User:Contribution/Collection of tunings: Difference between revisions

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{|class="wikitable sortable"
{|class="wikitable sortable"
|+ style="font-size: 105%;" | Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 1.6 and cents ≥ 6.0)
|+ style="font-size: 105%;" | Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 1.627 and cents ≥ 6.0)
!colspan="3"|Tuning
!colspan="3"|Tuning
!colspan="1"|Strength
!colspan="1"|Strength
Line 881: Line 881:
!Distinct
!Distinct
|-
|-
|[[no-2 no-3 36zpi (σ = 1)|'''no-2 no-3 36zpi (σ = 1)''']]
|[[no-2 no-3 36zpi (σ = 1)]]
|'''12.4660713853'''
|12.4660713853
|'''96.2612809531'''
|96.2612809531
|'''1.63006'''
|1.63006
|'''[[29ed5]]'''
|[[29ed5]]
|'''2791.57714764'''
|2791.57714764
|'''23'''
|23
|'''13'''
|-
|[[no-2 no-3 55zpi (σ = 1)]]
|16.7644794252
|71.5799142678
|1.61533
|[[39ed5]]
|2791.61665644
|13
|13
|13
|-
|-
|[[no-2 no-3 125zpi (σ = 1)]]
|[[no-2 no-3 186zpi (σ = 1)]]
|30.5978454621
|41.3464998527
|39.2184476350
|29.0230129340
|1.60272
|1.75534
|[[71ed5]]
|[[96ed5]]
|2784.50978208
|2786.20924167
|19
|35
|19
|-
|[[no-2 no-3 186zpi (σ = 1)|'''no-2 no-3 186zpi (σ = 1)''']]
|'''41.3464998527'''
|'''29.0230129340'''
|'''1.75534'''
|'''[[96ed5]]'''
|'''2786.20924167'''
|'''35'''
|'''23'''
|-
|[[no-2 no-3 262zpi (σ = 1)]]
|53.7853073038
|22.3109257928
|1.60529
|[[125ed5]]
|2788.86572410
|17
|17
|-
|[[no-2 no-3 284zpi (σ = 1)]]
|57.2735400587
|20.9520836109
|1.60690
|[[133ed5]]
|2786.62712024
|17
|17
|-
|[[no-2 no-3 298zpi (σ = 1)]]
|59.4886140169
|20.1719273483
|1.61011
|[[138ed5]]
|2783.72597407
|23
|23
|23
|-
|-
Line 970: Line 925:
|19
|19
|19
|19
|-
|[[no-2 no-3 394zpi (σ = 1)]]
|74.0800438156
|16.1986945227
|1.61352
|[[172ed5]]
|2786.17545791
|17
|17
|-
|-
|[[no-2 no-3 423zpi (σ = 1)]]
|[[no-2 no-3 423zpi (σ = 1)]]
Line 988: Line 934:
|19
|19
|19
|19
|-
|[[no-2 no-3 438zpi (σ = 1)]]
|80.4984134261
|14.9071261026
|1.60066
|[[187ed5]]
|2787.63258118
|7
|7
|-
|[[no-2 no-3 453zpi (σ = 1)]]
|82.6821657004
|14.5134079379
|1.62198
|[[192ed5]]
|2786.57432408
|25
|25
|-
|-
|[[no-2 no-3 465zpi (σ = 1)]]
|[[no-2 no-3 465zpi (σ = 1)]]
Line 1,024: Line 952:
|25
|25
|25
|25
|-
|[[no-2 no-3 507zpi (σ = 1)]]
|90.4604301285
|13.2654686507
|1.60322
|[[210ed5]]
|2785.74841665
|17
|17
|-
|-
|[[no-2 no-3 540zpi (σ = 1)]]
|[[no-2 no-3 540zpi (σ = 1)]]
Line 1,061: Line 980:
|25
|25
|-
|-
|[[no-2 no-3 659zpi (σ = 1)]]
|[[no-2 no-3 671zpi (σ = 1)]]
|111.567387279
|113.258011095
|10.7558313344
|10.5952769998
|1.61434
|1.77217
|[[259ed5]]
|[[263ed5]]
