31-limit

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In 31-limit Just Intonation, all ratios in the system will contain no primes higher than 31.

Intervals

Here is a table of all 31-limit intervals with at most two primes from 5 to 31 in the ratio (excepting 5-limit intervals, which have at most three) — vertical axis, on a chain of 53 harmonic fifths — horizontal axis.

For each intervals, you'll find: its factorization (ignoring octaves), its just ratio, its Functional Just System notation, and its cents value (rounded to 9 decimal places).

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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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31-limit
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3-26⋅312

4127463571456/2541865828329

dd731,31

839.241122428
3-25⋅312

1031865892864/847288609443

dd431,31

341.196123294
3-24⋅312

515932946432/282429536481

dd831,31

1043.151124159
3-23⋅312

128983236608/94143178827

dd531,31

545.106125025
3-22⋅312

32245809152/31381059609

d231,31

47.061125890
3-21⋅312

16122904576/10460353203

d631,31

749.016126755
3-20⋅312

4030726144/3486784401

d331,31

250.971127621
3-19⋅312

2015363072/1162261467

d731,31

952.926128486
3-18⋅312

503840768/387420489

d431,31

454.881129352
3-17⋅312

251920384/129140163

d831,31

1156.836130217
3-16⋅312

62980096/43046721

d531,31

658.791131082
3-15⋅312

15745024/14348907

m231,31

160.746131948
3-14⋅312

7872512/4782969

m631,31

862.701132813
3-13⋅312

1968128/1594323

m331,31

364.656133678
3-12⋅312

984064/531441

m731,31

1066.611134544
3-11⋅312

246016/177147

P431,31

568.566135409
3-10⋅312

61504/59049

P131,31

70.521136275
3-9⋅312

30752/19683

P531,31

772.476137140
3-8⋅312

7688/6561

M231,31

274.431138005
3-7⋅312

3844/2187

M631,31

976.386138871
3-6⋅312

961/729

M331,31

478.341139736
3-5⋅312

961/486

M731,31

1180.296140602
3-4⋅312

961/648

A431,31

682.251141467
3-3⋅312

961/864

A131,31

184.206142332
3-2⋅312

961/576

A531,31

886.161143198
3-1⋅312

961/768

A231,31

388.116144063
312

961/512

A631,31

1090.071144929
31⋅312

2883/2048

A331,31

592.026145794
32⋅312

8649/8192

-cA731,31

93.981146659
33⋅312

25947/16384

AA431,31

795.936147525
34⋅312

77841/65536

AA131,31

297.891148390
35⋅312

233523/131072

AA531,31

999.846149255
36⋅312

700569/524288

AA231,31

501.801150121
37⋅312

2101707/2097152

-cAA631,31

3.756150986
38⋅312

6305121/4194304

AA331,31

705.711151852
39⋅312

18915363/16777216

-cAA731,31

207.666152717
310⋅312

56746089/33554432

AAA431,31

909.621153582
311⋅312

170238267/134217728

AAA131,31

411.576154448
312⋅312

510714801/268435456

AAA531,31

1113.531155313
313⋅312

1532144403/1073741824

AAA231,31

615.486156179
314⋅312

4596433209/4294967296

-cAAA631,31

117.441157044
315⋅312

13789299627/8589934592

AAA331,31

819.396157909
316⋅312

41367898881/34359738368

-cAAA731,31

321.351158775
317⋅312

124103696643/68719476736

AAAA431,31

1023.306159640
318⋅312

372311089929/274877906944

AAAA131,31

525.261160505
319⋅312

1116933269787/1099511627776

-cAAAA531,31

27.216161371
320⋅312

3350799809361/2199023255552

AAAA231,31

729.171162236
321⋅312

10052399428083/8796093022208

-cAAAA631,31

231.126163102
322⋅312

30157198284249/17592186044416

AAAA331,31

933.081163967
323⋅312

90471594852747/70368744177664

-cAAAA731,31

