61-limit

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In 61-limit Just Intonation, all ratios in the system will contain no primes higher than 61.

Intervals

Here is the complete list of 61-limit intervals with at most six pythagorean fifths (prime 3) and at most two primes from 5 to 61 in the ratio.

Limit Ratio FJS name Cents value
1/1 P1 0.000000000
59 14337/14336 -cA7597 0.120757092
37 9477/9472 P11337 0.913628207
53 1537/1536 P129,53 1.126738764
47 1458/1457 P131,47 1.187810715
43 1377/1376 P11743 1.257707319
19 1216/1215 d2195 1.424297941
31 1024/1023 m211,31 1.691484306
61 262144/261873 d253,61 1.790645987
59 768/767 P113,59 2.255679862
13 729/728 -cA77,13 2.376436954
59 649/648 P111,59 2.669598137
61 14848/14823 d22961 2.917384751
47 4096/4089 P129,47 2.961182968
19 513/512 P119 3.378018728
47 262683/262144 -cA723,47 3.555973609
41 3328/3321 d21341 3.645252766
53 424/423 P15347 4.087921732
59 7552/7533 m25931 4.361082443
37 4107/4096 -cM737,37 4.643078375
41 369/368 P14123 4.698060004
43 731/729 d217,43 4.743109951
13 352/351 P11113 4.925277999
53 1701/1696 P1753 5.096365320
19 324/323 -cA717,19 5.351577829
31 22599/22528 -cA73111 5.447635292
7 5120/5103 P157 5.757802203
37 297/296 P11137 5.838906206
61 2097152/2090043 d247,61 5.878567719
59 65536/65313 d241,59 5.900932628
17 289/288 d217,17 6.000817270
41 288/287 P17,41 6.021689720
29 16443/16384 P17,29 6.223104084
47 3008/2997 m24737 6.342579797
61 31347/31232 P14361 6.362905759
29 1863/1856 -cA72329 6.517156577
31 32768/32643 m213,31 6.616762305
19 263169/262144 P119,19 6.756037457
17 256/255 P15,17 6.775875769
61 244/243 P161 7.109800749
11 243/242 A111,11 7.139119597
37 1152/1147 m231,37 7.530390512
53 32913/32768 -cA723,53 7.643895340
43 10496/10449 P14143 7.769695572
31 217/216 P17,31 7.996476337
43 216/215 P15,43 8.033583089
59 16461/16384 -cA731,59 8.117233429
23 208/207 d21323 8.343312770
61 8235/8192 -cA75,61 9.063521536
47 189/188 P1747 9.184287052
37 262144/260739 m229,37 9.303762765
61 184/183 P12361 9.434541328
11 896/891 m2711 9.687960643
59 177/176 A15911 9.808717735
43 172/171 P14319 10.094687779
31 837/832 -cA73113 10.372913291
53 27136/26973 m25337 10.430501529
37 32967/32768 -cM711,37 10.481984581
61 1159/1152 P119,61 10.487819477
59 4096/4071 d223,59 10.598992632
23 162/161 P17,23 10.719749724
53 160/159 P1553 10.854167524
47 1739/1728 P137,47 10.985658172
41 29889/29696 -cA74129 11.215216581
19 153/152 P11719 11.352395099
47 16497/16384 P113,47 11.899286379
29 145/144 P15,29 11.980906287
13 144/143 A111,13 12.064397597
37 999/992 P13731 12.173468887
41 21141/20992 P12941 12.244793804
53 38912/38637 d21953 12.278465464
53 528039/524288 -cA741,53 12.341955344
59 3481/3456 A159,59 12.478315872
17 136/135 d2175 12.776693039
23 5632/5589 d21123 13.268590769
43 129/128 P143 13.472706508
43 8192/8127 P17,43 13.791385292
37 2368/2349 P13729 13.946841140
41 124/123 P13141 14.018166057
61 123/122 P14161 14.132601332
59 3807/3776 P14759 14.154966241
17 4131/4096 P117 14.730413827
59 118/117 A15913 14.733995734
29 117/116 P11329 14.860469347
47 1280/1269 P1547 14.942089256
31 524288/519777 dd323,31 14.960075075
53 1961/1944 P137,53 15.073579904
47 1944/1927 P141,47 15.205976772
37 112/111 m2737 15.526866849
