User:Contribution/Collection of tunings

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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

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.5 and cents ≥ 40.0) or (height ≥ 2.8 and cents ≥ 12.0) or (height ≥ 3.25 and cents ≥ 10.0) or (height ≥ 3.6 and cents ≥ 6.0)
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 19zpi analog 8.18712929074 146.571521883 1.87661 13ed3 1905.42978449 15 11
no-2 29zpi analog 10.7334869381 111.799642271 1.95394 17ed3 1900.59391860 17 11
no-2 53zpi analog 16.4033618519 73.1557354420 2.01896 26ed3 1902.04912149 21 15
no-2 71zpi analog 20.2433432017 59.2787460076 2.00269 32ed3 1896.91987224 21 15
no-2 84zpi analog 22.7835155508 52.6696592247 1.89685 36ed3 1896.10773209 17 13
no-2 93zpi analog 24.5747239922 48.8306603314 2.12985 39ed3 1904.39575293 15 15
no-2 106zpi analog 27.1258094838 44.2383111448 1.97822 43ed3 1902.24737923 11 11
no-2 113zpi analog 28.4085507996 42.2408030759 1.96399 45ed3 1900.83613842 9 9
no-2 137zpi analog 32.7488975372 36.6424548685 2.02055 52ed3 1905.40765316 25 15
no-2 151zpi analog 35.3061077059 33.9884534992 2.08576 56ed3 1903.35339595 15 15
no-2 166zpi analog 37.8594891129 31.6961487891 1.97021 60ed3 1901.76892734 15 15
no-2 173zpi analog 39.1519961740 30.6497782301 1.99822 62ed3 1900.28625027 9 9
no-2 199zpi analog 43.5167998698 27.5755571088 2.05686 69ed3 1902.71344050 9 9
no-2 207zpi analog 44.8164999984 26.7758526445 2.10342 71ed3 1901.08553776 17 17
no-2 222zpi analog 47.3516876312 25.3422857776 2.11876 75ed3 1900.67143332 15 15
no-2 233zpi analog 49.1657210129 24.4072491012 2.07714 78ed3 1903.76542989 21 21
no-2 249zpi analog 51.6879877530 23.2162259002 2.03774 82ed3 1903.73052382 17 17
no-2 273zpi analog 55.5359583782 21.6076220712 2.19450 88ed3 1901.47074227 11 11
no-2 289zpi analog 58.0976839265 20.6548681272 1.99993 92ed3 1900.24786771 15 15
no-2 301zpi analog 59.8907003349 20.0364997118 1.93131 95ed3 1903.46747262 11 11
no-2 309zpi analog 61.2052267978 19.6061686686 1.96785 97ed3 1901.79836086 11 11
no-2 317zpi analog 62.4122030931 19.2270091509 2.07392 99ed3 1903.47390594 25 23
no-2 326zpi analog 63.7602215687 18.8205117623 2.05280 101ed3 1900.87168799 9 9
no-2 342zpi analog 66.2583876236 18.1109146033 2.06825 105ed3 1901.64603334 17 17
no-2 363zpi analog 69.4191721809 17.2862908372 2.08043 110ed3 1901.49199210 23 23
no-2 380zpi analog 71.9200195089 16.6852012582 2.07565 114ed3 1902.11294344 17 17
no-2 397zpi analog 74.4867252346 16.1102531521 1.92629 118ed3 1901.00987195 15 15
no-2 409zpi analog 76.2807590080 15.7313589378 1.97954 121ed3 1903.49443147 25 23
no-2 418zpi analog 77.5713604064 15.4696268534 1.90376 123ed3 1902.76410297 9 9
no-2 435zpi analog 80.1032694573 14.9806619396 1.99098 127ed3 1902.54406634 11 11
no-2 453zpi analog 82.6700405439 14.5155366092 2.38406 131ed3 1901.53529581 27 27
no-2 492zpi analog 88.3238806401 13.5863595587 2.12238 140ed3 1902.09033822 9 9
no-2 510zpi analog 90.8334979880 13.2109852266 2.23067 144ed3 1902.38187263 39 27
no-2 519zpi analog 92.1840749628 13.0174327885 1.99259 146ed3 1900.54518712 17 17
no-2 550zpi analog 96.5187261015 12.4328205362 2.24293 153ed3 1902.22154203 15 15
no-2 568zpi analog 99.0730275901 12.1122774704 2.00937 157ed3 1901.62756285 11 11
no-2 577zpi analog 100.316260311 11.9621684090 1.98584 159ed3 1901.98477703 11 11
no-2 596zpi analog 102.908364024 11.6608597502 1.96654 163ed3 1900.72013927 15 15
no-2 609zpi analog 104.713326539 11.4598594053 2.00635 166ed3 1902.33666128 11 11
no-2 614zpi analog 105.436045548 11.3813069692 1.92595 167ed3 1900.67826385 23 23
no-2 627zpi analog 107.244021785 11.1894348983 2.29774 170ed3 1902.20393272 15 15
