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. 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.
Notable Local Maxima of the Riemann Zeta Function
| Tuning | Strength | Closest EDO | Integer limit | ||||
|---|---|---|---|---|---|---|---|
| ZPI (σ = 1) | Steps per octave | Step size (cents) | Height | EDO | Octave (cents) | Consistent | Distinct |
| 15zpi (σ = 1) | 6.95688550773 | 172.490980147 | 2.55384 | 7edo | 1207.43686103 | 6 | 5 |
| 26zpi (σ = 1) | 10.0089746115 | 119.892401228 | 2.57426 | 10edo | 1198.92401228 | 8 | 5 |
| 34zpi (σ = 1) | 12.0220488259 | 99.8165967700 | 2.85866 | 12edo | 1197.79916124 | 10 | 6 |
| 42zpi (σ = 1) | 13.9020220557 | 86.3183783764 | 2.50514 | 14edo | 1208.45729727 | 7 | 5 |
| 47zpi (σ = 1) | 15.0534708836 | 79.7158349246 | 2.69313 | 15edo | 1195.73752387 | 8 | 7 |
| 56zpi (σ = 1) | 17.0432556931 | 70.4090827252 | 2.65741 | 17edo | 1196.95440633 | 4 | 4 |
| 65zpi (σ = 1) | 18.9489976130 | 63.3278880767 | 3.02387 | 19edo | 1203.22987346 | 10 | 7 |
| 80zpi (σ = 1) | 22.0251749360 | 54.4831086920 | 2.99601 | 22edo | 1198.62839122 | 12 | 8 |
| 90zpi (σ = 1) | 24.0053572889 | 49.9888414723 | 2.82476 | 24edo | 1199.73219533 | 6 | 6 |
| 100zpi (σ = 1) | 25.9356337472 | 46.2683893402 | 2.71167 | 26edo | 1202.97812285 | 14 | 9 |
| 106zpi (σ = 1) | 27.0853383248 | 44.3044124320 | 2.90524 | 27edo | 1196.21913566 | 10 | 8 |
| 116zpi (σ = 1) | 28.9431579907 | 41.4605759463 | 2.68561 | 29edo | 1202.35670244 | 8 | 7 |
| 127zpi (σ = 1) | 30.9779815456 | 38.7371913897 | 3.23190 | 31edo | 1200.85293308 | 12 | 9 |
| 144zpi (σ = 1) | 34.0437506778 | 35.2487600839 | 3.07414 | 34edo | 1198.45784285 | 6 | 6 |
| 155zpi (σ = 1) | 35.9827898689 | 33.3492762616 | 2.80355 | 36edo | 1200.57394542 | 8 | 8 |
| 184zpi (σ = 1) | 40.9880790756 | 29.2768050385 | 3.32966 | 41edo | 1200.34900658 | 16 | 10 |
| 214zpi (σ = 1) | 46.0106419996 | 26.0809227572 | 3.25119 | 46edo | 1199.72244683 | 14 | 11 |
| 238zpi (σ = 1) | 49.9382924730 | 24.0296562132 | 2.90274 | 50edo | 1201.48281066 | 10 | 9 |
| 257zpi (σ = 1) | 52.9969882711 | 22.6427961125 | 3.46399 | 53edo | 1200.06819396 | 10 | 10 |
| 289zpi (σ = 1) | 58.0645692462 | 20.6666477609 | 3.25823 | 58edo | 1198.66557013 | 16 | 12 |
| 301zpi (σ = 1) | 59.9223835273 | 20.0259056693 | 2.98826 | 60edo | 1201.55434016 | 10 | 10 |
| 321zpi (σ = 1) | 63.0197888699 | 19.0416378969 | 2.87513 | 63edo | 1199.62318750 | 8 | 8 |
| 334zpi (σ = 1) | 65.0145858034 | 18.4573966776 | 3.23462 | 65edo | 1199.73078404 | 6 | 6 |
| 354zpi (σ = 1) | 68.0496579343 | 17.6341812204 | 3.14200 | 68edo | 1199.12432299 | 10 | 10 |
| 380zpi (σ = 1) | 71.9512656175 | 16.6779554147 | 3.61665 | 72edo | 1200.81278986 | 18 | 13 |