59 4779/4736 A15937 15.647623941
41 4096/4059 m211,41 15.709650363
61 549/544 -cA76117 15.839397306
53 2067/2048 P113,53 15.987208111
23 736/729 P123 16.544342076
19 729/722 -cA719,19 16.703972928
47 517/512 P111,47 16.824564378
43 66177/65536 P119,43 16.850725236
13 512/507 A113,13 16.989675596
61 44469/44032 -cA76143 17.097104625
59 1728/1711 P129,59 17.116149209
53 8192/8109 P117,53 17.630043293
61 1664/1647 d21361 17.777854098
37 486/481 A113,37 17.903303803
31 8019/7936 P11131 18.012375093
19 96/95 P15,19 18.128270868
59 477/472 P15359 18.242887972
59 30208/29889 m25941 18.379248500
23 1035/1024 -cA75,23 18.498062864
47 94/93 m24731 18.516048684
31 93/92 P13123 18.716226061
29 464/459 P12917 18.756782057
53 2304/2279 P143,53 18.887750612
5 2048/2025 d25,5 19.552568809
31 972/961 m231,31 19.703859398
29 88/87 P11129 19.785747346
23 437/432 P119,23 19.922360805
43 87/86 P12943 20.014489376
61 32768/32391 d259,61 20.033533960
61 432/427 P17,61 20.154291052
37 259/256 P17,37 20.169945224
59 530469/524288 -cA737,59 20.290702316
23 256/253 m211,23 20.407710367
61 2623/2592 P143,61 20.582507256
41 131072/129519 m213,41 20.634928362
53 583/576 P111,53 20.912486110
13 66339/65536 P17,13 21.083573431
41 82/81 P141 21.242402080
31 8192/8091 m229,31 21.477231652
5 81/80 P15 21.506289597
41 1536/1517 m237,41 21.548556569
47 65536/64719 P117,47 21.717965025
43 31744/31347 P13143 21.787861629
61 1037/1024 P117,61 21.840214576
47 1062153/1048576 -cA731,47 22.272199670
37 26973/26624 -cA73713 22.546382178
53 848/837 m25331 22.603970415
61 45056/44469 d21161 22.703132097
47 2048/2021 P143,47 22.975672344
59 1888/1863 m25923 23.077308504
41 16605/16384 -cA75,41 23.196122868
53 4352/4293 d21753 23.630860563
19 32768/32319 P17,19 23.886073072
47 11421/11264 A14711 23.963683975
41 779/768 P119,41 24.620420809
29 493/486 d217,29 24.757599327
31 7533/7424 -cA73129 25.233382638
23 69/68 -cA72317 25.273938633
23 8192/8073 m213,23 25.332988366
37 740/729 P15,37 25.927747427
11 2079/2048 P17,11 26.008851430
41 333/328 P13741 26.191634944
37 864/851 m223,37 26.246616573
53 133083/131072 -cA731,53 26.360121402
47 329/324 m27,47 26.512525021
43 5589/5504 -cA72343 26.531645953
59 8319/8192 A147,59 26.633282113
13 65/64 P15,13 26.841375634
29 324/319 A111,29 26.924866944
43 16641/16384 P143,43 26.945413016
61 4941/4864 -cA76119 27.191792405
7 64/63 P17 27.264091800
31 63/62 m2731 27.700335736
47 34816/34263 d21747 27.718782295
53 1431/1408 A15311 28.051605707
61 62/61 P13161 28.150767389
43 532899/524288 P117,43 28.203120335
19 247/243 d213,19 28.265673575
61 61/60 P1615 28.616090345
47 423/416 A14713 28.888961975
53 1024/1007 P119,53 28.982438392
59 60/59 P1559 29.097055496
7 3645/3584 -cA757 29.217812588
43 131072/128871 m237,43 29.318252141
61 2916/2867 P147,61 29.338578104
29 7168/7047 m2729 29.473707989
59 59/58 A15929 29.594465081
61 2048/2013 m211,61 29.842251694
41 1107/1088 -cA74117 29.971998637
43 989/972 P123,43 30.017048584
29 58/57 P12919 30.109177155
53 65536/64395 d25,53 30.406736332
59 4864/4779 d21959 30.521353437
53 742/729 m27,53 30.600446753
19 57/56 P1197 30.642110529
59 3127/3072 A153,59 30.721203845
37 2997/2944 P13723 30.889694948
23 391/384 P117,23 31.274755903
29 3712/3645 d2295 31.533475096