435.036164832
324⋅312

271414784558241/140737488355328

AAAAA431,31

1136.991165698
325⋅312

814244353674723/562949953421312

AAAAA131,31

638.946166563
326⋅312

2442733061024169/2251799813685248

-cAAAAA531,31

140.901167429
3-26⋅29-1⋅311

136339441844224/73714109021541

cdd23129

1064.628355811
3-25⋅29-1⋅311

34084860461056/24571369673847

dd63129

566.583356676
3-24⋅29-1⋅311

8521215115264/8190456557949

dd33129

68.538357542
3-23⋅29-1⋅311

4260607557632/2730152185983

dd73129

770.493358407
3-22⋅29-1⋅311

1065151889408/910050728661

dd43129

272.448359273
3-21⋅29-1⋅311

532575944704/303350242887

dd83129

974.403360138
3-20⋅29-1⋅311

133143986176/101116747629

dd53129

476.358361003
3-19⋅29-1⋅311

66571993088/33705582543

cd23129

1178.313361869
3-18⋅29-1⋅311

16642998272/11235194181

d63129

680.268362734
3-17⋅29-1⋅311

4160749568/3745064727

d33129

182.223363600
3-16⋅29-1⋅311

2080374784/1248354909

d73129

884.178364465
3-15⋅29-1⋅311

520093696/416118303

d43129

386.133365330
3-14⋅29-1⋅311

260046848/138706101

d83129

1088.088366196
3-13⋅29-1⋅311

65011712/46235367

d53129

590.043367061
3-12⋅29-1⋅311

16252928/15411789

m23129

91.998367927
3-11⋅29-1⋅311

8126464/5137263

m63129

793.953368792
3-10⋅29-1⋅311

2031616/1712421

m33129

295.908369657
3-9⋅29-1⋅311

1015808/570807

m73129

997.863370523
3-8⋅29-1⋅311

253952/190269

P43129

499.818371388
3-7⋅29-1⋅311

63488/63423

P13129

1.773372253
3-6⋅29-1⋅311

31744/21141

P53129

703.728373119
3-5⋅29-1⋅311

7936/7047

M23129

205.683373984
3-4⋅29-1⋅311

3968/2349

M63129

907.638374850
3-3⋅29-1⋅311

992/783

M33129

409.593375715
3-2⋅29-1⋅311

496/261

M73129

1111.548376580
3-1⋅29-1⋅311

124/87

A43129

613.503377446
29-1⋅311

31/29

A13129

115.458378311
31⋅29-1⋅311

93/58

A53129

817.413379177
32⋅29-1⋅311

279/232

A23129

319.368380042
33⋅29-1⋅311

837/464

A63129

1021.323380907
34⋅29-1⋅311

2511/1856

A33129

523.278381773
35⋅29-1⋅311

7533/7424

-cA73129

25.233382638
36⋅29-1⋅311

22599/14848

AA43129

727.188383503
37⋅29-1⋅311

67797/59392

AA13129

229.143384369
38⋅29-1⋅311

203391/118784

AA53129

931.098385234
39⋅29-1⋅311

610173/475136

AA23129

433.053386100
310⋅29-1⋅311

1830519/950272

AA63129

1135.008386965
311⋅29-1⋅311

5491557/3801088

AA33129

636.963387830
312⋅29-1⋅311

16474671/15204352

-cAA73129

138.918388696
313⋅29-1⋅311

49424013/30408704

AAA43129

840.873389561
314⋅29-1⋅311

148272039/121634816

AAA13129

342.828390427
315⋅29-1⋅311

444816117/243269632

AAA53129

1044.783391292
316⋅29-1⋅311

1334448351/973078528

AAA23129

546.738392157
317⋅29-1⋅311

4003345053/3892314112

-cAAA63129

48.693393023
318⋅29-1⋅311

12010035159/7784628224

AAA33129

750.648393888
319⋅29-1⋅311

36030105477/31138512896

-cAAA73129

252.603394754
320⋅29-1⋅311

108090316431/62277025792

AAAA43129

954.558395619
321⋅29-1⋅311

324270949293/249108103168

AAAA13129

456.513396484
322⋅29-1⋅311

972812847879/498216206336

AAAA53129

1158.468397350
323⋅29-1⋅311

2918438543637/1992864825344

AAAA23129

660.423398215
324⋅29-1⋅311

8755315630911/7971459301376

-cAAAA63129

162.378399080
325⋅29-1⋅311

26265946892733/15942918602752

AAAA33129

864.333399946
326⋅29-1⋅311

78797840678199/63771674411008

-cAAAA73129

366.288400811
3-26⋅291⋅311

3861175599104/2541865828329

ddd729,31

723.782744117
3-25⋅291⋅311

965293899776/847288609443

ddd429,31

225.737744983