|2785.76031562
|2786.55785095
|19
|19
|19
|19
|-
|[[no-2 no-3 671zpi (σ = 1)|'''no-2 no-3 671zpi (σ = 1)''']]
|'''113.258011095'''
|'''10.5952769998'''
|'''1.77217'''
|'''[[263ed5]]'''
|'''2786.55785095'''
|'''19'''
|'''19'''
|-
|[[no-2 no-3 687zpi (σ = 1)]]
|115.394324373
|10.3991249701
|1.61876
|[[268ed5]]
|2786.96549199
|13
|13
|-
|-
|[[no-2 no-3 764zpi (σ = 1)]]
|[[no-2 no-3 764zpi (σ = 1)]]
Line 1,142: Line 1,043:
|41
|41
|-
|-
|[[no-2 no-3 938zpi (σ = 1)|'''no-2 no-3 938zpi (σ = 1)''']]
|[[no-2 no-3 938zpi (σ = 1)]]
|'''148.562870929'''
|148.562870929
|'''8.07738833059'''
|8.07738833059
|'''1.79949'''
|1.79949
|'''[[345ed5]]'''
|[[345ed5]]
|'''2786.69897405'''
|2786.69897405
|'''25'''
|25
|'''25'''
|25
|-
|[[no-2 no-3 951zpi (σ = 1)]]
|150.288484121
|7.98464371385
|1.62413
|[[349ed5]]
|2786.64065613
|17
|17
|-
|-
|[[no-2 no-3 985zpi (σ = 1)]]
|[[no-2 no-3 985zpi (σ = 1)]]
Line 1,231: Line 1,123:
|29
|29
|29
|29
|-
|[[no-2 no-3 1315zpi (σ = 1)]]
|195.943977306
|6.12419945997
|1.62667
|[[455ed5]]
|2786.51075429
|17
|17
|-
|-
|[[no-2 no-3 1343zpi (σ = 1)]]
|[[no-2 no-3 1343zpi (σ = 1)]]
Line 1,249: Line 1,132:
|37
|37
|37
|37
|-
|[[no-2 no-3 1403zpi (σ = 1)]]
|206.698324430
|5.80556230105
|1.66948
|[[480ed5]]
|2786.66990450
|37
|37
|-
|[[no-2 no-3 1431zpi (σ = 1)]]
|210.179102638
|5.70941632607
|1.71227
|[[488ed5]]
|2786.19516712
|29
|29
|-
|[[no-2 no-3 1446zpi (σ = 1)]]
|211.894497896
|5.66319565592
|1.65644
|[[492ed5]]
|2786.29226271
|19
|19
|-
|[[no-2 no-3 1481zpi (σ = 1)]]
|216.192309734
|5.55061371737
|1.71109
|[[502ed5]]
|2786.40808612
|17
|17
|-
|[[no-2 no-3 1535zpi (σ = 1)]]
|222.644199436
|5.38976538818
|1.69521
|[[517ed5]]
|2786.50870569
|25
|25
|-
|[[no-2 no-3 1549zpi (σ = 1)]]
|224.362977171
|5.34847600585
|1.69928
|[[521ed5]]
|2786.55599905
|29
|29
|-
|[[no-2 no-3 1553zpi (σ = 1)]]
|224.825007465
|5.33748453310
|1.70925
|[[522ed5]]
|2786.16692628
|37
|37
|-
|[[no-2 no-3 1582zpi (σ = 1)]]
|228.308062571
|5.25605616589
|1.66018
|[[530ed5]]
|2785.70976792
|35
|35
|-
|[[no-2 no-3 1585zpi (σ = 1)]]
|228.660777319
|5.24794857286
|1.63087
|[[531ed5]]
|2786.66069219
|13
|13
|-
|[[no-2 no-3 1657zpi (σ = 1)]]
|237.287583943
|5.05715461408
|1.64045
|[[551ed5]]
|2786.49219236
|19
|19
|-
|[[no-2 no-3 1675zpi (σ = 1)]]
|239.399620091
|5.01253928282
|1.64360
|[[556ed5]]
|2786.97184125
|13
|13
|-
|[[no-2 no-3 1687zpi (σ = 1)]]
|240.773124157
|4.98394496562
|1.78734
|[[559ed5]]
|2786.02523578
|35
|35
|-
|[[no-2 no-3 1723zpi (σ = 1)]]
|245.097933582
|4.89600211011
|1.70978
|[[569ed5]]
|2785.82520065
|25
|25
|-
