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


=== Equal divisions of a ratio & optimization ===
=== About this list ===
{| class="wikitable sortable"
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:
|+ style="font-size: 105%;" | EDRs collection & optimization
 
|- style="white-space: nowrap;"
* '''Prominent peak counts from the classic Riemann zeta function'''
! colspan="3" |EDRs
* '''Prominent peaks after removing the prime 2 from the zeta product'''
! colspan="3" |Optimization
* '''Prominent peaks after removing the prime 3'''
!Comments
* '''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
 
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
!Cents
!Step size (cents)
!Optimization
!Height
!Steps per octave
!EDO
!Cents
!Octave (cents)
!Why it matters
!Consistent
!Distinct
|-
|[[15zpi (σ = 1)]]
|6.95688550773
|172.490980147
|2.55384
|[[7edo]]
|1207.43686103
|6
|5
|-
|-
|7ed5/3
|[[26zpi (σ = 1)]]
|9.49840814199707
|10.0089746115
|126.336958999921
|119.892401228
|
|2.57426
|9.50583353877785
|[[10edo]]
|126.238272015258
|1198.92401228
|Alpha 5/3 analogue
|8
|5
|-
|-
|10edo
|[[34zpi (σ = 1)]]
|12.0220488259
|99.8165967700
|2.85866
|[[12edo]]
|1197.79916124
|10
|10
|120.
|6
| colspan="3" |None
|
|-
|-
|11edo
|[[42zpi (σ = 1)]]
|11
|13.9020220557
|109.090909090909
|86.3183783764
| colspan="3" |None
|2.50514
|
|[[14edo]]
|1208.45729727
|7
|5
|-
|-
|12edo
|[[47zpi (σ = 1)]]
|12
|15.0534708836
|100.
|79.7158349246
|34zpi
|2.69313
|12.0231830072926
|[[15edo]]
|99.8071807833375
|1195.73752387
|EDO ≤ 29, Has a strong zeta peak
|8
|7
|-
|-
|9ed5/3
|[[56zpi (σ = 1)]]
|12.2122390397105
|17.0432556931
|98.2620792221608
|70.4090827252
|
|2.65741
|12.2053823008782
|[[17edo]]
|98.3172808862904
|1196.95440633
|Beta 5/3 analogue
|4
|4
|-
|-
|13edo
|[[65zpi (σ = 1)]]
|13
|18.9489976130
|92.3076923076923
|63.3278880767
| colspan="3" |None
|3.02387
|
|[[19edo]]
|1203.22987346
|10
|7
|-
|-
|14edo
|[[80zpi (σ = 1)]]
|14
|22.0251749360
|85.7142857142857
|54.4831086920
|42zpi
|2.99601
|13.9002525327005
|[[22edo]]
|86.3293668353859
|1198.62839122
|EDO ≤ 29
|12
|8
|-
|-
|15edo
|[[90zpi (σ = 1)]]
|15
|24.0053572889
|80.
|49.9888414723
|47zpi
|2.82476
|15.0534898676781
|[[24edo]]
|79.7157343943591
|1199.73219533
|EDO ≤ 29
|6
|6
|-
|-
|9ed3/2
|[[100zpi (σ = 1)]]
|15.3856016221631
|25.9356337472
|77.9950000961542
|46.2683893402
|
|2.71167
|15.3915238996928
|[[26edo]]
|77.9649895501219
|1202.97812285
|Carlos Alpha 3/2
|14
|9
|-
|-
|16edo
|[[106zpi (σ = 1)]]
|16
|27.0853383248
|75.
|44.3044124320
| colspan="3" |None
|2.90524
|
|[[27edo]]
|1196.21913566
|10
|8
|-
|-
|17edo
|[[116zpi (σ = 1)]]
|17
|28.9431579907
|70.5882352941176
|41.4605759463
|56zpi
|2.68561
|17.0445886606675
|[[29edo]]
|70.4035764012981
|1202.35670244
|EDO ≤ 29
|8
|7
|-
|-
|18edo
|[[127zpi (σ = 1)]]
|18
|30.9779815456
