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== Equal-step tunings ==
== Equal-step tunings ==


=== Equal divisions of a ratio & optimization ===
=== 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:


{| class="wikitable sortable"
* '''Prominent peak counts from the classic Riemann zeta function'''
|+ style="font-size: 105%;" | EDRs collection & optimization
* '''Prominent peaks after removing the prime 2 from the zeta product'''
|- style="white-space: nowrap;"
* '''Prominent peaks after removing the prime 3'''
! colspan="4" | EDRs !! colspan="3" | Optimization
* '''Prominent peaks after simultaneously removing the primes 2 and 3'''
|- style="white-space: nowrap;"
* '''The α–β–γ family, with an equave sliding from 3/1 down to 4/3'''
! EDR !! Steps per octave !! Cents !! Why it matters !! Optimization !! Steps per octave !! Cents
 
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
!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
|-
|-
|7ed5/3
|[[155zpi (σ = 1)]]
|9.49840814199706858453236308296456727180164766756424872573514114916072877570
|35.9827898689
|
|33.3492762616
|Alpha 5/3 analogue
|2.80355
|
|[[36edo]]
|
|1200.57394542
|
|8
|8
|-
|-
|10edo
|[[184zpi (σ = 1)]]
|40.9880790756
|29.2768050385
|3.32966
|[[41edo]]
|1200.34900658
|16
|10
|10
|
|EDO ≤ 29
| colspan="3" |None
|-
|-
|11edo
|[[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
|11
|
|EDO ≤ 29
| colspan="3" |None
|-
|-
|12edo
|[[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
|12
|
|EDO ≤ 29, Has a strong zeta peak
|34zpi
|12.0231830072926
|99.8071807833375
|-
|-
|9ed5/3
|[[301zpi (σ = 1)]]
|12.2122390397105167515416096780973007780306898582968912188023243346352227116
|59.9223835273
|
|20.0259056693
|Beta 5/3 analogue
|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
|-
|-
|13edo
|[[380zpi (σ = 1)]]
|71.9512656175
|16.6779554147
|3.61665
|[[72edo]]
|1200.81278986
|18
|13
|13
|
|EDO ≤ 29
| colspan="3" |None
|-
|-
|14edo
|[[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
|14
|
|EDO ≤ 29
|42zpi
|13.9002525327005
|86.3293668353859
|-
|-
|15edo
|[[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
|15
|
|EDO ≤ 29
|47zpi
|15.0534898676781
|79.7157343943591
|-
|-
|9ed3/2
|[[655zpi (σ = 1)]]
|15.3856016221630929927857123595661272655350336171249650076700851565894672010
|111.058159333
|
|10.8051493669
|Carlos Alpha 3/2
|3.39509
|
|[[111edo]]
|
|1199.37157972
|
|22
|16
|-
|[[706zpi (σ = 1)]]
|117.971388652
|10.1719579104
|3.62695
|[[118edo]]
|1200.29103343
|12
|12
|-
|-
|16edo
|[[796zpi (σ = 1)]]
|130.004267285
|9.23046623824
|3.72487
|[[130edo]]
|1199.96061097
|16
|16
|
|16
|EDO ≤ 29
|-
| colspan="3" |None
|[[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
|-
|-
|17edo
|[[no-3 75zpi (σ = 1)]]
|21.0417134383
|57.0295762045
|2.60042
|[[21edo]]
|1197.62110029
|17
|17
|
|10
