User:Contribution/(31-Limit Factor Gradient) x (26 Tritaves Chain) x (Infinite Octaves or No-Twos) Comparison Table

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Purpose

See here.

Content

This table shows these sets:

  • ((5;7;0;3)-limit ∪ (5;13;0;2)-limit ∪ (5;31;0;1)-limit) × (3;3;0;26)-limit × (2;2;0;+∞)-limit
  • ((5;7;0;3)-limit ∪ (5;13;0;2)-limit ∪ (5;31;0;1)-limit) × (3;3;0;26)-limit


You'll find these columns:

  • Limit: the smaller maximal prime limit whose intervals belong
  • Factorization: exact factorization into primes of the second set, or factorization into primes ignoring octaves of the first set
  • Ratio: octave-reduced ratios of the first set
  • FJS name: octave-reduced FJS names of the first set
  • Cents value: octave-reduced cents values of the first set
  • Octaves: octave difference between the second set (no-twos just intonation) and the octave-reducing of the first set (infinite octaves just intonation). Intervals outside the human hearing range (10 octaves) are ignored.

Table

Limit Factorization Ratio FJS name Cents value Octaves
1 1/1 P1 0.000000000 0
13 3-20⋅131 3489660928/3486784401 ddd313 1.427644462
7 3-22⋅5-1⋅7-1 1099511627776/1098337086315 ddd35,7 1.850360628
5 38⋅51 32805/32768 -cA75 1.953720788
13 36⋅7-1⋅13-1 729/728 -cA77,13 2.376436954 3
11 3-9⋅71⋅111 19712/19683 d27,11 2.548841045 -8
13 325⋅5-1⋅131 11014751922759/10995116277760 -cAA6135 3.088969539
19 33⋅191 513/512 P119 3.378018728 9
13 3-12⋅51⋅131 532480/531441 d25,13 3.381365249
7 323⋅5-2⋅7-1 94143178827/93952409600 -cAA65,5,7 3.511685705
7 3-14⋅7-1 33554432/33480783 d27 3.804081415
5 316⋅52 1076168025/1073741824 -cAAA65,5 3.907441576
13 3-3⋅111⋅13-1 352/351 P11113 4.925277999 -5
7 3-6⋅51⋅7-1 5120/5103 P157 5.757802203 -10
5 324⋅53 35303692060125/35184372088832 -cAAAAA55,5,5 5.861162364
11 3-23⋅111 94489280512/94143178827 ddd311 6.352922461
13 313⋅132 269440587/268435456 -cM713,13 6.470334789
23 318⋅23-1 387420489/385875968 -cA723 6.915668309
11 35⋅11-2 243/242 A111,11 7.139119597 1
7 32⋅52⋅7-1 225/224 -cA75,57 7.711522991 5
11 322⋅5-1⋅111 345191655699/343597383680 -cAA6115 8.014247538
31 3-24⋅31-1 8796093022208/8755315630911 dddd431 8.044406766
11 3-15⋅51⋅111 14417920/14348907 d25,11 8.306643249
7 31⋅73 1029/1024 m27,7,7 8.432720273
17 37⋅17-1 2187/2176 -cA717 8.729596557 7
13 319⋅7-1⋅131 15109399071/15032385536 -cAA6137 8.846771743
7 317⋅5-1⋅7-2 129140163/128450560 -cAA65,7,7 9.269487909
7 3-20⋅51⋅7-2 171798691840/170852435649 d257,7 9.561883619
