User:Contribution/Collection of tunings: Difference between revisions

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Line 16: Line 16:
=== Notable Local Maxima of the Riemann Zeta Function ===
=== Notable Local Maxima of the Riemann Zeta Function ===
{|class="wikitable sortable"
{|class="wikitable sortable"
|+ style="font-size: 105%;" | 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)
|+ style="font-size: 105%;" |
|-
|-
!colspan="3"|Tuning
!colspan="3"|Tuning
Line 175: Line 175:
|16
|16
|10
|10
|-
|[[196zpi (σ = 1)]]
|43.0234004818
|27.8917981043
|2.78019
|[[43edo]]
|1199.34731849
|8
|8
|-
|-
|[[214zpi (σ = 1)]]
|[[214zpi (σ = 1)]]
Line 310: Line 319:
|24
|24
|15
|15
|-
|[[546zpi (σ = 1)]]
|95.9558568688
|12.5057504477
|2.93099
|[[96edo]]
|1200.55204298
|6
|6
|-
|-
|[[568zpi (σ = 1)]]
|[[568zpi (σ = 1)]]
Line 550: Line 550:


{|class="wikitable sortable"
{|class="wikitable sortable"
|+ style="font-size: 105%;" | Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 1.875 and cents ≥ 6.0)
|+ style="font-size: 105%;" | Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.075 and cents ≥ 6.0)
!colspan="3"|Tuning
!colspan="3"|Tuning
!colspan="1"|Strength
!colspan="1"|Strength
Line 564: Line 564:
!Consistent
!Consistent
!Distinct
!Distinct
|-
|[[no-2 19zpi (σ = 1)]]
|8.18712929074
|146.571521883
|1.87661
|[[13edt]]
|1905.42978449
|15
|11
|-
|[[no-2 29zpi (σ = 1)]]
|10.7334869381
|111.799642271
|1.95394
|[[17edt]]
|1900.59391860
|17
|11
|-
|[[no-2 53zpi (σ = 1)]]
|16.4033618519
|73.1557354420
|2.01896
|[[26edt]]
|1902.04912149
|21
|15
|-
|[[no-2 71zpi (σ = 1)]]
|20.2433432017
|59.2787460076
|2.00269
|[[32edt]]
|1896.91987224
|21
|15
|-
|[[no-2 84zpi (σ = 1)]]
|22.7835155508
|52.6696592247
|1.89685
|[[36edt]]
|1896.10773209
|17
|13
|-
|-
|[[no-2 93zpi (σ = 1)]]
|[[no-2 93zpi (σ = 1)]]
Line 617: Line 572:
|1904.39575293
|1904.39575293
|15
|15
|15
|-
|[[no-2 106zpi (σ = 1)]]
|27.1258094838
|44.2383111448
|1.97822
|[[43edt]]
|1902.24737923
|11
|11
|-
|[[no-2 113zpi (σ = 1)]]
|28.4085507996
|42.2408030759
|1.96399
|[[45edt]]
|1900.83613842
|9
|9
|-
|[[no-2 137zpi (σ = 1)]]
|32.7488975372
|36.6424548685
|2.02055
|[[52edt]]
|1905.40765316
|25
|15
|15
|-
|-
Line 654: Line 582:
|15
|15
|15
|15
|-
|[[no-2 166zpi (σ = 1)]]
|37.8594891129
|31.6961487891
|1.97021
|[[60edt]]
|1901.76892734
|15
|15
|-
|[[no-2 173zpi (σ = 1)]]
|39.1519961740
|30.6497782301
|1.99822
|[[62edt]]
|1900.28625027
|9
|9
|-
|[[no-2 199zpi (σ = 1)]]
|43.5167998698
|27.5755571088
|2.05686
|[[69edt]]
|1902.71344050
|9
|9
|-
|-
|[[no-2 207zpi (σ = 1)]]
|[[no-2 207zpi (σ = 1)]]
Line 708: Line 609:
|21
|21
|21
|21
|-
|[[no-2 249zpi (σ = 1)]]
|51.6879877530
|23.2162259002
|2.03774
|[[82edt]]
|1903.73052382
|17
|17
|-
|-
|[[no-2 273zpi (σ = 1)]]
|[[no-2 273zpi (σ = 1)]]
Line 726: Line 618:
|11
|11
|11
