User:Contribution/Collection of tunings
Equal-step tunings
About this list
The table that follows is not a “best-of” roster but a modest snapshot of equal-step tunings that happen to score highly under a few specific mathematical lenses. In particular, it gathers:
- Prominent peak counts from the classic Riemann zeta function
- Prominent peaks after removing the prime 2 from the zeta product
- Prominent peaks after removing the prime 3
- Prominent peaks after simultaneously removing the primes 2 and 3
- The α–β–γ family, with an equave sliding from 3/1 down to 4/3
These tunings earn the label “optimized” only relative to the limited set of zeta-derived functions explored here. In practice, you can make convincing music with any equal-step interval, every real-number step size repeated ad infinitum forms its own viable lattice. When you layer many differently pruned zeta functions in a tool such as Wolfram Mathematica, striking peaks emerge almost everywhere; the peaks simply shift as each combination of omitted primes reshapes the landscape. That ubiquity means there is no absolute “good” or “bad” equal-step tuning, only different alignments of primes that reveal different musical affordances.
Consequently, the list below is inherently biased toward a handful of functions and can only hint at the boundless diversity of xenharmonic equal-step systems. Treat it as a useful starting palette, not a definitive canon.
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
From the no-3 Riemann zeta: 16edo, 21edo, 25edo, (28edo), 35edo, 37edo, 43edo, 47edo, (52edo), 56edo, (66edo), 74edo, 78edo, 93edo, 109edo, 124edo
From the no-2 no-3 Riemann zeta:
From the alpha-beta-gamma set: 5ed2/1, 7ed2/1, 12ed2/1, 7ed5/3, 9ed5/3, 16ed5/3, 9ed3/2, 11ed3/2, 20ed3/2, 11ed7/5, 13ed7/5, 24ed7/5, 13ed4/3, 15ed4/3, 28ed4/3
| Tuning | Strength | Closest EDO | Integer limit | ||||||
|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per octave | Cents | Height | Integral | Gap | EDO | Octave | Consistent | Distinct |
| 34zpi | 12.0231830072926 | 99.8071807833375 | 5.193290 | 1.269599 | 15.899282 | 12edo | 1197.68616940005 | 10 | 6 |
| 42zpi | 13.9002525327005 | 86.3293668353859 | 4.592177 | 0.984037 | 14.097244 | 14edo | 1208.61113569540 | 7 | 5 |
| 47zpi | 15.0534898676781 | 79.7157343943591 | 5.050324 | 1.104057 | 14.918297 | 15edo | 1195.73601591539 | 8 | 7 |
| 56zpi | 17.0445886606675 | 70.4035764012981 | 5.056957 | 1.032175 | 14.269437 | 17edo | 1196.86079882207 | 4 | 4 |
| 65zpi | 18.9480867166984 | 63.3309324546460 | 5.980169 | 1.313799 | 16.699651 | 19edo | 1203.28771663827 | 10 | 7 |
| 80zpi | 22.0251467420146 | 54.4831784348982 | 6.062600 | 1.258178 | 16.213941 | 22edo | 1198.62992556776 | 12 | 8 |
| 90zpi | 24.0057421830853 | 49.9880399800983 | 5.721613 | 1.092055 | 14.821136 | 24edo | 1199.71295952236 | 6 | 6 |
| 100zpi | 25.9356996537225 | 46.2682717652372 | 5.545073 | 1.031155 | 14.793013 | 26edo | 1202.97506589617 | 14 | 9 |
| 106zpi | 27.0866140827635 | 44.3023257293579 | 6.069233 | 1.185939 | 16.215619 | 27edo | 1196.16279469266 | 10 | 8 |
| 116zpi | 28.9399661541990 | 41.4651487014917 | 5.566209 | 1.000619 | 14.904418 | 29edo | 1202.48931234326 | 8 | 7 |
| 127zpi | 30.9783816349790 | 38.7366910944446 | 7.003472 | 1.403777 | 17.739476 | 31edo | 1200.83742392778 | 12 | 9 |
| 144zpi | 34.0448410043159 | 35.2476312005063 | 6.685147 | 1.241437 | 16.236989 | 34edo | 1198.41946081721 | 6 | 6 |
| 155zpi | 35.9823877000425 | 33.3496490006021 | 6.027497 | 1.028887 | 14.706508 | 36edo | 1200.58736402167 | 8 | 8 |
| 184zpi | 40.9880783925993 | 29.2768055263764 | 7.570230 | 1.423937 | 17.722623 | 41edo | 1200.34902658143 | 16 | 10 |
| 214zpi | 46.0089748051542 | 26.0818678330031 | 7.495674 | 1.356067 | 17.747832 | 46edo | 1199.76592031814 | 14 | 11 |
| 238zpi | 49.9385162652878 | 24.0295485277387 | 6.655352 | 1.111229 | 15.942083 | 50edo | 1201.47742638693 | 10 | 9 |