no-2 646zpi analog 109.793603482 10.9295984642 1.96998 174ed3 1901.75013278 15 15
no-2 655zpi analog 111.085500608 10.8024899148 2.00672 176ed3 1901.23822501 21 21
no-2 659zpi analog 111.586744725 10.7539654729 1.88303 177ed3 1903.45188870 7 7
no-2 687zpi analog 115.412802617 10.3974600113 2.18983 183ed3 1902.73518207 15 15
no-2 697zpi analog 116.734850378 10.2797064983 2.15793 185ed3 1901.74570218 29 29
no-2 706zpi analog 117.949591604 10.1738376851 1.91643 187ed3 1902.50764711 11 11
no-2 725zpi analog 120.530724507 9.95596769960 1.89765 191ed3 1901.58983062 5 5
no-2 729zpi analog 121.102378223 9.90897138117 2.05767 192ed3 1902.52250518 17 17
no-2 748zpi analog 123.601895646 9.70858896401 1.91762 196ed3 1902.88343695 11 11
no-2 753zpi analog 124.304838560 9.65368696748 1.91680 197ed3 1901.77633259 21 21
no-2 767zpi analog 126.183698594 9.50994473428 2.05769 200ed3 1901.98894686 9 9
no-2 777zpi analog 127.486291223 9.41277676594 2.21095 202ed3 1901.38090672 17 17
no-2 810zpi analog 131.822840677 9.10312654342 2.25360 209ed3 1902.55344758 21 21
no-2 829zpi analog 134.373782790 8.93031345169 2.13475 213ed3 1902.15676521 29 29
no-2 839zpi analog 135.657892938 8.84578091263 2.11125 215ed3 1901.84289622 15 15
no-2 858zpi analog 138.196070465 8.68331491602 2.20051 219ed3 1901.64596661 11 11
no-2 878zpi analog 140.756053126 8.52538823977 1.91894 223ed3 1901.16157747 15 15
no-2 882zpi analog 141.320264620 8.49135121014 1.94097 224ed3 1902.06267107 17 17
no-2 902zpi analog 143.873905513 8.34063686336 2.09948 228ed3 1901.66520485 11 11
no-2 911zpi analog 145.102065664 8.27004077793 1.96452 230ed3 1902.10937892 23 23
no-2 921zpi analog 146.379932964 8.19784498941 1.96989 232ed3 1901.90003754 9 9
no-2 945zpi analog 149.470277594 8.02835198621 1.92855 237ed3 1902.71942073 19 19
no-2 965zpi analog 152.075713777 7.89080629768 2.10893 241ed3 1901.68431774 15 15
no-2 985zpi analog 154.604034485 7.76176381166 2.40811 245ed3 1901.63213386 21 21
no-2 995zpi analog 155.863142206 7.69906202978 1.88900 247ed3 1901.66832135 7 7
no-2 1019zpi analog 158.932236585 7.55038767329 1.94652 252ed3 1902.69769367 15 15
no-2 1029zpi analog 160.260260060 7.48782012177 2.17192 254ed3 1901.90631093 9 9
no-2 1049zpi analog 162.750022676 7.37327086209 2.14738 258ed3 1902.30388242 17 17
no-2 1069zpi analog 165.332187903 7.25811480039 2.19607 262ed3 1901.62607770 17 17
no-2 1083zpi analog 167.112289634 7.18080042243 1.93984 265ed3 1902.91211194 11 11
no-2 1104zpi analog 169.714157484 7.07071241310 1.92771 269ed3 1902.02163912 15 15
no-2 1114zpi analog 170.990381058 7.01793862657 1.91502 271ed3 1901.86136780 9 9
no-2 1134zpi analog 173.506549648 6.91616542681 2.26764 275ed3 1901.94549237 29 29
no-2 1145zpi analog 174.860916353 6.86259700012 1.98752 277ed3 1900.93936903 15 15
no-2 1159zpi analog 176.625850825 6.79402247404 2.14379 280ed3 1902.32629273 11 11
no-2 1179zpi analog 179.167803205 6.69763193238 2.29964 284ed3 1902.12746880 15 15
no-2 1200zpi analog 181.734924328 6.60302363146 1.98334 288ed3 1901.67080586 11 11
no-2 1210zpi analog 183.000523023 6.55735830793 1.88033 290ed3 1901.63390930 17 17
no-2 1225zpi analog 184.832854856 6.49235224405 1.92540 293ed3 1902.25920751 9 9
no-2 1245zpi analog 187.354933401 6.40495544056 2.28021 297ed3 1902.27176585 21 21
no-2 1266zpi analog 189.909845446 6.31878772364 2.17116 301ed3 1901.95510482 17 17
no-2 1297zpi analog 193.736743714 6.19397217583 2.12380 307ed3 1901.54945798 21 21
no-2 1301zpi analog 194.272130007 6.17690247159 1.87710 308ed3 1902.48596125 7 7
no-2 1312zpi analog 195.595668163 6.13510519569 1.92538 310ed3 1901.88261066 9 9
no-2 1332zpi analog 198.083101013 6.05806347873 2.07112 314ed3 1902.23193232 15 15
no-2 1343zpi analog 199.415414525 6.01758897555 2.36503 316ed3 1901.55811627 39 39