11 55/54 P15,11 31.766653633
29 384/377 A113,29 31.850144943
41 2673/2624 P11141 32.030541150
53 54/53 P153 32.360457120
61 15104/14823 m25961 32.511849832
47 376/369 m24741 32.534214741
53 53/52 A15313 32.976883707
47 8192/8037 P119,47 33.070360124
31 2560/2511 m2531 33.458137939
29 261/256 P129 33.487195884
17 52/51 P11317 33.617251404
23 131072/128547 dd323,23 33.676301136
41 1296/1271 m231,41 33.722025456
53 38637/37888 A15337 33.890511913
43 1184/1161 P13743 33.961330516
43 1763/1728 P141,43 34.715108588
61 65536/64233 m213,61 34.767529694
43 351/344 P11343 34.874958723
43 10449/10240 P1435 34.978996105
31 248/243 P131 35.260568137
41 32768/32103 m229,41 35.495397709
61 2304/2257 m237,61 35.681157901
7 49/48 m27,7 35.696812073
53 13851/13568 P11953 35.738475849
59 33453/32768 A17,59 35.817569165
59 1024/1003 P117,59 35.872931265
41 2091/2048 P117,41 35.972815907
17 243/238 -cA77,17 35.993688357
47 48/47 P147 36.448378852
53 3392/3321 m25341 36.622136473
23 621/608 -cA72319 36.626333732
29 133893/131072 P119,29 36.865214612
11 8192/8019 m211 36.952052443
59 2592/2537 P143,59 37.130638585
31 4185/4096 -cA75,31 37.214288925
47 47/46 m24723 37.232274745
61 33489/32768 -cA761,61 37.679611882
23 46/45 P1235 38.050631673
13 1701/1664 A1713 38.073249027
31 729/713 m223,31 38.420085460
17 1408/1377 P11117 38.542529403
31 589/576 P119,31 38.638586866
11 45/44 A1511 38.905773231
37 524288/512487 m219,37 39.412939921
61 488/477 P16153 39.470257869
17 221/216 d213,17 39.618068674
43 44/43 P11143 39.800236722
47 1539/1504 P11947 39.826397581
23 16767/16384 -cA723 40.004352461
29 2048/2001 m223,29 40.193457713
31 131072/128061 m217,31 40.234013708
61 999/976 P13761 40.324236275
19 304/297 m21911 40.330071171
61 1769/1728 P129,61 40.596996632
59 44032/43011 d24359 40.616041216
43 43/42 P1437 40.736798308
47 536787/524288 A147,47 40.788248353
53 2097152/2047761 d253,53 41.260903856
53 212/207 m25323 41.320196477
41 9963/9728 -cA74119 41.324393736
43 4096/3999 m231,43 41.491721028
59 295/288 A15,59 41.575371368
41 42/41 m2741 41.718501793
59 43011/41984 A15941 41.839258885
59 544/531 d21759 41.873748535
17 459/448 P1177 41.994505627
41 41/40 P1415 42.748691677
37 1024/999 m237 42.790958649
61 1048576/1022787 dd323,61 43.110842464
61 3904/3807 P16147 43.558179601
47 3807/3712 A14729 43.749431322
13 40/39 A1513 43.831051230
31 279/272 -cA73117 43.990164695
37 4736/4617 P13719 44.056018296
17 748/729 d211,17 44.543346673
37 1215/1184 A1537 44.744679437
53 67257/65536 A147,53 44.876170085
19 39/38 P11319 44.969646502
43 22599/22016 -cA73143 45.247872014
19 9728/9477 m21913 45.255349171
17 192/187 A111,17 45.681649000
53 3483/3392 P14353 45.833163628
61 8019/7808 P11161 46.163142482
37 38/37 m21937 46.168977378
13 416/405 d2135 46.393944443
23 189/184 m2723 46.416561797
61 188/183 m24761 46.666816072
29 864/841 A129,29 46.710614290
43 486/473 A111,43 46.939356320
43 33669/32768 P129,43 46.959902392
61 4209/4096 -cA723,61 47.114153209
59 1152/1121 P119,59 47.225326364
37 37/36 P137 47.434037024
41 10240/9963 m2541 47.476303996
41 1728/1681 m241,41 47.740191513
53 477/464 A15329 47.837353054
61 1944/1891 m231,61 47.854626787
13 1053/1024 P113 48.347665231
59 472/459 A15917 48.351247138
41 5248/5103 P1417 48.506493880
