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List of superparticular intervals: Difference between revisions

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This is a list of [[superparticular]] [[interval]]s ordered by [[prime limit]]. It reaches to the 101-limit and is complete up to the [[29-limit]].
This is a list of [[superparticular]] [[interval]]s ordered by [[prime limit]]. It reaches to the 107-limit and is complete up to the [[29-limit]].


[[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).
[[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).
Line 596: Line 596:
| (5×17)/(2<sup>2</sup>×3×7)
| (5×17)/(2<sup>2</sup>×3×7)
| {{Monzo| -2 -1 1 -1 0 0 1 }}
| {{Monzo| -2 -1 1 -1 0 0 1 }}
| Septendecimal comma (?)
| Monk comma
|
|
|-
|-
Line 673: Line 673:
| (3×11×17)/(2<sup>4</sup>×5×7)
| (3×11×17)/(2<sup>4</sup>×5×7)
| {{Monzo| -4 1 -1 -1 1 0 1 }}
| {{Monzo| -4 1 -1 -1 1 0 1 }}
| Monardisma, tsaharukma
| Monardisma, tsaharuk comma
|
|
|-
|-
Line 966: Line 966:
| S20
| S20
|-
|-
| 456/455
| [[456/455]]
| 3.8007
| 3.8007
| (2<sup>3</sup>×3×19)/(5×7×13)
| (2<sup>3</sup>×3×19)/(5×7×13)
| {{Monzo| 3 1 -1 -1 0 -1 0 1 }}
| {{Monzo| 3 1 -1 -1 0 -1 0 1 }}
|  
| Abnobisma
|
|
|-
|-
| 476/475
| [[476/475]]
| 3.6409
| 3.6409
| (2<sup>2</sup>×7×17)/(5<sup>2</sup>×19)
| (2<sup>2</sup>×7×17)/(5<sup>2</sup>×19)
| {{Monzo| 2 0 -2 1 0 0 1 -1 }}
| {{Monzo| 2 0 -2 1 0 0 1 -1 }}
|  
| Hedwigma
|
|
|-
|-
| 495/494
| [[495/494]]
| 3.5010
| 3.5010
| (3<sup>2</sup>×5×11)/(2×13×19)
| (3<sup>2</sup>×5×11)/(2×13×19)
| {{Monzo| -1 2 1 0 1 -1 0 -1 }}
| {{Monzo| -1 2 1 0 1 -1 0 -1 }}
|  
| Eulalisma
|
|
|-
|-
Line 2,547: Line 2,547:
| (3<sup>2</sup>×7)/(2×31)
| (3<sup>2</sup>×7)/(2×31)
| 2.3.7.31 {{monzo| -1 2 1 -1 }}
| 2.3.7.31 {{monzo| -1 2 1 -1 }}
|  
| tricesimoprimal 1/8-tone
|
|
|-
|-
Line 2,564: Line 2,564:
|
|
|-
|-
| [[621/620]]
| [[155/154]]
| 2.7901
| 11.205
| (3<sup>3</sup>×23)/(2<sup>2</sup>×5×31)
| (5×31)/(2×7×11)
| 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }}
| 2.5.7.11.31 {{monzo| -1 1 -1 -1 1 }}
| Owowhatsthisma
|
|
|-
| [[156/155]]
| 11.133
| (2<sup>2</sup>×3×13)/(5×31)
| 2.3.5.13.31 {{monzo| 2 1 -1 1 -1 }}
|  
|
|
|-  
|-
| [[900/899]]
| [[187/186]]
| 1.9247
| 9.2828
| (2×3×5)<sup>2</sup>/(29×31)
| (11×17)/(2×3×31)
| 2.3.5.29.31 {{monzo| 2 2 2 -1 -1 }}
| 2.3.11.17.31 {{monzo| -1 -1 1 1 -1 }}
|  
|  
| S30
|
|-
|-
| [[961/960]]
| [[217/216]]
| 1.8024
| 7.9965
| 31<sup>2</sup>/(2<sup>6</sup>×3×5)
| (7×31)/(2×3)<sup>3</sup>
| 2.3.5.31 {{monzo| -6 -1 -1 2 }}
| 2.3.7.31 {{monzo| -3 -3 1 1 }}
|
|
|
| S31
|-
|-  
| [[248/247]]
| [[1024/1023]]
| 6.9949
| 1.9247
| (2<sup>3</sup>×31)/(13×19)
| 2<sup>10</sup>/(3×11×31)
| 2.13.19.31 {{monzo| 3 -1 -1 1 }}
| 2.3.11.31 {{monzo| 10 -1 -1 -1 }}
|  
|  
| S32
|
|-
|-
| [[3969/3968]]
| [[280/279]]
| 0.43624
| 6.1940
| (3<sup>2</sup>×7)<sup>2</sup>/(2<sup>7</sup>×31)
| (2<sup>3</sup>×5×7)/(3<sup>2</sup>×31)
| 2.3.7.31 {{monzo| -7 4 2 -1 }}
| 2.3.5.7.31 {{monzo| 3 -2 1 1 -1 }}
| Yunzee comma
|
| S63
|
|}
|-
 
