma
99/98 17.576 (3/7)2*(11/2) [-1 2 0 -2 1⟩ small undecimal comma, mothwellsma
100/99 17.399 (2*5/3)2/11) [2 -2 2 0 -1⟩ Ptolemy's comma, ptolemisma
121/120 14.376 112/(23*3*5) [-3 -1 -1 0 2⟩ undecimal seconds comma, biyatisma
176/175 9.8646 (24*11)/(52*7) [4 0 -2 -1 1⟩ valinorsma
243/242 7.1391 35/(2*112) [-1 5 0 0 -2⟩ neutral third comma, rastma
385/384 4.5026 (5*7*11)/(27*3) [-7 -1 1 1 1⟩ keenanisma
441/440 3.9302 (3*7)2/(23*5*11) [-3 2 -1 2 -1⟩ Werckmeister's undecimal septenarian schisma, werckisma
540/539 3.2090 (2/7)2*33*5/11 [2 3 1 -2 -1⟩ Swets' comma, swetisma
3025/3024 0.57240 (5*11)2/(24*32*7) [-4 -3 2 -1 2⟩ Lehmerisma
9801/9800 0.17665 [11/(5*7)]2*34/23 [-3 4 -2 -2 2⟩ Gauss comma, kalisma
13-limit (complete)
13/12 138.573 13/(22*3) [-2 -1 0 0 0 1⟩ tridecimal 2/3-tone
14/13 128.298 (2*7)/13 [1 0 0 1 0 -1⟩ 2/3-tone, trienthird
26/25 67.900 (2*13)/52 [1 0 -2 0 0 1⟩ tridecimal 1/3-tone
27/26 65.337 33/(2*13) [-1 3 0 0 0 -1⟩ tridecimal comma
40/39 43.831 (23*5)/(3*13) [3 -1 1 0 0 -1⟩ tridecimal minor diesis
65/64 26.841 (5*13)/26 [-6 0 1 0 0 1⟩ wilsorma, 13th-partial chroma
66/65 26.432 (2*3*11)/(5*13) [1 1 -1 0 1 -1⟩ winmeanma
78/77 22.339 (2*3*13)/(7*11) [1 1 0 -1 -1 1⟩ negustma
91/90 19.130 (7*13)/(2*32*5) [-1 -2 -1 1 0 1⟩ Biome comma, superleap comma
105/104 16.567 (3*5*7)/(23*13) [-3 1 1 1 0 -1⟩ small tridecimal comma, animist comma
144/143 12.064 (22*3)2/(11*13) [4 2 0 0 -1 -1⟩ grossma
169/168 10.274 132/(23*3*7) [-3 -1 0 -1 0 2⟩ buzurgisma, dhanvantarisma
196/195 8.8554 (2*7)2/(3*5*13) [2 -1 -1 2 0 -1⟩ marveltwin comma
325/324 5.3351 (52*13)/(22*34) [-2 -4 2 0 0 1⟩
351/350 4.9393 (3/5)2*13/(2*7) [-1 3 -2 -1 0 1⟩ ratwolfsma
352/351 4.9253 (25*11)/(32*13) [5 -3 0 0 1 -1⟩ minthma
364/363 4.7627 (2/11)2*7*13/3 [2 -1 0 1 -2 1⟩ gentle comma
625/624 2.7722 [-4 -1 4 0 0 -1⟩ tunbarsma
676/675 2.5629 [2 -3 -2 0 0 2⟩ island comma
729/728 2.3764 [-3 6 0 -1 0 -1⟩ squbema
1001/1000 1.7304 [-3 0 -3 1 1 1⟩ sinbadma
1716/1715 1.0092 [2 1 -1 -3 1 1⟩ lummic comma
2080/2079 0.83252 [5 -3 1 -1 -1 1⟩ ibnsinma
4096/4095 0.42272 [12 -2 -1 -1 0 -1⟩ tridecimal schisma, Sagittal schismina
4225/4224 0.40981 [-7 -1 2 0 -1 2⟩ leprechaun comma
6656/6655 0.26012 [9 0 -1 0 -3 1⟩ jacobin comma
10648/10647 0.16260 [3 -2 0 -1 3 -2⟩ harmonisma
123201/123200 0.014052 [-6 6 -2 -1 -1 2⟩ chalmersia
17-limit (complete)
17/16 104.955 17/24 [-4 0 0 0 0 0 1⟩ 17th harmonic (octave reduced)
18/17 98.955 (2*32)/17 [1 2 0 0 0 0 -1⟩ Arabic lute index finger
34/33 51.682 (2*17)/(3*11) [1 -1 0 0 -1 0 1⟩
