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User:Contribution/Collection of tunings

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Revision as of 15:57, 24 June 2025 by Contribution (talk | contribs) (→Equal-step tunings)

Equal-step tunings

  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, add more ED7 from the no-2 no-3 no-5 Riemann zeta (327ed7 is enough for ED7s). Add ED4s, add ED6s (detect those with a bad 2 and a bad 3, but with good results on other primes), add ED8s.

From the original Riemann zeta: 12edo, 14edo, 15edo, 17edo, 19edo, 22edo, 24edo, 26edo, 27edo, 29edo, 31edo, 34edo, 36edo, 41edo, 46edo, 50edo, 53edo, 58edo, 60edo, 65edo, 68edo, 72edo, 77edo, 80edo, 84edo, 87edo, 94edo, 99edo, 103edo, 111edo, 118edo, 130edo, 140edo, 152edo, 171edo From the no-2 Riemann zeta: 39edt, 56edt, 69edt, 71edt, 75edt, 78edt, 82edt, 88edt, 99edt, 101edt, 105edt, 110edt, 131edt, 140edt, 144edt, 153edt, 170edt, 183edt, 185edt, 202edt, 209edt, 213edt, 215edt, 219edt, 245edt

Equal divisions of a ratio & optimization

EDRs collection & 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
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