User:Contribution/31-limit
♥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.
The rules to get this table are:
- 1/1 is the center of symmetry between reciprocals.
- The positive abscissa is the positive Pythagorean scale. The negative abscissa is deducted by rule 1.
- The positive ordinate is the positive quasi-prime scale (quasi-prime means one or two primes in the ratio, positive means the highest prime is in the numerator) from 5 to 31, in this logical order:
- a ; a*a ; 5-limit allows also a*a*a to show in its reciprocal the worth mentioning diesis ;
- b ; a*b ; a-1*b ; b*b ;
- c ; a*c ; a-1*c ; b*c ; b-1*c ; c*c ;
- d ; a*d ; a-1*d ; b*d ; b-1*d ; c*d ; c-1*d ; d*d ;
- e ; a*e ; a-1*e ; b*e ; b-1*e ; c*e ; c-1*e ; d*e ; d-1*e ; e*e ;
- etc...
- The negative ordinate is deducted by rule 1.
For each interval, 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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4127463571456/2541865828329 dd731,31 839.241122428 |
1031865892864/847288609443 dd431,31 341.196123294 |
515932946432/282429536481 dd831,31 1043.151124159 |
128983236608/94143178827 dd531,31 545.106125025 |
32245809152/31381059609 d231,31 47.061125890 |
16122904576/10460353203 d631,31 749.016126755 |
4030726144/3486784401 d331,31 250.971127621 |
2015363072/1162261467 d731,31 952.926128486 |
503840768/387420489 d431,31 454.881129352 |
251920384/129140163 d831,31 1156.836130217 |
62980096/43046721 d531,31 658.791131082 |
15745024/14348907 m231,31 160.746131948 |
7872512/4782969 m631,31 862.701132813 |
1968128/1594323 m331,31 364.656133678 |
984064/531441 m731,31 1066.611134544 |
246016/177147 P431,31 568.566135409 |
61504/59049 P131,31 70.521136275 |
30752/19683 P531,31 772.476137140 |
7688/6561 M231,31 274.431138005 |
3844/2187 M631,31 976.386138871 |
961/729 M331,31 478.341139736 |
961/486 M731,31 1180.296140602 |
961/648 A431,31 682.251141467 |
961/864 A131,31 184.206142332 |
961/576 A531,31 886.161143198 |
961/768 A231,31 388.116144063 |
961/512 A631,31 1090.071144929 |
2883/2048 A331,31 592.026145794 |
8649/8192 -cA731,31 93.981146659 |
25947/16384 AA431,31 795.936147525 |
77841/65536 AA131,31 297.891148390 |
233523/131072 AA531,31 999.846149255 |
700569/524288 AA231,31 501.801150121 |
2101707/2097152 -cAA631,31 3.756150986 |
6305121/4194304 AA331,31 705.711151852 |
18915363/16777216 -cAA731,31 207.666152717 |
56746089/33554432 AAA431,31 909.621153582 |
170238267/134217728 AAA131,31 411.576154448 |
510714801/268435456 AAA531,31 1113.531155313 |
1532144403/1073741824 AAA231,31 615.486156179 |
4596433209/4294967296 -cAAA631,31 117.441157044 |
13789299627/8589934592 AAA331,31 819.396157909 |
41367898881/34359738368 -cAAA731,31 321.351158775 |
124103696643/68719476736 AAAA431,31 1023.306159640 |
372311089929/274877906944 AAAA131,31 525.261160505 |
1116933269787/1099511627776 -cAAAA531,31 27.216161371 |
3350799809361/2199023255552 AAAA231,31 729.171162236 |
10052399428083/8796093022208 -cAAAA631,31 231.126163102 |
30157198284249/17592186044416 AAAA331,31 933.081163967 |
90471594852747/70368744177664 -cAAAA731,31 435.036164832 |
271414784558241/140737488355328 AAAAA431,31 1136.991165698 |
814244353674723/562949953421312 AAAAA131,31 638.946166563 |
2442733061024169/2251799813685248 -cAAAAA531,31 140.901167429 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
136339441844224/73714109021541 cdd23129 1064.628355811 |
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8521215115264/8190456557949 dd33129 68.538357542 |
