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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
|-
|-
|5edo
|[[106zpi (σ = 1)]]
|
|27.0853383248
|
|44.3044124320
|EDO ≤ 29, Alpha 2/1 analogue
|2.90524
|[[27edo]]
|1196.21913566
|10
|8
|-
|-
|6edo
|[[116zpi (σ = 1)]]
|
|28.9431579907
|
|41.4605759463
|EDO ≤ 29
|2.68561
|[[29edo]]
|1202.35670244
|8
|7
|-
|-
|7edo
|[[127zpi (σ = 1)]]
|
|30.9779815456
|
|38.7371913897
|EDO ≤ 29, Beta 2/1 analogue
|3.23190
|[[31edo]]
|1200.85293308
|12
|9
|-
|-
|8edo
|[[144zpi (σ = 1)]]
|
|34.0437506778
|
|35.2487600839
|EDO ≤ 29
|3.07414
|[[34edo]]
|1198.45784285
|6
|6
|-
|-
|9edo
|[[155zpi (σ = 1)]]
|
|35.9827898689
|
|33.3492762616
|EDO ≤ 29
|2.80355
|[[36edo]]
|1200.57394542
|8
|8
|-
|-
|7ed5/3
|[[184zpi (σ = 1)]]
|
|40.9880790756
|
|29.2768050385
|Alpha 5/3 analogue
|3.32966
|[[41edo]]
|1200.34900658
|16
|10
|-
|-
|10edo
|[[196zpi (σ = 1)]]
|
|43.0234004818
|
|27.8917981043
|EDO ≤ 29
|2.78019
|[[43edo]]
|1199.34731849
|8
|8
|-
|-
|26zpi
|[[214zpi (σ = 1)]]
|
|46.0106419996
|
|26.0809227572
|
|3.25119
|[[46edo]]
|1199.72244683
|14
|11
|-
|-
|11edo
|[[238zpi (σ = 1)]]
|
|49.9382924730
|
|24.0296562132
|EDO ≤ 29
|2.90274
|[[50edo]]
|1201.48281066
|10
|9
|-
|-
|12edo
|[[257zpi (σ = 1)]]
|
|52.9969882711
|
|22.6427961125
|EDO ≤ 29, Has a strong zeta peak, Gamma 2/1 analogue
|3.46399
|[[53edo]]
|1200.06819396
|10
|10
|-
|-
|34zpi
|[[289zpi (σ = 1)]]
|
|58.0645692462
|
|20.6666477609
|
|3.25823
|[[58edo]]
|1198.66557013
|16
|12
|-
|-
|9ed5/3
|[[301zpi (σ = 1)]]
|
|59.9223835273
|
|20.0259056693
|Beta 5/3 analogue
|2.98826
|[[60edo]]
|1201.55434016
|10
|10
|-
|-
|13edo
|[[321zpi (σ = 1)]]
|
|63.0197888699
|
|19.0416378969
|EDO ≤ 29
|2.87513
|[[63edo]]
|1199.62318750
|8
|8
|-
|-
|42zpi
|[[334zpi (σ = 1)]]
|
|65.0145858034
|
|18.4573966776
|
|3.23462
|[[65edo]]
|1199.73078404
|6
|6
|-
|-
|14edo
|[[354zpi (σ = 1)]]
|
|68.0496579343
|
|17.6341812204
|EDO ≤ 29
|3.14200
|[[68edo]]
|1199.12432299
|10
|10
|-
|-
|15edo
|[[380zpi (σ = 1)]]
|
|71.9512656175
|
|16.6779554147
|EDO ≤ 29
|3.61665
|[[72edo]]
|1200.81278986
|18
|13
|-
|-
|47zpi
|[[414zpi (σ = 1)]]
|
|76.9924672555
|
|15.5859403235
|
|3.28825
|[[77edo]]
|1200.11740491
|10
|10
|-
|-
|9ed3/2
|[[435zpi (σ = 1)]]
|
|80.0733926855
|
|14.9862514845
|Carlos Alpha 3/2
|3.14833
|[[80edo]]
|1198.90011876
|12
|12
|-
|-
|51zpi
|[[462zpi (σ = 1)]]
|
|83.9950884037
|
|14.2865496400
|
|3.19687
|[[84edo]]
|1200.07016976
|10
|10
|-
|-
|16edo
|[[483zpi (σ = 1)]]
|
|87.0139579095
|
|13.7908908965
|EDO ≤ 29
