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In 61-limit [[Just intonation|Just Intonation]], all ratios in the system will contain no primes higher than 61.
{{Prime limit navigation|61}}
The '''61-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 61. It is the 18th [[prime limit]] and is a superset of the [[59-limit]] and a subset of the [[67-limit]].  


== Intervals ==
These things are contained by the 61-limit, but not the 59-limit:
{| class="wikitable"
* The [[61-odd-limit|61-]], [[63-odd-limit|63-]], and [[65-odd-limit]];
|+
* Mode 31, 32 and 33 of the harmonic or subharmonic series.  
!Limit
!Ratio
!FJS name
!Cents Value
|-
-0
-1/1
-P1
-  0.000000000
|-
-59
-14337/14336
--cA7<sup>59</sup><sub>7</sub>
-  0.120757092
|-
-37
-9477/9472
-P1<sup>13</sup><sub>37</sub>
-  0.913628207
|-
-53
-1537/1536
-P1<sup>29,53</sup>
-  1.126738764
|-
-47
-1458/1457
-P1<sub>31,47</sub>
-  1.187810715
|-
-43
-1377/1376
-P1<sup>17</sup><sub>43</sub>
-  1.257707319
|-
-19
-1216/1215
-d2<sup>19</sup><sub>5</sub>
-  1.424297941
|-
-31
-1024/1023
-m2<sub>11,31</sub>
-  1.691484306
|-
-61
-262144/261873
-d2<sub>53,61</sub>
-  1.790645987
|-
-59
-768/767
-P1<sub>13,59</sub>
-  2.255679862
|-
-13
-729/728
--cA7<sub>7,13</sub>
-  2.376436954
|-
-59
-649/648
-P1<sup>11,59</sup>
-  2.669598137
|-
-61
-14848/14823
-d2<sup>29</sup><sub>61</sub>
-  2.917384751
|-
-47
-4096/4089
-P1<sub>29,47</sub>
-  2.961182968
|-
-19
-513/512
-P1<sup>19</sup>
-  3.378018728
|-
-47
-262683/262144
--cA7<sup>23,47</sup>
-  3.555973609
|-
-41
-3328/3321
-d2<sup>13</sup><sub>41</sub>
-  3.645252766
|-
-53
-424/423
-P1<sup>53</sup><sub>47</sub>
-  4.087921732
|-
-59
-7552/7533
-m2<sup>59</sup><sub>31</sub>
-  4.361082443
|-
-37
-4107/4096
--cM7<sup>37,37</sup>
-  4.643078375
|-
-41
-369/368
-P1<sup>41</sup><sub>23</sub>
-  4.698060004
|-
-43
-731/729
-d2<sup>17,43</sup>
-  4.743109951
|-
-13
-352/351
-P1<sup>11</sup><sub>13</sub>
-  4.925277999
|-
-53
-1701/1696
-P1<sup>7</sup><sub>53</sub>
-  5.096365320
|-
-19
-324/323
--cA7<sub>17,19</sub>
-  5.351577829
|-
-31
-22599/22528
--cA7<sup>31</sup><sub>11</sub>
-  5.447635292
|-
-7
-5120/5103
-P1<sup>5</sup><sub>7</sub>
-  5.757802203
|-
-37
-297/296
-P1<sup>11</sup><sub>37</sub>
-  5.838906206
|-
-61
-2097152/2090043
-d2<sub>47,61</sub>
-  5.878567719
|-
-59
-65536/65313
-d2<sub>41,59</sub>
-  5.900932628
|-
-17
-289/288
-d2<sup>17,17</sup>
-  6.000817270
|-
-41
-288/287
-P1<sub>7,41</sub>
-  6.021689720
|-
-29
-16443/16384
-P1<sup>7,29</sup>
-  6.223104084
|-
-47
-3008/2997
-m2<sup>47</sup><sub>37</sub>
-  6.342579797