3-24⋅291⋅311

482646949888/282429536481

ddd829,31

927.692745848
3-23⋅291⋅311

120661737472/94143178827

ddd529,31

429.647746713
3-22⋅291⋅311

60330868736/31381059609

cdd229,31

1131.602747579
3-21⋅291⋅311

15082717184/10460353203

dd629,31

633.557748444
3-20⋅291⋅311

3770679296/3486784401

dd329,31

135.512749310
3-19⋅291⋅311

1885339648/1162261467

dd729,31

837.467750175
3-18⋅291⋅311

471334912/387420489

dd429,31

339.422751040
3-17⋅291⋅311

235667456/129140163

dd829,31

1041.377751906
3-16⋅291⋅311

58916864/43046721

dd529,31

543.332752771
3-15⋅291⋅311

14729216/14348907

d229,31

45.287753637
3-14⋅291⋅311

7364608/4782969

d629,31

747.242754502
3-13⋅291⋅311

1841152/1594323

d329,31

249.197755367
3-12⋅291⋅311

920576/531441

d729,31

951.152756233
3-11⋅291⋅311

230144/177147

d429,31

453.107757098
3-10⋅291⋅311

115072/59049

d829,31

1155.062757963
3-9⋅291⋅311

28768/19683

d529,31

657.017758829
3-8⋅291⋅311

7192/6561

m229,31

158.972759694
3-7⋅291⋅311

3596/2187

m629,31

860.927760560
3-6⋅291⋅311

899/729

m329,31

362.882761425
3-5⋅291⋅311

899/486

m729,31

1064.837762290
3-4⋅291⋅311

899/648

P429,31

566.792763156
3-3⋅291⋅311

899/864

P129,31

68.747764021
3-2⋅291⋅311

899/576

P529,31

770.702764887
3-1⋅291⋅311

899/768

M229,31

272.657765752
291⋅311

899/512

M629,31

974.612766617
31⋅291⋅311

2697/2048

M329,31

476.567767483
32⋅291⋅311

8091/4096

M729,31

1178.522768348
33⋅291⋅311

24273/16384

A429,31

680.477769213
34⋅291⋅311

72819/65536

A129,31

182.432770079
35⋅291⋅311

218457/131072

A529,31

884.387770944
36⋅291⋅311

655371/524288

A229,31

386.342771810
37⋅291⋅311

1966113/1048576

A629,31

1088.297772675
38⋅291⋅311

5898339/4194304

A329,31

590.252773540
39⋅291⋅311

17695017/16777216

-cA729,31

92.207774406
310⋅291⋅311

53085051/33554432

AA429,31

794.162775271
311⋅291⋅311

159255153/134217728

AA129,31

296.117776137
312⋅291⋅311

477765459/268435456

AA529,31

998.072777002
313⋅291⋅311

1433296377/1073741824

AA229,31

500.027777867
314⋅291⋅311

4299889131/4294967296

-cAA629,31

1.982778733
315⋅291⋅311

12899667393/8589934592

AA329,31

703.937779598
316⋅291⋅311

38699002179/34359738368

-cAA729,31

205.892780464
317⋅291⋅311

116097006537/68719476736

AAA429,31

907.847781329
318⋅291⋅311

348291019611/274877906944

AAA129,31

409.802782194
319⋅291⋅311

1044873058833/549755813888

AAA529,31

1111.757783060
320⋅291⋅311

3134619176499/2199023255552

AAA229,31

613.712783925
321⋅291⋅311

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-cAAA629,31

115.667784790
322⋅291⋅311

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AAA329,31

817.622785656
323⋅291⋅311

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-cAAA729,31

319.577786521
324⋅291⋅311

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AAAA429,31

1021.532787387
325⋅291⋅311

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AAAA129,31

523.487788252
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-cAAAA529,31

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3-26⋅23-1⋅311

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dddd53123

265.931202696
3-25⋅23-1⋅311

34084860461056/19487638017189

cddd23123

967.886203561
3-24⋅23-1⋅311

8521215115264/6495879339063

ddd63123

469.841204427
3-23⋅23-1⋅311

4260607557632/2165293113021

cddd33123

1171.796205292
3-22⋅23-1⋅311

1065151889408/721764371007