|[[no-2 no-3 1763zpi (σ = 1)]]
|249.738577955
|4.80502455738
|1.68685
|[[580ed5]]
|2786.91424328
|25
|25
|-
|[[no-2 no-3 1792zpi (σ = 1)]]
|253.230037362
|4.73877432750
|1.67948
|[[588ed5]]
|2786.39930457
|19
|19
|-
|[[no-2 no-3 1829zpi (σ = 1)]]
|257.550656754
|4.65927757717
|1.87016
|[[598ed5]]
|2786.24799115
|41
|41
|-
|[[no-2 no-3 1899zpi (σ = 1)]]
|265.713323283
|4.51614538997
|1.66869
|[[617ed5]]
|2786.46170561
|17
|17
|-
|[[no-2 no-3 1966zpi (σ = 1)]]
|273.500307403
|4.38756362431
|1.72141
|[[635ed5]]
|2786.10290144
|25
|25
|-
|[[no-2 no-3 1973zpi (σ = 1)]]
|274.308106457
|4.37464286237
|1.71720
|[[637ed5]]
|2786.64750333
|25
|25
|-
|[[no-2 no-3 2025zpi (σ = 1)]]
|280.352563990
|4.28032468447
|1.68610
|[[651ed5]]
|2786.49136959
|11
|11
|-
|[[no-2 no-3 2040zpi (σ = 1)]]
|282.106226052
|4.25371682432
|1.65977
|[[655ed5]]
|2786.18451993
|25
|25
|}
|}


Revision as of 16:47, 27 September 2025

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

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 ≥ 1.627 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 36zpi (σ = 1) 12.4660713853 96.2612809531 1.63006 29ed5 2791.57714764 23 13
no-2 no-3 186zpi (σ = 1) 41.3464998527 29.0230129340 1.75534 96ed5 2786.20924167 35 23
no-2 no-3 312zpi (σ = 1) 61.6053540989 19.4788264357 1.69262 143ed5 2785.47218030 25 23
no-2 no-3 340zpi (σ = 1) 65.8959418265 18.2105296129 1.70245 153ed5 2786.21103077 13 13
no-2 no-3 368zpi (σ = 1) 70.2130992609 17.0908279599 1.69532 163ed5 2785.80495746 19 19
no-2 no-3 423zpi (σ = 1) 78.3584494159 15.3142387189 1.68605 182ed5 2787.19144685 19 19
no-2 no-3 465zpi (σ = 1) 84.4093692514 14.2164313114 1.66499 196ed5 2786.42053703 17 17
no-2 no-3 477zpi (σ = 1) 86.1785294210 13.9245820051 1.67898 200ed5 2784.91640101 25 25
no-2 no-3 540zpi (σ = 1) 95.1233580316 12.6151980421 1.65279 221ed5 2787.95876731 23 23
no-2 no-3 565zpi (σ = 1) 98.6253027359 12.1672630320 1.74188 229ed5 2786.30323433 29 29
no-2 no-3 581zpi (σ = 1) 100.799606439 11.9048083856 1.71723 234ed5 2785.72516223 25 25
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 810zpi (σ = 1) 131.804682622 9.10438063447 1.63433 306ed5 2785.94047415 25 25
no-2 no-3 823zpi (σ = 1) 133.549370751 8.98544106384 1.63157 310ed5 2785.48672979 25 25
no-2 no-3 845zpi (σ = 1) 136.480899907 8.79243909456 1.62731 317ed5 2787.20319298 19 19
no-2 no-3 888zpi (σ = 1) 142.134887689 8.44268440710 1.65729 330ed5 2786.08585434 25 25
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 985zpi (σ = 1) 154.617025672 7.76111165495 1.66586 359ed5 2786.23908413 19 19