Notable Local Maxima of the Riemann Zeta Function after removing the prime 3 from the zeta product

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.5 and cents ≥ 40.0) or (height ≥ 2.6 and cents ≥ 15.0) or (height ≥ 2.8 and cents ≥ 12.0) or (height ≥ 3.1 and cents ≥ 6.0)
Tuning Strength Closest EDO No-3 Integer limit
No-3 ZPI analog Steps per octave Cents Height EDO Octave Consistent Distinct
no-3 51zpi (σ = 1) 15.9687074547 75.1469712502 2.56677 16edo 1202.35154000 26 8
no-3 75zpi (σ = 1) 21.0417134383 57.0295762045 2.60042 21edo 1197.62110029 17 10
no-3 95zpi (σ = 1) 24.9617781085 48.0734984016 2.64675 25edo 1201.83746004 14 11
no-3 127zpi (σ = 1) 31.0146799866 38.6913552073 2.60405 31edo 1199.43201143 11 11
no-3 161zpi (σ = 1) 37.0135086000 32.4205957606 2.92705 37edo 1199.56204314 22 16
no-3 196zpi (σ = 1) 43.0494972034 27.8748900209 2.71380 43edo 1198.62027090 22 19
no-3 220zpi (σ = 1) 47.0043385196 25.5295582875 2.69328 47edo 1199.88923951 10 10
no-3 276zpi (σ = 1) 55.9891415481 21.4327272543 2.76321 56edo 1200.23272624 20 19
no-3 340zpi (σ = 1) 65.9204029312 18.2037722259 2.65263 66edo 1201.44896691 16 16
no-3 354zpi (σ = 1) 68.0229453080 17.6411061674 2.76285 68edo 1199.59521939 11 11
no-3 394zpi (σ = 1) 74.0566473758 16.2038121158 2.76672 74edo 1199.08209657 16 16
no-3 421zpi (σ = 1) 78.0097604150 15.3826904943 2.81219 78edo 1199.84985856 17 16
no-3 525zpi (σ = 1) 93.0066513531 12.9023030347 2.97919 93edo 1199.91418223 35 19
no-3 751zpi (σ = 1) 124.013627761 9.67635591079 3.13747 124edo 1199.86813294 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

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
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