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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 107-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 127-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 1,795: Line 1,795:
| 29/(2<sup>2</sup>×7)
| 29/(2<sup>2</sup>×7)
| 2.7.29 {{monzo| -2 -1 1 }}
| 2.7.29 {{monzo| -2 -1 1 }}
| Vicesimononal thirdtone, vicesimononal subchroma, large vicesimononal 2/7-tone
| Vicesimononal thirdtone, vicesimononal subchroma, large vicesimononal quartertone, large vicesimononal 2/7-tone
|
|
|-
|-
Line 1,802: Line 1,802:
| (2×3×5)/29
| (2×3×5)/29
| 2.3.5.29 {{monzo| 1 1 1 -1 }}
| 2.3.5.29 {{monzo| 1 1 1 -1 }}
| Small vicesimononal 2/7-tone
| Small vicesimononal quartertone, small vicesimononal 2/7-tone
|
|
|-
|-
Line 3,361: Line 3,361:
| 2.13.53 {{monzo| -2 -1 1 }}
| 2.13.53 {{monzo| -2 -1 1 }}
|  
|  
|
|  
|-
|-
| [[54/53]]
| [[54/53]]
Line 3,368: Line 3,368:
| 2.3.53 {{monzo| 1 3 -1 }}
| 2.3.53 {{monzo| 1 3 -1 }}
|  
|  
|
|  
|-
|-
| [[106/105]]
| [[106/105]]
Line 3,375: Line 3,375:
| 2.3.5.7.53 {{monzo| 1 -1 -1 -1 1 }}
| 2.3.5.7.53 {{monzo| 1 -1 -1 -1 1 }}
|  
|  
|
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|-
| [[160/159]]
| 10.854
| (2<sup>5</sup>×5)/(3×53)
| 2.3.5.53 {{monzo| 5 -1 1 -1 }}
|
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|-
| [[265/264]]
| 6.5453
| (5×53)/(2<sup>3</sup>×3×11)
| 2.3.5.11.53 {{monzo| -3 -1 1 -1 1 }}
|
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|-
| [[266/265]]
| 6.5207
| (2×7×19)/(5×53)
| 2.5.7.19.53 {{monzo| 1 -1 1 1 -1 }}
|
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|-
| [[319/318]]
| 5.4356
| (11×29)/(2×3×53)
| 2.3.11.29.53 {{monzo| -1 -1 1 1 -1 }}
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|-
| [[371/370]]
| 4.6727
| (7×53)/(2×5×37)
| 2.5.7.37.53 {{monzo| -1 -1 1 -1 1 }}
|
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|-
| [[372/371]]
| 4.6601
| (2<sup>2</sup>×3×31)/(7×53)
| 2.3.7.31.53 {{monzo| 2 1 -1 1 -1 }}
|
|
|-
| [[424/423]]
| 4.0879
| (2<sup>3</sup>×53)/(3<sup>2</sup>×47)
| 2.3.47.53 {{monzo| 3 -2 -1 1 }}
|
|
|-
| [[425/424]]
| 4.0783
| (5<sup>2</sup>×17)/(2<sup>3</sup>×53)
| 2.5.17.53 {{monzo| -3 2 1 -1 }}
|
|
|-
| [[477/476]]
| 3.6332
| (3<sup>2</sup>×53)/(2<sup>2</sup>×7×17)
| 2.3.7.17.53 {{monzo| -2 2 -1 -1 1 }}
|
|  
|-
|-
| [[2809/2808]]
| [[2809/2808]]
Line 3,381: Line 3,444:
| 53<sup>2</sup>/(2<sup>3</sup>×3<sup>3</sup>×13)
| 53<sup>2</sup>/(2<sup>3</sup>×3<sup>3</sup>×13)
| 2.3.13.53 {{monzo| -3 -3 -1 2 }}
| 2.3.13.53 {{monzo| -3 -3 -1 2 }}
|
|  
| S53
| S53
|-
|-
Line 3,388: Line 3,451:
| (7×11×53)/(2<sup>4</sup>×3×5×17)
| (7×11×53)/(2<sup>4</sup>×3×5×17)
| 2.3.5.7.11.17.53 {{monzo| -4 -1 -1 1 1 -1 1 }}
| 2.3.5.7.11.17.53 {{monzo| -4 -1 -1 1 1 -1 1 }}
|
|  
|
|  
|}
|}