|-
-61
-31347/31232
-P1<sup>43</sup><sub>61</sub>
-  6.362905759
|-
-29
-1863/1856
--cA7<sup>23</sup><sub>29</sub>
-  6.517156577
|-
-31
-32768/32643
-m2<sub>13,31</sub>
-  6.616762305
|-
-19
-263169/262144
-P1<sup>19,19</sup>
-  6.756037457
|-
-17
-256/255
-P1<sub>5,17</sub>
-  6.775875769
|-
-61
-244/243
-P1<sup>61</sup>
-  7.109800749
|-
-11
-243/242
-A1<sub>11,11</sub>
-  7.139119597
|-
-37
-1152/1147
-m2<sub>31,37</sub>
-  7.530390512
|-
-53
-32913/32768
--cA7<sup>23,53</sup>
-  7.643895340
|-
-43
-10496/10449
-P1<sup>41</sup><sub>43</sub>
-  7.769695572
|-
-31
-217/216
-P1<sup>7,31</sup>
-  7.996476337
|-
-43
-216/215
-P1<sub>5,43</sub>
-  8.033583089
|-
-59
-16461/16384
--cA7<sup>31,59</sup>
-  8.117233429
|-
-23
-208/207
-d2<sup>13</sup><sub>23</sub>
-  8.343312770
|-
-61
-8235/8192
--cA7<sup>5,61</sup>
-  9.063521536
|-
-47
-189/188
-P1<sup>7</sup><sub>47</sub>
-  9.184287052
|-
-37
-262144/260739
-m2<sub>29,37</sub>
-  9.303762765
|-
-61
-184/183
-P1<sup>23</sup><sub>61</sub>
-  9.434541328
|-
-11
-896/891
-m2<sup>7</sup><sub>11</sub>
-  9.687960643
|-
-59
-177/176
-A1<sup>59</sup><sub>11</sub>
-  9.808717735
|-
-43
-172/171
-P1<sup>43</sup><sub>19</sub>
-  10.094687779
|-
-31
-837/832
--cA7<sup>31</sup><sub>13</sub>
-  10.372913291
|-
-53
-27136/26973
-m2<sup>53</sup><sub>37</sub>
-  10.430501529
|-
-37
-32967/32768
--cM7<sup>11,37</sup>
-  10.481984581
|-
-61
-1159/1152
-P1<sup>19,61</sup>
-  10.487819477
|-
-59
-4096/4071
-d2<sub>23,59</sub>
-  10.598992632
|-
-23
-162/161
-P1<sub>7,23</sub>
-  10.719749724
|-
-53
-160/159
-P1<sup>5</sup><sub>53</sub>
-  10.854167524
|-
-47
-1739/1728
-P1<sup>37,47</sup>
-  10.985658172
|-
-41
-29889/29696
--cA7<sup>41</sup><sub>29</sub>
-  11.215216581
|-
-19
-153/152
-P1<sup>17</sup><sub>19</sub>
-  11.352395099
|-
-47
-16497/16384
-P1<sup>13,47</sup>
-  11.899286379
|-
-29
-145/144
-P1<sup>5,29</sup>
-  11.980906287
|-
-13
-144/143
-A1<sub>11,13</sub>
-  12.064397597
|-
-37
-999/992
-P1<sup>37</sup><sub>31</sub>
-  12.173468887
|-
-41
-21141/20992
-P1<sup>29</sup><sub>41</sub>
-  12.244793804
|-
-53
-38912/38637
-d2<sup>19</sup><sub>53</sub>
-  12.278465464
|-
-53
-528039/524288
--cA7<sup>41,53</sup>
-  12.341955344
|-
-59
-3481/3456
-A1<sup>59,59</sup>
-  12.478315872
|-
-17
-136/135
-d2<sup>17</sup><sub>5</sub>
-  12.776693039
|-
-23
-5632/5589
-d2<sup>11</sup><sub>23</sub>
-  13.268590769
|-
-43
-129/128
-P1<sup>43</sup>
-  13.472706508
|-
-43
-8192/8127
-P1<sub>7,43</sub>
-  13.791385292
|-
-37
-2368/2349
-P1<sup>37</sup><sub>29</sub>
-  13.946841140
|-
-41
-124/123