| [[341/340]]
=== 37-limit (incomplete) ===
| 5.0844
{| class="wikitable center-6" style="width:100%"
| (11×31)/(2<sup>2</sup>×5×17)
! width="10%" | [[Ratio]]
| 2.5.11.17.31 {{monzo| -2 -1 1 -1 1 }}
! width="10%" | [[Cent]]s
|
! width="15%" | Factorization
|
! width="15%" | [[Monzo]]
|-
! width="45%" | Name(s)
| [[342/341]]
| 5.0695
| (2×3<sup>2</sup>×19)/(11×31)
| 2.3.11.19.31 {{monzo| 1 2 -1 1 -1 }}
|
|
|-
| [[435/434]]
| 3.9844
| (3×5×29)/(2×7×31)
| 2.3.5.7.29.31 {{monzo| -1 1 1 -1 1 -1 }}
|
|
|-
| [[465/464]]
| 3.7271
| (3×5×31)/(2<sup>4</sup>×29)
| 2.3.5.29.31 {{monzo| -4 1 1 -1 -1 }}
|
|
|-
| [[496/495]]
| 3.4939
| (2<sup>4</sup>×31)/(3<sup>2</sup>×5×11)
| 2.3.5.11.31 {{monzo| 4 -2 -1 -1 1 }}
|
|
|-
| [[528/527]]
| 3.2820
| (2<sup>4</sup>×3×11)/(17×31)
| 2.3.11.17.31 {{monzo| 4 1 1 -1 -1 }}
|
|
|-
| [[589/588]]
| 2.9418
| (19×31)/(2<sup>2</sup>×3×7<sup>2</sup>)
| 2.3.7.19.31 {{monzo| -2 -1 -2 1 1 }}
|
|
|-
| [[621/620]]
| 2.7901
| (3<sup>3</sup>×23)/(2<sup>2</sup>×5×31)
| 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }}
| Owowhatsthisma
|
|-
| [[900/899]]
| 1.9247
| (2×3×5)<sup>2</sup>/(29×31)
| 2.3.5.29.31 {{monzo| 2 2 2 -1 -1 }}
|
| S30
|-
| [[961/960]]
| 1.8024
| 31<sup>2</sup>/(2<sup>6</sup>×3×5)
| 2.3.5.31 {{monzo| -6 -1 -1 2 }}
|
| S31
|-
| [[1024/1023]]
| 1.9247
| 2<sup>10</sup>/(3×11×31)
| 2.3.11.31 {{monzo| 10 -1 -1 -1 }}
|
| S32
|-
| [[3969/3968]]
| 0.43624
| (3<sup>2</sup>×7)<sup>2</sup>/(2<sup>7</sup>×31)
| 2.3.7.31 {{monzo| -7 4 2 -1 }}
| Yunzee comma
| S63
|}
 