|EDO ≤ 29
|-
|56zpi
|[[no-3 95zpi (σ = 1)]]
|17.0445886606675
|24.9617781085
|70.4035764012981
|48.0734984016
|2.64675
|[[25edo]]
|1201.83746004
|14
|11
|-
|-
|18edo
|[[no-3 127zpi (σ = 1)]]
|18
|31.0146799866
|
|38.6913552073
|EDO ≤ 29
|2.60405
| colspan="3" |None
|[[31edo]]
|1199.43201143
|11
|11
|-
|-
|11ed3/2
|[[no-3 161zpi (σ = 1)]]
|18.8046242048660025467380928839141555467650410875971794538189929691649043568
|37.0135086000
|
|32.4205957606
|Carlos Beta 3/2
|2.92705
|
|[[37edo]]
|
|1199.56204314
|
|22
|16
|-
|-
|19edo
|[[no-3 196zpi (σ = 1)]]
|43.0494972034
|27.8748900209
|2.71380
|[[43edo]]
|1198.62027090
|22
|19
|19
|
|EDO ≤ 29, Has a strong zeta peak
|65zpi
|18.9480867166984
|63.3309324546460
|-
|-
|20edo
|[[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
|20
|
|19
|EDO ≤ 29
|-
| colspan="3" |None
|[[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
|-
|-
|21edo
|[[no-2 233zpi (σ = 1)]]
|49.1657210129
|24.4072491012
|2.07714
|[[78edt]]
|1903.76542989
|21
|21
|
|21
|EDO ≤ 29
|-
| colspan="3" |None
|[[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
|-
|-
|16ed5/3
|[[no-2 777zpi (σ = 1)]]
|21.7106471817075853360739727610618680498323375258611399445374654837959514873
|127.486291223
|
|9.41277676594
|Gamma 5/3 analogue
|2.21095
|
|[[202edt]]
|
|1901.38090672
|
|17
|17
|-
|-
|22edo
|[[no-2 810zpi (σ = 1)]]
|22
|131.822840677
|
|9.10312654342
|EDO ≤ 29, Has a strong zeta peak
|2.25360
|80zpi
|[[209edt]]
|22.0251467420146
|1902.55344758
|54.4831784348982
|21
|21
|-
|-
|11ed7/5
|[[no-2 829zpi (σ = 1)]]
|22.6604698881675975718539343001772210291924199534946609650608950558236849973
|134.373782790
|
|8.93031345169
|Alpha 7/5 analogue
|2.13475
|
|[[213edt]]
|
|1902.15676521
|
|29
|29
|-
|-
|23edo
|[[no-2 839zpi (σ = 1)]]
|23
|135.657892938
|
|8.84578091263
|EDO ≤ 29
|2.11125
| colspan="3" |None
|[[215edt]]
|1901.84289622
|15
|15
|-
|-
|24edo
|[[no-2 858zpi (σ = 1)]]
|24
|138.196070465
|
|8.68331491602
|EDO ≤ 29
|2.20051
|90zpi
|[[219edt]]
|24.0057421830853
|1901.64596661
|49.9880399800983
|11
|11
|-
|-
|39edt
|[[no-2 902zpi (σ = 1)]]
|24.6062603892868400468815574593676733176838399651433366869555428097087228538
|143.873905513
|
|8.34063686336
|Has a strong no-2 zeta peak
|2.09948
|93zpi no-2 analogue
|[[228edt]]
|24.5738316304204445883184323365600165414701853056787276394517489970293
|1901.66520485
|48.8324335434322607337830293873763777285246843568212813459755275268185837463
|11
|11
|-
|-
|25edo
|[[no-2 965zpi (σ = 1)]]
|25
|152.075713777
|
|7.89080629768
|EDO ≤ 29
|2.10893
| colspan="3" |None
|[[241edt]]
|1901.68431774
|15
|15
|-
|-
|26edo
|[[no-2 985zpi (σ = 1)]]
|26
|154.604034485
|
|7.76176381166
|EDO ≤ 29
|2.40811
|100zpi
|[[245edt]]
|25.9356996537225
|1901.63213386
|46.2682717652372
|21
|21
|-
|-
|13ed7/5
|[[no-2 1029zpi (σ = 1)]]