|11
|-
|[[no-2 289zpi (σ = 1)]]
|58.0976839265
|20.6548681272
|1.99993
|[[92edt]]
|1900.24786771
|15
|15
|-
|[[no-2 301zpi (σ = 1)]]
|59.8907003349
|20.0364997118
|1.93131
|[[95edt]]
|1903.46747262
|11
|11
|-
|[[no-2 309zpi (σ = 1)]]
|61.2052267978
|19.6061686686
|1.96785
|[[97edt]]
|1901.79836086
|11
|11
|-
|[[no-2 317zpi (σ = 1)]]
|62.4122030931
|19.2270091509
|2.07392
|[[99edt]]
|1903.47390594
|25
|23
|-
|[[no-2 326zpi (σ = 1)]]
|63.7602215687
|18.8205117623
|2.05280
|[[101edt]]
|1900.87168799
|9
|9
|-
|[[no-2 342zpi (σ = 1)]]
|66.2583876236
|18.1109146033
|2.06825
|[[105edt]]
|1901.64603334
|17
|17
|-
|-
|[[no-2 363zpi (σ = 1)]]
|[[no-2 363zpi (σ = 1)]]
Line 798: Line 636:
|17
|17
|17
|17
|-
|[[no-2 397zpi (σ = 1)]]
|74.4867252346
|16.1102531521
|1.92629
|[[118edt]]
|1901.00987195
|15
|15
|-
|[[no-2 409zpi (σ = 1)]]
|76.2807590080
|15.7313589378
|1.97954
|[[121edt]]
|1903.49443147
|25
|23
|-
|[[no-2 418zpi (σ = 1)]]
|77.5713604064
|15.4696268534
|1.90376
|[[123edt]]
|1902.76410297
|9
|9
|-
|[[no-2 435zpi (σ = 1)]]
|80.1032694573
|14.9806619396
|1.99098
|[[127edt]]
|1902.54406634
|11
|11
|-
|-
|[[no-2 453zpi (σ = 1)]]
|[[no-2 453zpi (σ = 1)]]
Line 861: Line 663:
|39
|39
|27
|27
|-
|[[no-2 519zpi (σ = 1)]]
|92.1840749628
|13.0174327885
|1.99259
|[[146edt]]
|1900.54518712
|17
|17
|-
|-
|[[no-2 550zpi (σ = 1)]]
|[[no-2 550zpi (σ = 1)]]
Line 879: Line 672:
|15
|15
|15
|15
|-
|[[no-2 568zpi (σ = 1)]]
|99.0730275901
|12.1122774704
|2.00937
|[[157edt]]
|1901.62756285
|11
|11
|-
|[[no-2 577zpi (σ = 1)]]
|100.316260311
|11.9621684090
|1.98584
|[[159edt]]
|1901.98477703
|11
|11
|-
|[[no-2 596zpi (σ = 1)]]
|102.908364024
|11.6608597502
|1.96654
|[[163edt]]
|1900.72013927
|15
|15
|-
|[[no-2 609zpi (σ = 1)]]
|104.713326539
|11.4598594053
|2.00635
|[[166edt]]
|1902.33666128
|11
|11
|-
|[[no-2 614zpi (σ = 1)]]
|105.436045548
|11.3813069692
|1.92595
|[[167edt]]
|1900.67826385
|23
|23
|-
|-
|[[no-2 627zpi (σ = 1)]]
|[[no-2 627zpi (σ = 1)]]
Line 933: Line 681:
|15
|15
|15
|15
|-
|[[no-2 646zpi (σ = 1)]]
|109.793603482
|10.9295984642
|1.96998
|[[174edt]]
|1901.75013278
|15
|15
|-
|[[no-2 655zpi (σ = 1)]]
|111.085500608
|10.8024899148
|2.00672
|[[176edt]]
|1901.23822501
|21
|21
|-
|[[no-2 659zpi (σ = 1)]]
|111.586744725
|10.7539654729
|1.88303
|[[177edt]]
|1903.45188870
|7
|7
|-
|-
|[[no-2 687zpi (σ = 1)]]
|[[no-2 687zpi (σ = 1)]]
Line 978: Line 699:
|29
|29
|29
|29
|-
|[[no-2 706zpi (σ = 1)]]
|117.949591604
|10.1738376851
|1.91643
|[[187edt]]
|1902.50764711
|11
|11
|-
|[[no-2 725zpi (σ = 1)]]
|120.530724507
|9.95596769960
|1.89765
|[[191edt]]
|1901.58983062
|5
|5
|-
|[[no-2 729zpi (σ = 1)]]
|121.102378223
|9.90897138117
|2.05767
|[[192edt]]
|1902.52250518
|17
|17
|-
|[[no-2 748zpi (σ = 1)]]
|123.601895646
|9.70858896401
|1.91762
|[[196edt]]
|1902.88343695
|11
|11
|-
