User:Contribution/Collection of tunings
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Equal-step tunings
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Todo: Add other EDRs add more EDOs from the no-3 Riemann zeta, the no-5 Riemann zeta, the no-3 no-5 Riemann zeta, add more EDTs from the no-2 Riemann zeta, the no-2 no-5 Riemann zeta, add more ED5 from the no-2 no-3 Riemann zeta, |
Equal divisions of a ratio & optimization
| EDRs | Optimization | Comments | ||||
|---|---|---|---|---|---|---|
| EDR | Steps per octave | Cents | Optimization | Steps per octave | Cents | Why it matters |
| 7ed5/3 | 9.49840814199707 | 126.336958999921 | Benson Alpha 5/3 | 9.50583353877785 | 126.238272015258 | Alpha 5/3 |
| 10edo | 10 | 120. | None | EDO ≤ 29 | ||
| 11edo | 11 | 109.090909090909 | None | EDO ≤ 29 | ||
| 12edo | 12 | 100. | 34zpi | 12.0231830072926 | 99.8071807833375 | EDO ≤ 29, strong zeta peak |
| 9ed5/3 | 12.2122390397105 | 98.2620792221608 | Benson Beta 5/3 | 12.2053823008782 | 98.3172808862904 | Beta 5/3 |
| 13edo | 13 | 92.3076923076923 | None | EDO ≤ 29 | ||
| 14edo | 14 | 85.7142857142857 | 42zpi | 13.9002525327005 | 86.3293668353859 | EDO ≤ 29, medium zeta peak |
| 15edo | 15 | 80. | 47zpi | 15.0534898676781 | 79.7157343943591 | EDO ≤ 29, medium zeta peak |
| 9ed3/2 | 15.3856016221631 | 77.9950000961542 | Benson Alpha 3/2 | 15.3915238996928 | 77.9649895501219 | Alpha 3/2 |
| 16edo | 16 | 75. | None | EDO ≤ 29 | ||
| 17edo | 17 | 70.5882352941176 | 56zpi | 17.0445886606675 | 70.4035764012981 | EDO ≤ 29, medium zeta peak |
| 18edo | 18 | 66.6666666666667 | None | EDO ≤ 29 | ||
| 11ed3/2 | 18.8046242048660 | 63.8140909877625 | Benson Beta 3/2 | 18.7990736394111 | 63.8329325698408 | Beta 3/2 |
| 19edo | 19 | 63.1578947368421 | 65zpi | 18.9480867166984 | 63.3309324546460 | EDO ≤ 29, strong zeta peak |
| 20edo | 20 | 60. | None | EDO ≤ 29 | ||
| 21edo | 21 | 57.1428571428571 | None | EDO ≤ 29 | ||
| 16ed5/3 | 21.7106471817076 | 55.2724195624655 | Benson Gamma 5/3 | 21.7094399215509 | 55.2754932571412 | Gamma 5/3 |
| 22edo | 22 | 54.5454545454545 | 80zpi | 22.0251467420146 | 54.4831784348982 | EDO ≤ 29, strong zeta peak |
| 11ed7/5 | 22.6604698881676 | 52.9556538731173 | Benson Alpha 7/5 | 22.6653911133366 | 52.9441558718088 | Alpha 7/5 |
| 23edo | 23 | 52.1739130434783 | None | EDO ≤ 29 | ||
| 24edo | 24 | 50. | 90zpi | 24.0057421830853 | 49.9880399800983 | EDO ≤ 29, medium zeta peak |
| 39edt | 24.6062603892868 | 48.7680769452663 | 93zpi no-2 analogue | 24.5738316304204 | 48.8324335434323 | strong no-2 zeta peak |
| 25edo | 25 | 48. | None | EDO ≤ 29 | ||
| 26edo | 26 | 46.1538461538462 | 100zpi | 25.9356996537225 | 46.2682717652372 | EDO ≤ 29, medium zeta peak |
| 13ed7/5 | 26.7805553223799 | 44.8086302003300 | Benson Beta 7/5 | 26.7758951088566 | 44.8164289231577 | Beta 7/5 |
| 27edo | 27 | 44.4444444444444 | 106zpi | 27.0866140827635 | 44.3023257293579 | EDO ≤ 29, strong zeta peak |
| 28edo | 28 | 42.8571428571429 | None | EDO ≤ 29 | ||
| 29edo | 29 | 41.3793103448276 | 116zpi | 28.9399661541990 | 41.4651487014917 | EDO ≤ 29, medium zeta peak |
| 31edo | 31 | 38.7096774193548 | 127zpi | 30.9783816349790 | 38.7366910944446 | strong zeta peak |
| 13ed4/3 | 31.3224709154917 | 38.3111537795856 | Benson Alpha 4/3 | 31.3266790320926 | 38.3060074376432 | Alpha 4/3 |
| 34edo | 34 | 35.2941176470588 | 144zpi | 34.0448410043159 | 35.2476312005063 | strong zeta peak |
| 20ed3/2 | 34.1902258270291 | 35.0977500432694 | Benson Gamma 3/2 | 34.1894540921914 | 35.0985422804417 | Gamma 3/2 |
| 56edt | 35.3320662000016 | 33.9634821583105 | 151zpi no-2 analogue | 35.3059427335609 | 33.9886123153798 | strong no-2 zeta peak |
| 36edo | 36 | 33.3333333333333 | 155zpi no-5 analogue | 35.9775957344990 | 33.3540909419168 | strong no-5 zeta peak |
| 15ed4/3 | 36.1413125947981 | 33.2029999423075 | Benson Beta 4/3 | 36.1372975038827 | 33.2066890135066 | Beta 4/3 |
| 37edo | 37 | 32.4324324324324 | 161zpi no-3 analogue | 37.0117501336435 | 32.4221360964286 | strong no-3 zeta peak |