11 3-4⋅71⋅11-1 896/891 m2711 9.687960643 -7
29 3-10⋅291 59392/59049 d229 10.027185499
7 3-21⋅5-1⋅72 52613349376/52301766015 dddd47,75 10.283080900
7 325⋅7-2 847288609443/841813590016 -cAAAA57,7 11.223208696
13 310⋅111⋅131 8444007/8388608 -cM711,13 11.395612788
11 3-26⋅5-1⋅11-1 140737488355328/139802620558095 dddd45,11 11.538321270
31 3-17⋅311 130023424/129140163 d231 11.800557753
13 32⋅11-1⋅13-1 144/143 A111,13 12.064397597 -4
7 3-13⋅72 1605632/1594323 dd37,7 12.236801688
29 322⋅29-1 31381059609/31138512896 -cAAA629 13.432824885
11 3-18⋅11-1 4294967296/4261625379 dd311 13.492042058
11 316⋅7-1⋅111 473513931/469762048 -cAA6117 13.772049742
7 3-5⋅51⋅72 245/243 m25,7,7 14.190522476 0
13 3-7⋅71⋅13-1 28672/28431 m2713 14.613238642
17 35⋅171 4131/4096 P117 14.730413827
7 311⋅7-3 177147/175616 -cAA67,7,7 15.027290112 9
11 3-10⋅51⋅11-1 655360/649539 m2511 15.445762846
7 310⋅5-2⋅71 413343/409600 P175,5 15.748487393
11 37⋅112 264627/262144 -cM711,11 16.320890787
23 3-6⋅231 736/729 P123 16.544342076 -5
13 3-1⋅13-2 512/507 A113,13 16.989675596 -9
5 3-12⋅5-3 67108864/66430125 ddd35,5,5 17.598848021
7 318⋅5-1⋅71 2711943423/2684354560 -cA775 17.702208181
7 3-19⋅51⋅71 1174405120/1162261467 dd35,7 17.994603892
13 3-21⋅13-1 137438953472/135984591639 dd313 18.417320058
13 315⋅11-1⋅131 186535791/184549376 -cA71311 18.534732385
5 3-4⋅5-2 2048/2025 d25,5 19.552568809
7 326⋅71 17793060798303/17592186044416 -cAAA67 19.655928969
7 3-11⋅52⋅71 179200/177147 m25,5,7 19.948324680 -10
13 324⋅5-1⋅13-1 282429536481/279172874240 -cAAA65,13 20.078645135
19 39⋅19-1 19683/19456 -cA719 20.081991656
13 3-13⋅51⋅13-1 20971520/20726199 m2513 20.371040845
11 321⋅7-1⋅11-1 10460353203/10334765056 -cAAA67,11 20.911169339
13 36⋅71⋅131 66339/65536 P17,13 21.083573431
5 34⋅5-1 81/80 P15 21.506289597 4
13 3-16⋅5-1⋅131 218103808/215233605 ddd3135 22.933934058
7 3-18⋅5-2⋅7-1 68719476736/67798585575 ddd35,5,7 23.356650224
3 312 531441/524288 -cA7 23.460010385
5 3-25⋅52 858993459200/847288609443 dd35,5 23.752406095
13 3-8⋅131 6656/6561 d213 24.887654846 -9
7 3-10⋅5-1⋅7-1 2097152/2066715 d25,7 25.310371012
5 320⋅51 17433922005/17179869184 -cAAA65 25.413731173
5 3-17⋅53 131072000/129140163 m25,5,5 25.706126883
13 318⋅7-1⋅13-1 387420489/381681664 -cAAA67,13 25.836447339
11 33⋅71⋅111 2079/2048 P17,11 26.008851430
23 3-23⋅23-1 2199023255552/2165293113021 dddd423 26.760632828
19 315⋅191 272629233/268435456 -cA719 26.838029113
13 51⋅131 65/64 P15,13 26.841375634 6
7 3-2⋅7-1 64/63 P17 27.264091800 -6