| 257zpi | 52.9968291550147 | 22.6428640945673 | 8.249774 | 1.486620 | 18.069918 | 53edo | 1200.07179701207 | 10 | 10 |
| 289zpi | 58.0667185533159 | 20.6658827964969 | 7.814035 | 1.358357 | 18.056292 | 58edo | 1198.62120219682 | 16 | 12 |
| 301zpi | 59.9201656607655 | 20.0266469020418 | 7.046396 | 1.131000 | 15.932359 | 60edo | 1201.59881412251 | 10 | 10 |
| 334zpi | 65.0158450885860 | 18.4570391781413 | 7.813349 | 1.269821 | 16.514861 | 65edo | 1199.70754657919 | 6 | 6 |
| 354zpi | 68.0493056282519 | 17.6342725163943 | 7.666604 | 1.254592 | 17.034505 | 68edo | 1199.13053111481 | 10 | 10 |
| 380zpi | 71.9506065993786 | 16.6781081733140 | 9.157547 | 1.625363 | 19.964746 | 72edo | 1200.82378847861 | 18 | 13 |
| 414zpi | 76.9918536925042 | 15.5860645308353 | 8.194847 | 1.311364 | 17.029289 | 77edo | 1200.12696887432 | 10 | 10 |
| 435zpi | 80.0731374302484 | 14.9862992572924 | 7.873146 | 1.247325 | 17.087322 | 80edo | 1198.90394058339 | 12 | 12 |
| 462zpi | 83.9972142607288 | 14.2861880666087 | 8.020965 | 1.241945 | 16.733121 | 84edo | 1200.03979759513 | 10 | 10 |
| 483zpi | 87.0139255957575 | 13.7908960178956 | 8.869041 | 1.439474 | 18.061741 | 87edo | 1199.80795355692 | 16 | 14 |
| 532zpi | 93.9836761074943 | 12.7681747480009 | 8.806201 | 1.394050 | 17.832744 | 94edo | 1200.20842631208 | 24 | 15 |
| 568zpi | 99.0473345956631 | 12.1154194093028 | 9.406495 | 1.510412 | 18.536483 | 99edo | 1199.42652152097 | 12 | 12 |
| 596zpi | 102.936629522070 | 11.6576577800491 | 8.543510 | 1.340775 | 18.270998 | 103edo | 1200.73875134506 | 15 | 15 |
| 655zpi | 111.059577998833 | 10.8050113427643 | 9.038544 | 1.394739 | 18.041165 | 111edo | 1199.35625904684 | 22 | 16 |
| 706zpi | 117.969513574257 | 10.1721195895637 | 9.850823 | 1.544280 | 18.861062 | 118edo | 1200.31011156852 | 12 | 12 |
| 796zpi | 130.003910460506 | 9.23049157328654 | 10.355108 | 1.634018 | 19.594551 | 130edo | 1199.96390452725 | 16 | 16 |
| 872zpi | 139.990541024216 | 8.57200773152536 | 10.076688 | 1.548424 | 19.514765 | 140edo | 1200.08108241355 | 10 | 10 |
| 965zpi | 152.052848107925 | 7.89199291517551 | 10.468420 | 1.593855 | 19.487224 | 152edo | 1199.58292310668 | 15 | 15 |
| 1114zpi | 170.995891689006 | 7.01771246166817 | 11.076998 | 1.652856 | 19.091741 | 171edo | 1200.02883094526 | 14 | 14 |
| Optimization | Equal division of a ratio | |||
|---|---|---|---|---|
| Proposed name | Steps per octave | Cents | Optimization method | |
| Alpha 3/1 | 1.90739592696007 | 629.130000247254 | Dave Benson | 3ed3/1 |
| Beta 3/1 | 3.14186231690763 | 381.939079106782 | Dave Benson | 5ed3/1 |
| Alpha 2/1 | 5.00991270509077 | 239.525131601721 | Dave Benson | 5ed2/1 |
| Gamma 3/1 | 5.04255621376059 | 237.974540913462 | Dave Benson | 8ed3/1 |
| Beta 2/1 | 6.99104980248710 | 171.648040552235 | Dave Benson | 7ed2/1 |
| Alpha 5/3 | 9.50583353877785 | 126.238272015258 | Dave Benson | 7ed5/3 |
| Gamma 2/1 | 11.9978480914311 | 100.017935787756 | Dave Benson | 12ed2/1 |
| Beta 5/3 | 12.2053823008782 | 98.3172808862904 | Dave Benson | 9ed5/3 |
| Alpha 3/2 | 15.3915238996928 | 77.9649895501219 | Dave Benson | 9ed3/2 |
| Beta 3/2 | 18.7990736394111 | 63.8329325698408 | Dave Benson | 11ed3/2 |
| Gamma 5/3 | 21.7094399215509 | 55.2754932571412 | Dave Benson | 16ed5/3 |
| Alpha 7/5 | 22.6653911133366 | 52.9441558718088 | Dave Benson | 11ed7/5 |
| Beta 7/5 | 26.7758951088566 | 44.8164289231577 | Dave Benson | 13ed7/5 |
| Alpha 4/3 | 31.3266790320926 | 38.3060074376432 | Dave Benson | 13ed4/3 |
| Gamma 3/2 | 34.1894540921914 | 35.0985422804417 | Dave Benson | 20ed3/2 |
| Beta 4/3 | 36.1372975038827 | 33.2066890135065 | Dave Benson | 15ed4/3 |
| Gamma 7/5 | 49.4404896216012 | 24.2716042900130 | Dave Benson | 24ed7/5 |
| Gamma 4/3 | 67.4633901646646 | 17.7874251067289 | Dave Benson | 28ed4/3 |
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 |