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

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
196zpi (σ = 1) 43.0234004818 27.8917981043 2.78019 43edo 1199.34731849 8 8
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
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

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 prime 2 from the zeta product

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.075 and cents ≥ 6.0)
Tuning Strength Closest EDT No-2 Integer limit
No-2 ZPI (σ = 1) Steps per octave Cents Height EDT Tritave Consistent Distinct
no-2 93zpi (σ = 1) 24.5747239922 48.8306603314 2.12985 39edt 1904.39575293 15 15
no-2 151zpi (σ = 1) 35.3061077059 33.9884534992 2.08576 56edt 1903.35339595 15 15
no-2 207zpi (σ = 1) 44.8164999984 26.7758526445 2.10342 71edt 1901.08553776 17 17
no-2 222zpi (σ = 1) 47.3516876312 25.3422857776 2.11876 75edt 1900.67143332 15 15
no-2 233zpi (σ = 1) 49.1657210129 24.4072491012 2.07714 78edt 1903.76542989 21 21
no-2 273zpi (σ = 1) 55.5359583782 21.6076220712 2.19450 88edt 1901.47074227 11 11
no-2 363zpi (σ = 1) 69.4191721809 17.2862908372 2.08043 110edt 1901.49199210 23 23
no-2 380zpi (σ = 1) 71.9200195089 16.6852012582 2.07565 114edt 1902.11294344 17 17
no-2 453zpi (σ = 1) 82.6700405439 14.5155366092 2.38406 131edt 1901.53529581 27 27
no-2 492zpi (σ = 1) 88.3238806401 13.5863595587 2.12238 140edt 1902.09033822 9 9
no-2 510zpi (σ = 1) 90.8334979880 13.2109852266 2.23067 144edt 1902.38187263 39 27
no-2 550zpi (σ = 1) 96.5187261015 12.4328205362 2.24293 153edt 1902.22154203 15 15
no-2 627zpi (σ = 1) 107.244021785 11.1894348983 2.29774 170edt 1902.20393272 15 15
no-2 687zpi (σ = 1) 115.412802617 10.3974600113 2.18983 183edt 1902.73518207 15 15
no-2 697zpi (σ = 1) 116.734850378 10.2797064983 2.15793 185edt 1901.74570218 29 29
no-2 777zpi (σ = 1) 127.486291223 9.41277676594 2.21095 202edt 1901.38090672 17 17
no-2 810zpi (σ = 1) 131.822840677 9.10312654342 2.25360 209edt 1902.55344758 21 21
no-2 829zpi (σ = 1) 134.373782790 8.93031345169 2.13475 213edt 1902.15676521 29 29
no-2 839zpi (σ = 1) 135.657892938 8.84578091263 2.11125 215edt 1901.84289622 15 15
no-2 858zpi (σ = 1) 138.196070465 8.68331491602 2.20051 219edt 1901.64596661 11 11
no-2 902zpi (σ = 1) 143.873905513 8.34063686336 2.09948 228edt 1901.66520485 11 11
no-2 965zpi (σ = 1) 152.075713777 7.89080629768 2.10893 241edt 1901.68431774 15 15
no-2 985zpi (σ = 1) 154.604034485 7.76176381166 2.40811 245edt 1901.63213386 21 21
no-2 1029zpi (σ = 1) 160.260260060 7.48782012177 2.17192 254edt 1901.90631093 9 9
no-2 1049zpi (σ = 1) 162.750022676 7.37327086209 2.14738 258edt 1902.30388242 17 17
no-2 1069zpi (σ = 1) 165.332187903 7.25811480039 2.19607 262edt 1901.62607770 17 17
no-2 1134zpi (σ = 1) 173.506549648 6.91616542681 2.26764 275edt 1901.94549237 29 29
no-2 1159zpi (σ = 1) 176.625850825 6.79402247404 2.14379 280edt 1902.32629273 11 11
no-2 1179zpi (σ = 1) 179.167803205 6.69763193238 2.29964 284edt 1902.12746880 15 15
no-2 1245zpi (σ = 1) 187.354933401 6.40495544056 2.28021 297edt 1902.27176585 21 21
no-2 1266zpi (σ = 1) 189.909845446 6.31878772364 2.17116 301edt 1901.95510482 17 17
no-2 1297zpi (σ = 1) 193.736743714 6.19397217583 2.12380 307edt 1901.54945798 21 21
no-2 1343zpi (σ = 1) 199.415414525 6.01758897555 2.36503 316edt 1901.55811627 39 39

Notable Local Maxima of the Riemann Zeta Function after removing the primes 2 and 3 from the zeta product

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 1.725 and cents ≥ 6.0)
Tuning Strength Closest ED5 No-2 No-3 Integer limit
No-2 No-3 ZPI analog Steps per octave Cents Height ED5 Pentave Consistent Distinct
no-2 no-3 186zpi (σ = 1) 41.3464998527 29.0230129340 1.75534 96ed5 2786.20924167 35 23
no-2 no-3 565zpi (σ = 1) 98.6253027359 12.1672630320 1.74188 229ed5 2786.30323433 29 29
no-2 no-3 671zpi (σ = 1) 113.258011095 10.5952769998 1.77217 263ed5 2786.55785095 19 19
no-2 no-3 764zpi (σ = 1) 125.745000550 9.54312294522 1.75634 292ed5 2786.59190001 37 37
no-2 no-3 905zpi (σ = 1) 144.297529480 8.31615069448 1.73926 335ed5 2785.91048265 43 41
no-2 no-3 938zpi (σ = 1) 148.562870929 8.07738833059 1.79949 345ed5 2786.69897405 25 25
no-2 no-3 1046zpi (σ = 1) 162.414291729 7.38851234841 1.73251 377ed5 2785.46915535 23 23
no-2 no-3 1145zpi (σ = 1) 174.880594782 6.86182478678 1.74084 406ed5 2785.90086343 25 25
no-2 no-3 1196zpi (σ = 1) 181.292147244 6.61915046096 1.77770 421ed5 2786.66234406 35 35
no-2 no-3 1280zpi (σ = 1) 191.632570168 6.26198353937 1.75036 445ed5 2786.58267502 29 29

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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