11 3-19⋅5-1⋅111 5905580032/5811307335 ddd3115 27.859212058
13 39⋅111⋅13-1 216513/212992 -cA71113 28.385288384
13 3-22⋅7-1⋅131 223338299392/219667417263 ddd3137 28.691736262
7 3-24⋅5-1⋅7-2 70368744177664/69195236437845 ddd35,7,7 29.114452428
7 36⋅51⋅7-1 3645/3584 -cA757 29.217812588 9
11 3-11⋅111 180224/177147 d211 29.812932845
13 325⋅132 143191774995867/140737488355328 -cAA613,13 29.930345173
11 317⋅11-2 129140163/126877696 -cAA711,11 30.599129982
7 3-16⋅7-2 2147483648/2109289329 d27,7 31.068173216
7 314⋅52⋅7-1 119574225/117440512 -cAAA65,57 31.171533376
31 3-12⋅31-1 16777216/16474671 dd331 31.504417151
11 3-3⋅51⋅111 55/54 P15,11 31.766653633 1
7 313⋅73 546852789/536870912 A17,7,7 31.892730657
17 319⋅17-1 1162261467/1140850688 -cAAA617 32.189606942
7 3-8⋅51⋅7-2 327680/321489 P157,7 33.021894003
11 38⋅71⋅11-1 45927/45056 A1711 33.147971027
29 3-19⋅29-1 34359738368/33705582543 dd329 33.277789405
29 32⋅291 261/256 P129 33.487195884 8
11 3-25⋅7-1⋅111 6047313952768/5931020266101 ddd3117 33.617014261
7 3-9⋅5-1⋅72 100352/98415 dd37,75 33.743091285
13 322⋅111⋅131 4487491524087/4398046511104 -cAA611,13 34.855623173
11 3-14⋅5-1⋅11-1 268435456/263063295 dd35,11 34.998331655
31 3-5⋅311 248/243 P131 35.260568137 -3
13 314⋅11-1⋅13-1 4782969/4685824 -cAA711,13 35.524407981
7 3-1⋅72 49/48 m27,7 35.696812073 4
11 3-6⋅11-1 8192/8019 m211 36.952052443
7 3-23⋅5-1⋅71 481036337152/470715894135 dddd475 37.547172700
7 37⋅51⋅72 535815/524288 A15,7,7 37.650532861
13 35⋅71⋅13-1 1701/1664 A1713 38.073249027 7
17 317⋅171 2195382771/2147483648 -cA717 38.190424212
13 3-26⋅11-1⋅131 28587302322176/27960524111619 dddd41311 38.379696904
7 323⋅7-3 94143178827/92073361408 -cAAAA57,7,7 38.487300497
11 32⋅51⋅11-1 45/44 A1511 38.905773231 2
7 322⋅5-2⋅71 219667417263/214748364800 -cA775,5 39.208497778
7 3-15⋅71 14680064/14348907 dd37 39.500893488
11 319⋅112 140633637507/137438953472 -cAA611,11 39.780901172
13 3-17⋅5-1⋅13-1 8589934592/8394110595 dd35,13 39.923609654
23 36⋅231 16767/16384 -cA723 40.004352461
13 311⋅13-2 177147/173056 -cAA713,13 40.449685981
11 3-20⋅7-1⋅11-1 274877906944/268482398877 dd37,11 40.756133858
5 5-3 128/125 d25,5,5 41.058858405 -7
7 3-7⋅51⋅71 2240/2187 m25,7 41.454614276 -6
13 3-9⋅13-1 262144/255879 m213 41.877330442
5 38⋅5-2 6561/6400 P15,5 43.012579193 8
7 31⋅52⋅71 525/512 A15,5,7 43.408335064 9
19 321⋅19-1 10460353203/10200547328 -cAAA619 43.542002041
13 3-1⋅51⋅13-1 40/39 A1513 43.831051230 -3
13 318⋅71⋅131 35255264499/34359738368 -cA77,13 44.543583815
5 316⋅5-1 43046721/41943040 -cA75 44.966299981
