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m move section as this is used in other parts of the page including for deriving the general S-expression equivalence
- monzos. Consolidate prime limit and subgroup columns. Unify the precision
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=== Table of ultraparticulars ===
=== Table of ultraparticulars ===
{| class="wikitable center-all"
{| class="wikitable center-all right-4 left-5"
|-
|-
! S-expression
! S-expression
! Cube relation
! Cube relation
! Comma
! Ratio
!Monzo
! Cents
! Cents
! colspan="2" |Prime limit - Subgroup
! Subgroup
|-
|-
| S2/S3 = ([[4/3]])/([[9/8]])
| S2/S3 = ([[4/3]])/([[9/8]])
| ([[4/1]])/([[3/2]])<sup>3</sup>
| ([[4/1]])/([[3/2]])<sup>3</sup>
| [[32/27]]
| [[32/27]]
|{{Monzo|4 -3}}
| 294.135
| <small>294.135</small>
| 2.3
| colspan="2" |3
|-
|-
| S3/S4 = ([[9/8]])/([[16/15]])
| S3/S4 = ([[9/8]])/([[16/15]])
| ([[5/2]])/([[4/3]])<sup>3</sup>
| ([[5/2]])/([[4/3]])<sup>3</sup>
| [[135/128]]
| [[135/128]]
|{{Monzo|-7 3 1}}
| 92.179
| <small>92.179</small>
| 2.3.5
| colspan="2" |5
|-
|-
| S4/S5 = ([[16/15]])/([[25/24]])
| S4/S5 = ([[16/15]])/([[25/24]])
| ([[2/1]])/([[5/4]])<sup>3</sup>
| ([[2/1]])/([[5/4]])<sup>3</sup>
| [[128/125]]
| [[128/125]]
|{{Monzo|7 0 -3}}
| 41.059
| <small>41.059</small>
| 2.5
|5
|2.5
|-
|-
| S5/S6 = ([[25/24]])/([[36/35]])
| S5/S6 = ([[25/24]])/([[36/35]])
| ([[7/4]])/([[6/5]])<sup>3</sup>
| ([[7/4]])/([[6/5]])<sup>3</sup>
| [[875/864]]
| [[875/864]]
|{{Monzo|-5 -3 3 1}}
| 21.902
| <small>21.902</small>
| 2.3.5.7
| colspan="2" |7
|-
|-
| S6/S7 = ([[36/35]])/([[49/48]])
| S6/S7 = ([[36/35]])/([[49/48]])
| ([[8/5]])/([[7/6]])<sup>3</sup>
| ([[8/5]])/([[7/6]])<sup>3</sup>
| [[1728/1715]]
| [[1728/1715]]
|{{Monzo|6 3 -1 -3}}
| 13.074
| <small>13.074</small>
| 2.3.5.7
| colspan="2" |7
|-
|-
| S7/S8 = ([[49/48]])/([[64/63]])
| S7/S8 = ([[49/48]])/([[64/63]])
| ([[3/2]])/([[8/7]])<sup>3</sup>
| ([[3/2]])/([[8/7]])<sup>3</sup>
| [[1029/1024]]
| [[1029/1024]]
|{{Monzo|-10 1 0 3}}
| 8.433
| 8.433
|7
| 2.3.7
|2.3.7
|-
|-
| S8/S9 = ([[64/63]])/([[81/80]])
| S8/S9 = ([[64/63]])/([[81/80]])
| ([[10/7]])/([[9/8]])<sup>3</sup>
| ([[10/7]])/([[9/8]])<sup>3</sup>
| [[5120/5103]]
| [[5120/5103]]
|{{Monzo|10 -6 1 -1}}
| 5.758
| 5.758
| colspan="2" |7
| 2.3.5.7
|-
|-
| S9/S10 = ([[81/80]])/([[100/99]])
| S9/S10 = ([[81/80]])/([[100/99]])
| ([[11/8]])/([[10/9]])<sup>3</sup>
| ([[11/8]])/([[10/9]])<sup>3</sup>
| [[8019/8000]]
| [[8019/8000]]
|{{Monzo|-6 6 -3 0 1}}
| 4.107
| 4.107
|11
| 2.3.5.11
|2.3.5.11
|-
|-
| S10/S11 = ([[100/99]])/([[121/120]])
| S10/S11 = ([[100/99]])/([[121/120]])
| ([[4/3]])/([[11/10]])<sup>3</sup>
| ([[4/3]])/([[11/10]])<sup>3</sup>
| [[4000/3993]]
| [[4000/3993]]
|{{Monzo|5 -1 3 0 -3}}
| 3.032
| 3.032