=== 37-limit (incomplete) ===
{| 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"/>
|-
| [[37/36]]
| 47.434
| 37/(2<sup>2</sup>×3<sup>2</sup>)
| 2.3.37 {{monzo| -2 -2 1 }}
| Large tricesimoseptimal 1/4-tone, 37th-partial chroma
|
|-
| [[38/37]]
| 46.169
| (2×19)/37
| 2.19.37 {{monzo| 1 1 -1 }}
| Small tricesimoseptimal 1/4-tone
|
|-
| [[75/74]]
| 23.238
| (3×5<sup>2</sup>)/(2×37)
| 2.3.5.37 {{monzo| -1 1 2 -1 }}
|
|
|-
| [[1296/1295]]
| 1.3363
| (2×3)<sup>4</sup>/(5×7×37)
| 2.3.5.7.37 {{monzo| 4 4 -1 -1 -1 }}
|
| S36
|-
| [[1369/1368]]
| 1.2651
| 37<sup>2</sup>/(2<sup>3</sup>×3<sup>2</sup>×19)
| 2.3.19.37 {{monzo| -3 -2 -1 2 }}
|
| S37
|-
| [[1444/1443]]
| 1.1993
| (2×19)<sup>2</sup>/(3×13×37)
| 2.3.13.19.37 {{monzo| 2 -1 -1 2 -1 }}
|
| S38
|-
| [[5292/5291]]
| 0.32717
| (2<sup>2</sup>×3<sup>3</sup>×7<sup>2</sup>)/(11×13×37)
| 2.3.7.11.13.37 {{monzo| 2 3 2 -1 -1 -1 }}
| Bullionisma
|
|}
 
=== 41-limit (incomplete) ===
{| 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"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[37/36]]
| [[41/40]]
| 47.434
| 42.749
| 37/(2<sup>2</sup>×3<sup>2</sup>)
| 41/(2<sup>3</sup>×5)
| 2.3.37 {{monzo| -2 -2 1 }}
| 2.5.41 {{monzo| -3 -1 1 }}
| Large tricesimoseptimal 1/4-tone, 37th-partial chroma
| Large quadracesimoprimal 1/5-tone
|
|
|-
|-
| [[38/37]]
| [[42/41]]
| 46.169
| 41.719
| (2×19)/37
| (2×3×7)/41
| 2.19.37 {{monzo| 1 1 -1 }}
| 2.3.7.41 {{monzo| 1 1 1 -1 }}
| Small tricesimoseptimal 1/4-tone
| Small quadracesimoprimal 1/5-tone
|
|
|-
|-
| [[75/74]]
| [[82/81]]
| 23.238
| 21.242
| (3×5<sup>2</sup>)/(2×37)
| (2×41)/3<sup>4</sup>
| 2.3.5.37 {{monzo| -1 1 2 -1 }}
| 2.3.41 {{monzo| 1 -4 1 }}
|  
| 41st-partial chroma
|
|
|-
|-
| [[1296/1295]]
| [[1600/1599]]
| 1.3363
| 1.0824
| (2×3)<sup>4</sup>/(5×7×37)
| (2<sup>3</sup>×5)<sup>2</sup>/(3×13×41)
| 2.3.5.7.37 {{monzo| 4 4 -1 -1 -1 }}
| 2.3.5.13.41 {{monzo| 6 -1 2 -1 -1 }}
|
|
| S36
| S40
|-
|-
| [[1369/1368]]
| [[1681/1680]]
| 1.2651
| 1.0302
| 37<sup>2</sup>/(2<sup>3</sup>×3<sup>2</sup>×19)
| 41<sup>2</sup>/(2<sup>4</sup>×3×5×7)
| 2.3.19.37 {{monzo| -3 -2 -1 2 }}
| 2.3.5.7.41 {{monzo| -4 -1 -1 -1 2 }}
|
|
| S37
| S41
|-
|-
| [[1444/1443]]
| [[1682/1681]]
| 1.1993
| 1.0296
| (2×19)<sup>2</sup>/(3×13×37)
| (2×29<sup>2</sup>)/41<sup>2</sup>
| 2.3.13.19.37 {{monzo| 2 -1 -1 2 -1 }}
| 2.29.41 {{monzo| 1 2 -2 }}
|
| Shaftesburisma
| S38
|  
|-
| [[5292/5291]]
| 0.32717
| (2<sup>2</sup>×3<sup>3</sup>×7<sup>2</sup>)/(11×13×37)
| 2.3.7.11.13.37 {{monzo| 2 3 2 -1 -1 -1 }}
| Bullionisma
|  
|}
|}