5 3-21⋅51 10737418240/10460353203 dd35 45.258695692
13 3-23⋅7-1⋅13-1 8796093022208/8567029273257 dd37,13 45.681411858
13 3-4⋅5-1⋅131 416/405 d2135 46.393944443 -5
19 3-26⋅191 2611340115968/2541865828329 dddd419 46.682993632
7 3-6⋅5-2⋅7-1 131072/127575 d25,5,7 46.816660609
3 324 282429536481/274877906944 -cAAA6 46.920020769
5 3-13⋅52 1638400/1594323 m25,5 47.212416480
13 34⋅131 1053/1024 P113 48.347665231
7 32⋅5-1⋅7-1 36/35 P15,7 48.770381397 -2
5 3-5⋅53 250/243 A15,5,5 49.166137268 -1
11 315⋅71⋅111 1104865839/1073741824 -cA77,11 49.468861815
13 3-16⋅132 44302336/43046721 ddd313,13 49.775309692
23 3-11⋅23-1 4194304/4074381 dd323 50.220643212
13 312⋅51⋅131 34543665/33554432 -cA75,13 50.301386019
11 3-24⋅11-2 35184372088832/34173973914201 ddd411,11 50.444094501
7 310⋅7-1 59049/57344 -cA77 50.724102185
11 3-7⋅5-1⋅111 11264/10935 d2115 51.319222442 -10
13 321⋅111⋅13-1 115063885233/111669149696 -cAAA61113 51.845298769
17 3-22⋅17-1 549755813888/533478013353 dd317 52.034571461
13 3-10⋅7-1⋅131 425984/413343 d2137 52.151746646
7 3-12⋅5-1⋅7-2 134217728/130203045 d25,7,7 52.574462812
7 318⋅51⋅7-1 1937102445/1879048192 -cAAA657 52.677822973
11 31⋅111 33/32 P111 53.272943230 5
7 3-4⋅7-2 4096/3969 P17,7 54.528183600
7 326⋅52⋅7-1 63546645708225/61572651155456 -cAAAAA55,57 54.631543761
13 3-19⋅111⋅131 1199570944/1162261467 ddd311,13 54.700587692
31 31-1 32/31 m231 54.964427536 -5
11 39⋅51⋅111 1082565/1048576 -cA75,11 55.226664018
7 325⋅73 290619993038949/281474976710656 -cAA77,7,7 55.352741042
7 34⋅51⋅7-2 405/392 -cA757,7 56.481904388 3
11 320⋅71⋅11-1 24407490807/23622320128 -cAA7711 56.607981412
29 3-7⋅29-1 65536/63423 m229 56.737799789
29 314⋅291 138706101/134217728 -cA729 56.947206269
11 3-13⋅7-1⋅111 11534336/11160261 d2117 57.077024646
7 33⋅5-1⋅72 1323/1280 m27,75 57.203101670 8
17 3-24⋅171 292057776128/282429536481 dddd417 58.035388731
7 3-18⋅7-3 137438953472/132885227727 d27,7,7 58.332265016
11 3-2⋅5-1⋅11-1 512/495 m25,11 58.458342040 -9
31 37⋅311 67797/65536 -cA731 58.720578522
13 326⋅11-1⋅13-1 2541865828329/2456721293312 -cAAAA611,13 58.984418366
7 3-19⋅5-2⋅71 30064771072/29056536675 dddd475,5 59.053462297
7 311⋅72 8680203/8388608 A17,7 59.156822458
11 3-22⋅112 32480690176/31381059609 ddd311,11 59.625865691
11 36⋅11-1 729/704 A111 60.412062828 6
7 3-11⋅5-1⋅71 917504/885735 dd375 61.007183085
7 319⋅51⋅72 284754059415/274877906944 -cAA75,7,7 61.110543245
13 317⋅71⋅13-1 903981141/872415232 -cAA7713 61.533259411
13 3-14⋅11-1⋅131 54525952/52612659 dd31311 61.839707289