35/34 50.184 (5*7)/(2*17) [-1 0 1 1 0 0 -1⟩ septendecimal 1/4-tone
51/50 34.283 (3*17)/(2*52) [-1 1 -2 0 0 0 1⟩ 17th-partial chroma
52/51 33.617 (22*13)/(3*17) [2 -1 0 0 0 1 -1⟩
85/84 20.488 (5*17)/(22*3*7) [-2 -1 1 -1 0 0 1⟩
120/119 14.487 (23*3*5)/(7*17) [3 1 1 -1 0 0 -1⟩
136/135 12.777 (2/3)3*17/5 [3 -3 -1 0 0 0 1⟩
154/153 11.278 (2*7*11)/(32*17) [1 -2 0 1 1 0 -1⟩
170/169 10.214 (2*5*17)/132 [1 0 1 0 0 -2 1⟩
221/220 7.8514 (13*17)/(22*5*11) [-2 0 -1 0 -1 1 1⟩
256/255 6.7759 (28)/(3*5*17) [8 -1 -1 0 0 0 -1⟩ 255th subharmonic
273/272 6.3532 (3*7*13)/(24*17) [-4 1 0 1 0 1 -1⟩
289/288 6.0008 (17/3)2/25 [-5 -2 0 0 0 0 2⟩
375/374 4.6228 (3*53)/(2*11*17) [-1 1 3 0 -1 0 -1⟩
442/441 3.9213 (2*13*17)/(3*7)2 [1 -2 0 -2 0 1 1⟩
561/560 3.0887 (3*11*17)/(24*5*7) [-4 1 -1 -1 1 0 1⟩
595/594 2.9121 (5*7*17)/(2*33*11) [-1 -3 1 1 -1 0 1⟩
715/714 2.4230 (5*11*13)/(2*3*7*17) [-1 -1 1 -1 1 1 -1⟩
833/832 2.0796 (72*17)/(26*13) [-6 0 0 2 0 -1 1⟩
936/935 1.8506 (23*32*13)/(5*11*17) [3 2 -1 0 -1 1 -1⟩
1089/1088 1.5905 (32*112)/(26*17) [-6 2 0 0 2 0 -1⟩ twosquare comma
1156/1155 1.4983 (22*172)/(3*5*7*11) [2 -1 -1 -1 -1 0 2⟩
1225/1224 1.4138 (52*72)/(23*32*17) [-3 -2 2 2 0 0 -1⟩
1275/1274 1.3584 (3*52*17)/(2*72*13) [-1 1 2 -2 0 -1 1⟩
1701/1700 1.0181 (35*7)/[(2*5)2*17] [-2 5 -2 1 0 0 -1⟩
2058/2057 0.8414 (2*3*73)/(112*17) [1 1 0 3 -2 0 -1⟩ xenisma
2431/2430 0.7123 (11*13*17)/(2*35*5) [-1 -5 -1 0 1 1 1⟩
2500/2499 0.6926 (22*54)/(3*72*17) [2 -1 4 -2 0 0 -1⟩
2601/2600 0.6657 (32*172)/(23*52*13) [-3 2 -2 0 0 -1 2⟩
4914/4913 0.3523 (2*33*7*13)/(173) [1 3 0 1 0 1 -3⟩
5832/5831 0.2969 (23*36)/(73*17) [3 6 0 -3 0 0 -1⟩
12376/12375 0.1399 (23*7*13*17)/(32*53*11) [3 -2 -3 1 -1 1 1⟩
14400/14399 0.1202 (26*32*52)/(7*112*17) [6 2 2 -1 -2 0 -1⟩
28561/28560 0.0606 (134)/(24*3*5*7*17) [-4 -1 -1 -1 0 4 -1⟩
31213/31212 0.0555 (74*13)/(22*33*172) [-2 -3 0 4 0 1 -2⟩
37180/37179 0.0466 (22*5*11*132)/(37*17) [2 -7 1 0 1 2 -1⟩
194481/194480 0.0089 (34*74)/(24*5*11*13*17) [-4 4 -1 4 -1 -1 -1⟩ scintillisma
336141/336140 0.0052 (32*133*17)/(22*5*75) [-2 2 -1 -5 0 3 1⟩
19-limit (incomplete)
19/18 93.603 19/(2*32) [-1 -2 0 0 0 0 0 1⟩ undevicesimal semitone
20/19 88.801 (22*5)/19 [2 0 1 0 0 0 0 -1⟩ small undevicesimal semitone
39/38 44.970 (3*13)/(2*19) [-1 1 0 0 0 1 0 -1⟩
57/56 30.642 (3*19)/(23*7) [-3 1 0 -1 0 0 0 1⟩
76/75 22.931 (22*19)/(3*52) [2 -1 -2 0 0 0 0 1⟩
77/76 22.631 (7*11)/(22*19) [-2 0 0 1 1 0 0 -1⟩
96/95 18.128 (25*3)/(5*19) [5 1 -1 0 0 0 0 -1⟩
133/132 13.066 (19*7)/(22*3*11) [-2 -1 0 1 -1 0 0 1⟩
153/152 11.352 (32*17)/(23*19) [-3 2 0 0 0 0 1 -1⟩