Line 3,406: Line 3,469:
| 2.29.59 {{monzo| -1 -1 1 }}
| 2.29.59 {{monzo| -1 -1 1 }}
|  
|  
|
|  
|-
|-
| [[60/59]]
| [[60/59]]
Line 3,413: Line 3,476:
| 2.3.5.59 {{monzo| 2 1 1 -1 }}
| 2.3.5.59 {{monzo| 2 1 1 -1 }}
|  
|  
|
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|-
| [[118/117]]
| 14.734
| (2×59)/(3<sup>2</sup>×13)
| 2.3.13.59 {{monzo| 1 -2 -1 1 }}
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|-
| [[119/118]]
| 14.610
| (7×17)/(2×59)
| 2.7.17.59 {{monzo| -1 1 1 -1 }}
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|-
| [[177/176]]
| 9.8087
| (3×59)/(2<sup>4</sup>×11)
| 2.3.11.59 {{monzo| -4 1 -1 1 }}
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|-
| [[236/235]]
| 7.3513
| (2<sup>2</sup>×59)/(5×47)
| 2.5.47.59 {{monzo| 2 -1 -1 1 }}
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|-
| [[295/294]]
| 5.8786
| (5×59)/(2×3×7<sup>2</sup>)
| 2.3.5.7.59 {{monzo| -1 -1 1 -2 1 }}
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|-
| [[296/295]]
| 5.8587
| (2<sup>3</sup>×37)/(5×59)
| 2.5.37.59 {{monzo| 3 -1 1 -1 }}
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|-
| [[414/413]]
| 4.1868
| (2×3<sup>2</sup>×23)/(7×59)
| 2.3.7.23.59 {{monzo| 1 2 -1 1 -1 }}
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|-
| [[473/472]]
| 3.6640
| (11×43)/(2<sup>3</sup>×59)
| 2.11.43.59 {{monzo| -3 1 1 -1 }}
|
|  
|-
|-
| [[3481/3480]]
| [[3481/3480]]
Line 3,437: Line 3,556:
| 2.3.5.61 {{monzo| -2 -1 -1 1 }}
| 2.3.5.61 {{monzo| -2 -1 -1 1 }}
|  
|  
|
|  
|-
|-
| [[62/61]]
| [[62/61]]
Line 3,444: Line 3,563:
| 2.31.61 {{monzo| 1 1 -1 }}
| 2.31.61 {{monzo| 1 1 -1 }}
|  
|  
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|  
|-
|-
| [[672/671]]
| [[122/121]]
| 2.5782
| 14.249
| (2<sup>5</sup>×3×7)/(11×61)
| (2×61)/(11<sup>2</sup>)
| 2.3.7.11.61 {{monzo| 5 1 1 -1 -1 }}
| 2.11.61 {{monzo| 1 -2 1 }}
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|-
| [[123/122]]
| 14.133
| (3×41)/(2×61)
| 2.3.41.61 {{monzo| -1 1 1 -1 }}
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|-
| [[183/182]]
| 9.4862
| (3×61)/(2×7×13)
| 2.3.7.13.61 {{monzo| -1 1 -1 -1 1 }}
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|-
| [[184/183]]
| 9.4345
| (2<sup>3</sup>×23)/(3×61)
| 2.3.23.61 {{monzo| 3 -1 1 -1 }}
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|-
| [[244/243]]
| 7.1098
| (2<sup>2</sup>×61)/3<sup>5</sup>
| 2.3.61 {{monzo| 2 -5 1 }}
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|-
| [[245/244]]
| 7.0807
| (5×7<sup>2</sup>)/(2<sup>2</sup>×61)
| 2.5.7.61 {{monzo| -2 1 2 -1 }}
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|-
| [[305/304]]
| 5.6855
| (5×61)/(2<sup>4</sup>×19)
| 2.5.19.61 {{monzo| -4 1 -1 1 }}
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|-
| [[306/305]]
| 5.6669
| (2×3<sup>2</sup>×17)/(5×61)
| 2.3.5.17.61 {{monzo| 1 2 -1 1 -1 }}
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|-
| [[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]]
| [[1404/1403]]
Line 3,482: Line 3,657:
| 2.3.11.67 {{monzo| -1 -1 -1 1 }}
| 2.3.11.67 {{monzo| -1 -1 -1 1 }}
|  
|  
|
|  
|-
|-
| [[68/67]]
| [[68/67]]
Line 3,489: Line 3,664:
| 2.17.67 {{monzo| 2 1 -1 }}
| 2.17.67 {{monzo| 2 1 -1 }}
|  
|  
|
|  
|}
|}