|26.7805553223798880394637405365730793981364963086755084132537850659734459059
|160.260260060
|
|7.48782012177
|Beta 7/5 analogue
|2.17192
|
|[[254edt]]
|
|1901.90631093
|
|9
|9
|-
|-
|27edo
|[[no-2 1049zpi (σ = 1)]]
|27
|162.750022676
|
|7.37327086209
|EDO ≤ 29, Has a strong zeta peak
|2.14738
|106zpi
|[[258edt]]
|27.0866140827635
|1902.30388242
|44.3023257293579
|17
|17
|-
|-
|28edo
|[[no-2 1069zpi (σ = 1)]]
|28
|165.332187903
|
|7.25811480039
|EDO ≤ 29
|2.19607
| colspan="3" |None
|[[262edt]]
|1901.62607770
|17
|17
|-
|-
|29edo
|[[no-2 1134zpi (σ = 1)]]
|173.506549648
|6.91616542681
|2.26764
|[[275edt]]
|1901.94549237
|29
|29
|
|29
|EDO ≤ 29
|-
|116zpi
|[[no-2 1159zpi (σ = 1)]]
|28.9399661541990
|176.625850825
|41.4651487014917
|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
|-
|-
|31edo
|[[no-2 1266zpi (σ = 1)]]
|31
|189.909845446
|
|6.31878772364
|Has a strong zeta peak
|2.17116
|127zpi
|[[301edt]]
|30.9783816349790
|1901.95510482
|38.7366910944446
|17
|17
|-
|-
|13ed4/3
|[[no-2 1297zpi (σ = 1)]]
|31.3224709154917170595712563005616739301927219730991737328958039656595105508
|193.736743714
|
|6.19397217583
|Alpha 4/3 analogue
|2.12380
|
|[[307edt]]
|
|1901.54945798
|
|21
|21
|-
|-
|34edo
|[[no-2 1343zpi (σ = 1)]]
|34
|199.415414525
|
|6.01758897555
|Has a strong zeta peak
|2.36503
|144zpi
|[[316edt]]
|34.0448410043159
|1901.55811627
|35.2476312005063
|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
|-
|-
|20ed3/2
!No-2 No-3 ZPI analog
|34.1902258270290955395238052434802828123000747047221444614890781257543715579
!Steps per octave
|
!Cents
|Carlos Gamma 3/2
!Height
|
!ED5
|
!Pentave
|
!Consistent
!Distinct
|-
|-
|56edt
|[[no-2 no-3 186zpi (σ = 1)]]
|35.3320662000016164775735184031946078407767958473853039607566768549663712773
|41.3464998527
|
|29.0230129340
|Has a strong no-2 zeta peak
|1.75534
|151zpi no-2 analogue
|[[96ed5]]
|35.3059427335608633586867709728239574896988978536248407971925774931920
|2786.20924167
|33.9886123153797795726938859938695575674205028551304634432771826217692714955
|35
|23
|-
|-
|36edo
|[[no-2 no-3 565zpi (σ = 1)]]
|36
|98.6253027359
|
|12.1672630320
|Has a strong no-5 zeta peak
|1.74188
|155zpi no-5 analogue
|[[229ed5]]
|35.9775957344990354876843659181629374042162799645238247644819739175425
|2786.30323433
|33.3540909419168338960282298282173036675588854165862895775989035929190051321
|29
|29
|-
|-
|15ed4/3
|[[no-2 no-3 671zpi (σ = 1)]]
|36.1413125947981350687360649621865468425300638151144312302643891911455890971
|113.258011095
|
|10.5952769998
|Beta 4/3 analogue
|1.77217
|
|[[263ed5]]
|
|2786.55785095
|
|19
|19
|-
|-
|37edo
|[[no-2 no-3 764zpi (σ = 1)]]
|125.745000550
|9.54312294522
|1.75634
|[[292ed5]]
|2786.59190001
|37
|37
|37
|
|Has a strong no-3 zeta peak
|161zpi no-3 analogue
|37.0117501336435252522939269985227920601261578745487306336979972897294