| 41edo | 41 | 29.2682926829268 | 184zpi | 40.9880783925993 | 29.2768055263764 | strong zeta peak |
| 96ed5 | 41.3449495750457 | 29.0241011860920 | 186zpi no-2 no-3 analogue | 41.3477989230936 | 29.0221010852836 | strong no-2 no-3 zeta peak |
| 66edt | 41.6413637357162 | 28.8175000131119 | 188zpi no-2 no-5 analogue | 41.6281274155763 | 28.8266629920756 | strong no-2 no-5 zeta peak |
| 46edo | 46 | 26.0869565217391 | 214zpi | 46.0089748051542 | 26.0818678330031 | strong zeta peak |
| 24ed7/5 | 49.4410252105475 | 24.2713413585121 | Benson Gamma 7/5 | 49.4404896216012 | 24.2716042900130 | Gamma 7/5 |
| 50edo | 50 | 24.0 | 238zpi | 49.9385162652878 | 24.0295485277387 | medium zeta peak |
| 53edo | 53 | 22.6415094339623 | 257zpi | 52.9968291550147 | 22.6428640945673 | strong zeta peak |
| 57edo | 57 | 21.0526315789474 | 282zpi no-3 no-5 analogue | 56.9949885079207 | 21.0544827083040 | strong no-3 no-5 zeta peak |
| 58edo | 58 | 20.6896551724138 | 289zpi | 58.0667185533159 | 20.6658827964969 | strong zeta peak |
| 60edo | 60 | 20. | 301zpi | 59.9201656607655 | 20.0266469020418 | medium zeta peak |
| 65edo | 65 | 18.4615384615385 | 334zpi | 65.0158450885860 | 18.4570391781413 | strong zeta peak |
| 28ed4/3 | 67.4637835102899 | 17.7873213976647 | Benson Gamma 4/3 | 67.4633901646646 | 17.7874251067289 | Gamma 4/3 |
| 68edo | 68 | 17.6470588235294 | 354zpi | 68.0493056282519 | 17.6342725163943 | strong zeta peak |
| 72edo | 72 | 16.6666666666667 | 380zpi | 71.9506065993786 | 16.6781081733140 | strong zeta peak |
| 77edo | 77 | 15.5844155844156 | 414zpi | 76.9918536925042 | 15.5860645308353 | strong zeta peak |
| 80edo | 80 | 15. | 435zpi | 80.0731374302484 | 14.9862992572924 | medium zeta peak |
| 131edt | 82.6517977178609 | 14.5187404646213 | 453zpi no-2 analogue | 82.6705208991009 | 14.5154522670130 | strong no-2 zeta peak |
| 83edo | 83 | 14.4578313253012 | 455zpi no-3 no-5 analogue | 82.9585473728587 | 14.4650555970632 | strong no-3 no-5 zeta peak |
| 84edo | 84 | 14.2857142857143 | 462zpi | 83.9972142607288 | 14.2861880666087 | medium zeta peak |
| 87edo | 87 | 13.7931034482759 | 483zpi | 87.0139255957575 | 13.7908960178956 | strong zeta peak |
| 94edo | 94 | 12.7659574468085 | 532zpi | 93.9836761074943 | 12.7681747480009 | strong zeta peak |
| 99edo | 99 | 12.1212121212121 | 568zpi | 99.0473345956631 | 12.1154194093028 | strong zeta peak |
| 103edo | 103 | 11.6504854368932 | 596zpi | 102.936629522070 | 11.6576577800491 | medium zeta peak |
| 111edo | 111 | 10.8108108108108 | 655zpi | 111.059577998833 | 10.8050113427643 | medium zeta peak |
| 327ed7 | 116.479750184323 | 10.3022198974591 | 695zpi no-2 no-3 no-5 analogue | 116.481879086492 | 10.3020316070705 | strong no-2 no-3 no-5 zeta peak |
| 118edo | 118 | 10.1694915254237 | 706zpi | 117.969513574257 | 10.1721195895637 | strong zeta peak |
| 130edo | 130 | 9.23076923076923 | 796zpi | 130.003910460506 | 9.23049157328654 | strong zeta peak |
| 140edo | 140 | 8.57142857142857 | 872zpi | 139.990541024216 | 8.57200773152536 | strong zeta peak |
| 152edo | 152 | 7.89473684210526 | 965zpi | 152.052848107925 | 7.89199291517551 | strong zeta peak |
| 171edo | 171 | 7.01754385964912 | 1114zpi | 170.995891689006 | 7.01771246166817 | exceptionally strong zeta peak |
| 270edo | 270 | 4.44444444444444 | 1936zpi | 270.017794631965 | 4.44415154799558 | exceptionally strong zeta peak |
| 311edo | 311 | 3.85852090032154 | 2293zpi | 311.004029926555 | 3.85847090239759 | exceptionally strong zeta peak |
| 342edo | 342 | 3.50877192982456 | None | 171*2^n family | ||
| 684edo | 684 | 1.75438596491228 | None | 171*2^n family | ||
Unequal-step tunings
Unequal-step tunings from equal divisions of a ratio
| Tuning | Period | Mode | Why it matters |
|---|---|---|---|
| Stretched hemififth | 94\93<2/1> | 16 11 16 12 16 11 12 | |
| 833 Cent Acoustic Golden Scale [11] | 25\36<2/1> | 3 1 3 3 1 3 1 3 3 1 3 | |
| 833 Cent Logarithmic Golden Scale [8] | ϕ | ϕ 1 ϕ ϕ 1 ϕ 1 ϕ |