4260607557632/2730152185983 dd73129 770.493358407 |
1065151889408/910050728661 dd43129 272.448359273 |
532575944704/303350242887 dd83129 974.403360138 |
133143986176/101116747629 dd53129 476.358361003 |
66571993088/33705582543 cd23129 1178.313361869 |
16642998272/11235194181 d63129 680.268362734 |
4160749568/3745064727 d33129 182.223363600 |
2080374784/1248354909 d73129 884.178364465 |
520093696/416118303 d43129 386.133365330 |
260046848/138706101 d83129 1088.088366196 |
65011712/46235367 d53129 590.043367061 |
16252928/15411789 m23129 91.998367927 |
8126464/5137263 m63129 793.953368792 |
2031616/1712421 m33129 295.908369657 |
1015808/570807 m73129 997.863370523 |
253952/190269 P43129 499.818371388 |
63488/63423 P13129 1.773372253 |
31744/21141 P53129 703.728373119 |
7936/7047 M23129 205.683373984 |
3968/2349 M63129 907.638374850 |
992/783 M33129 409.593375715 |
496/261 M73129 1111.548376580 |
124/87 A43129 613.503377446 |
31/29 A13129 115.458378311 |
93/58 A53129 817.413379177 |
279/232 A23129 319.368380042 |
837/464 A63129 1021.323380907 |
2511/1856 A33129 523.278381773 |
7533/7424 -cA73129 25.233382638 |
22599/14848 AA43129 727.188383503 |
67797/59392 AA13129 229.143384369 |
203391/118784 AA53129 931.098385234 |
610173/475136 AA23129 433.053386100 |
1830519/950272 AA63129 1135.008386965 |
5491557/3801088 AA33129 636.963387830 |
16474671/15204352 -cAA73129 138.918388696 |
49424013/30408704 AAA43129 840.873389561 |
148272039/121634816 AAA13129 342.828390427 |
444816117/243269632 AAA53129 1044.783391292 |
1334448351/973078528 AAA23129 546.738392157 |
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2918438543637/1992864825344 AAAA23129 660.423398215 |
8755315630911/7971459301376 -cAAAA63129 162.378399080 |
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78797840678199/63771674411008 -cAAAA73129 366.288400811 |
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965293899776/847288609443 ddd429,31 225.737744983 |
482646949888/282429536481 ddd829,31 927.692745848 |
120661737472/94143178827 ddd529,31 429.647746713 |
60330868736/31381059609 cdd229,31 1131.602747579 |
15082717184/10460353203 dd629,31 633.557748444 |
3770679296/3486784401 dd329,31 135.512749310 |
1885339648/1162261467 dd729,31 837.467750175 |
471334912/387420489 dd429,31 339.422751040 |
235667456/129140163 dd829,31 1041.377751906 |
58916864/43046721 dd529,31 543.332752771 |
14729216/14348907 d229,31 45.287753637 |
7364608/4782969 d629,31 747.242754502 |
1841152/1594323 d329,31 249.197755367 |
920576/531441 d729,31 951.152756233 |
230144/177147 d429,31 453.107757098 |
115072/59049 d829,31 1155.062757963 |
28768/19683 d529,31 657.017758829 |
7192/6561 m229,31 158.972759694 |
3596/2187 m629,31 860.927760560 |
899/729 m329,31 362.882761425 |
899/486 m729,31 1064.837762290 |
899/648 P429,31 566.792763156 |
899/864 P129,31 68.747764021 |
899/576 P529,31 770.702764887 |
899/768 M229,31 272.657765752 |
899/512 M629,31 974.612766617 |
2697/2048 M329,31 476.567767483 |
8091/4096 M729,31 1178.522768348 |
24273/16384 A429,31 680.477769213 |
72819/65536 A129,31 182.432770079 |
218457/131072 A529,31 884.387770944 |
655371/524288 A229,31 386.342771810 |
1966113/1048576 A629,31 1088.297772675 |
5898339/4194304 A329,31 590.252773540 |
17695017/16777216 -cA729,31 92.207774406 |
53085051/33554432 AA429,31 794.162775271 |
159255153/134217728 AA129,31 296.117776137 |
477765459/268435456 AA529,31 998.072777002 |
1433296377/1073741824 AA229,31 500.027777867 |