| 414zpi (σ = 1) | 76.9924672555 | 15.5859403235 | 3.28825 | 77edo | 1200.11740491 | 10 | 10 |
| 435zpi (σ = 1) | 80.0733926855 | 14.9862514845 | 3.14833 | 80edo | 1198.90011876 | 12 | 12 |
| 462zpi (σ = 1) | 83.9950884037 | 14.2865496400 | 3.19687 | 84edo | 1200.07016976 | 10 | 10 |
| 483zpi (σ = 1) | 87.0139579095 | 13.7908908965 | 3.44872 | 87edo | 1199.80750799 | 16 | 14 |
| 497zpi (σ = 1) | 89.0215260329 | 13.4798857476 | 3.02681 | 89edo | 1199.70983154 | 12 | 12 |
| 532zpi (σ = 1) | 93.9843698073 | 12.7680805059 | 3.39762 | 94edo | 1200.19956756 | 24 | 15 |
| 546zpi (σ = 1) | 95.9558568688 | 12.5057504477 | 2.93099 | 96edo | 1200.55204298 | 6 | 6 |
| 568zpi (σ = 1) | 99.0456175574 | 12.1156294402 | 3.56676 | 99edo | 1199.44731458 | 12 | 12 |
| 596zpi (σ = 1) | 102.936325452 | 11.6576922163 | 3.25007 | 103edo | 1200.74229828 | 15 | 15 |
| 655zpi (σ = 1) | 111.058159333 | 10.8051493669 | 3.39509 | 111edo | 1199.37157972 | 22 | 16 |
| 706zpi (σ = 1) | 117.971388652 | 10.1719579104 | 3.62695 | 118edo | 1200.29103343 | 12 | 12 |
| 796zpi (σ = 1) | 130.004267285 | 9.23046623824 | 3.72487 | 130edo | 1199.96061097 | 16 | 16 |
| 872zpi (σ = 1) | 139.992781938 | 8.57187051639 | 3.60746 | 140edo | 1200.06187229 | 10 | 10 |
| 965zpi (σ = 1) | 152.050659206 | 7.89210652729 | 3.68901 | 152edo | 1199.60019215 | 15 | 15 |
| 1114zpi (σ = 1) | 170.995049914 | 7.01774700849 | 3.82285 | 171edo | 1200.03473845 | 14 | 14 |
| 1210zpi (σ = 1) | 183.000273182 | 6.55736726036 | 3.76064 | 183edo | 1199.99820865 | 18 | 18 |
|
Todo: use sigma 1.0
instead of sigma 1/2 |
Notable Local Maxima of the Riemann Zeta Function after removing the prime 2 from the zeta product
| Tuning | Strength | Closest EDT | No-2 Integer limit | ||||||
|---|---|---|---|---|---|---|---|---|---|
| No-2 ZPI analog | Steps per octave | Cents | Height | Integral | Gap | EDT | Tritave | Consistent | Distinct |
| no-2 93zpi analog | 24.5738316304204 | 48.8324335434323 | 4.665720 | 0.766618 | 13.261693 | 39edt | 1904.46490819386 | 15 | 15 |
| no-2 151zpi analog | 35.3059427335609 | 33.9886123153798 | 4.738265 | 0.709543 | 13.081926 | 56edt | 1903.36228966127 | 15 | 15 |
| no-2 199zpi analog | 43.5176229677494 | 27.5750355411028 | 4.824506 | 0.678480 | 12.871286 | 69edt | 1902.67745233609 | 9 | 9 |
| no-2 207zpi analog | 44.8152489207676 | 26.7766001282638 | 4.819120 | 0.732965 | 14.719415 | 71edt | 1901.13860910673 | 17 | 17 |
| no-2 222zpi analog | 47.3521317910583 | 25.3420480686067 | 5.059485 | 0.721113 | 13.412098 | 75edt | 1900.65360514550 | 15 | 15 |
| no-2 233zpi analog | 49.1685275266548 | 24.4058559481869 | 4.790248 | 0.736865 | 15.624024 | 78edt | 1903.65676395858 | 21 | 21 |
| no-2 249zpi analog | 51.6860577447882 | 23.2170928168922 | 4.848916 | 0.664134 | 13.043858 | 82edt | 1903.80161098516 | 17 | 17 |