11 314⋅51⋅11-1 23914845/23068672 -cAA7511 62.365783616
7 3-3⋅71 28/27 m27 62.960903873 -2
13 3-5⋅5-1⋅13-1 16384/15795 m25,13 63.383620039
19 3-20⋅19-1 68719476736/66248903619 dd319 63.386966560
23 318⋅231 8910671247/8589934592 -cAAA623 63.464362845
13 323⋅13-2 94143178827/90731184128 -cAAAA613,13 63.909696365
11 3-8⋅7-1⋅11-1 524288/505197 m27,11 64.216144243
13 3-23⋅71⋅131 97710505984/94143178827 dddd47,13 64.388548335
5 312⋅5-3 531441/512000 P15,5,5 64.518868790
5 3-25⋅5-1 4398046511104/4236443047215 dddd45 64.811264500
7 35⋅51⋅71 8505/8192 A15,7 64.914624661
13 33⋅13-1 27/26 A113 65.337340827 1
5 320⋅5-2 3486784401/3355443200 -cA75,5 66.472589578
3 3-17 134217728/129140163 dd3 66.764985288
7 313⋅52⋅71 279006525/268435456 -cAA75,5,7 66.868345449
13 311⋅51⋅13-1 885735/851968 -cAA7513 67.291061615
5 3-9⋅51 20480/19683 m25 68.718706076
13 3-11⋅7-1⋅13-1 16777216/16120377 m27,13 69.141422242
11 3-26⋅71⋅111 2645699854336/2541865828329 dddd47,11 69.313826334
13 38⋅5-1⋅131 85293/81920 P1135 69.853954828
19 3-14⋅191 4980736/4782969 dd319 70.143004017
7 36⋅5-2⋅7-1 729/700 P15,5,7 70.276670994 2
5 3-1⋅52 25/24 A15,5 70.672426864 3
13 3-20⋅111⋅13-1 47244640256/45328197213 dd31113 71.690263288
13 316⋅131 559607373/536870912 -cA713 71.807675616
7 314⋅5-1⋅7-1 4782969/4587520 -cA75,7 72.230391781
7 3-23⋅51⋅7-1 687194767360/659002251789 dd357 72.522787492
5 37⋅53 273375/262144 -cAA75,5,5 72.626147652
13 3-4⋅132 169/162 d213,13 73.235320077 1
23 31⋅23-1 24/23 m223 73.680653597 -3
13 324⋅51⋅131 18357919871265/17592186044416 -cAAA65,13 73.761396403
11 3-12⋅11-2 67108864/64304361 d311,11 73.904104886
7 322⋅7-1 31381059609/30064771072 -cAAA67 74.184112569
7 3-15⋅52⋅7-1 104857600/100442349 m25,57 74.476508280
11 35⋅5-1⋅111 2673/2560 P1115 74.779232827 9
7 3-16⋅73 44957696/43046721 ddd47,7,7 75.197705561
17 3-10⋅17-1 1048576/1003833 m217 75.494581846
13 32⋅7-1⋅131 117/112 P1137 75.611757031 4
7 5-1⋅7-2 256/245 P15,7,7 76.034473197 -8
11 3-21⋅71⋅11-1 120259084288/115063885233 ddd4711 76.452945931
11 313⋅111 17537553/16777216 -cA711 76.732953615
7 38⋅7-2 6561/6272 -cA77,7 77.988193985 7
13 3-7⋅111⋅131 2288/2187 d211,13 78.160598076 -4
31 312⋅31-1 531441/507904 A131 78.424437920
11 321⋅51⋅111 575319426165/549755813888 -cAAA65,11 78.686674403
13 3-15⋅11-1⋅13-1 2147483648/2051893701 d311,13 78.829382885
7 316⋅51⋅7-2 215233605/205520896 -cAAA657,7 79.941914773
29 35⋅29-1 243/232 A129 80.197810174 3
29 326⋅291 73714109021541/70368744177664 -cAAA629 80.407216653