4299889131/4294967296 -cAA629,31 1.982778733 |
12899667393/8589934592 AA329,31 703.937779598 |
38699002179/34359738368 -cAA729,31 205.892780464 |
116097006537/68719476736 AAA429,31 907.847781329 |
348291019611/274877906944 AAA129,31 409.802782194 |
1044873058833/549755813888 AAA529,31 1111.757783060 |
3134619176499/2199023255552 AAA229,31 613.712783925 |
9403857529497/8796093022208 -cAAA629,31 115.667784790 |
28211572588491/17592186044416 AAA329,31 817.622785656 |
84634717765473/70368744177664 -cAAA729,31 319.577786521 |
253904153296419/140737488355328 AAAA429,31 1021.532787387 |
761712459889257/562949953421312 AAAA129,31 523.487788252 |
2285137379667771/2251799813685248 -cAAAA529,31 25.442789117 |
68169720922112/58462914051567 dddd53123 265.931202696 |
34084860461056/19487638017189 cddd23123 967.886203561 |
8521215115264/6495879339063 ddd63123 469.841204427 |
4260607557632/2165293113021 cddd33123 1171.796205292 |
1065151889408/721764371007 ddd73123 673.751206157 |
266287972352/240588123669 ddd43123 175.706207023 |
133143986176/80196041223 ddd83123 877.661207888 |
33285996544/26732013741 ddd53123 379.616208753 |
16642998272/8910671247 cdd23123 1081.571209619 |
4160749568/2970223749 dd63123 583.526210484 |
1040187392/990074583 dd33123 85.481211350 |
520093696/330024861 dd73123 787.436212215 |
130023424/110008287 dd43123 289.391213080 |
65011712/36669429 dd83123 991.346213946 |
16252928/12223143 dd53123 493.301214811 |
8126464/4074381 cd23123 1195.256215677 |
2031616/1358127 d63123 697.211216542 |
507904/452709 d33123 199.166217407 |
253952/150903 d73123 901.121218273 |
63488/50301 d43123 403.076219138 |
31744/16767 d83123 1105.031220004 |
7936/5589 d53123 606.986220869 |
1984/1863 m23123 108.941221734 |
992/621 m63123 810.896222600 |
248/207 m33123 312.851223465 |
124/69 m73123 1014.806224330 |
31/23 P43123 516.761225196 |
93/92 P13123 18.716226061 |
279/184 P53123 720.671226927 |
837/736 M23123 222.626227792 |
2511/1472 M63123 924.581228657 |
7533/5888 M33123 426.536229523 |
22599/11776 M73123 1128.491230388 |
67797/47104 A43123 630.446231254 |
203391/188416 A13123 132.401232119 |
610173/376832 A53123 834.356232984 |
1830519/1507328 A23123 336.311233850 |
5491557/3014656 A63123 1038.266234715 |
16474671/12058624 A33123 540.221235580 |
49424013/48234496 -cA73123 42.176236446 |
148272039/96468992 AA43123 744.131237311 |
444816117/385875968 AA13123 246.086238177 |
1334448351/771751936 AA53123 948.041239042 |
4003345053/3087007744 AA23123 449.996239907 |
12010035159/6174015488 AA63123 1151.951240773 |
36030105477/24696061952 AA33123 653.906241638 |
108090316431/98784247808 -cAA73123 155.861242504 |
324270949293/197568495616 AAA43123 857.816243369 |
972812847879/790273982464 AAA13123 359.771244234 |
2918438543637/1580547964928 AAA53123 1061.726245100 |
8755315630911/6322191859712 AAA23123 563.681245965 |
26265946892733/25288767438848 -cAAA63123 65.636246831 |
78797840678199/50577534877696 AAA33123 767.591247696 |
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1531155841024/847288609443 dd823,31 1024.434898098 |
382788960256/282429536481 dd523,31 526.389898963 |
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11962155008/10460353203 d323,31 232.254901560 |
5981077504/3486784401 d723,31 934.209902425 |
1495269376/1162261467 d423,31 436.164903290 |
747634688/387420489 d823,31 1138.119904156 |
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2920448/1594323 m723,31 1047.894908483 |