|[[no-2 753zpi (σ = 1)]]
|124.304838560
|9.65368696748
|1.91680
|[[197edt]]
|1901.77633259
|21
|21
|-
|[[no-2 767zpi (σ = 1)]]
|126.183698594
|9.50994473428
|2.05769
|[[200edt]]
|1901.98894686
|9
|9
|-
|-
|[[no-2 777zpi (σ = 1)]]
|[[no-2 777zpi (σ = 1)]]
Line 1,077: Line 744:
|11
|11
|11
|11
|-
|[[no-2 878zpi (σ = 1)]]
|140.756053126
|8.52538823977
|1.91894
|[[223edt]]
|1901.16157747
|15
|15
|-
|[[no-2 882zpi (σ = 1)]]
|141.320264620
|8.49135121014
|1.94097
|[[224edt]]
|1902.06267107
|17
|17
|-
|-
|[[no-2 902zpi (σ = 1)]]
|[[no-2 902zpi (σ = 1)]]
Line 1,104: Line 753:
|11
|11
|11
|11
|-
|[[no-2 911zpi (σ = 1)]]
|145.102065664
|8.27004077793
|1.96452
|[[230edt]]
|1902.10937892
|23
|23
|-
|[[no-2 921zpi (σ = 1)]]
|146.379932964
|8.19784498941
|1.96989
|[[232edt]]
|1901.90003754
|9
|9
|-
|[[no-2 945zpi (σ = 1)]]
|149.470277594
|8.02835198621
|1.92855
|[[237edt]]
|1902.71942073
|19
|19
|-
|-
|[[no-2 965zpi (σ = 1)]]
|[[no-2 965zpi (σ = 1)]]
Line 1,149: Line 771:
|21
|21
|21
|21
|-
|[[no-2 995zpi (σ = 1)]]
|155.863142206
|7.69906202978
|1.88900
|[[247edt]]
|1901.66832135
|7
|7
|-
|[[no-2 1019zpi (σ = 1)]]
|158.932236585
|7.55038767329
|1.94652
|[[252edt]]
|1902.69769367
|15
|15
|-
|-
|[[no-2 1029zpi (σ = 1)]]
|[[no-2 1029zpi (σ = 1)]]
Line 1,194: Line 798:
|17
|17
|17
|17
|-
|[[no-2 1083zpi (σ = 1)]]
|167.112289634
|7.18080042243
|1.93984
|[[265edt]]
|1902.91211194
|11
|11
|-
|[[no-2 1104zpi (σ = 1)]]
|169.714157484
|7.07071241310
|1.92771
|[[269edt]]
|1902.02163912
|15
|15
|-
|[[no-2 1114zpi (σ = 1)]]
|170.990381058
|7.01793862657
|1.91502
|[[271edt]]
|1901.86136780
|9
|9
|-
|-
|[[no-2 1134zpi (σ = 1)]]
|[[no-2 1134zpi (σ = 1)]]
Line 1,230: Line 807:
|29
|29
|29
|29
|-
|[[no-2 1145zpi (σ = 1)]]
|174.860916353
|6.86259700012
|1.98752
|[[277edt]]
|1900.93936903
|15
|15
|-
|-
|[[no-2 1159zpi (σ = 1)]]
|[[no-2 1159zpi (σ = 1)]]
Line 1,257: Line 825:
|15
|15
|15
|15
|-
|[[no-2 1200zpi (σ = 1)]]
|181.734924328
|6.60302363146
|1.98334
|[[288edt]]
|1901.67080586
|11
|11
|-
|[[no-2 1210zpi (σ = 1)]]
|183.000523023
|6.55735830793
|1.88033
|[[290edt]]
|1901.63390930
|17
|17
|-
|[[no-2 1225zpi (σ = 1)]]
|184.832854856
|6.49235224405
|1.92540
|[[293edt]]
|1902.25920751
|9
|9
|-
|-
|[[no-2 1245zpi (σ = 1)]]
|[[no-2 1245zpi (σ = 1)]]
Line 1,311: Line 852:
|21
|21
|21
|21
|-
|[[no-2 1301zpi (σ = 1)]]
|194.272130007
|6.17690247159
|1.87710
|[[308edt]]
|1902.48596125
|7
|7
|-
|[[no-2 1312zpi (σ = 1)]]
|195.595668163
|6.13510519569
|1.92538
|[[310edt]]
|1901.88261066
|9
|9
|-
|[[no-2 1332zpi (σ = 1)]]
|198.083101013
|6.05806347873
|2.07112
|[[314edt]]
|1902.23193232
|15
|15
|-
|-
|[[no-2 1343zpi (σ = 1)]]
|[[no-2 1343zpi (σ = 1)]]
Line 1,352: Line 866:


{|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 ≥ 1.725 and cents ≥ 6.0)
!colspan="3"|Tuning
!colspan="3"|Tuning
!colspan="1"|Strength