ddd73123

673.751206157
3-21⋅23-1⋅311

266287972352/240588123669

ddd43123

175.706207023
3-20⋅23-1⋅311

133143986176/80196041223

ddd83123

877.661207888
3-19⋅23-1⋅311

33285996544/26732013741

ddd53123

379.616208753
3-18⋅23-1⋅311

16642998272/8910671247

cdd23123

1081.571209619
3-17⋅23-1⋅311

4160749568/2970223749

dd63123

583.526210484
3-16⋅23-1⋅311

1040187392/990074583

dd33123

85.481211350
3-15⋅23-1⋅311

520093696/330024861

dd73123

787.436212215
3-14⋅23-1⋅311

130023424/110008287

dd43123

289.391213080
3-13⋅23-1⋅311

65011712/36669429

dd83123

991.346213946
3-12⋅23-1⋅311

16252928/12223143

dd53123

493.301214811
3-11⋅23-1⋅311

8126464/4074381

cd23123

1195.256215677
3-10⋅23-1⋅311

2031616/1358127

d63123

697.211216542
3-9⋅23-1⋅311

507904/452709

d33123

199.166217407
3-8⋅23-1⋅311

253952/150903

d73123

901.121218273
3-7⋅23-1⋅311

63488/50301

d43123

403.076219138
3-6⋅23-1⋅311

31744/16767

d83123

1105.031220004
3-5⋅23-1⋅311

7936/5589

d53123

606.986220869
3-4⋅23-1⋅311

1984/1863

m23123

108.941221734
3-3⋅23-1⋅311

992/621

m63123

810.896222600
3-2⋅23-1⋅311

248/207

m33123

312.851223465
3-1⋅23-1⋅311

124/69

m73123

1014.806224330
23-1⋅311

31/23

P43123

516.761225196
31⋅23-1⋅311

93/92

P13123

18.716226061
32⋅23-1⋅311

279/184

P53123

720.671226927
33⋅23-1⋅311

837/736

M23123

222.626227792
34⋅23-1⋅311

2511/1472

M63123

924.581228657
35⋅23-1⋅311

7533/5888

M33123

426.536229523
36⋅23-1⋅311

22599/11776

M73123

1128.491230388
37⋅23-1⋅311

67797/47104

A43123

630.446231254
38⋅23-1⋅311

203391/188416

A13123

132.401232119
39⋅23-1⋅311

610173/376832

A53123

834.356232984
310⋅23-1⋅311

1830519/1507328

A23123

336.311233850
311⋅23-1⋅311

5491557/3014656

A63123

1038.266234715
312⋅23-1⋅311

16474671/12058624

A33123

540.221235580
313⋅23-1⋅311

49424013/48234496

-cA73123

42.176236446
314⋅23-1⋅311

148272039/96468992

AA43123

744.131237311
315⋅23-1⋅311

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AA13123

246.086238177
316⋅23-1⋅311

1334448351/771751936

AA53123

948.041239042
317⋅23-1⋅311

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AA23123

449.996239907
318⋅23-1⋅311

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AA63123

1151.951240773
319⋅23-1⋅311

36030105477/24696061952

AA33123

653.906241638
320⋅23-1⋅311

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-cAA73123

155.861242504
321⋅23-1⋅311

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AAA43123

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322⋅23-1⋅311

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AAA13123

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323⋅23-1⋅311

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AAA53123

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324⋅23-1⋅311

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AAA23123

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-cAAA63123

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326⋅23-1⋅311

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AAA33123

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3-26⋅231⋅311

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dd423,31

322.479897233
3-25⋅231⋅311

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dd823,31

1024.434898098
3-24⋅231⋅311

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See also

Harmonic limit