|32.4221360964286053986540281462323756320027885683144327873809896041665053646
|-
|-
|41edo
|[[no-2 no-3 905zpi (σ = 1)]]
|144.297529480
|8.31615069448
|1.73926
|[[335ed5]]
|2785.91048265
|43
|41
|41
|
|Has a strong zeta peak
|184zpi
|40.9880783925993
|29.2768055263764
|-
|-
|96ed5
|[[no-2 no-3 938zpi (σ = 1)]]
|41.3449495750457328643302306013407006787000254796570031349096996442235599236
|148.562870929
|
|8.07738833059
|Has a strong no-2 no-3 zeta peak
|1.79949
|186zpi no-2 no-3 analogue
|[[345ed5]]
|41.3477989230936
|2786.69897405
|29.0221010852836
|25
|25
|-
|[[no-2 no-3 1046zpi (σ = 1)]]
|162.414291729
|7.38851234841
|1.73251
|[[377ed5]]
|2785.46915535
|23
|23
|-
|-
|66edt
|[[no-2 no-3 1145zpi (σ = 1)]]
|41.6413637357161908485687895466222163837726522487041082394632262933532232911
|174.880594782
|
|6.86182478678
|Has a strong no-2 no-5 zeta peak
|1.74084
|188zpi no-2 no-5 analogue
|[[406ed5]]
|41.6281274155763001275416027845619755345480144939422820248677372190977
|2785.90086343
|28.8266629920755754571831740158108063867663530357929200798818480023767470361
|25
|25
|-
|-
|46edo
|[[no-2 no-3 1196zpi (σ = 1)]]
|46
|181.292147244
|
|6.61915046096
|Has a strong zeta peak
|1.77770
|214zpi
|[[421ed5]]
|46.0089748051542
|2786.66234406
|26.0818678330031
|35
|35
|-
|-
|24ed7/5
|[[no-2 no-3 1280zpi (σ = 1)]]
|49.4410252105474856113176748367503004273289162621701693783146801217971309033
|191.632570168
|
|6.26198353937
|Gamma 7/5 analogue
|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
|-
|-
|53edo
|[[Alpha 3/1]]
|53
|1.90739592696007
|
|629.130000247254
|Has a strong zeta peak
|Dave Benson
|257zpi
|[[3edt|3ed3/1]]
|52.9968291550147
|22.6428640945673
|-
|-
|57edo
|[[Beta 3/1]]
|57
|3.14186231690763
|
|381.939079106782
|Has a strong no-3 no-5 zeta peak
|Dave Benson
|282zpi no-3 no-5 analogue
|[[5edt|5ed3/1]]
|56.9949885079206769176514037038198357725273287855611008976484058072516
|21.0544827083039808806917479490481104480956786904618314090632684997207904034
|-
|-
|58edo
|[[Alpha 2/1]]
|58
|5.00991270509077
|
|239.525131601721
|Has a strong zeta peak
|Dave Benson
|289zpi
|[[5edo|5ed2/1]]
|58.0667185533159
|20.6658827964969
|-
|-
|60edo
|[[Gamma 3/1]]
|60
|5.04255621376059
|
|237.974540913462
|Has a strong zeta peak
|Dave Benson
|301zpi
|[[8edt|8ed3/1]]
|59.9201656607655
|20.0266469020418
|-
|-
|65edo
|[[Beta 2/1]]
|65
|6.99104980248710
|
|171.648040552235
|Has a strong zeta peak
|Dave Benson
|334zpi
|[[7edo|7ed2/1]]
|65.0158450885860
|18.4570391781413
|-
|-
|28ed4/3
|[[Alpha 5/3]]
|67.4637835102898521283073212627482207727227857882136049631601931568050996480
|9.50583353877785
|
|126.238272015258
|Gamma 4/3 analogue
|Dave Benson
|
|[[7ed5/3]]
|
|
|-
|-
|68edo
|[[Gamma 2/1]]
|68
|11.9978480914311
|
|100.017935787756
|Has a strong zeta peak
|Dave Benson
|354zpi
|[[12edo|12ed2/1]]
|68.0493056282519
|17.6342725163943