|3.44872
|[[87edo]]
|1199.80750799
|16
|14
|-
|-
|17edo
|[[497zpi (σ = 1)]]
|
|89.0215260329
|
|13.4798857476
|EDO ≤ 29
|3.02681
|[[89edo]]
|1199.70983154
|12
|12
|-
|-
|56zpi
|[[532zpi (σ = 1)]]
|
|93.9843698073
|
|12.7680805059
|
|3.39762
|[[94edo]]
|1200.19956756
|24
|15
|-
|-
|18edo
|[[568zpi (σ = 1)]]
|
|99.0456175574
|
|12.1156294402
|EDO ≤ 29
|3.56676
|[[99edo]]
|1199.44731458
|12
|12
|-
|-
|11ed3/2
|[[596zpi (σ = 1)]]
|
|102.936325452
|
|11.6576922163
|Carlos Beta 3/2
|3.25007
|[[103edo]]
|1200.74229828
|15
|15
|-
|-
|65zpi
|[[655zpi (σ = 1)]]
|
|111.058159333
|
|10.8051493669
|
|3.39509
|[[111edo]]
|1199.37157972
|22
|16
|-
|-
|19edo
|[[706zpi (σ = 1)]]
|
|117.971388652
|
|10.1719579104
|EDO ≤ 29, Has a strong zeta peak
|3.62695
|[[118edo]]
|1200.29103343
|12
|12
|-
|-
|20edo
|[[796zpi (σ = 1)]]
|
|130.004267285
|
|9.23046623824
|EDO ≤ 29
|3.72487
|[[130edo]]
|1199.96061097
|16
|16
|-
|-
|
|[[872zpi (σ = 1)]]
|
|139.992781938
|
|8.57187051639
|
|3.60746
|[[140edo]]
|1200.06187229
|10
|10
|-
|-
|21edo
|[[965zpi (σ = 1)]]
|
|152.050659206
|
|7.89210652729
|EDO ≤ 29
|3.68901
|[[152edo]]
|1199.60019215
|15
|15
|-
|-
|75zpi
|[[1114zpi (σ = 1)]]
|
|170.995049914
|
|7.01774700849
|
|3.82285
|[[171edo]]
|1200.03473845
|14
|14
|-
|-
|16ed5/3
|[[1210zpi (σ = 1)]]
|
|183.000273182
|
|6.55736726036
|Gamma 5/3 analogue
|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
|-
|-
|22edo
!No-3 ZPI analog
|
!Steps per octave
|
!Cents
|EDO ≤ 29, Has a strong zeta peak
!Height
!EDO
!Octave
!Consistent
!Distinct
|-
|-
|80zpi
|[[no-3 51zpi (σ = 1)]]
|
|15.9687074547
|
|75.1469712502
|
|2.56677
|[[16edo]]
|1202.35154000
|26
|8
|-
|-
|11ed7/5
|[[no-3 75zpi (σ = 1)]]
|
|21.0417134383
|
|57.0295762045
|Alpha 7/5 analogue
|2.60042
|[[21edo]]
|1197.62110029
|17
|10
|-
|-
|23edo
|[[no-3 95zpi (σ = 1)]]
|
|24.9617781085
|
|48.0734984016
|EDO ≤ 29
|2.64675
|[[25edo]]
|1201.83746004
|14
|11
|-
|-
|24edo
|[[no-3 127zpi (σ = 1)]]
|
|31.0146799866
|
|38.6913552073
|EDO ≤ 29
|2.60405
|[[31edo]]
|1199.43201143
|11
|11
|-
|-
|90zpi
|[[no-3 161zpi (σ = 1)]]
|
|37.0135086000
|
|32.4205957606
|
|2.92705
|[[37edo]]
|1199.56204314
|22
|16
|-
|-
|93zpi no-2 analogue
|[[no-3 196zpi (σ = 1)]]
|24.5738316304204445883184323365600165414701853056787276394517489970293
|43.0494972034
|
|27.8748900209
|
|2.71380
|[[43edo]]
|1198.62027090
|22
|19
|-
|-
|93zpi
|[[no-3 220zpi (σ = 1)]]
|
|47.0043385196
|
|25.5295582875
|
|2.69328
|[[47edo]]
|1199.88923951
|10
|10
|-
|-
|39edt
|[[no-3 276zpi (σ = 1)]]
|
|55.9891415481