=== 41-limit (incomplete) ===
=== 43-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 2,668: Line 2,811:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[41/40]]
| [[43/42]]
| 42.749
| 40.737
| 41/(2<sup>3</sup>×5)
| 43/(2×3×7)
| 2.5.41 {{monzo| -3 -1 1 }}
| 2.3.7.43 {{monzo| -1 -1 -1 1 }}
| Large quadracesimoprimal 1/5-tone
| Large quadracesimotertial 1/5-tone
|
|
|-
|-
| [[42/41]]
| [[44/43]]
| 41.719
| 39.800
| (2×3×7)/41
| (2<sup>2</sup>×11)/43
| 2.3.7.41 {{monzo| 1 1 1 -1 }}
| 2.11.43 {{monzo| 2 1 -1 }}
| Small quadracesimoprimal 1/5-tone
| Small quadracesimotertial 1/5-tone
|
|
|-
|-
| [[82/81]]
| [[86/85]]
| 21.242
| 20.249
| (2×41)/3<sup>4</sup>
| (2×43)/(5×17)
| 2.3.41 {{monzo| 1 -4 1 }}
| 2.5.17.43 {{monzo| 1 -1 -1 1 }}
| 41st-partial chroma
|
|
|-
| [[87/86]]
| 20.014
| (3×29)/(2×43)
| 2.3.29.43 {{monzo| -1 1 1 -1 }}
|  
|
|
|-
|-
| [[1600/1599]]
| [[129/128]]
| 1.0824
| 13.473
| (2<sup>3</sup>×5)<sup>2</sup>/(3×13×41)
| (3×43)/2<sup>7
| 2.3.5.13.41 {{monzo| 6 -1 2 -1 -1 }}
| 2.3.43 {{monzo| -7 1 1 }}
| 43rd-partial chroma
|
|
| S40
|-
|-
| [[1681/1680]]
| [[1764/1763]]
| 1.0302
| 0.98170
| 41<sup>2</sup>/(2<sup>4</sup>×3×5×7)
| (2×3×7)<sup>2</sup>/(41×43)
| 2.3.5.7.41 {{monzo| -4 -1 -1 -1 2 }}
| 2.3.7.41.43 {{monzo| 2 2 2 -1 -1 }}
|
|
| S41
| S42
|-
|-
| [[1682/1681]]
| [[1849/1848]]
| 1.0296
| 0.93656
| (2×29<sup>2</sup>)/41<sup>2</sup>
| 43<sup>2</sup>/(2<sup>3</sup>×3×7×11)
| 2.29.41 {{monzo| 1 2 -2 }}
| 2.3.7.11.43 {{monzo| -3 -1 -1 -1 2 }}
| Shaftesburisma
|
|  
| S43
|}
|-
| [[1936/1935]]
| 0.98170
| (2<sup>2</sup>×11)<sup>2</sup>/(3<sup>2</sup>×5×43)
| 2.3.5.11.43 {{monzo| 4 -2 -1 2 -1 }}
|
| S44
|}