|11
| 2.3.5.11
|2.3.5.11
|-
|-
| S11/S12 = ([[121/120]])/([[144/143]])
| S11/S12 = ([[121/120]])/([[144/143]])
| ([[13/10]])/([[12/11]])<sup>3</sup>
| ([[13/10]])/([[12/11]])<sup>3</sup>
| [[17303/17280]]
| [[17303/17280]]
|{{Monzo|-7 -3 -1 0 3 1}}
| 2.303
| 2.303
|11
| 2.3.5.11.13
|2.3.5.11.13
|-
|-
| S12/S13 = ([[144/143]])/([[169/168]])
| S12/S13 = ([[144/143]])/([[169/168]])
| ([[14/11]])/([[13/12]])<sup>3</sup>
| ([[14/11]])/([[13/12]])<sup>3</sup>
| [[24192/24167]]
| [[24192/24167]]
|{{Monzo|7 3 0 1 -1 -3}}
| 1.790
| 1.79
| 2.3.7.11.13
|13
|2.3.7.11.13
|-
|-
| S13/S14 = ([[169/168]])/([[196/195]])
| S13/S14 = ([[169/168]])/([[196/195]])
| ([[5/4]])/([[14/13]])<sup>3</sup>
| ([[5/4]])/([[14/13]])<sup>3</sup>
| [[10985/10976]]
| [[10985/10976]]
|{{Monzo|-5 0 1 -3 0 3}}
| 1.419
| 1.419
|13
| 2.5.7.13
|2.5.7.13
|-
|-
| S14/S15 = ([[196/195]])/([[225/224]])
| S14/S15 = ([[196/195]])/([[225/224]])
| ([[16/13]])/([[15/14]])<sup>3</sup>
| ([[16/13]])/([[15/14]])<sup>3</sup>
| [[43904/43875]]
| [[43904/43875]]
|{{Monzo|7 -3 -3 3 0 -1}}
| 1.144
| 1.144
|13
| 2.3.5.7.13
|2.3.5.7.13
|-
|-
| S15/S16 = ([[225/224]])/([[256/255]])
| S15/S16 = ([[225/224]])/([[256/255]])
| ([[17/14]])/([[16/15]])<sup>3</sup>
| ([[17/14]])/([[16/15]])<sup>3</sup>
| [[57375/57344]]
| [[57375/57344]]
|{{Monzo|-13 3 3 -1 0 0 1}}
| 0.936
| 0.936
|17
| 2.3.5.7.17
|2.3.5.7.17
|-
|-
| S16/S17 = ([[256/255]])/([[289/288]])
| S16/S17 = ([[256/255]])/([[289/288]])
| ([[6/5]])/([[17/16]])<sup>3</sup>
| ([[6/5]])/([[17/16]])<sup>3</sup>
| [[24576/24565]]
| [[24576/24565]]
|{{Monzo|13 1 -1 0 0 0 -3}}
| 0.775
| 0.775
|17
| 2.3.5.17
|2.3.5.17
|-
|-
| S17/S18 = ([[289/288]])/([[324/323]])
| S17/S18 = ([[289/288]])/([[324/323]])
| ([[19/16]])/([[18/17]])<sup>3</sup>
| ([[19/16]])/([[18/17]])<sup>3</sup>
| [[93347/93312]]
| [[93347/93312]]
|{{Monzo|-7 -6 0 0 0 0 3 1}}
| 0.649
| 0.649
|19
| 2.3.17.19
|2.3.17.19
|-
|-
| S18/S19 = ([[324/323]])/([[361/360]])
| S18/S19 = ([[324/323]])/([[361/360]])
| ([[20/17]])/([[19/18]])<sup>3</sup>
| ([[20/17]])/([[19/18]])<sup>3</sup>
| [[116640/116603]]
| [[116640/116603]]
|{{Monzo|5 6 1 0 0 0 -1 -3}}
| 0.549
| 0.549
|19
| 2.3.5.17.19
|2.3.5.17.19
|-
|-
| S19/S20 = ([[361/360]])/([[400/399]])
| S19/S20 = ([[361/360]])/([[400/399]])
| ([[7/6]])/([[20/19]])<sup>3</sup>
| ([[7/6]])/([[20/19]])<sup>3</sup>
| [[48013/48000]]
| [[48013/48000]]
|{{Monzo|-7 -1 -3 1 0 0 0 3}}
| 0.469
| 0.469
|19
| 2.3.5.7.19
|2.3.5.7.19
|-
|-
| S20/S21 = ([[400/399]])/([[441/440]])
| S20/S21 = ([[400/399]])/([[441/440]])
| ([[22/19]])/([[21/20]])<sup>3</sup>
| ([[22/19]])/([[21/20]])<sup>3</sup>
| [[176000/175959]]
| [[176000/175959]]
|{{Monzo|7 -3 3 -3 1 0 0 -1}}
| 0.403
| 0.403
|19
| 2.3.5.7.11.19
|2.3.5.7.11.19
|-
|-
| S21/S22 = ([[441/440]])/([[484/483]])