|
|21.4327272543
|Has a strong no-2 zeta peak
|2.76321
|[[56edo]]
|1200.23272624
|20
|19
|-
|-
|25edo
|[[no-3 340zpi (σ = 1)]]
|
|65.9204029312
|
|18.2037722259
|EDO ≤ 29
|2.65263
|[[66edo]]
|1201.44896691
|16
|16
|-
|-
|26edo
|[[no-3 354zpi (σ = 1)]]
|
|68.0229453080
|
|17.6411061674
|EDO ≤ 29
|2.76285
|[[68edo]]
|1199.59521939
|11
|11
|-
|-
|13ed7/5
|[[no-3 394zpi (σ = 1)]]
|
|74.0566473758
|
|16.2038121158
|Beta 7/5 analogue
|2.76672
|[[74edo]]
|1199.08209657
|16
|16
|-
|-
|27edo
|[[no-3 421zpi (σ = 1)]]
|
|78.0097604150
|
|15.3826904943
|EDO ≤ 29, Has a strong zeta peak
|2.81219
|[[78edo]]
|1199.84985856
|17
|16
|-
|-
|106zpi
|[[no-3 525zpi (σ = 1)]]
|
|93.0066513531
|
|12.9023030347
|
|2.97919
|[[93edo]]
|1199.91418223
|35
|19
|-
|-
|28edo
|[[no-3 751zpi (σ = 1)]]
|
|124.013627761
|
|9.67635591079
|EDO ≤ 29
|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
|-
|-
|116zpi
!No-2 ZPI (σ = 1)
|
!Steps per octave
|
!Cents
|
!Height
!EDT
!Tritave
!Consistent
!Distinct
|-
|-
|29edo
|[[no-2 93zpi (σ = 1)]]
|
|24.5747239922
|
|48.8306603314
|EDO ≤ 29
|2.12985
|[[39edt]]
|1904.39575293
|15
|15
|-
|-
|127zpi
|[[no-2 151zpi (σ = 1)]]
|
|35.3061077059
|
|33.9884534992
|
|2.08576
|[[56edt]]
|1903.35339595
|15
|15
|-
|-
|31edo
|[[no-2 207zpi (σ = 1)]]
|
|44.8164999984
|
|26.7758526445
|Has a strong zeta peak
|2.10342
|[[71edt]]
|1901.08553776
|17
|17
|-
|-
|13ed4/3
|[[no-2 222zpi (σ = 1)]]
|
|47.3516876312
|
|25.3422857776
|Alpha 4/3 analogue
|2.11876
|[[75edt]]
|1900.67143332
|15
|15
|-
|-
|34edo
|[[no-2 233zpi (σ = 1)]]
|
|49.1657210129
|
|24.4072491012
|Has a strong zeta peak
|2.07714
|[[78edt]]
|1903.76542989
|21
|21
|-
|-
|144zpi
|[[no-2 273zpi (σ = 1)]]
|
|55.5359583782
|
|21.6076220712
|
|2.19450
|[[88edt]]
|1901.47074227
|11
|11
|-
|-
|20ed3/2
|[[no-2 363zpi (σ = 1)]]
|
|69.4191721809
|
|17.2862908372
|Carlos Gamma 3/2
|2.08043
|[[110edt]]
|1901.49199210
|23
|23
|-
|-
|151zpi
|[[no-2 380zpi (σ = 1)]]
|
|71.9200195089
|
|16.6852012582
|
|2.07565
|[[114edt]]
|1902.11294344
|17
|17
|-
|-
|151zpi no-2 analogue
|[[no-2 453zpi (σ = 1)]]
|35.3059427335608633586867709728239574896988978536248407971925774931920
|82.6700405439
|
|14.5155366092
|
|2.38406
|[[131edt]]
|1901.53529581
|27
|27
|-
|-
|56edt
|[[no-2 492zpi (σ = 1)]]
|
|88.3238806401
|
|13.5863595587
|Has a strong no-2 zeta peak
|2.12238
|[[140edt]]
|1902.09033822
|9
|9
|-
|-
|155zpi no-5 analogue
|[[no-2 510zpi (σ = 1)]]
|35.9775957344990354876843659181629374042162799645238247644819739175425
|90.8334979880
|