=== 43-limit (incomplete) ===
=== 47-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 2,720: Line 2,877:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[43/42]]
| [[47/46]]
| 40.737
| 37.232
| 43/(2×3×7)
| 47/(2×23)
| 2.3.7.43 {{monzo| -1 -1 -1 1 }}
| 2.23.47 {{monzo| -1 -1 1 }}
| Large quadracesimotertial 1/5-tone
|  
|
|
|-
|-
| [[44/43]]
| [[48/47]]
| 39.800
| 36.448
| (2<sup>2</sup>×11)/43
| (2<sup>4</sup>×3)/47
| 2.11.43 {{monzo| 2 1 -1 }}
| 2.3.47 {{monzo| 4 1 -1 }}
| Small quadracesimotertial 1/5-tone
|  
|
|
|-
|-
| [[86/85]]
| [[94/93]]
| 20.249
| 18.516
| (2×43)/(5×17)
| (2×47)/(3×31)
| 2.5.17.43 {{monzo| 1 -1 -1 1 }}
| 2.3.31.47 {{monzo| 1 -1 -1 1 }}
|  
|  
|
|
|-
|-
| [[87/86]]
| [[95/94]]
| 20.014
| 18.320
| (3×29)/(2×43)
| (5×19)/(2×47)
| 2.3.29.43 {{monzo| -1 1 1 -1 }}
| 2.5.19.47 {{monzo| -1 1 1 -1 }}
|  
|  
|
|
|-
|-
| [[129/128]]
| [[2116/2115]]
| 13.473
| 0.81836
| (3×43)/2<sup>7
| (2×23)<sup>2</sup>/(3<sup>2</sup>×5×47)
| 2.3.43 {{monzo| -7 1 1 }}
| 2.3.5.23.47 {{monzo| 2 -2 -1 2 -1 }}
| 43rd-partial chroma
|
|
| S46
|-
|-
| [[1764/1763]]
| [[2209/2208]]
| 0.98170
| 0.78390
| (2×3×7)<sup>2</sup>/(41×43)
| 47<sup>2</sup>/(2<sup>5</sup>×3×5)
| 2.3.7.41.43 {{monzo| 2 2 2 -1 -1 }}
| 2.3.23.47 {{monzo| -5 -1 -1 2 }}
|
|  
| S42
| S47
|-
|-
| [[1849/1848]]
| [[2304/2303]]
| 0.93656
| 0.75157
| 43<sup>2</sup>/(2<sup>3</sup>×3×7×11)
| (2<sup>4</sup>×3)<sup>2</sup>/(7<sup>2</sup>×47)
| 2.3.7.11.43 {{monzo| -3 -1 -1 -1 2 }}
| 2.3.7.47 {{monzo| 8 2 -2 -1 }}
|  
|  
| S43
| S48
|-
| [[1936/1935]]
| 0.98170
| (2<sup>2</sup>×11)<sup>2</sup>/(3<sup>2</sup>×5×43)
| 2.3.5.11.43 {{monzo| 4 -2 -1 2 -1 }}
|
| S44
|}
|}


=== 47-limit (incomplete) ===
=== 53-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 2,786: Line 2,936:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[47/46]]
| [[53/52]]
| 37.232
| 32.977
| 47/(2×23)
| 53/(2<sup>2</sup>×13)
| 2.23.47 {{monzo| -1 -1 1 }}
| 2.13.53 {{monzo| -2 -1 1 }}
|  
|  
|
|
|-
|-
| [[48/47]]
| [[54/53]]
| 36.448
| 32.360
| (2<sup>4</sup>×3)/47
| (2×3<sup>3</sup>)/53
| 2.3.47 {{monzo| 4 1 -1 }}
| 2.3.53 {{monzo| 1 3 -1 }}
|  
|  
|
|
|-
|-
| [[94/93]]
| [[106/105]]
| 18.516
| 16.410
| (2×47)/(3×31)
| (2×53)/(3×5×7)
| 2.3.31.47 {{monzo| 1 -1 -1 1 }}
| 2.3.5.7.53 {{monzo| 1 -1 -1 -1 1 }}
|  
|  
|
|
|-
|-
| [[95/94]]
| [[2809/2808]]
| 18.320
| 0.61643
| (5×19)/(2×47)
| 53<sup>2</sup>/(2<sup>3</sup>×3<sup>3</sup>×13)
| 2.5.19.47 {{monzo| -1 1 1 -1 }}
| 2.3.13.53 {{monzo| -3 -3 -1 2 }}
|
|
|
| S53
|-
|-
| [[2116/2115]]
| [[4081/4080]]
| 0.81836
| 0.42427
| (2×23)<sup>2</sup>/(3<sup>2</sup>×5×47)
| (7×11×53)/(2<sup>4</sup>×3×5×17)
| 2.3.5.23.47 {{monzo| 2 -2 -1 2 -1 }}
| 2.3.5.7.11.17.53 {{monzo| -4 -1 -1 1 1 -1 1 }}
|
|
|
| S46
|}
 