59 8192/7965 d25,59 48.649624305
43 1333/1296 P131,43 48.733274645
7 36/35 P15,7 48.770381397
53 1472/1431 P12353 48.904799196
53 8427/8192 A153,53 48.964091817
37 134865/131072 -cA75,37 49.387757812
43 3584/3483 m2743 49.488197365
59 177/172 A15943 49.608954457
61 16384/15921 m229,61 49.627999041
19 176/171 P11119 49.894924502
47 387/376 P14347 49.921085360
31 527/512 P117,31 49.990981965
29 667/648 P123,29 50.031537960
43 2752/2673 m24311 50.424758951
59 243/236 P159 50.603345093
17 2048/1989 A113,17 50.606927000
61 1696/1647 m25361 50.754737804
37 2109/2048 P119,37 50.812055752
47 12393/12032 P11747 51.178792680
37 648/629 A117,37 51.520555206
31 16384/15903 m219,31 51.586408807
17 34/33 m21711 51.682466270
61 67527/65536 -cA741,61 51.812213213
43 576/559 A113,43 51.864634319
23 640/621 m2523 52.174364000
41 972/943 m223,41 52.438251517
47 11776/11421 P12347 52.992720928
11 33/32 P111 53.272943230
53 164/159 P14153 53.602859200
47 752/729 m247 53.776616821
53 270459/262144 A17,53 54.060457137
41 524288/508113 m217,41 54.252179766
59 1003/972 m217,59 54.352064408
7 4096/3969 P17,7 54.528183600
41 1189/1152 P129,41 54.729597964
31 32/31 m231 54.964427536
29 21141/20480 P1295 54.993485481
31 2511/2432 -cA73119 55.342559793
61 793/768 P113,61 55.457465979
43 16384/15867 m241,43 55.509887085
47 2115/2048 A15,47 55.730337609
61 63/61 m2761 55.851103124
37 8991/8704 -cA73717 56.163633581
43 344/333 m24337 56.263665157
23 529/512 -cA723,23 56.548694537
17 1088/1053 m21713 56.607744270
19 16929/16384 P111,19 56.650961959
31 31/30 P1315 56.766857734
19 216/209 A111,19 57.034044099
47 893/864 m219,47 57.154635549
53 13312/12879 d21353 57.248111967
37 153/148 m21737 57.521372476
47 1312/1269 P14147 57.690780932
61 61/59 P16159 57.713145841
61 14823/14336 -cA7617 57.833902933
47 243/235 P15,47 57.954668449
31 513/496 m21931 58.342446264
29 754/729 d213,29 58.374850730
11 512/495 m25,11 58.458342040
29 30/29 A1529 58.691520577
23 32768/31671 m217,23 58.950239770
59 262144/253287 d253,59 59.503791828
19 14336/13851 m2719 59.582885144
53 384/371 P17,53 59.624548920
59 59/57 A15919 59.703642237
53 265/256 A15,53 59.818259341
29 2432/2349 m21929 60.115818518
43 1024/989 m223,43 60.207947089
61 671/648 P111,61 60.382743979
11 729/704 A111 60.412062828
59 14848/14337 d22959 60.630530592
29 29/28 P1297 60.751287684
43 1591/1536 P137,43 60.906743532
59 3776/3645 m2595 61.127940177
53 1007/972 m219,53 61.242557281
41 8487/8192 -cA723,41 61.246754541
61 5120/4941 m2561 61.608905328
43 135837/131072 P113,43 61.820371739
61 2592/2501 m241,61 61.872792844
19 256/247 A113,19 61.959322098
61 7808/7533 m26131 62.074228284
37 16983/16384 P117,37 62.164450851
47 141/136 A14717 62.506213378
37 729/703 A119,37 62.872950305
7 28/27 m27 62.960903873
59 531/512 A159 63.081660965
13 16384/15795 m25,13 63.383620039
59 2097152/2021517 d247,59 63.591713560
47 1024/987 P17,47 63.712470652
47 11421/11008 A14743 63.763920698
59 31347/30208 P14359 64.076051601
37 192/185 m25,37 64.297248246
53 2916/2809 P153,53 64.720914241
23 299/288 P113,23 64.892007307
7 8505/8192 A15,7 64.914624661
23 17408/16767 dd31723 64.951057040
13 27/26 A113 65.337340827
29 512/493 A117,29 65.467396347
41 65536/63099 m219,41 65.604574864
61 17019/16384 -cA731,61 65.830379270