|13.2109852266
|
|2.23067
|[[144edt]]
|1902.38187263
|39
|27
|-
|-
|155zpi
|[[no-2 550zpi (σ = 1)]]
|
|96.5187261015
|
|12.4328205362
|
|2.24293
|[[153edt]]
|1902.22154203
|15
|15
|-
|-
|36edo
|[[no-2 627zpi (σ = 1)]]
|
|107.244021785
|
|11.1894348983
|Has a strong no-5 zeta peak
|2.29774
|[[170edt]]
|1902.20393272
|15
|15
|-
|-
|15ed4/3
|[[no-2 687zpi (σ = 1)]]
|
|115.412802617
|
|10.3974600113
|Beta 4/3 analogue
|2.18983
|[[183edt]]
|1902.73518207
|15
|15
|-
|-
|37edo
|[[no-2 697zpi (σ = 1)]]
|
|116.734850378
|
|10.2797064983
|Has a strong no-3 zeta peak
|2.15793
|[[185edt]]
|1901.74570218
|29
|29
|-
|-
|161zpi no-3 analogue
|[[no-2 777zpi (σ = 1)]]
|37.0117501336435252522939269985227920601261578745487306336979972897294
|127.486291223
|
|9.41277676594
|
|2.21095
|[[202edt]]
|1901.38090672
|17
|17
|-
|-
|161zpi
|[[no-2 810zpi (σ = 1)]]
|
|131.822840677
|
|9.10312654342
|
|2.25360
|[[209edt]]
|1902.55344758
|21
|21
|-
|-
|41edo
|[[no-2 829zpi (σ = 1)]]
|
|134.373782790
|
|8.93031345169
|Has a strong zeta peak
|2.13475
|[[213edt]]
|1902.15676521
|29
|29
|-
|-
|184zpi
|[[no-2 839zpi (σ = 1)]]
|
|135.657892938
|
|8.84578091263
|
|2.11125
|[[215edt]]
|1901.84289622
|15
|15
|-
|-
|186zpi
|[[no-2 858zpi (σ = 1)]]
|41.3438354846780
|138.196070465
|
|8.68331491602
|
|2.20051
|[[219edt]]
|1901.64596661
|11
|11
|-
|-
|96ed5
|[[no-2 902zpi (σ = 1)]]
|
|143.873905513
|
|8.34063686336
|Has a strong no-2 no-3 zeta peak
|2.09948
|[[228edt]]
|1901.66520485
|11
|11
|-
|-
|186zpi no-2 no-3 analogue
|[[no-2 965zpi (σ = 1)]]
|41.3477989230936
|152.075713777
|
|7.89080629768
|
|2.10893
|[[241edt]]
|1901.68431774
|15
|15
|-
|-
|188zpi no-2 no-5 analogue
|[[no-2 985zpi (σ = 1)]]
|41.6281274155763001275416027845619755345480144939422820248677372190977
|154.604034485
|
|7.76176381166
|
|2.40811
|[[245edt]]
|1901.63213386
|21
|21
|-
|-
|66edt
|[[no-2 1029zpi (σ = 1)]]
|
|160.260260060
|
|7.48782012177
|Has a strong no-2 no-5 zeta peak
|2.17192
|[[254edt]]
|1901.90631093
|9
|9
|-
|-
|188zpi
|[[no-2 1049zpi (σ = 1)]]
|
|162.750022676
|
|7.37327086209
|
|2.14738
|[[258edt]]
|1902.30388242
|17
|17
|-
|-
|46edo
|[[no-2 1069zpi (σ = 1)]]
|
|165.332187903
|
|7.25811480039
|Has a strong zeta peak
|2.19607
|[[262edt]]
|1901.62607770
|17
|17
|-
|-
|214zpi
|[[no-2 1134zpi (σ = 1)]]
|
|173.506549648
|
|6.91616542681
|
|2.26764
|[[275edt]]
|1901.94549237
|29
|29
|-
|-
|24ed7/5
|[[no-2 1159zpi (σ = 1)]]
|
|176.625850825
|
|6.79402247404
|Gamma 7/5 analogue
|2.14379
|[[280edt]]
|1902.32629273
|11
|11
|-
|-
|53edo