| S21/S22 = ([[441/440]])/([[484/483]])
| ([[23/20]])/([[22/21]])<sup>3</sup>
| ([[23/20]])/([[22/21]])<sup>3</sup>
| [[213003/212960]]
| [[213003/212960]]
|{{Monzo|-5 3 -1 3 -3 0 0 0 1}}
| 0.350
| 0.35
| 2.3.5.7.11.23
|23
|2.3.5.7.11.23
|-
|-
| S22/S23 = ([[484/483]])/([[529/528]])
| S22/S23 = ([[484/483]])/([[529/528]])
| ([[8/7]])/([[23/22]])<sup>3</sup>
| ([[8/7]])/([[23/22]])<sup>3</sup>
| [[85184/85169]]
| [[85184/85169]]
|{{Monzo|6 0 0 -1 3 0 0 0 -3}}
| 0.305
| 0.305
|23
| 2.7.11.23
|2.7.11.23
|-
|-
| S23/S24 = ([[529/528]])/([[576/575]])
| S23/S24 = ([[529/528]])/([[576/575]])
| ([[25/22]])/([[24/23]])<sup>3</sup>
| ([[25/22]])/([[24/23]])<sup>3</sup>
| [[304175/304128]]
| [[304175/304128]]
|{{Monzo|-10 -3 2 0 -1 0 0 0 3}}
| 0.268
| 0.268
|23
| 2.3.5.11.23
|2.3.5.11.23
|-
|-
| S24/S25 = ([[576/575]])/([[625/624]])
| S24/S25 = ([[576/575]])/([[625/624]])
| ([[26/23]])/([[25/24]])<sup>3</sup>
| ([[26/23]])/([[25/24]])<sup>3</sup>
| [[359424/359375]]
| [[359424/359375]]
|{{Monzo|10 3 -6 0 0 1 0 0 -1}}
| 0.236
| 0.236
|23
| 2.3.5.13.23
|2.3.5.13.23
|-
|-
| S25/S26 = ([[625/624]])/([[676/675]])
| S25/S26 = ([[625/624]])/([[676/675]])
| ([[9/8]])/([[26/25]])<sup>3</sup>
| ([[9/8]])/([[26/25]])<sup>3</sup>
| [[140625/140608]]
| [[140625/140608]]
|{{Monzo|-6 2 6 0 0 -3}}
| 0.209
| 0.209
|13
| 2.3.5.13
|2.3.5.13
|-
|-
| S26/S27 = ([[676/675]])/([[729/728]])
| S26/S27 = ([[676/675]])/([[729/728]])
| ([[28/25]])/([[27/26]])<sup>3</sup>
| ([[28/25]])/([[27/26]])<sup>3</sup>
| [[492128/492075]]
| [[492128/492075]]
|{{Monzo|5 -9 -2 1 0 3}}
| 0.186
| 0.186
|13
| 2.3.5.7.13
|2.3.5.7.13
|-
|-
| S27/S28 = ([[729/728]])/([[784/783]])
| S27/S28 = ([[729/728]])/([[784/783]])
| ([[29/26]])/([[28/27]])<sup>3</sup>
| ([[29/26]])/([[28/27]])<sup>3</sup>
| [[570807/570752]]
| [[570807/570752]]
|
| 0.167
| 0.167
|29
| 29
|
|-
|-
| S28/S29 = ([[784/783]])/([[841/840]])
| S28/S29 = ([[784/783]])/([[841/840]])
| ([[10/9]])/([[29/28]])<sup>3</sup>
| ([[10/9]])/([[29/28]])<sup>3</sup>
| [[219520/219501]]
| [[219520/219501]]
|
| 0.150
| 0.15
| 29
|29
|
|-
|-
| <small>S31/S32 = ([[961/960]])/([[1024/1023]])</small>
| <small>S31/S32 = ([[961/960]])/([[1024/1023]])</small>
| ([[11/10]])/([[32/31]])<sup>3</sup>
| ([[11/10]])/([[32/31]])<sup>3</sup>
| [[327701/327680]]
| [[327701/327680]]
|
| 0.111
| 0.111
|31
| 31
|
|-
|-
| <small>S33/S34 = ([[1089/1088]])/([[1156/1155]])</small>
| <small>S33/S34 = ([[1089/1088]])/([[1156/1155]])</small>
| ([[35/32]])/([[34/33]])<sup>3</sup>
| ([[35/32]])/([[34/33]])<sup>3</sup>
| <small>[[1257795/1257728]]</small>
| <small>[[1257795/1257728]]</small>
|
| 0.092
| 0.092
|17
| 17
|
|-
|-
| <small>S34/S35 = ([[1156/1155]])/([[1225/1224]])</small>
| <small>S34/S35 = ([[1156/1155]])/([[1225/1224]])</small>