-P1<sup>31</sup><sub>41</sub>
-  14.018166057
|-
-61
-123/122
-P1<sup>41</sup><sub>61</sub>
-  14.132601332
|-
-59
-3807/3776
-P1<sup>47</sup><sub>59</sub>
-  14.154966241
|-
-17
-4131/4096
-P1<sup>17</sup>
-  14.730413827
|-
-59
-118/117
-A1<sup>59</sup><sub>13</sub>
-  14.733995734
|-
-29
-117/116
-P1<sup>13</sup><sub>29</sub>
-  14.860469347
|-
-47
-1280/1269
-P1<sup>5</sup><sub>47</sub>
-  14.942089256
|-
-31
-524288/519777
-dd3<sub>23,31</sub>
-  14.960075075
|-
-53
-1961/1944
-P1<sup>37,53</sup>
-  15.073579904
|-
-47
-1944/1927
-P1<sub>41,47</sub>
-  15.205976772
|-
-37
-112/111
-m2<sup>7</sup><sub>37</sub>
-  15.526866849
|-
-59
-4779/4736
-A1<sup>59</sup><sub>37</sub>
-  15.647623941
|-
-41
-4096/4059
-m2<sub>11,41</sub>
-  15.709650363
|-
-61
-549/544
--cA7<sup>61</sup><sub>17</sub>
-  15.839397306
|-
-53
-2067/2048
-P1<sup>13,53</sup>
-  15.987208111
|-
-23
-736/729
-P1<sup>23</sup>
-  16.544342076
|-
-19
-729/722
--cA7<sub>19,19</sub>
-  16.703972928
|-
-47
-517/512
-P1<sup>11,47</sup>
-  16.824564378
|-
-43
-66177/65536
-P1<sup>19,43</sup>
-  16.850725236
|-
-13
-512/507
-A1<sub>13,13</sub>
-  16.989675596
|-
-61
-44469/44032
--cA7<sup>61</sup><sub>43</sub>
-  17.097104625
|-
-59
-1728/1711
-P1<sub>29,59</sub>
-  17.116149209
|-
-53
-8192/8109
-P1<sub>17,53</sub>
-  17.630043293
|-
-61
-1664/1647
-d2<sup>13</sup><sub>61</sub>
-  17.777854098
|-
-37
-486/481
-A1<sub>13,37</sub>
-  17.903303803
|-
-31
-8019/7936
-P1<sup>11</sup><sub>31</sub>
-  18.012375093
|-
-19
-96/95
-P1<sub>5,19</sub>
-  18.128270868
|-
-59
-477/472
-P1<sup>53</sup><sub>59</sub>
-  18.242887972
|-
-59
-30208/29889
-m2<sup>59</sup><sub>41</sub>
-  18.379248500
|-
-23
-1035/1024
--cA7<sup>5,23</sup>
-  18.498062864
|-
-47
-94/93
-m2<sup>47</sup><sub>31</sub>
-  18.516048684
|-
-31
-93/92
-P1<sup>31</sup><sub>23</sub>
-  18.716226061
|-
-29
-464/459
-P1<sup>29</sup><sub>17</sub>
-  18.756782057
|-
-53
-2304/2279
-P1<sub>43,53</sub>
-  18.887750612
|-
-5
-2048/2025
-d2<sub>5,5</sub>
-  19.552568809
|-
-31
-972/961
-m2<sub>31,31</sub>
-  19.703859398
|-
-29
-88/87
-P1<sup>11</sup><sub>29</sub>
-  19.785747346
|-
-23
-437/432
-P1<sup>19,23</sup>
-  19.922360805
|-
-43
-87/86
-P1<sup>29</sup><sub>43</sub>
-  20.014489376
|-
-61
-32768/32391
-d2<sub>59,61</sub>
-  20.033533960
|-
-61
-432/427
-P1<sub>7,61</sub>
-  20.154291052
|-
-37
-259/256
-P1<sup>7,37</sup>
-  20.169945224
|-
-59
-530469/524288
--cA7<sup>37,59</sup>
-  20.290702316
|-
-23
-256/253
-m2<sub>11,23</sub>
-  20.407710367
|-
-61
-2623/2592
-P1<sup>43,61</sup>
-  20.582507256
|-
-41
-131072/129519