|[[no-2 1179zpi (σ = 1)]]
|
|179.167803205
|
|6.69763193238
|Has a strong zeta peak
|2.29964
|[[284edt]]
|1902.12746880
|15
|15
|-
|-
|257zpi
|[[no-2 1245zpi (σ = 1)]]
|
|187.354933401
|
|6.40495544056
|
|2.28021
|[[297edt]]
|1902.27176585
|21
|21
|-
|-
|
|[[no-2 1266zpi (σ = 1)]]
|56.9949885079206769176514037038198357725273287855611008976484058072516
|189.909845446
|
|6.31878772364
|
|2.17116
|[[301edt]]
|1901.95510482
|17
|17
|-
|-
|57edo
|[[no-2 1297zpi (σ = 1)]]
|
|193.736743714
|
|6.19397217583
|Has a strong no-3 no-5 zeta peak
|2.12380
|[[307edt]]
|1901.54945798
|21
|21
|-
|-
|58edo
|[[no-2 1343zpi (σ = 1)]]
|
|199.415414525
|
|6.01758897555
|Has a strong zeta peak
|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
|-
|-
|289zpi
!No-2 No-3 ZPI analog
|
!Steps per octave
|
!Cents
|
!Height
!ED5
!Pentave
!Consistent
!Distinct
|-
|-
|301zpi
|[[no-2 no-3 186zpi (σ = 1)]]
|
|41.3464998527
|
|29.0230129340
|
|1.75534
|[[96ed5]]
|2786.20924167
|35
|23
|-
|-
|60edo
|[[no-2 no-3 565zpi (σ = 1)]]
|
|98.6253027359
|
|12.1672630320
|Has a strong zeta peak
|1.74188
|[[229ed5]]
|2786.30323433
|29
|29
|-
|-
|65edo
|[[no-2 no-3 671zpi (σ = 1)]]
|
|113.258011095
|
|10.5952769998
|Has a strong zeta peak
|1.77217
|[[263ed5]]
|2786.55785095
|19
|19
|-
|-
|28ed4/3
|[[no-2 no-3 764zpi (σ = 1)]]
|
|125.745000550
|
|9.54312294522
|Gamma 4/3 analogue
|1.75634
|[[292ed5]]
|2786.59190001
|37
|37
|-
|-
|68edo
|[[no-2 no-3 905zpi (σ = 1)]]
|
|144.297529480
|
|8.31615069448
|Has a strong zeta peak
|1.73926
|[[335ed5]]
|2785.91048265
|43
|41
|-
|-
|354zpi
|[[no-2 no-3 938zpi (σ = 1)]]
|
|148.562870929
|
|8.07738833059
|
|1.79949
|[[345ed5]]
|2786.69897405
|25
|25
|-
|-
|380zpi
|[[no-2 no-3 1046zpi (σ = 1)]]
|
|162.414291729
|
|7.38851234841
|
|1.73251
|[[377ed5]]
|2785.46915535
|23
|23
|-
|-
|72edo
|[[no-2 no-3 1145zpi (σ = 1)]]
|
|174.880594782
|
|6.86182478678
|Has a strong zeta peak
|1.74084
|[[406ed5]]
|2785.90086343
|25
|25
|-
|-
|414zpi
|[[no-2 no-3 1196zpi (σ = 1)]]
|
|181.292147244
|
|6.61915046096
|
|1.77770
|[[421ed5]]
|2786.66234406
|35
|35
|-
|-
|77edo
|[[no-2 no-3 1280zpi (σ = 1)]]
|
|191.632570168
|
|6.26198353937
|Has a strong zeta peak
|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
|-
|-
|80edo
|[[Alpha 3/1]]
|
|1.90739592696007
|
|629.130000247254
|Has a strong zeta peak
|Dave Benson
|[[3edt|3ed3/1]]
|-
|-
|
|[[Beta 3/1]]
|82.9585473728587401934282446836610895074185494886540503684148508037660
|3.14186231690763
|
|381.939079106782
|
|Dave Benson
|[[5edt|5ed3/1]]
|-
|-