=== 59-limit (incomplete) ===
{| 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"/>
|-
|-
| [[2209/2208]]
| [[59/58]]
| 0.78390
| 29.594
| 47<sup>2</sup>/(2<sup>5</sup>×3×5)
| 59/(2×29)
| 2.3.23.47 {{monzo| -5 -1 -1 2 }}
| 2.29.59 {{monzo| -1 -1 1 }}
|
|
|-
| [[60/59]]
| 29.097
| (2<sup>2</sup>×3×5)/59
| 2.3.5.59 {{monzo| 2 1 1 -1 }}
|  
|  
| S47
|
|-
|-
| [[2304/2303]]
| [[3481/3480]]
| 0.75157
| 0.49741
| (2<sup>4</sup>×3)<sup>2</sup>/(7<sup>2</sup>×47)
| 59<sup>2</sup>/(2<sup>3</sup>×3×5×29)
| 2.3.7.47 {{monzo| 8 2 -2 -1 }}
| 2.3.5.29.59 {{monzo| -3 -1 -1 -1 2}}
|  
|  
| S48
| S59
|}
|}


=== 53-limit (incomplete) ===
=== 61-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 2,845: Line 3,012:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[53/52]]
| [[61/60]]
| 32.977
| 28.616
| 53/(2<sup>2</sup>×13)
| 61/(2<sup>2</sup>×3×5)
| 2.13.53 {{monzo| -2 -1 1 }}
| 2.3.5.61 {{monzo| -2 -1 -1 1 }}
|  
|  
|
|
|-
|-
| [[54/53]]
| [[62/61]]
| 32.360
| 28.151
| (2×3<sup>3</sup>)/53
| (2×31)/61
| 2.3.53 {{monzo| 1 3 -1 }}
| 2.31.61 {{monzo| 1 1 -1 }}
|  
|  
|
|
|-
|-
| [[2809/2808]]
| [[672/671]]
| 0.61643
| 2.5782
| 53<sup>2</sup>/(2<sup>3</sup>×3<sup>3</sup>×13)
| (2<sup>5</sup>×3×7)/(11×61)
| 2.3.13.53 {{monzo| -3 -3 -1 2 }}
| 2.3.7.11.61 {{monzo| 5 1 1 -1 -1 }}
|
|
|
| S53
|-
|-
| [[4081/4080]]
| [[1404/1403]]
| 0.42427
| 1.2335
| (7×11×53)/(2<sup>4</sup>×3×5×17)
| (2<sup>2</sup>×3<sup>3</sup>×13)/(23×61)
| 2.3.5.7.11.17.53 {{monzo| -4 -1 -1 1 1 -1 1 }}
| 2.3.13.23.61 {{monzo| 2 3 1 -1 -1 }}
|
|
|
|
|-
| [[3721/3720]]
| 0.46532
| 61<sup>2</sup>/(2<sup>3</sup>×3×5×31)
| 2.3.5.31.61 {{monzo| -3 -1 -1 -1 2 }}
|  
| S61
|}
|}


=== 59-limit (incomplete) ===
=== 67-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 2,883: Line 3,057:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[59/58]]
| [[67/66]]
| 29.594
| 26.034
| 59/(2×29)
| 67/(2×3×11)
| 2.29.59 {{monzo| -1 -1 1 }}
| 2.3.11.67 {{monzo| -1 -1 -1 1 }}
|  
|  
|
|
|-
|-
| [[60/59]]
| [[68/67]]
| 29.097
| 25.648
| (2<sup>2</sup>×3×5)/59
| (2<sup>2</sup>×17)/67
| 2.3.5.59 {{monzo| 2 1 1 -1 }}
| 2.17.67 {{monzo| 2 1 -1 }}
|  
|  
|
|
|-
| [[3481/3480]]
| 0.49741
| 59<sup>2</sup>/(2<sup>3</sup>×3×5×29)
| 2.3.5.29.59 {{monzo| -3 -1 -1 -1 2}}
|
| S59
|}
|}