53 7047/6784 P12953 65.847653004
41 136161/131072 -cA741,41 65.944814545
19 133/128 m27,19 66.338922601
59 272403/262144 A119,59 66.459679693
61 1458/1403 m223,61 66.570852848
53 53/51 A15317 66.594135110
43 1296/1247 A129,43 66.725103666
43 4257/4096 P111,43 66.745649738
29 68121/65536 P129,29 66.974391768
59 184/177 P12359 67.147687169
47 2021/1944 m243,47 67.249323329
23 1863/1792 -cA7237 67.268444261
61 183/176 A16111 67.521863576
53 496/477 P13153 67.621025258
53 1431/1376 A15343 67.851842429
17 136323/131072 P111,17 68.003357057
61 262144/251991 m217,61 68.384781097
43 129/124 m24331 68.437134044
47 799/768 m217,47 68.507030648
41 1517/1458 P137,41 68.676439104
19 13851/13312 A11913 68.715359555
31 899/864 P129,31 68.747764021
53 2592/2491 P147,53 68.808835973
37 333/320 P1375 68.940326621
41 128/123 m241 68.982593593
53 16384/15741 m211,53 69.312509563
41 533/512 P113,41 69.590067311
61 8192/7869 m243,61 69.642488417
31 4131/3968 m21731 69.694841363
23 253/243 P111,23 69.817285306
47 783/752 P12947 69.935574736
37 65536/62937 m27,37 70.055050449
61 3599/3456 A159,61 70.191461714
43 22016/21141 m24329 70.210506297
23 4096/3933 m219,23 70.302634869
29 928/891 m22911 70.439248327
5 25/24 A15,5 70.672426864
29 272/261 m21729 71.468213617
31 23552/22599 m22331 71.508769612
17 567/544 A1717 71.690500430
47 3968/3807 P13147 71.708946990
59 123/118 P14159 71.845747173
41 29889/28672 -cA7417 71.966504265
19 760/729 m25,19 72.096724805
41 171/164 m21941 72.360612321
61 122/117 A16113 72.447141575
53 901/864 m217,53 72.594952380
47 2304/2209 P147,47 72.896757704
13 169/162 d213,13 73.235320077
61 4941/4736 A16137 73.360769782
47 131072/125631 m211,47 73.400431295
23 24/23 m223 73.680653597
47 1269/1216 A14719 73.858608477
53 524288/502281 m213,53 74.237787563
41 451/432 P111,41 74.515345310
37 592/567 P1377 74.698128824
11 2673/2560 P1115 74.779232827
53 2048/1961 m237,53 75.151415769
31 2139/2048 -cA723,31 75.264920598
47 47/45 m2475 75.282906418
59 1664/1593 d21359 75.490999939
13 117/112 P1137 75.611757031
61 3888/3721 m261,61 76.005394176
61 31232/29889 m26141 76.092394341
41 2624/2511 m24131 76.206829616
37 116/111 m22937 76.278154533
43 301/288 m27,43 76.433610381
31 162/155 m25,31 76.470717132
59 68499/65536 A143,59 76.554367473
43 32768/31347 m243 76.752289165
29 576/551 A119,29 76.819791445
23 23/22 A12311 76.956404904
23 1539/1472 m21923 77.058672325
47 34263/32768 A147 77.236627205
17 160/153 A1517 77.448302634
17 1071/1024 m27,17 77.691317700
41 729/697 A117,41 77.712190150
59 32768/31329 d259,59 77.746679801
59 432/413 P17,59 77.867436893
53 159/152 A15319 77.946530209
61 548451/524288 -cA737,61 78.003848157
29 4096/3915 m25,29 78.244089386
43 45/43 A1543 78.706009953
43 3483/3328 A14313 78.810047335
47 16384/15651 m237,47 79.239337501
53 424/405 m2535 79.370828149
59 1357/1296 A123,59 79.626003041
61 32768/31293 m219,61 79.737176196
53 111/106 P13753 79.794494144
41 34317/32768 -cA731,41 79.962980602
43 1216/1161 m21943 80.130307894
29 243/232 A129 80.197810174
59 45056/43011 d21159 80.416277938
11 22/21 P1117 80.537035030
61 1952/1863 m26123 80.790454345
37 1073/1024 P129,37 80.921232908
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See also

Harmonic limit