-m2<sub>13,41</sub>
-  20.634928362
|-
-53
-583/576
-P1<sup>11,53</sup>
-  20.912486110
|-
-13
-66339/65536
-P1<sup>7,13</sup>
-  21.083573431
|-
-41
-82/81
-P1<sup>41</sup>
-  21.242402080
|-
-31
-8192/8091
-m2<sub>29,31</sub>
-  21.477231652
|-
-5
-81/80
-P1<sub>5</sub>
-  21.506289597
|-
-41
-1536/1517
-m2<sub>37,41</sub>
-  21.548556569
|-
-47
-65536/64719
-P1<sub>17,47</sub>
-  21.717965025
|-
-43
-31744/31347
-P1<sup>31</sup><sub>43</sub>
-  21.787861629
|-
-61
-1037/1024
-P1<sup>17,61</sup>
-  21.840214576
|-
-47
-1062153/1048576
--cA7<sup>31,47</sup>
-  22.272199670
|-
-37
-26973/26624
--cA7<sup>37</sup><sub>13</sub>
-  22.546382178
|-
-53
-848/837
-m2<sup>53</sup><sub>31</sub>
-  22.603970415
|-
-61
-45056/44469
-d2<sup>11</sup><sub>61</sub>
-  22.703132097
|-
-47
-2048/2021
-P1<sub>43,47</sub>
-  22.975672344
|-
-59
-1888/1863
-m2<sup>59</sup><sub>23</sub>
-  23.077308504
|-
-41
-16605/16384
--cA7<sup>5,41</sup>
-  23.196122868
|-
-53
-4352/4293
-d2<sup>17</sup><sub>53</sub>
-  23.630860563
|-
-19
-32768/32319
-P1<sub>7,19</sub>
-  23.886073072
|-
-47
-11421/11264
-A1<sup>47</sup><sub>11</sub>
-  23.963683975
|-
-41
-779/768
-P1<sup>19,41</sup>
-  24.620420809
|-
-29
-493/486
-d2<sup>17,29</sup>
-  24.757599327
|-
-31
-7533/7424
--cA7<sup>31</sup><sub>29</sub>
-  25.233382638
|-
-23
-69/68
--cA7<sup>23</sup><sub>17</sub>
-  25.273938633
|-
-23
-8192/8073
-m2<sub>13,23</sub>
-  25.332988366
|-
-37
-740/729
-P1<sup>5,37</sup>
-  25.927747427
|-
-11
-2079/2048
-P1<sup>7,11</sup>
-  26.008851430
|-
-41
-333/328
-P1<sup>37</sup><sub>41</sub>
-  26.191634944
|-
-37
-864/851
-m2<sub>23,37</sub>
-  26.246616573
|-
-53
-133083/131072
--cA7<sup>31,53</sup>
-  26.360121402
|-
-47
-329/324
-m2<sup>7,47</sup>
-  26.512525021
|-
-43
-5589/5504
--cA7<sup>23</sup><sub>43</sub>
-  26.531645953
|-
-59
-8319/8192
-A1<sup>47,59</sup>
-  26.633282113
|-
-13
-65/64
-P1<sup>5,13</sup>
-  26.841375634
|-
-29
-324/319
-A1<sub>11,29</sub>
-  26.924866944
|-
-43
-16641/16384
-P1<sup>43,43</sup>
-  26.945413016
|-
-61
-4941/4864
--cA7<sup>61</sup><sub>19</sub>
-  27.191792405
|-
-7
-64/63
-P1<sub>7</sub>
-  27.264091800
|-
-31
-63/62
-m2<sup>7</sup><sub>31</sub>
-  27.700335736
|-
-47
-34816/34263
-d2<sup>17</sup><sub>47</sub>
-  27.718782295
|-
-53
-1431/1408
-A1<sup>53</sup><sub>11</sub>
-  28.051605707
|-
-61
-62/61
-P1<sup>31</sup><sub>61</sub>
-  28.150767389
|-
-43
-532899/524288
-P1<sup>17,43</sup>
-  28.203120335
|-
-19
-247/243
-d2<sup>13,19</sup>
-  28.265673575
|-
-61
-61/60
-P1<sup>61</sup><sub>5</sub>
-  28.616090345
|-
-47
