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== Equal-step tunings ==
== Equal-step tunings ==
{| class="wikitable"
 
|+
=== About this list ===
!Name
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 ===
{|class="wikitable sortable"
|+ style="font-size: 105%;" |
|-
!colspan="3"|Tuning
!colspan="1"|Strength
!colspan="2"|Closest EDO
!colspan="2"|Integer limit
|-
!ZPI (σ = 1)
!Steps per octave
!Steps per octave
!Step size (cents)
!Step size (cents)
!Why it matters
!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 ===
 
{|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 ≥ 2.8 and cents ≥ 12.0) or (height ≥ 3.1 and cents ≥ 6.0)
!colspan="3"|Tuning
!colspan="1"|Strength
!colspan="2"|Closest EDO
!colspan="2"|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 ===
 
{|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-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 ===
 
{|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="1"|Strength
!colspan="2"|Closest ED5
!colspan="2"|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
|-
|-
|5edo
|[[no-2 no-3 1046zpi (σ = 1)]]
|
|162.414291729
|
|7.38851234841
|EDO ≤ 29
|1.73251
|[[377ed5]]
|2785.46915535
|23
|23
|-
|-
|6edo
|[[no-2 no-3 1145zpi (σ = 1)]]
|
|174.880594782
|
|6.86182478678
|EDO ≤ 29
|1.74084
|[[406ed5]]
|2785.90086343
|25
|25
|-
|-
|7edo
|[[no-2 no-3 1196zpi (σ = 1)]]
|
|181.292147244
|
|6.61915046096
|EDO ≤ 29
|1.77770
|[[421ed5]]
|2786.66234406
|35
|35
|-
|-
|8edo
|[[no-2 no-3 1280zpi (σ = 1)]]
|
|191.632570168
|
|6.26198353937
|EDO ≤ 29
|1.75036
|[[445ed5]]
|2786.58267502
|29
|29
|}
 
=== The α–β–γ family ===
{| class="wikitable sortable"
|+ style="font-size: 105%;" | α–β–γ family
|- style="white-space: nowrap;"
! colspan="4" |Optimization
! rowspan="2" |Equal division of a ratio
|- style="white-space: nowrap;"
!Proposed name
!Steps per octave
!Cents
!Optimization method
|-
|-
|9edo
|[[Alpha 3/1]]
|
|1.90739592696007
|
|629.130000247254
|EDO ≤ 29
|Dave Benson
|[[3edt|3ed3/1]]
|-
|-
|10edo
|[[Beta 3/1]]
|
|3.14186231690763
|
|381.939079106782
|EDO ≤ 29
|Dave Benson
|[[5edt|5ed3/1]]
|-
|-
|11edo
|[[Alpha 2/1]]
|
|5.00991270509077
|
|239.525131601721
|EDO ≤ 29
|Dave Benson
|[[5edo|5ed2/1]]
|-
|-
|12edo
|[[Gamma 3/1]]
|
|5.04255621376059
|
|237.974540913462
|EDO ≤ 29
|Dave Benson
|[[8edt|8ed3/1]]
|-
|-
|13edo
|[[Beta 2/1]]
|
|6.99104980248710
|
|171.648040552235
|EDO ≤ 29
|Dave Benson
|[[7edo|7ed2/1]]
|-
|-
|14edo
|[[Alpha 5/3]]
|
|9.50583353877785
|
|126.238272015258
|EDO ≤ 29
|Dave Benson
|[[7ed5/3]]
|-
|-
|15edo
|[[Gamma 2/1]]
|
|11.9978480914311
|
|100.017935787756
|EDO ≤ 29
|Dave Benson
|[[12edo|12ed2/1]]
|-
|-
|16edo
|[[Beta 5/3]]
|
|12.2053823008782
|
|98.3172808862904
|EDO ≤ 29
|Dave Benson
|[[9ed5/3]]
|-
|-
|17edo
|[[Carlos Alpha|Alpha 3/2]]
|
|15.3915238996928
|
|77.9649895501219
|EDO ≤ 29
|Dave Benson
|[[9edf|9ed3/2]]
|-
|-
|18edo
|[[Carlos Beta|Beta 3/2]]
|
|18.7990736394111
|
|63.8329325698408
|EDO ≤ 29
|Dave Benson
|[[11edf|11ed3/2]]
|-
|-
|19edo
|[[Gamma 5/3]]
|
|21.7094399215509
|
|55.2754932571412
|EDO ≤ 29
|Dave Benson
|[[16ed5/3]]
|-
|-
|20edo
|[[Alpha 7/5]]
|
|22.6653911133366
|
|52.9441558718088
|EDO ≤ 29
|Dave Benson
|[[11ed7/5]]
|-
|-
|21edo
|[[Beta 7/5]]
|
|26.7758951088566
|
|44.8164289231577
|EDO ≤ 29
|Dave Benson
|[[13ed7/5]]
|-
|-
|22edo
|[[Alpha 4/3]]
|
|31.3266790320926
|
|38.3060074376432
|EDO ≤ 29
|Dave Benson
|[[13ed4/3]]
|-
|-
|23edo
|[[Carlos Gamma|Gamma 3/2]]
|
|34.1894540921914
|
|35.0985422804417
|EDO ≤ 29
|Dave Benson
|[[20edf|20ed3/2]]
|-
|-
|24edo
|[[Beta 4/3]]
|
|36.1372975038827
|
|33.2066890135065
|EDO ≤ 29
|Dave Benson
|[[15ed4/3]]
|-
|-
|25edo
|[[Gamma 7/5]]
|
|49.4404896216012
|
|24.2716042900130
|EDO ≤ 29
|Dave Benson
|[[24ed7/5]]
|-
|-
|26edo
|[[Gamma 4/3]]
|
|67.4633901646646
|
|17.7874251067289
|EDO ≤ 29
|Dave Benson
|[[28ed4/3]]
|}
 
== Unequal-step tunings ==
 
=== Unequal-step tunings from equal divisions of a ratio ===
{| class="wikitable"
|+
!Tuning
!Period
!Mode
!Why it matters
|-
|-
|27edo
|[[93edo and stretched hemififths|Stretched hemififth]]
|
|94\93<2/1>
|16 11 16 12 16 11 12
|
|
|EDO ≤ 29
|-
|-
|28edo
|[[36edo|833 Cent Acoustic Golden Scale [11]]]
|
|25\36<2/1>
|3 1 3 3 1 3 1 3 3 1 3
|
|
|EDO ≤ 29
|-
|-
|29edo
|833 Cent Logarithmic Golden Scale [8]
|
|ϕ
|ϕ 1 ϕ ϕ 1 ϕ 1 ϕ
|
|
|EDO ≤ 29
|}
|}
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