!colspan="1"|Strength
Line 1,366: Line 880:
!Consistent
!Consistent
!Distinct
!Distinct
|-
|[[no-2 no-3 36zpi (σ = 1)]]
|12.4660713853
|96.2612809531
|1.63006
|[[29ed5]]
|2791.57714764
|23
|13
|-
|[[no-2 no-3 55zpi (σ = 1)]]
|16.7644794252
|71.5799142678
|1.61533
|[[39ed5]]
|2791.61665644
|13
|13
|-
|[[no-2 no-3 125zpi (σ = 1)]]
|30.5978454621
|39.2184476350
|1.60272
|[[71ed5]]
|2784.50978208
|19
|19
|-
|-
|[[no-2 no-3 186zpi (σ = 1)]]
|[[no-2 no-3 186zpi (σ = 1)]]
Line 1,401: Line 888:
|2786.20924167
|2786.20924167
|35
|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
|-
|[[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 394zpi (σ = 1)]]
|74.0800438156
|16.1986945227
|1.61352
|[[172ed5]]
|2786.17545791
|17
|17
|-
|[[no-2 no-3 423zpi (σ = 1)]]
|78.3584494159
|15.3142387189
|1.68605
|[[182ed5]]
|2787.19144685
|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)]]
|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 507zpi (σ = 1)]]
|90.4604301285
|13.2654686507
|1.60322
|[[210ed5]]
|2785.74841665
|17
|17
|-
|[[no-2 no-3 540zpi (σ = 1)]]
|95.1233580316
|12.6151980421
|1.65279
|[[221ed5]]
|2787.95876731
|23
|23
|23
|-
|-
Line 1,537: Line 898:
|29
|29
|29
|29
|-
|[[no-2 no-3 581zpi (σ = 1)]]
|100.799606439
|11.9048083856
|1.71723
|[[234ed5]]
|2785.72516223
|25
|25
|-
|[[no-2 no-3 659zpi (σ = 1)]]
|111.567387279
|10.7558313344
|1.61434
|[[259ed5]]
|2785.76031562
|19
|19
|-
|-
|[[no-2 no-3 671zpi (σ = 1)]]
|[[no-2 no-3 671zpi (σ = 1)]]
Line 1,564: Line 907:
|19
|19
|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 1,582: Line 916:
|37
|37
|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)]]
|[[no-2 no-3 905zpi (σ = 1)]]
Line 1,636: Line 934:
|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)]]
|154.617025672
|7.76111165495
|1.66586
|[[359ed5]]
|2786.23908413
|19
|19
|-
|-
|[[no-2 no-3 1046zpi (σ = 1)]]
|[[no-2 no-3 1046zpi (σ = 1)]]
Line 1,663: Line 943:
|23
|23
|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)]]
|[[no-2 no-3 1145zpi (σ = 1)]]
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|35
|35
|35
|35
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|[[no-2 no-3 1214zpi (σ = 1)]]
|183.477053621
|6.54032739419
|1.68165
|[[426ed5]]
|2786.17946993
|17
|17
|-
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|[[no-2 no-3 1280zpi (σ = 1)]]
|[[no-2 no-3 1280zpi (σ = 1)]]
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|[[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)]]
|199.431052743
|6.01711711137
|1.70966
|[[463ed5]]
|2785.92522256
|37
|37
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|}


Latest revision as of 18:31, 14 December 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

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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