| no-2 273zpi analog | 55.5353711835277 | 21.6078505360910 | 5.441186 | 0.771944 | 14.061502 | 88edt | 1901.49084717601 | 11 | 11 |
| no-2 317zpi analog | 62.4092182976906 | 19.2279287055965 | 5.154539 | 0.705887 | 14.235540 | 99edt | 1903.56494185405 | 25 | 23 |
| no-2 326zpi analog | 63.7619933650274 | 18.8199887843874 | 4.961196 | 0.662970 | 13.437518 | 101edt | 1900.81886722313 | 9 | 9 |
| no-2 342zpi analog | 66.2581615380500 | 18.1109764011620 | 5.073625 | 0.677884 | 13.529076 | 105edt | 1901.65252212201 | 17 | 17 |
| no-2 363zpi analog | 69.4221749409126 | 17.2855431426825 | 5.247825 | 0.705262 | 14.276498 | 110edt | 1901.40974569508 | 23 | 23 |
| no-2 453zpi analog | 82.6705208991009 | 14.5154522670130 | 6.410342 | 0.925687 | 16.646686 | 131edt | 1901.52424697870 | 27 | 27 |
| no-2 492zpi analog | 88.3242305963095 | 13.5863057271867 | 5.480169 | 0.696272 | 13.636687 | 140edt | 1902.08280180614 | 9 | 9 |
| no-2 510zpi analog | 90.8297848520406 | 13.2115252937654 | 5.712975 | 0.810755 | 16.378662 | 144edt | 1902.45964230221 | 39 | 27 |
| no-2 550zpi analog | 96.5193707902430 | 12.4327374927449 | 6.047703 | 0.795582 | 14.790729 | 153edt | 1902.20883638997 | 15 | 15 |
| no-2 627zpi analog | 107.244707551072 | 11.1893633485693 | 6.217266 | 0.828658 | 15.375247 | 170edt | 1902.19176925679 | 15 | 15 |
| no-2 687zpi analog | 115.410497106759 | 10.3976677172610 | 5.985004 | 0.754232 | 14.631506 | 183edt | 1902.77319225877 | 15 | 15 |
| no-2 697zpi analog | 116.733331758968 | 10.2798402300191 | 5.835644 | 0.746180 | 15.041001 | 185edt | 1901.77044255353 | 29 | 29 |
| no-2 777zpi analog | 127.487421022497 | 9.41269334947362 | 6.134922 | 0.758067 | 14.474624 | 202edt | 1901.36405659367 | 17 | 17 |
| no-2 810zpi analog | 131.820548689719 | 9.10328482112888 | 6.140639 | 0.820704 | 16.484428 | 209edt | 1902.58652761594 | 21 | 21 |
| no-2 829zpi analog | 134.375301622234 | 8.93021251311149 | 5.870928 | 0.707721 | 14.252150 | 213edt | 1902.13526529275 | 29 | 29 |
| no-2 839zpi analog | 135.657235331861 | 8.84582379306507 | 5.733350 | 0.672634 | 13.637550 | 215edt | 1901.85211550899 | 15 | 15 |
| no-2 858zpi analog | 138.196733558228 | 8.68327325185579 | 5.998270 | 0.762777 | 15.383590 | 219edt | 1901.63684215642 | 11 | 11 |
| no-2 985zpi analog | 154.604938100947 | 7.76171844664157 | 7.104335 | 0.924588 | 16.674411 | 245edt | 1901.62101942718 | 21 | 21 |
Notable Local Maxima of the Riemann Zeta Function after removing the prime 3 from the zeta product
| Tuning | Strength | Closest EDO | No-3 Integer limit | ||||||
|---|---|---|---|---|---|---|---|---|---|
| No-3 ZPI analog | Steps per octave | Cents | Height | Integral | Gap | EDO | Octave | Consistent | Distinct |