no-2 no-3 1046zpi (σ = 1) 162.414291729 7.38851234841 1.73251 377ed5 2785.46915535 23 23
no-2 no-3 1083zpi (σ = 1) 167.090722171 7.18172729405 1.64644 388ed5 2786.51019009 17 17
no-2 no-3 1097zpi (σ = 1) 168.816431308 7.10831280284 1.70949 392ed5 2786.45861871 29 29
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 1214zpi (σ = 1) 183.477053621 6.54032739419 1.68165 426ed5 2786.17946993 17 17
no-2 no-3 1280zpi (σ = 1) 191.632570168 6.26198353937 1.75036 445ed5 2786.58267502 29 29
no-2 no-3 1343zpi (σ = 1) 199.431052743 6.01711711137 1.70966 463ed5 2785.92522256 37 37
no-2 no-3 1403zpi (σ = 1) 206.698324430 5.80556230105 1.66948 480ed5 2786.66990450 37 37
no-2 no-3 1431zpi (σ = 1) 210.179102638 5.70941632607 1.71227 488ed5 2786.19516712 29 29
no-2 no-3 1446zpi (σ = 1) 211.894497896 5.66319565592 1.65644 492ed5 2786.29226271 19 19
no-2 no-3 1481zpi (σ = 1) 216.192309734 5.55061371737 1.71109 502ed5 2786.40808612 17 17
no-2 no-3 1535zpi (σ = 1) 222.644199436 5.38976538818 1.69521 517ed5 2786.50870569 25 25
no-2 no-3 1549zpi (σ = 1) 224.362977171 5.34847600585 1.69928 521ed5 2786.55599905 29 29
no-2 no-3 1553zpi (σ = 1) 224.825007465 5.33748453310 1.70925 522ed5 2786.16692628 37 37
no-2 no-3 1582zpi (σ = 1) 228.308062571 5.25605616589 1.66018 530ed5 2785.70976792 35 35
no-2 no-3 1585zpi (σ = 1) 228.660777319 5.24794857286 1.63087 531ed5 2786.66069219 13 13
no-2 no-3 1657zpi (σ = 1) 237.287583943 5.05715461408 1.64045 551ed5 2786.49219236 19 19
no-2 no-3 1675zpi (σ = 1) 239.399620091 5.01253928282 1.64360 556ed5 2786.97184125 13 13
no-2 no-3 1687zpi (σ = 1) 240.773124157 4.98394496562 1.78734 559ed5 2786.02523578 35 35
no-2 no-3 1723zpi (σ = 1) 245.097933582 4.89600211011 1.70978 569ed5 2785.82520065 25 25
no-2 no-3 1763zpi (σ = 1) 249.738577955 4.80502455738 1.68685 580ed5 2786.91424328 25 25
no-2 no-3 1792zpi (σ = 1) 253.230037362 4.73877432750 1.67948 588ed5 2786.39930457 19 19
no-2 no-3 1829zpi (σ = 1) 257.550656754 4.65927757717 1.87016 598ed5 2786.24799115 41 41
no-2 no-3 1899zpi (σ = 1) 265.713323283 4.51614538997 1.66869 617ed5 2786.46170561 17 17
no-2 no-3 1966zpi (σ = 1) 273.500307403 4.38756362431 1.72141 635ed5 2786.10290144 25 25
no-2 no-3 1973zpi (σ = 1) 274.308106457 4.37464286237 1.71720 637ed5 2786.64750333 25 25
no-2 no-3 2025zpi (σ = 1) 280.352563990 4.28032468447 1.68610 651ed5 2786.49136959 11 11
no-2 no-3 2040zpi (σ = 1) 282.106226052 4.25371682432 1.65977 655ed5 2786.18451993 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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