=== 61-limit (incomplete) ===
=== 71-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 2,914: Line 3,081:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[61/60]]
| [[71/70]]
| 28.616
| 24.557
| 61/(2<sup>2</sup>×3×5)
| 71/(2×5×7)
| 2.3.5.61 {{monzo| -2 -1 -1 1 }}
| 2.5.7.71 {{monzo| -1 -1 -1 1 }}
|  
|  
|
|
|-
|-
| [[62/61]]
| [[72/71]]
| 28.151
| 24.213
| (2×31)/61
| (2<sup>3</sup>×3<sup>2</sup>)/71
| 2.31.61 {{monzo| 1 1 -1 }}
| 2.3.71 {{monzo| 3 2 -1 }}
|  
|  
|
|
|-
| [[672/671]]
| 2.5782
| (2<sup>5</sup>×3×7)/(11×61)
| 2.3.7.11.61 {{monzo| 5 1 1 -1 -1 }}
|
|
|-
| [[1404/1403]]
| 1.2335
| (2<sup>2</sup>×3<sup>3</sup>×13)/(23×61)
| 2.3.13.23.61 {{monzo| 2 3 1 -1 -1 }}
|
|
|-
| [[3721/3720]]
| 0.46532
| 61<sup>2</sup>/(2<sup>3</sup>×3×5×31)
| 2.3.5.31.61 {{monzo| -3 -1 -1 -1 2 }}
|
| S61
|}
|}


=== 67-limit (incomplete) ===
=== 73-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 2,959: Line 3,105:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[67/66]]
| [[73/72]]
| 26.034
| 23.879
| 67/(2×3×11)
| 73/(2<sup>3</sup>×3<sup>2</sup>)
| 2.3.11.67 {{monzo| -1 -1 -1 1 }}
| 2.3.73 {{monzo| -3 -2 1 }}
|  
|  
|
|
|-
|-
| [[68/67]]
| [[74/73]]
| 25.648
| 23.555
| (2<sup>2</sup>×17)/67
| (2×37)/73
| 2.17.67 {{monzo| 2 1 -1 }}
| 2.37.73 {{monzo| 1 1 -1 }}
|  
|  
|
|
|}
|}


=== 71-limit (incomplete) ===
=== 79-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 2,983: Line 3,129:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[71/70]]
| [[79/78]]
| 24.557
| 22.054
| 71/(2×5×7)
| 79/(2×3×13)
| 2.5.7.71 {{monzo| -1 -1 -1 1 }}
| 2.3.13.79 {{monzo| -1 -1 -1 1 }}
|  
|  
|
|
|-
|-
| [[72/71]]
| [[80/79]]
| 24.213
| 21.777
| (2<sup>3</sup>×3<sup>2</sup>)/71
| (2<sup>4</sup>×5)/79
| 2.3.71 {{monzo| 3 2 -1 }}
| 2.5.79 {{monzo| 4 1 -1 }}
|  
|  
|
|
|}
|}


=== 73-limit (incomplete) ===
=== 83-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 3,007: Line 3,153:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[73/72]]
| [[83/82]]
| 23.879
| 20.985
| 73/(2<sup>3</sup>×3<sup>2</sup>)
| 83/(2×41)
| 2.3.73 {{monzo| -3 -2 1 }}
| 2.41.83 {{monzo| -1 -1 1 }}
|  
|  
|
|
|-
|-
| [[74/73]]
| [[84/83]]
| 23.555
| 20.734
| (2×37)/73
| (2<sup>2</sup>×3×7)/83
| 2.37.73 {{monzo| 1 1 -1 }}
| 2.3.7.83 {{monzo| 2 1 1 -1 }}
|  
|  
|
|
|}
|}


=== 79-limit (incomplete) ===
=== 89-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 3,031: Line 3,177:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[79/78]]
| [[89/88]]
| 22.054
| 19.562
| 79/(2×3×13)
| 89/(2<sup>3</sup>×11)
| 2.3.13.79 {{monzo| -1 -1 -1 1 }}
| 2.11.89 {{monzo| -3 -1 1 }}
| Sky comma
|  
|  
|
|-
|-
| [[80/79]]
| [[90/89]]
| 21.777
| 19.344
| (2<sup>4</sup>×5)/79
| (2×3<sup>2</sup>×5)/89
| 2.5.79 {{monzo| 4 1 -1 }}
| 2.3.5.89 {{monzo| 1 2 1 -1 }}
|  
|  
|
|
|}
|}