| no-3 51zpi analog | 15.9698898591818 | 75.1414073973756 | 5.367776 | 0.953376 | 13.070433 | 16edo | 1202.26251835801 | 26 | 8 |
| no-3 75zpi analog | 21.0437746046821 | 57.0239903507143 | 5.752828 | 0.956754 | 12.853639 | 21edo | 1197.50379736500 | 17 | 10 |
| no-3 95zpi analog | 24.9596545948521 | 48.0775883912872 | 6.060198 | 0.954994 | 12.605015 | 25edo | 1201.93970978218 | 14 | 11 |
| no-3 111zpi analog | 28.0369867749215 | 42.8006051304121 | 5.701943 | 0.838390 | 11.937782 | 28edo | 1198.41694365154 | 16 | 8 |
| no-3 149zpi analog | 34.9357059709719 | 34.3488121006365 | 6.001080 | 0.875916 | 12.775820 | 35edo | 1202.20842352228 | 14 | 11 |
| no-3 161zpi analog | 37.0117501336435 | 32.4221360964286 | 7.215934 | 1.160421 | 15.095854 | 37edo | 1199.61903556786 | 22 | 16 |
| no-3 196zpi analog | 43.0546167485686 | 27.8715754690789 | 6.495142 | 1.018487 | 15.545919 | 43edo | 1198.47774517039 | 22 | 19 |
| no-3 220zpi analog | 47.0058691719873 | 25.5287269683150 | 6.758393 | 0.939366 | 13.012654 | 47edo | 1199.85016751081 | 10 | 10 |
| no-3 251zpi analog | 52.0433965143593 | 23.0576803277801 | 6.442846 | 0.856289 | 12.619985 | 52edo | 1198.99937704456 | 11 | 11 |
| no-3 276zpi analog | 55.9872265526305 | 21.4334603424577 | 6.932381 | 1.003267 | 14.804703 | 56edo | 1200.27377917763 | 20 | 19 |
| no-3 340zpi analog | 65.9172827630736 | 18.2046338941664 | 7.029648 | 0.948492 | 13.998526 | 66edo | 1201.50583701498 | 16 | 16 |
| no-3 394zpi analog | 74.0597618189548 | 16.2031306950932 | 7.464214 | 1.007842 | 14.386154 | 74edo | 1199.03167143690 | 16 | 16 |
| no-3 421zpi analog | 78.0110209886063 | 15.3824419267024 | 7.592394 | 1.008960 | 14.204322 | 78edo | 1199.83047028279 | 17 | 16 |
| no-3 525zpi analog | 93.0076810773635 | 12.9021601882735 | 8.466134 | 1.133255 | 15.018535 | 93edo | 1199.90089750944 | 35 | 19 |
| no-3 640zpi analog | 108.976082315502 | 11.0115905665045 | 8.633826 | 1.182085 | 16.319873 | 109edo | 1200.26337174899 | 16 | 16 |
| no-3 751zpi analog | 124.014367753602 | 9.67629817203298 | 9.498846 | 1.276085 | 16.564895 | 124edo | 1199.86097333209 | 28 | 26 |
Notable Local Maxima of the Riemann Zeta Function after removing the primes 2 and 3 from the zeta product
| Tuning | Strength | Closest ED5 | No-2 No-3 Integer limit | ||||||
|---|---|---|---|---|---|---|---|---|---|
| No-2 No-3 ZPI analog | Steps per octave | Cents | Height | Integral | Gap | ED5 | Pentave | Consistent | Distinct |
| no-2 no-3 55zpi analog | 16.7630030425585 | 71.5862185882446 | 3.480299 | 0.477759 | 9.649416 | 39ed5 | 2791.86252494154 | 13 | 13 |
| no-2 no-3 125zpi analog | 30.5974484926723 | 39.2189564527704 | 3.769318 | 0.448541 | 9.828199 | 71ed5 | 2784.54590814670 | 19 | 19 |