|83edo
|[[Alpha 2/1]]
|
|5.00991270509077
|
|239.525131601721
|Has a strong no-3 no-5 zeta peak
|Dave Benson
|[[5edo|5ed2/1]]
|-
|-
|462zpi
|[[Gamma 3/1]]
|
|5.04255621376059
|
|237.974540913462
|
|Dave Benson
|[[8edt|8ed3/1]]
|-
|-
|84edo
|[[Beta 2/1]]
|
|6.99104980248710
|
|171.648040552235
|Has a strong zeta peak
|Dave Benson
|[[7edo|7ed2/1]]
|-
|-
|87edo
|[[Alpha 5/3]]
|
|9.50583353877785
|
|126.238272015258
|Has a strong zeta peak
|Dave Benson
|[[7ed5/3]]
|-
|-
|483zpi
|[[Gamma 2/1]]
|
|11.9978480914311
|
|100.017935787756
|
|Dave Benson
|[[12edo|12ed2/1]]
|-
|-
|532zpi
|[[Beta 5/3]]
|
|12.2053823008782
|
|98.3172808862904
|
|Dave Benson
|[[9ed5/3]]
|-
|-
|94edo
|[[Carlos Alpha|Alpha 3/2]]
|
|15.3915238996928
|
|77.9649895501219
|Has a strong zeta peak
|Dave Benson
|[[9edf|9ed3/2]]
|-
|-
|99edo
|[[Carlos Beta|Beta 3/2]]
|
|18.7990736394111
|
|63.8329325698408
|Has a strong zeta peak
|Dave Benson
|[[11edf|11ed3/2]]
|-
|-
|568zpi
|[[Gamma 5/3]]
|
|21.7094399215509
|
|55.2754932571412
|
|Dave Benson
|[[16ed5/3]]
|-
|-
|327ed7
|[[Alpha 7/5]]
|
|22.6653911133366
|
|52.9441558718088
|Has a strong no-2 no-3 no-5 zeta peak
|Dave Benson
|[[11ed7/5]]
|-
|-
|
|[[Beta 7/5]]
|116.481879086491562246584713240674074197523436163157572853694264779074
|26.7758951088566
|
|44.8164289231577
|
|Dave Benson
|[[13ed7/5]]
|-
|-
|1114zpi
|[[Alpha 4/3]]
|
|31.3266790320926
|
|38.3060074376432
|
|Dave Benson
|[[13ed4/3]]
|-
|-
|171edo
|[[Carlos Gamma|Gamma 3/2]]
|
|34.1894540921914
|
|35.0985422804417
|Exceptionally strong zeta peak
|Dave Benson
|[[20edf|20ed3/2]]
|-
|-
|270edo
|[[Beta 4/3]]
|
|36.1372975038827
|
|33.2066890135065
|Exceptionally strong zeta peak
|Dave Benson
|[[15ed4/3]]
|-
|-
|1936zpi
|[[Gamma 7/5]]
|
|49.4404896216012
|
|24.2716042900130
|
|Dave Benson
|[[24ed7/5]]
|-
|-
|311edo
|[[Gamma 4/3]]
|
|67.4633901646646
|
|17.7874251067289
|Exceptionally strong zeta peak
|Dave Benson
|[[28ed4/3]]
|}
 
== Unequal-step tunings ==
 
=== Unequal-step tunings from equal divisions of a ratio ===
{| class="wikitable"
|+
!Tuning
!Period
!Mode
!Why it matters
|-
|-
|2293zpi
|[[93edo and stretched hemififths|Stretched hemififth]]
|
|94\93<2/1>
|
|16 11 16 12 16 11 12
|
|
|-
|-
|342edo
|[[36edo|833 Cent Acoustic Golden Scale [11]]]
|
|25\36<2/1>
|3 1 3 3 1 3 1 3 3 1 3
|
|
|171*2^n family
|-
|-
|684edo
|833 Cent Logarithmic Golden Scale [8]
|
|ϕ
|ϕ 1 ϕ ϕ 1 ϕ 1 ϕ
|
|
|171*2^n family
|}
|}

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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