Xenharmonic Wiki

List of superparticular intervals: Difference between revisions

← Older editNewer edit →
VisualWikitext
Dummy index (talk | contribs)
autopatrolled, emailconfirmed
561 edits
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
4,516 edits
complete 37-limit comma list
Tags: Mobile edit Mobile web edit
Line 1: Line 1:
This is a list of [[superparticular]] [[interval]]s ordered by [[prime limit]]. It reaches to the 127-limit and is complete up to the [[31-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 [[37-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 3,489: Line 3,489:
|}
|}


=== 37-limit (incomplete) ===
=== 37-limit ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 4,510: Line 4,510:
| (3<sup>3</sup>×11<sup>2</sup>×17×19)/(2<sup>3</sup>×5×23×31×37)
| (3<sup>3</sup>×11<sup>2</sup>×17×19)/(2<sup>3</sup>×5×23×31×37)
| 2.3.5.11.17.19.23.31.37 {{monzo| -3 3 -1 2 1 1 -1 -1 -1 }}
| 2.3.5.11.17.19.23.31.37 {{monzo| -3 3 -1 2 1 1 -1 -1 -1 }}
|
|
|-
| 1341250/1341249
| 0.0012908
| (2×5<sup>4</sup>×29×37)/(3×7×13×17<sup>3</sup>)
| 2.3.5.7.13.17.29.37 {{monzo| 1 -1 4 -1 -1 -3 1 1 }}
|
|
|-
| 1510785/1510784
| 0.0011459
| (3<sup>3</sup>×5×19<sup>2</sup>×31)/(2<sup>7</sup>×11×29×37)
| 2.3.5.11.19.29.31.37 {{monzo| -7 3 1 -1 2 -1 1 -1 }}
|  
|  
|  
|  
|-
|-
| 1763125/1763124
| 1763125/1763124
| 9.8191×10<sup>-4</sup>
| 0.00098191
| (5<sup>4</sup>×7×13×31)/(2<sup>2</sup>×3×11×19<sup>2</sup>×37)
| (5<sup>4</sup>×7×13×31)/(2<sup>2</sup>×3×11×19<sup>2</sup>×37)
| 2.3.5.7.11.13.19.31.37 {{monzo| -2 -1 4 1 -1 1 -2 1 -1 }}
| 2.3.5.7.11.13.19.31.37 {{monzo| -2 -1 4 1 -1 1 -2 1 -1 }}
|  
|  
|  
|  
|-
| 1771561/1771560
| 0.00097724
| 11<sup>6</sup>/(2<sup>3</sup>×3<sup>2</sup>×5×7×19×37)
| 2.3.5.7.11.19.37 {{monzo| -3 -2 -1 -1 6 -1 -1 }}
|
| S1331
|-
| 2085136/2085135
| 0.00083027
| (2×19)<sup>4</sup>/(3×5×13×17<sup>2</sup>×37)
| 2.3.5.13.17.19.37 {{monzo| 4 -1 -1 -1 -2 4 -1 }}
|
| S1444
|-
| 2417876/2417875
| 0.00071601
| (2<sup>2</sup>×17×31<sup>2</sup>×37)/(5<sup>3</sup>×23×29<sup>2</sup>)
| 2.5.17.23.29.31.37 {{monzo| 2 -3 1 -1 -2 2 1 }}
|
|
|-
| 2560845/2560844
| 0.00067604
| (3×5×7×29<sup>3</sup>)/(2<sup>2</sup>×11<sup>3</sup>×13×37)
| 2.3.5.7.11.13.29.37 {{monzo| -2 1 1 1 -3 -1 3 -1 }}
|
|
|-
| 2598400/2598399
| 0.00066627
| (2<sup>9</sup>×5<sup>2</sup>×7×29)/(3<sup>5</sup>×17<sup>2</sup>×37)
| 2.3.5.7.17.29.37 {{monzo| 9 -5 2 1 -2 1 -1 }}
|
|
|-
| 2772225/2772224
| 0.00062449
| (3<sup>2</sup>×5×37)<sup>2</sup>/(2<sup>8</sup>×7<sup>2</sup>×13×17)
| 2.3.5.7.13.17.37 {{monzo| -8 4 2 -2 -1 -1 2 }}
|
| S1665
|-
| 2893401/2893400
| 0.00059834
| (3<sup>5</sup>×7)<sup>2</sup>/(2<sup>3</sup>×5<sup>2</sup>×17×23×37)
| 2.3.5.7.17.23.37 {{monzo| -3 10 -2 2 -1 -1 -1 }}
|
| S1701
|-
| 3930400/3930399
| 0.00044047
| (2<sup>5</sup>×5<sup>2</sup>×17<sup>3</sup>)/(3<sup>2</sup>×11×29×37<sup>2</sup>)
| 2.3.5.11.17.29.37 {{monzo| 5 -2 2 -1 3 -1 -2 }}
|
|
|-