| ([[12/11]])/([[35/34]])<sup>3</sup>
| ([[12/11]])/([[35/34]])<sup>3</sup>
| [[471648/471625]]
| [[471648/471625]]
|
| 0.084
| 0.084
|17
| 17
|
|-
|-
| <small>S37/S38 = ([[1369/1368]])/([[1444/1443]])</small>
| <small>S37/S38 = ([[1369/1368]])/([[1444/1443]])</small>
| ([[13/12]])/([[38/37]])<sup>3</sup>
| ([[13/12]])/([[38/37]])<sup>3</sup>
| [[658489/658464]]
| [[658489/658464]]
|
| 0.066
| 0.066
|37
| 37
|
|-
|-
| <small>S40/S41 = ([[1600/1599]])/([[1681/1680]])</small>
| <small>S40/S41 = ([[1600/1599]])/([[1681/1680]])</small>
| ([[14/13]])/([[41/40]])<sup>3</sup>
| ([[14/13]])/([[41/40]])<sup>3</sup>
| [[896000/895973]]
| [[896000/895973]]
|
| 0.052
| 0.052
|41
| 41
|
|-
|-
| <small>S43/S44 = ([[1849/1848]])/([[1936/1935]])</small>
| <small>S43/S44 = ([[1849/1848]])/([[1936/1935]])</small>
| ([[15/14]])/([[44/43]])<sup>3</sup>
| ([[15/14]])/([[44/43]])<sup>3</sup>
| <small>[[1192605/1192576]]</small>
| <small>[[1192605/1192576]]</small>
|
| 0.042
| 0.042
|43
| 43
|
|-
|-
| <small>S46/S47 = ([[2116/2115]])/([[2209/2208]])</small>
| <small>S46/S47 = ([[2116/2115]])/([[2209/2208]])</small>
| ([[16/15]])/([[47/46]])<sup>3</sup>
| ([[16/15]])/([[47/46]])<sup>3</sup>
| <small>[[1557376/1557345]]</small>
| <small>[[1557376/1557345]]</small>
|
| 0.034
| 0.034
|47
| 47
|
|-
|-
| <small>S49/S50 = ([[2401/2400]])/([[2500/2499]])</small>
| <small>S49/S50 = ([[2401/2400]])/([[2500/2499]])</small>
| ([[17/16]])/([[50/49]])<sup>3</sup>
| ([[17/16]])/([[50/49]])<sup>3</sup>
| <small>[[2000033/2000000]]</small>
| <small>[[2000033/2000000]]</small>
|{{Monzo|-7 0 -6 6 0 0 1}}
| 0.029
| 0.029
|17
| 2.5.7.17
|2.5.7.17
|-
|-
| <small>S50/S51 = ([[2500/2499]])/([[2601/2600]])</small>
| <small>S50/S51 = ([[2500/2499]])/([[2601/2600]])</small>
| ([[52/49]])/([[51/50]])<sup>3</sup>
| ([[52/49]])/([[51/50]])<sup>3</sup>
| <small>[[6500000/6499899]]</small>
| <small>[[6500000/6499899]]</small>
|{{Monzo|5 -3 6 -2 0 1 -3}}
| 0.027
| 0.027
|17
| 2.3.5.7.13.17
|2.3.5.7.13.17
|-
|-
| <small>S55/S56 = ([[3025/3024]])/([[3136/3135]])</small>
| <small>S55/S56 = ([[3025/3024]])/([[3136/3135]])</small>
| ([[19/18]])/([[56/55]])<sup>3</sup>
| ([[19/18]])/([[56/55]])<sup>3</sup>
| <small>[[3161125/3161088]]</small>
| <small>[[3161125/3161088]]</small>
|{{Monzo|-10 -2 3 -3 3 0 0 1}}
| 0.020
| 0.02
| 2.3.5.7.11.19
|19
|2.3.5.7.11.19
|-
|-
| <small>S64/S65 = ([[4096/4095]])/([[4225/4224]])</small>
| <small>S64/S65 = ([[4096/4095]])/([[4225/4224]])</small>
| ([[22/21]])/([[65/64]])<sup>3</sup>
| ([[22/21]])/([[65/64]])<sup>3</sup>
| <small>[[5767168/5767125]]</small>
| <small>[[5767168/5767125]]</small>
|{{Monzo|19 -1 -3 -1 1 -3}}
| 0.013
| 0.013
| colspan="2" |13
| 2.3.5.7.11.13
|}
|}


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