=== 83-limit (incomplete) ===
=== 97-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 3,055: Line 3,201:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[83/82]]
| [[97/96]]
| 20.985
| 17.940
| 83/(2×41)
| 97/(2<sup>5</sup>×3)
| 2.41.83 {{monzo| -1 -1 1 }}
| 2.3.97 {{monzo| -5 -1 1 }}
|  
|  
|
|
|-
|-
| [[84/83]]
| [[98/97]]
| 20.734
| 17.756
| (2<sup>2</sup>×3×7)/83
| (2×7<sup>2</sup>)/97
| 2.3.7.83 {{monzo| 2 1 1 -1 }}
| 2.7.97 {{monzo| 1 2 -1 }}
|  
|  
|
|
|}
|}


=== 89-limit (incomplete) ===
=== 101-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 3,079: Line 3,225:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[89/88]]
| [[101/100]]
| 19.562
| 17.226
| 89/(2<sup>3</sup>×11)
| 101/(2×5)<sup>2</sup>
| 2.11.89 {{monzo| -3 -1 1 }}
| 2.5.101 {{monzo| -2 -2 1 }}
| Sky comma
|
|
|-
| [[102/101]]
| 17.057
| (2×3×17)/101
| 2.3.17.101 {{monzo| 1 1 1 -1 }}
|  
|  
|  
|-
|-
| [[90/89]]
| [[7777/7776]]
| 19.344
| 0.223
| (2×3<sup>2</sup>×5)/89
| 7×11×101/(2×3)<sup>5</sup>
| 2.3.5.89 {{monzo| 1 2 1 -1 }}
| 2.3.7.11.101 {{monzo| -5 -5 1 1 1 }}
| Pulsar comma
|  
|  
|
|}
|}


=== 97-limit (incomplete) ===
=== 103-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 3,103: Line 3,256:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[97/96]]
| [[103/102]]
| 17.940
| 16.890
| 97/(2<sup>5</sup>×3)
| 103/(2×3×17)
| 2.3.97 {{monzo| -5 -1 1 }}
| 2.3.17.103 {{monzo| -1 -1 -1 1 }}
|  
|  
|
|
|-
|-
| [[98/97]]
| [[104/103]]
| 17.756
| 16.727
| (2×7<sup>2</sup>)/97
| (2<sup>3</sup>×13)/103
| 2.7.97 {{monzo| 1 2 -1 }}
| 2.13.103 {{monzo| 3 1 -1 }}
|
|  
|  
|
|}
|}


=== 101-limit (incomplete) ===
=== 107-limit (incomplete) ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 3,127: Line 3,280:
! width="5%" | Meta<ref name="ssp"/>
! width="5%" | Meta<ref name="ssp"/>
|-
|-
| [[101/100]]
| [[107/106]]
| 17.226
| 16.256
| 101/(2×5)<sup>2</sup>
| 107/(2×53)
| 2.5.101 {{monzo| -2 -2 1 }}
| 2.53.107 {{monzo| -1 -1 1 }}
|  
|  
|
|
|-
|-
| [[102/101]]
| [[108/107]]
| 17.057
| 16.105
| (2×3×17)/101
| (2<sup>2</sup>×3<sup>3</sup>)/107
| 2.3.17.101 {{monzo| 1 1 1 -1 }}
| 2.3.107 {{monzo| 2 3 -1 }}
|  
|  
|  
|  
|-
|-
| [[7777/7776]]
| [[750/749]]
| 0.223
| 2.3099
| 7×11×101/(2×3)<sup>5</sup>
| (2×3×5<sup>3</sup>)/(7×107)
| 2.3.7.11.101 {{monzo| -5 -5 1 1 1 }}
| 2.3.5.7.107 {{monzo| 1 1 3 -1 -1 }}
| Pulsar comma
| Ancient Chinese tempering comma
|  
|
|}
|}


Retrieved from "https://en.xen.wiki/w/List_of_superparticular_intervals"