730112/531441 P423,31 549.849909348 |
182528/177147 P123,31 51.804910213 |
91264/59049 P523,31 753.759911079 |
22816/19683 M223,31 255.714911944 |
11408/6561 M623,31 957.669912810 |
2852/2187 M323,31 459.624913675 |
1426/729 M723,31 1161.579914540 |
713/486 A423,31 663.534915406 |
713/648 A123,31 165.489916271 |
713/432 A523,31 867.444917137 |
713/576 A223,31 369.399918002 |
713/384 A623,31 1071.354918867 |
713/512 A323,31 573.309919733 |
2139/2048 -cA723,31 75.264920598 |
6417/4096 AA423,31 777.219921463 |
19251/16384 AA123,31 279.174922329 |
57753/32768 AA523,31 981.129923194 |
173259/131072 AA223,31 483.084924060 |
519777/262144 AA623,31 1185.039924925 |
1559331/1048576 AA323,31 686.994925790 |
4677993/4194304 -cAA723,31 188.949926656 |
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1040187392/817887699 d43119 416.242542486 |
520093696/272629233 d83119 1118.197543351 |
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507904/373977 P43119 529.927548543 |
126976/124659 P13119 31.882549409 |
63488/41553 P53119 733.837550274 |
15872/13851 M23119 235.792551140 |
7936/4617 M63119 937.747552005 |
1984/1539 M33119 439.702552870 |
992/513 M73119 1141.657553736 |
248/171 A43119 643.612554601 |
62/57 A13119 145.567555467 |
31/19 A53119 847.522556332 |
93/76 A23119 349.477557197 |
279/152 A63119 1051.432558063 |
837/608 A33119 553.387558928 |
2511/2432 -cA73119 55.342559793 |
7533/4864 AA43119 757.297560659 |
22599/19456 AA13119 259.252561524 |
67797/38912 AA53119 961.207562390 |
203391/155648 AA23119 463.162563255 |
610173/311296 AA63119 1165.117564120 |
1830519/1245184 AA33119 667.072564986 |
5491557/4980736 -cAA73119 169.027565851 |
16474671/9961472 AAA43119 870.982566717 |
49424013/39845888 AAA13119 372.937567582 |
148272039/79691776 AAA53119 1074.892568447 |
444816117/318767104 AAA23119 576.847569313 |
1334448351/1275068416 -cAAA63119 78.802570178 |
4003345053/2550136832 AAA33119 780.757571044 |
12010035159/10200547328 -cAAA73119 282.712571909 |
36030105477/20401094656 AAAA43119 984.667572774 |
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324270949293/163208757248 AAAA53119 1188.577574505 |
972812847879/652835028992 AAAA23119 690.532575370 |
2918438543637/2611340115968 -cAAAA63119 192.487576236 |
8755315630911/5222680231936 AAAA33119 894.442577101 |
26265946892733/20890720927744 -cAAAA73119 396.397577967 |
78797840678199/41781441855488 AAAAA43119 1098.352578832 |
5059471474688/2541865828329 cddd319,31 1191.718566096 |
1264867868672/847288609443 ddd719,31 693.673566962 |
316216967168/282429536481 ddd419,31 195.628567827 |
158108483584/94143178827 ddd819,31 897.583568693 |
39527120896/31381059609 ddd519,31 399.538569558 |
19763560448/10460353203 cdd219,31 1101.493570423 |
4940890112/3486784401 dd619,31 603.448571289 |
1235222528/1162261467 dd319,31 105.403572154 |
617611264/387420489 dd719,31 807.358573020 |
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24571369673847/17592186044416 AAA229 578.452215788 |
73714109021541/70368744177664 -cAAA629 80.407216653 |
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~~ Limit Border ~~ ↓ 23-limit ↓ |
~~ Limit Border ~~ ↓ 23-limit ↓ |
~~ Limit Border ~~ ↓ 23-limit ↓ |
~~ Limit Border ~~ ↓ 23-limit ↓ |
~~ Limit Border ~~ ↓ 23-limit ↓ |
~~ Limit Border ~~ ↓ 23-limit ↓ |
~~ Limit Border ~~ ↓ 23-limit ↓ |
~~ Limit Border ~~ ↓ 23-limit ↓ |
~~ Limit Border ~~ ↓ 23-limit ↓ |
~~ Limit Border ~~ ↓ 23-limit ↓ |
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