11 3-1⋅7-1⋅111 22/21 P1117 80.537035030 -1
7 315⋅5-1⋅72 703096443/671088640 A17,75 80.663112054
7 3-22⋅51⋅72 32883343360/31381059609 ddd45,7,7 80.955507765
13 3-24⋅71⋅13-1 3848290697216/3671583974253 ddd4713 81.378223931
17 3-12⋅171 557056/531441 dd317 81.495399116
7 3-6⋅7-3 262144/250047 P17,7,7 81.792275400
11 310⋅5-1⋅11-1 59049/56320 A15,11 81.918352424
31 319⋅311 36030105477/34359738368 -cAAA631 82.180588907
7 3-7⋅5-2⋅71 57344/54675 dd375,5 82.513472682
7 323⋅72 4613015762523/4398046511104 -cAA77,7 82.616832842
11 3-10⋅112 61952/59049 d211,11 83.085876076 -9
23 3-23⋅231 98784247808/94143178827 dd323 83.309327365
13 3-18⋅13-2 68719476736/65474062641 d313,13 83.754660884
11 318⋅11-1 387420489/369098752 -cAA711 83.872073212
7 31⋅5-1⋅71 21/20 m275 84.467193470 2
13 3-2⋅11-1⋅131 104/99 m21311 85.299717674 -3
11 326⋅51⋅11-1 12709329141645/12094627905536 -cAAAA6511 85.825794000
5 3-21⋅5-2 274877906944/261508830075 dddd45,5 86.317554097
7 39⋅71 137781/131072 A17 86.420914258
13 37⋅5-1⋅13-1 2187/2080 A15,13 86.843630424 5
19 3-8⋅19-1 131072/124659 m219 86.846976945
11 34⋅7-1⋅11-1 81/77 A17,11 87.676154628 0
13 3-11⋅71⋅131 186368/177147 dd37,13 87.848558719
5 324⋅5-3 282429536481/268435456000 -cA75,5,5 87.978879175
5 3-13⋅5-1 8388608/7971615 dd35 88.271274885
7 317⋅51⋅71 4519905705/4294967296 -cAA75,7 88.374635046
13 315⋅13-1 14348907/13631488 -cAA713 88.797351212
3 3-5 256/243 m2 90.224995673 -8
7 325⋅52⋅71 148275506652525/140737488355328 -cAAAA65,5,7 90.328355833
13 323⋅51⋅13-1 470715894135/446676598784 -cAAAA6513 90.751071999
13 3-25⋅131 893353197568/847288609443 dddd413 91.652640135
5 33⋅51 135/128 A15 92.178716461 7
13 31⋅7-1⋅13-1 96/91 A17,13 92.601432627 -5
11 3-14⋅71⋅111 5046272/4782969 dd37,11 92.773836718
13 320⋅5-1⋅131 45328197213/42949672960 -cA7135 93.313965212
19 3-2⋅191 19/18 m219 93.603014402 1
13 3-17⋅51⋅131 136314880/129140163 dd35,13 93.606360923
7 318⋅5-2⋅7-1 387420489/367001600 -cA75,5,7 93.736681378
7 3-19⋅7-1 8589934592/8135830269 dd37 94.029077089
5 311⋅52 4428675/4194304 -cAA75,5 94.132437249
13 3-8⋅111⋅13-1 90112/85293 m21113 95.150273672
7 326⋅5-1⋅7-1 2541865828329/2405181685760 -cAAA65,7 95.690402166
7 3-11⋅51⋅7-1 1310720/1240029 m257 95.982797876
5 319⋅53 145282683375/137438953472 -cAAAA65,5,5 96.086158037
13 38⋅132 1108809/1048576 P113,13 96.695330462
23 313⋅23-1 1594323/1507328 A123 97.140663982
11 11-2 128/121 M211,11 97.364115270 -7
7 3-3⋅52⋅7-1 200/189 A15,57 97.936518664 -3
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