|-
|-
|72edo
|[[Beta 5/3]]
|72
|12.2053823008782
|
|98.3172808862904
|Has a strong zeta peak
|Dave Benson
|380zpi
|[[9ed5/3]]
|71.9506065993786
|16.6781081733140
|-
|-
|77edo
|[[Carlos Alpha|Alpha 3/2]]
|77
|15.3915238996928
|
|77.9649895501219
|Has a strong zeta peak
|Dave Benson
|414zpi
|[[9edf|9ed3/2]]
|76.9918536925042
|15.5860645308353
|-
|-
|80edo
|[[Carlos Beta|Beta 3/2]]
|80
|18.7990736394111
|
|63.8329325698408
|Has a strong zeta peak
|Dave Benson
|435zpi
|[[11edf|11ed3/2]]
|80.0731374302484
|14.9862992572924
|-
|-
|83edo
|[[Gamma 5/3]]
|83
|21.7094399215509
|
|55.2754932571412
|Has a strong no-3 no-5 zeta peak
|Dave Benson
|455zpi no-3 no-5 analogue
|[[16ed5/3]]
|82.9585473728587401934282446836610895074185494886540503684148508037660
|14.4650555970631644892614919440394905869155594072293855522604093941309631517
|-
|-
|84edo
|[[Alpha 7/5]]
|84
|22.6653911133366
|
|52.9441558718088
|Has a strong zeta peak
|Dave Benson
|462zpi
|[[11ed7/5]]
|83.9972142607288
|14.2861880666087
|-
|-
|87edo
|[[Beta 7/5]]
|87
|26.7758951088566
|
|44.8164289231577
|Has a strong zeta peak
|Dave Benson
|483zpi
|[[13ed7/5]]
|87.0139255957575
|13.7908960178956
|-
|-
|94edo
|[[Alpha 4/3]]
|94
|31.3266790320926
|
|38.3060074376432
|Has a strong zeta peak
|Dave Benson
|532zpi
|[[13ed4/3]]
|93.9836761074943
|12.7681747480009
|-
|-
|99edo
|[[Carlos Gamma|Gamma 3/2]]
|99
|34.1894540921914
|
|35.0985422804417
|Has a strong zeta peak
|Dave Benson
|568zpi
|[[20edf|20ed3/2]]
|99.0473345956631
|12.1154194093028
|-
|-
|327ed7
|[[Beta 4/3]]
|116.479750184323251720135904506422003080366592226710079912941501697613698777
|36.1372975038827
|
|33.2066890135065
|Has a strong no-2 no-3 no-5 zeta peak
|Dave Benson
|695zpi no-2 no-3 no-5 analogue
|[[15ed4/3]]
|116.481879086491562246584713240674074197523436163157572853694264779074
|10.3020316070704971370763940790472253291607124811459581948058543261473166559
|-
|-
|171edo
|[[Gamma 7/5]]
|171
|49.4404896216012
|
|24.2716042900130
|Exceptionally strong zeta peak
|Dave Benson
|1114zpi
|[[24ed7/5]]
|170.995891689006
|7.01771246166817
|-
|-
|270edo
|[[Gamma 4/3]]
|270
|67.4633901646646
|
|17.7874251067289
|Exceptionally strong zeta peak
|Dave Benson
|1936zpi
|[[28ed4/3]]
|270.017794631965
|}
|4.44415154799558
 
== Unequal-step tunings ==
 
=== Unequal-step tunings from equal divisions of a ratio ===
{| class="wikitable"
|+
!Tuning
!Period
!Mode
!Why it matters
|-
|-
|311edo
|[[93edo and stretched hemififths|Stretched hemififth]]
|311
|94\93<2/1>
|16 11 16 12 16 11 12
|
|
|Exceptionally strong zeta peak
|2293zpi
|311.004029926555
|3.85847090239759
|-
|-
|342edo
|[[36edo|833 Cent Acoustic Golden Scale [11]]]
|342
|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]
|684
|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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