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 400: Line 400:
|18
|18
|18
|18
|}
{{todo|use sigma 1.0|inline=1|comment=instead of sigma 1/2}}
=== Notable Local Maxima of the Riemann Zeta Function after removing the prime 2 from the zeta product ===
{|class="wikitable sortable"
!colspan="3"|Tuning
!colspan="3"|Strength
!colspan="2"|Closest EDT
!colspan="2"|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
|}
|}


Line 702: Line 405:


{|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 ≥ 2.6 and cents ≥ 15.0) or (height ≥ 2.65 and cents ≥ 12.0)
|+ style="font-size: 105%;" | 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)
!colspan="3"|Tuning
!colspan="3"|Tuning
!colspan="1"|Strength
!colspan="1"|Strength
Line 823: Line 526:
|1199.84985856
|1199.84985856
|17
|17
|16
|-
|[[no-3 462zpi (σ = 1)]]
|84.0209403971
|14.2821538813
|2.70980
|[[84edo]]
|1199.70092603
|16
|16
|16
|-
|-
Line 842: Line 536:
|35
|35
|19
|19
|-
|[[no-3 596zpi (σ = 1)]]
|102.960307695
|11.6549768242
|2.88566
|[[103edo]]
|1200.46261290
|17
|17
|-
|[[no-3 640zpi (σ = 1)]]
|108.978652930
|11.0113308225
|3.02482
|[[109edo]]
|1200.23505965
|16
|16
|-
|[[no-3 684zpi (σ = 1)]]
|115.020054265
|10.4329632573
|2.86867
|[[115edo]]
|1199.79077459
|16
|16
|-
|-
|[[no-3 751zpi (σ = 1)]]
|[[no-3 751zpi (σ = 1)]]
Line 878: Line 545:
|28
|28
|26
|26
|}
=== Notable Local Maxima of the Riemann Zeta Function after removing the prime 2 from the zeta product ===
{|class="wikitable sortable"
|+ style="font-size: 105%;" | Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.075 and cents ≥ 6.0)
!colspan="3"|Tuning
!colspan="1"|Strength
!colspan="2"|Closest EDT
!colspan="2"|No-2 Integer limit
|-
!No-2 ZPI (σ = 1)
!Steps per octave
!Cents
!Height
!EDT
!Tritave
!Consistent
!Distinct
|-
|-
|[[no-3 826zpi (σ = 1)]]
|[[no-2 93zpi (σ = 1)]]
|133.978395805
|24.5747239922
|8.95666792242
|48.8306603314
|2.68076
|2.12985
|[[134edo]]
|[[39edt]]
|1200.19350160
|1904.39575293
|10
|15
|10
|15
|-
|-
|[[no-3 849zpi (σ = 1)]]
|[[no-2 151zpi (σ = 1)]]
|137.014168071
|35.3061077059
|8.75821834261
|33.9884534992
|2.65309
|2.08576
|[[137edo]]
|[[56edt]]
|1199.87591294
|1903.35339595
|8
|15
|8
|15
|-
|-
|[[no-3 872zpi (σ = 1)]]
|[[no-2 207zpi (σ = 1)]]
|139.969011238
|44.8164999984
|8.57332626265
|26.7758526445
|2.95738
|2.10342
|[[140edo]]
|[[71edt]]
|1200.26567677
|1901.08553776
|17
|17
|17
|17
|-
|-
|[[no-3 918zpi (σ = 1)]]
|[[no-2 222zpi (σ = 1)]]
|145.988163227
|47.3516876312
|8.21984449614
|25.3422857776
|2.96655
|2.11876
|[[146edo]]
|[[75edt]]
|1200.09729644
|1900.67143332
|22
|15
|22
|15
|-
|[[no-2 233zpi (σ = 1)]]
|49.1657210129
|24.4072491012
|2.07714
|[[78edt]]
|1903.76542989
|21
|21
|-
|-
|[[no-3 949zpi (σ = 1)]]
|[[no-2 273zpi (σ = 1)]]
|149.967271540
|55.5359583782
|8.00174589878
|21.6076220712
|2.75244
|2.19450
|[[150edo]]
|[[88edt]]
|1200.26188482
|1901.47074227
|22
|11
|22
|11
|-
|-
|[[no-3 1035zpi (σ = 1)]]
|[[no-2 363zpi (σ = 1)]]
|161.022597516
|69.4191721809
|7.45237015495
|17.2862908372
|3.03830
|2.08043
|[[161edo]]
|[[110edt]]
|1199.83159495
|1901.49199210
|22
|23
|22
|23
|-
|-
|[[no-3 1083zpi (σ = 1)]]
|[[no-2 380zpi (σ = 1)]]
|167.067779937
|71.9200195089
|7.18271350975
|16.6852012582
|2.70357
|2.07565
|[[167edo]]
|[[114edt]]
|1199.51315613
|1902.11294344
|17
|17
|17
|17
|-
|-
|[[no-3 1138zpi (σ = 1)]]
|[[no-2 453zpi (σ = 1)]]
|173.986832234
|82.6700405439
|6.89707367272
|14.5155366092
|2.71583
|2.38406
|[[174edo]]
|[[131edt]]
|1200.09081905
|1901.53529581
|10
|27
|10
|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-3 1162zpi (σ = 1)]]
|[[no-2 687zpi (σ = 1)]]
|177.011493724
|115.412802617
|6.77922079947
|10.3974600113
|2.97004
|2.18983
|[[177edo]]
|[[183edt]]
|1199.92208151
|1902.73518207
|16
|15
|16
|15
|-
|-
|[[no-3 1242zpi (σ = 1)]]
|[[no-2 697zpi (σ = 1)]]
|186.981915298
|116.734850378
|6.41773295610
|10.2797064983
|2.83950
|2.15793
|[[187edo]]
|[[185edt]]
|1200.11606279
|1901.74570218
|28
|29
|22
|29