-423/416
-A1<sup>47</sup><sub>13</sub>
-  28.888961975
|-
-53
-1024/1007
-P1<sub>19,53</sub>
-  28.982438392
|-
-59
-60/59
-P1<sup>5</sup><sub>59</sub>
-  29.097055496
|-
-7
-3645/3584
--cA7<sup>5</sup><sub>7</sub>
-  29.217812588
|-
-43
-131072/128871
-m2<sub>37,43</sub>
-  29.318252141
|-
-61
-2916/2867
-P1<sub>47,61</sub>
-  29.338578104
|-
-29
-7168/7047
-m2<sup>7</sup><sub>29</sub>
-  29.473707989
|-
-59
-59/58
-A1<sup>59</sup><sub>29</sub>
-  29.594465081
|-
-61
-2048/2013
-m2<sub>11,61</sub>
-  29.842251694
|-
-41
-1107/1088
--cA7<sup>41</sup><sub>17</sub>
-  29.971998637
|-
-43
-989/972
-P1<sup>23,43</sup>
-  30.017048584
|-
-29
-58/57
-P1<sup>29</sup><sub>19</sub>
-  30.109177155
|-
-53
-65536/64395
-d2<sub>5,53</sub>
-  30.406736332
|-
-59
-4864/4779
-d2<sup>19</sup><sub>59</sub>
-  30.521353437
|-
-53
-742/729
-m2<sup>7,53</sup>
-  30.600446753
|-
-19
-57/56
-P1<sup>19</sup><sub>7</sub>
-  30.642110529
|-
-59
-3127/3072
-A1<sup>53,59</sup>
-  30.721203845
|-
-37
-2997/2944
-P1<sup>37</sup><sub>23</sub>
-  30.889694948
|-
-23
-391/384
-P1<sup>17,23</sup>
-  31.274755903
|-
-29
-3712/3645
-d2<sup>29</sup><sub>5</sub>
-  31.533475096
|-
-11
-55/54
-P1<sup>5,11</sup>
-  31.766653633
|-
-29
-384/377
-A1<sub>13,29</sub>
-  31.850144943
|-
-41
-2673/2624
-P1<sup>11</sup><sub>41</sub>
-  32.030541150
|-
-53
-54/53
-P1<sub>53</sub>
-  32.360457120
|-
-61
-15104/14823
-m2<sup>59</sup><sub>61</sub>
-  32.511849832
|-
-47
-376/369
-m2<sup>47</sup><sub>41</sub>
-  32.534214741
|-
-53
-53/52
-A1<sup>53</sup><sub>13</sub>
-  32.976883707
|-
-47
-8192/8037
-P1<sub>19,47</sub>
-  33.070360124
|-
-31
-2560/2511
-m2<sup>5</sup><sub>31</sub>
-  33.458137939
|-
-29
-261/256
-P1<sup>29</sup>
-  33.487195884
|-
-17
-52/51
-P1<sup>13</sup><sub>17</sub>
-  33.617251404
|-
-23
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|-
-59
-767/384
-P8<sup>13,59</sup>
-1197.744320138
|-
-61
-261873/131072
-A7<sup>53,61</sup>
-1198.209354013
|-
-31
-1023/512
-M7<sup>11,31</sup>
-1198.308515694
|-
-19
-1215/608
-A7<sup>5</sup><sub>19</sub>
-1198.575702059
|-
-43
-2752/1377
-P8<sup>43</sup><sub>17</sub>
-1198.742292681
|-
-47
-1457/729
-P8<sup>31,47</sup>
-1198.812189285
|-
-53
-3072/1537
-P8<sub>29,53</sub>
-1198.873261236
|-
-37
-18944/9477
-P8<sup>37</sup><sub>13</sub>
-1199.086371793
|-
-59
-28672/14337
-cd2<sup>7</sup><sub>59</sub>
-1199.879242908


|}
== See also ==
* [[User:Contribution/61-limit]] – Includes a large table of 61-limit intervals
 
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[[Category:61-limit| ]] <!-- main article -->