171/170 10.154 (32*19)/(2*5*17) [-1 2 -1 0 0 0 -1 1⟩
190/189 9.1358 (2*5*19)/(33*7) [1 -3 1 -1 0 0 0 1⟩
209/208 8.3033 (11*19)/(24*13) [-4 0 0 0 1 -1 0 1⟩
210/209 8.2637 (2*3*5*7)/(11*19) [1 1 1 1 -1 0 0 -1⟩
286/285 6.0639 (2*11*13)/(3*5*19) [1 -1 -1 0 1 1 0 -1⟩
324/323 5.3516 (22*34)/(17*19) [2 4 0 0 0 0 -1 -1⟩
343/342 5.0547 74/(2*33*19) [-1 -2 0 3 0 0 0 -1⟩
361/360 4.8023 192/(23*32*5) [-3 -2 -1 0 0 0 0 2⟩
400/399 4.3335 (24*52)/(3*7*19) [4 -1 2 -1 0 0 0 -1⟩
456/455 3.8007 (23*3*19)/(5*7*13) [3 1 -1 -1 0 -1 0 1⟩
476/475 3.6409 (22*7*17)/(52*19) [2 0 -2 1 0 0 1 -1⟩
495/494 3.501 (32*5*11)/(2*13*19) [-1 2 1 0 1 -1 0 -1⟩
513/512 3.378 (33*19)/29 [-9 3 0 0 0 0 0 1⟩ 513th harmonic
23-limit (incomplete)
23/22 76.956 23/(2*11)
24/23 73.681 (23*3)/23
46/45 38.051 (2*23)/(32*5)
69/68 25.274 (3*23)/(22*17)
70/69 24.910 (2*5*7)/(3*23)
92/91 18.921 (22*23)/(7*13)
115/114 15.120 (5*23)/(2*3*19)
161/160 10.7865 (7*23)/(25*5)
162/161 10.720 (2*34)/(7*23)
208/207 8.343 (24*13)/(23*9)
576/575 3.008 (26*32)/(23*25)
29-limit (incomplete)
29/28 60.751 29/(22*7)
30/29 58.692 (2*3*5)/29
58/57 30.109 (2*29)/(3*19)
88/87 19.786 (23*11)/(3*29)
31-limit (incomplete)
31/30 56.767 31/(2*3*5)
32/31 54.964 25/31 31st subharmonic
63/62 27.700 (32*7)/(2*31)
93/92 18.716 (3*31)/(22*23)
37-limit (incomplete)
37/36 47.434 37/(22*32)
38/37 46.169 (2*19)/37
75/74 23.238 (3*52)/(2*37)
41-limit (incomplete)
41/40 42.749 41/(23*5)
42/41 41.719 (2*3*7)/41
82/81 21.242 (2*41)/34
43-limit (incomplete)
43/42 40.737 43/(2*3*7)
44/43 39.800 (22*11)/43
86/85 20.249 (2*43)/(5*17)
87/86 20.014 (3*29)/(2*43)
47-limit (incomplete)
47/46 37.232 47/(2*23)
48/47 36.448 (24*3)/47
94/93 18.516 (2*47)/(3*31)
95/94 18.320 (5*19)/(2*47)
53-limit (incomplete)
53/52 32.977 53/(22*13)
54/53 32.360 (2*33)/53
59-limit (incomplete)
59/58 29.594 59/(2*29)
60/59 29.097 (22*3*5)/59
61-limit (incomplete)
61/60 28.616 61/(22*3*5)
62/61 28.151 (2*31)/61
67-limit (incomplete)
67/66 26.034 67/(2*3*11)
68/67 25.648 (22*17)/67
71-limit (incomplete)
71/70 24.557 71/(2*5*7)
72/71 24.213 (23*32)/71
73-limit (incomplete)
73/72 23.879 73/(23*32)
74/73 23.555 (2*37)/73
79-limit (incomplete)
79/78 22.054 79/(2*3*13)
80/79 21.777 (24*5)/79
83-limit (incomplete)
83/82 20.985 83/(2*41)
84/83 20.734 (22*3*7)/83
89-limit (incomplete)
89/88 19.562 89/(23*11)
90/89 19.344 (2*32*5)/89
97-limit (incomplete)
97/96 17.940 97/(25*3)
98/97 17.756 (2*72)/97
101-limit (incomplete)
101/100 17.226 101/(22*52)
102/101 17.057 (2*3*17)/101
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