| no-2 no-3 176zpi analog | 39.5828667040955 | 30.3161468564337 | 3.603524 | 0.421674 | 10.452207 | 92ed5 | 2789.08551079190 | 11 | 11 |
| no-2 no-3 186zpi analog | 41.3477989230936 | 29.0221010852836 | 4.469823 | 0.556068 | 11.567493 | 96ed5 | 2786.12170418722 | 35 | 23 |
| no-2 no-3 212zpi analog | 45.6783815054539 | 26.2706330752267 | 3.818225 | 0.433470 | 10.611042 | 106ed5 | 2784.68710597403 | 13 | 13 |
| no-2 no-3 235zpi analog | 49.4631517377883 | 24.2604839732289 | 3.853032 | 0.428042 | 10.508697 | 115ed5 | 2789.95565692132 | 25 | 25 |
| no-2 no-3 284zpi analog | 57.2705618247184 | 20.9531731794898 | 3.913350 | 0.465932 | 11.922515 | 133ed5 | 2786.77203287214 | 17 | 17 |
| no-2 no-3 298zpi analog | 59.4923782274424 | 20.1706510271339 | 4.083075 | 0.465782 | 11.463643 | 138ed5 | 2783.54984174448 | 23 | 23 |
| no-2 no-3 312zpi analog | 61.6047959566046 | 19.4790029147292 | 4.416896 | 0.501431 | 11.339301 | 143ed5 | 2785.49741680628 | 25 | 23 |
| no-2 no-3 340zpi analog | 65.8904943328257 | 18.2120351676004 | 4.092923 | 0.526694 | 13.998526 | 153ed5 | 2786.44138064287 | 13 | 13 |
| no-2 no-3 368zpi analog | 70.2158409653819 | 17.0901606176251 | 4.382540 | 0.518334 | 12.481351 | 163ed5 | 2785.69618067290 | 19 | 19 |
| no-2 no-3 423zpi analog | 78.3601842342727 | 15.3138996765548 | 4.270381 | 0.502072 | 12.963711 | 182ed5 | 2787.12974113297 | 19 | 19 |
| no-2 no-3 438zpi analog | 80.4944089071946 | 14.9078677176639 | 4.243838 | 0.450422 | 11.371118 | 187ed5 | 2787.77126320314 | 7 | 7 |
| no-2 no-3 465zpi analog | 84.4075187897342 | 14.2167429774745 | 4.301350 | 0.486089 | 12.332303 | 196ed5 | 2786.48162358500 | 17 | 17 |
| no-2 no-3 477zpi analog | 86.1814871554687 | 13.9241041157161 | 4.459348 | 0.505570 | 12.446285 | 200ed5 | 2784.82082314323 | 25 | 25 |
| no-2 no-3 565zpi analog | 98.6257548378926 | 12.1672072570942 | 4.883729 | 0.545550 | 12.639964 | 229ed5 | 2786.29046187457 | 29 | 29 |
| no-2 no-3 581zpi analog | 100.797128599965 | 11.9051010347969 | 4.579796 | 0.536282 | 13.693791 | 234ed5 | 2785.79364214247 | 25 | 25 |
| no-2 no-3 671zpi analog | 113.256639862217 | 10.5954052800778 | 5.104294 | 0.563708 | 12.937931 | 263ed5 | 2786.59158866045 | 19 | 19 |
| no-2 no-3 764zpi analog | 125.745930952370 | 9.54305233506547 | 5.001815 | 0.548008 | 12.976730 | 292ed5 | 2786.57128183912 | 37 | 37 |
| no-2 no-3 905zpi analog | 144.300058486204 | 8.31600494545005 | 5.030210 | 0.539592 | 13.254432 | 335ed5 | 2785.86165672577 | 43 | 41 |
| no-2 no-3 938zpi analog | 148.561761173834 | 8.07744866861039 | 5.510552 | 0.600083 | 13.846076 | 345ed5 | 2786.71979067058 | 25 | 25 |
The α–β–γ family
| 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 |
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 |