Line 3,506: Line 3,681:
| 2.5.7.71 {{monzo| -1 -1 -1 1 }}
| 2.5.7.71 {{monzo| -1 -1 -1 1 }}
|  
|  
|
|  
|-
|-
| [[72/71]]
| [[72/71]]
Line 3,513: Line 3,688:
| 2.3.71 {{monzo| 3 2 -1 }}
| 2.3.71 {{monzo| 3 2 -1 }}
|  
|  
|
|  
|}
|}


Line 3,530: Line 3,705:
| 2.3.73 {{monzo| -3 -2 1 }}
| 2.3.73 {{monzo| -3 -2 1 }}
|  
|  
|
|  
|-
|-
| [[74/73]]
| [[74/73]]
Line 3,537: Line 3,712:
| 2.37.73 {{monzo| 1 1 -1 }}
| 2.37.73 {{monzo| 1 1 -1 }}
|  
|  
|
|  
|}
|}


Line 3,554: Line 3,729:
| 2.3.13.79 {{monzo| -1 -1 -1 1 }}
| 2.3.13.79 {{monzo| -1 -1 -1 1 }}
|  
|  
|
|  
|-
|-
| [[80/79]]
| [[80/79]]
Line 3,561: Line 3,736:
| 2.5.79 {{monzo| 4 1 -1 }}
| 2.5.79 {{monzo| 4 1 -1 }}
|  
|  
|
|  
|}
|}


Line 3,578: Line 3,753:
| 2.41.83 {{monzo| -1 -1 1 }}
| 2.41.83 {{monzo| -1 -1 1 }}
|  
|  
|
|  
|-
|-
| [[84/83]]
| [[84/83]]
Line 3,585: Line 3,760:
| 2.3.7.83 {{monzo| 2 1 1 -1 }}
| 2.3.7.83 {{monzo| 2 1 1 -1 }}
|  
|  
|
|  
|}
|}


Line 3,609: Line 3,784:
| 2.3.5.89 {{monzo| 1 2 1 -1 }}
| 2.3.5.89 {{monzo| 1 2 1 -1 }}
|  
|  
|
|  
|}
|}


Line 3,626: Line 3,801:
| 2.3.97 {{monzo| -5 -1 1 }}
| 2.3.97 {{monzo| -5 -1 1 }}
|  
|  
|
|  
|-
|-
| [[98/97]]
| [[98/97]]
Line 3,633: Line 3,808:
| 2.7.97 {{monzo| 1 2 -1 }}
| 2.7.97 {{monzo| 1 2 -1 }}
|  
|  
|
|  
|}
|}


Line 3,650: Line 3,825:
| 2.5.101 {{monzo| -2 -2 1 }}
| 2.5.101 {{monzo| -2 -2 1 }}
|  
|  
|
|  
|-
|-
| [[102/101]]
| [[102/101]]
Line 3,681: Line 3,856:
| 2.3.17.103 {{monzo| -1 -1 -1 1 }}
| 2.3.17.103 {{monzo| -1 -1 -1 1 }}
|  
|  
|
|  
|-
|-
| [[104/103]]
| [[104/103]]
Line 3,705: Line 3,880:
| 2.53.107 {{monzo| -1 -1 1 }}
| 2.53.107 {{monzo| -1 -1 1 }}
|  
|  
|
|  
|-
|-
| [[108/107]]
| [[108/107]]
Line 3,719: Line 3,894:
| 2.3.5.7.107 {{monzo| 1 1 3 -1 -1 }}
| 2.3.5.7.107 {{monzo| 1 1 3 -1 -1 }}
| Ancient Chinese tempering comma
| Ancient Chinese tempering comma
|
|
|}
 
=== 109-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"/>
|-
| [[109/108]]
| 15.956
| 109/(2<sup>2</sup>×3<sup>3</sup>)
| 2.3.109 {{monzo| -2 -3 1 }}
|
|
|-
| [[110/109]]
| 15.810
| (2×5×11)/109
| 2.5.11.109 {{monzo| 1 1 1 -1 }}
|
|
|}
 
=== 113-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"/>
|-
| [[113/112]]
| 15.389
| 113/(2<sup>4</sup>×7)
| 2.7.113 {{monzo| -4 -1 1 }}
|
|
|-
| [[114/113]]
| 15.253
| (2×3×19)/113
| 2.3.19.113 {{monzo| 1 1 1 -1 }}
|
|
|}
 
=== 127-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"/>
|-
| [[127/126]]
| 13.686
| 127/(2×3<sup>2</sup>×7)
| 2.3.7.127 {{monzo| -1 -2 -1 1 }}
|
|
|-
| [[128/127]]
| 13.578
| 2<sup>7</sup>/127
| 2.127 {{monzo| 7 -1 }}
|
|  
|}
|}


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