|-
|-
|[[no-3 1315zpi (σ = 1)]]
|[[no-2 777zpi (σ = 1)]]
|195.960897036
|127.486291223
|6.12367068200
|9.41277676594
|3.00252
|2.21095
|[[196edo]]
|[[202edt]]
|1200.23945367
|1901.38090672
|17
|17
|17
|17
|-
|-
|[[no-3 1332zpi (σ = 1)]]
|[[no-2 810zpi (σ = 1)]]
|198.038062648
|131.822840677
|6.05944122030
|9.10312654342
|2.65687
|2.25360
|[[198edo]]
|[[209edt]]
|1199.76936162
|1902.55344758
|16
|21
|16
|21
|-
|-
|[[no-3 1364zpi (σ = 1)]]
|[[no-2 829zpi (σ = 1)]]
|201.995809149
|134.373782790
|5.94071731021
|8.93031345169
|2.81223
|2.13475
|[[202edo]]
|[[213edt]]
|1200.02489666
|1902.15676521
|11
|29
|11
|29
|-
|-
|[[no-3 1381zpi (σ = 1)]]
|[[no-2 839zpi (σ = 1)]]
|204.057376735
|135.657892938
|5.88069894458
|8.84578091263
|2.71886
|2.11125
|[[204edo]]
|[[215edt]]
|1199.66258470
|1901.84289622
|23
|15
|23
|15
|-
|-
|[[no-3 1414zpi (σ = 1)]]
|[[no-2 858zpi (σ = 1)]]
|208.049926903
|138.196070465
|5.76784629470
|8.68331491602
|2.68595
|2.20051
|[[208edo]]
|[[219edt]]
|1199.71202930
|1901.64596661
|16
|16
|-
|[[no-3 1463zpi (σ = 1)]]
|214.026411191
|5.60678466421
|2.83329
|[[214edo]]
|1199.85191814
|10
|10
|-
|[[no-3 1496zpi (σ = 1)]]
|217.987761194
|5.50489620806
|3.05781
|[[218edo]]
|1200.06737336
|11
|11
|11
|11
|-
|-
|[[no-3 1546zpi (σ = 1)]]
|[[no-2 902zpi (σ = 1)]]
|224.015057895
|143.873905513
|5.35678275949
|8.34063686336
|2.65007
|2.09948
|[[224edo]]
|[[228edt]]
|1199.91933813
|1901.66520485
|16
|16
|-
|[[no-3 1571zpi (σ = 1)]]
|226.974561802
|5.28693607985
|2.83428
|[[227edo]]
|1200.13449013
|28
|26
|-
|[[no-3 1621zpi (σ = 1)]]
|232.982861825
|5.15059344109
|3.18230
|[[233edo]]
|1200.08827177
|17
|17
|-
|[[no-3 1672zpi (σ = 1)]]
|239.007141842
|5.02077047050
|3.03922
|[[239edo]]
|1199.96414245
|11
|11
|11
|11
|-
|-
|[[no-3 1723zpi (σ = 1)]]
|[[no-2 965zpi (σ = 1)]]
|245.078464843
|152.075713777
|4.89639104263
|7.89080629768
|2.70419
|2.10893
|[[245edo]]
|[[241edt]]
|1199.61580544
|1901.68431774
|16
|15
|16
|15
|-
|-
|[[no-3 1748zpi (σ = 1)]]
|[[no-2 985zpi (σ = 1)]]
|248.041899078
|154.604034485
|4.83789232569
|7.76176381166
|2.65779
|2.40811
|[[248edo]]
|[[245edt]]
|1199.79729677
|1901.63213386
|13
|21
|13
|21
|-
|-
|[[no-3 1799zpi (σ = 1)]]
|[[no-2 1029zpi (σ = 1)]]
|254.007436921
|160.260260060
|4.72427112586
|7.48782012177
|2.77515
|2.17192
|[[254edo]]
|[[254edt]]
|1199.96486597
|1901.90631093
|16
|9
|16
|9
|-
|-
|[[no-3 1859zpi (σ = 1)]]
|[[no-2 1049zpi (σ = 1)]]
|261.029447254
|162.750022676
|4.59718247356
|7.37327086209
|3.06231
|2.14738
|[[261edo]]
|[[258edt]]
|1199.86462560
|1902.30388242
|40
|17
|34
|17
|-
|-
|[[no-3 1884zpi (σ = 1)]]
|[[no-2 1069zpi (σ = 1)]]
|263.969408305
|165.332187903
|4.54598132301
|7.25811480039
|3.02070
|2.19607
|[[264edo]]
|[[262edt]]
|1200.13906928
|1901.62607770
|17
|17
|17
|17
|-
|-
|[[no-3 1936zpi (σ = 1)]]
|[[no-2 1134zpi (σ = 1)]]
|270.004586009
|173.506549648
|4.44436895587
|6.91616542681
|2.86256
|2.26764
|[[270edo]]
|[[275edt]]
|1199.97961808
|1901.94549237
|16
|29
|16
|29
|-
|-
|[[no-3 1970zpi (σ = 1)]]
|[[no-2 1159zpi (σ = 1)]]
|273.982199321
|176.625850825
|4.37984658483
|6.79402247404
|2.79132
|2.14379
|[[274edo]]
|[[280edt]]
|1200.07796424
|1902.32629273
|22
|22
|-
|[[no-3 1988zpi (σ = 1)]]
|276.049541463
|4.34704580069
|2.74028
|[[276edo]]
|1199.78464099
|11
|11
|11
|11
|-
|-
|[[no-3 2022zpi (σ = 1)]]
|[[no-2 1179zpi (σ = 1)]]
|279.959012818
|179.167803205
|4.28634173239
|6.69763193238
|2.75619
|2.29964
|[[280edo]]
|[[284edt]]
|1200.17568507
|1902.12746880
|10
|15
|10
|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-3 2048zpi (σ = 1)]]
|[[no-2 1297zpi (σ = 1)]]
|282.978336346
|193.736743714
|4.24060730405
|6.19397217583
|2.66363
|2.12380
|[[283edo]]
|[[307edt]]
|1200.09186705
|1901.54945798
|14
|21
|14
|21
|-
|-
|[[no-3 2074zpi (σ = 1)]]
|[[no-2 1343zpi (σ = 1)]]
|285.991584632
|199.415414525
|4.19592765830
|6.01758897555
|2.84732
|2.36503
|[[286edo]]
|[[316edt]]
|1200.03531028
|1901.55811627
|22
|39
|22
|39
|}
|}