| 4765600/4765599
| 0.00036328
| (2<sup>5</sup>×5<sup>2</sup>×7×23×37)/(3<sup>2</sup>×19×29×31<sup>2</sup>)
| 2.3.5.7.19.23.29.31.37 {{monzo| 5 -2 2 1 -1 1 -1 -2 1 }}
|
|
|-
| 5538975/5538974
| 0.00031255
| (3×(5×13)<sup>2</sup>×19×23)/(2×7×(17×37)<sup>2</sup>)
| 2.3.5.7.13.17.19.23.37 {{monzo| -1 1 2 -1 2 -2 1 1 -2 }}
|
|
|-
| 6615675/6615674
| 0.00026169
| (3<sup>7</sup>×(5×11)<sup>2</sup>)/(2×(13×23)<sup>2</sup>×37)
| 2.3.5.11.13.23.37 {{monzo| -1 7 2 2 -2 -2 -1 }}
|
|
|-
| 6770556/6770555
| 0.00025570
| ((2×3)<sup>2</sup>×13×17×23×37)/(5×(11×19)<sup>2</sup>×31)
| 2.3.5.11.13.17.19.23.31.37 {{monzo| 2 2 -1 -2 1 1 -2 1 -1 1 }}
|
|
|-
| 7105000/7104999
| 0.00024366
| (2<sup>3</sup>×5<sup>4</sup>×7<sup>2</sup>×29)/(3×(11×23)<sup>2</sup>×37)
| 2.3.5.7.11.23.29.37 {{monzo| 3 -1 4 2 -2 -2 1 -1 }}
|
|
|-
| 7475000/7474999
| 0.00023160
| (2<sup>3</sup>×5<sup>5</sup>×13×23)/(7<sup>3</sup>×19×31×37)
| 2.5.7.13.19.23.31.37 {{monzo| 3 5 -3 1 -1 1 -1 -1 }}
|
|
|-
| 7491169/7491168
| 0.00023110
| (7×17×23)<sup>2</sup>/(2<sup>5</sup>×3<sup>2</sup>×19×37<sup>2</sup>)
| 2.3.7.17.19.23.37 {{monzo| -5 -2 2 2 -1 2 -2 }}
|
| S2737
|-
| <font style="font-size:0.88em">13147876/13147875</font>
| 0.00013167
| (2×7<sup>2</sup>×37)<sup>2</sup>/(3<sup>2</sup>×5<sup>3</sup>×13×29×31)
| 2.3.5.7.13.29.31.37 {{monzo| 2 -2 -3 4 -1 -1 -1 2 }}
|
| S3626
|-
| <font style="font-size:0.88em">14080573/14080572</font>
| 0.00012295
| (13<sup>4</sup>×17×29)/((2×3)<sup>2</sup>×11×31<sup>2</sup>×37)
| 2.3.11.13.17.29.31.37 {{monzo| -2 -2 -1 4 1 1 -2 -1 }}
|
|
|-
| <font style="font-size:0.88em">21386001/21386000</font>
| 8.0952×10<sup>-5</sup>
| (3×7<sup>2</sup>×13×19<sup>2</sup>×31)/(2<sup>4</sup>×5<sup>3</sup>×17<sup>2</sup>×37)
| 2.3.5.7.13.17.19.31.37 {{monzo| -4 1 -3 2 1 -2 2 1 -1 }}
|
|
|-
| <font style="font-size:0.88em">27994681/27994680</font>
| 6.1842×10<sup>-5</sup>
| (11×13×37)<sup>2</sup>/((2×3)<sup>3</sup>×5×(7×23)<sup>2</sup>)
| 2.3.5.7.11.13.23.37 {{monzo| -3 -3 -1 -2 2 2 -2 2 }}
|
| S5291
|-
|-
| <font style="font-size:0.88em">50481025/50481024</font>
| <font style="font-size:0.88em">50481025/50481024</font>
Line 4,526: Line 4,673:
|  
|  
| S7105
| S7105
|-
| <font style="font-size:0.88em">71843751/71843750</font>
| 2.4097×10<sup>-5</sup>
| (3<sup>2</sup>×7<sup>3</sup>×17×37<sup>2</sup>)/(2×5<sup>6</sup>×11<sup>2</sup>×19)
| 2.3.5.7.11.17.19.37 {{monzo| -1 2 -6 3 -2 1 -1 2 }}
|
|
|-
| <font style="font-size:0.79em">308915776/308915775</font>
| 5.6042×10<sup>-6</sup>
| (2×13)<sup>6</sup>/(3<sup>4</sup>×5<sup>2</sup>×7×19×31×37)
| 2.3.5.7.13.19.31.37 {{monzo| 6 -4 -2 -1 6 -1 -1 -1 }}
|
| S17576
|-
| <font style="font-size:0.72em">3463200000/3463199999</font>
| 4.9989×10<sup>-7</sup>
| (2<sup>8</sup>×3<sup>2</sup>×5<sup>5</sup>×13×37)/(7<sup>5</sup>×17<sup>2</sup>×23×31)
| 2.3.5.7.13.17.23.31.37 {{monzo| 8 2 5 -5 1 -2 -1 -1 1 }}
|
|
|}
|}


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