Line 1,171: Line 866:


{|class="wikitable sortable"
{|class="wikitable sortable"
|+ 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="3"|Strength
!colspan="1"|Strength
!colspan="2"|Closest ED5
!colspan="2"|Closest ED5
!colspan="2"|No-2 No-3 Integer limit
!colspan="2"|No-2 No-3 Integer limit
Line 1,180: Line 876:
!Cents
!Cents
!Height
!Height
!Integral
!Gap
!ED5
!ED5
!Pentave
!Pentave
Line 1,187: Line 881:
!Distinct
!Distinct
|-
|-
|[[no-2 no-3 55zpi analog]]
|[[no-2 no-3 186zpi (σ = 1)]]
|16.7630030425585
|41.3464998527
|71.5862185882446
|29.0230129340
|3.480299
|1.75534
|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]]
|[[96ed5]]
|2786.12170418722
|2786.20924167
|35
|35
|23
|23
|-
|-
|[[no-2 no-3 212zpi analog]]
|[[no-2 no-3 565zpi (σ = 1)]]
|45.6783815054539
|98.6253027359
|26.2706330752267
|12.1672630320
|3.818225
|1.74188
|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]]
|[[229ed5]]
|2786.29046187457
|2786.30323433
|29
|29
|29
|29
|-
|-
|[[no-2 no-3 581zpi analog]]
|[[no-2 no-3 671zpi (σ = 1)]]
|100.797128599965
|113.258011095
|11.9051010347969
|10.5952769998
|4.579796
|1.77217
|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]]
|[[263ed5]]
|2786.59158866045
|2786.55785095
|19
|19
|19
|19
|-
|-
|[[no-2 no-3 764zpi analog]]
|[[no-2 no-3 764zpi (σ = 1)]]
|125.745930952370
|125.745000550
|9.54305233506547
|9.54312294522
|5.001815
|1.75634
|0.548008
|12.976730
|[[292ed5]]
|[[292ed5]]
|2786.57128183912
|2786.59190001
|37
|37
|37
|37
|-
|-
|[[no-2 no-3 905zpi analog]]
|[[no-2 no-3 905zpi (σ = 1)]]
|144.300058486204
|144.297529480
|8.31600494545005
|8.31615069448
|5.030210
|1.73926
|0.539592
|13.254432
|[[335ed5]]
|[[335ed5]]
|2785.86165672577
|2785.91048265
|43
|43
|41
|41
|-
|-
|[[no-2 no-3 938zpi analog]]
|[[no-2 no-3 938zpi (σ = 1)]]
|148.561761173834
|148.562870929
|8.07744866861039
|8.07738833059
|5.510552
|1.79949
|0.600083
|13.846076
|[[345ed5]]
|[[345ed5]]
|2786.71979067058
|2786.69897405
|25
|25
|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
|}
|}


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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