|66.6666666666667
|38.7371913897
| colspan="3" |None
|3.23190
|
|[[31edo]]
|1200.85293308
|12
|9
|-
|-
|11ed3/2
|[[144zpi (σ = 1)]]
|18.8046242048660
|34.0437506778
|63.8140909877625
|35.2487600839
|
|3.07414
|18.7990736394111
|[[34edo]]
|63.8329325698408
|1198.45784285
|Carlos Beta 3/2
|6
|6
|-
|-
|19edo
|[[155zpi (σ = 1)]]
|19
|35.9827898689
|63.1578947368421
|33.3492762616
|65zpi
|2.80355
|18.9480867166984
|[[36edo]]
|63.3309324546460
|1200.57394542
|EDO ≤ 29, Has a strong zeta peak
|8
|8
|-
|-
|20edo
|[[184zpi (σ = 1)]]
|20
|40.9880790756
|60.
|29.2768050385
| colspan="3" |None
|3.32966
|
|[[41edo]]
|1200.34900658
|16
|10
|-
|-
|21edo
|[[196zpi (σ = 1)]]
|21
|43.0234004818
|57.1428571428571
|27.8917981043
| colspan="3" |None
|2.78019
|
|[[43edo]]
|1199.34731849
|8
|8
|-
|-
|16ed5/3
|[[214zpi (σ = 1)]]
|21.7106471817076
|46.0106419996
|55.2724195624655
|26.0809227572
|
|3.25119
|21.7094399215509
|[[46edo]]
|55.2754932571412
|1199.72244683
|Gamma 5/3 analogue
|14
|11
|-
|-
|22edo
|[[238zpi (σ = 1)]]
|22
|49.9382924730
|54.5454545454545
|24.0296562132
|80zpi
|2.90274
|22.0251467420146
|[[50edo]]
|54.4831784348982
|1201.48281066
|EDO ≤ 29, Has a strong zeta peak
|10
|9
|-
|-
|11ed7/5
|[[257zpi (σ = 1)]]
|22.6604698881676
|52.9969882711
|52.9556538731173
|22.6427961125
|
|3.46399
|22.6653911133366
|[[53edo]]
|52.9441558718088
|1200.06819396
|Alpha 7/5 analogue
|10
|10
|-
|-
|23edo
|[[289zpi (σ = 1)]]
|23
|58.0645692462
|52.1739130434783
|20.6666477609
| colspan="3" |None
|3.25823
|
|[[58edo]]
|1198.66557013
|16
|12
|-
|-
|24edo
|[[301zpi (σ = 1)]]
|24
|59.9223835273
|50.
|20.0259056693
|90zpi
|2.98826
|24.0057421830853
|[[60edo]]
|49.9880399800983
|1201.55434016
|EDO ≤ 29
|10
|10
|-
|-
|39edt
|[[321zpi (σ = 1)]]
|24.6062603892868
|63.0197888699
|48.7680769452663
|19.0416378969
|93zpi no-2 analogue
|2.87513
|24.5738316304204
|[[63edo]]
|48.8324335434323
|1199.62318750
|Has a strong no-2 zeta peak
|8
|8
|-
|-
|25edo
|[[334zpi (σ = 1)]]
|25
|65.0145858034
|48.
|18.4573966776
| colspan="3" |None
|3.23462
|
|[[65edo]]
|1199.73078404
|6
|6
|-
|-
|26edo
|[[354zpi (σ = 1)]]
|26
|68.0496579343
|46.1538461538462
|17.6341812204
|100zpi
|3.14200
|25.9356996537225
|[[68edo]]
|46.2682717652372
|1199.12432299
|EDO ≤ 29
|10
|10
|-
|-
|13ed7/5
|[[380zpi (σ = 1)]]
|26.7805553223799
|71.9512656175
|44.8086302003300
|16.6779554147
|
|3.61665
|26.7758951088566
|[[72edo]]
|44.8164289231577
|1200.81278986
|Beta 7/5 analogue
|18
|13
|-
|-
|27edo
|[[414zpi (σ = 1)]]
|27
|76.9924672555
|44.4444444444444
|15.5859403235
|106zpi
|3.28825
|27.0866140827635
|[[77edo]]
|44.3023257293579
|1200.11740491
|EDO ≤ 29, Has a strong zeta peak
|10
|10
|-
|-
|28edo
|[[435zpi (σ = 1)]]
|28
|80.0733926855
|42.8571428571429
|14.9862514845
| colspan="3" |None
|3.14833
|
|[[80edo]]
|1198.90011876
|12
|12
|-
|-
|29edo
|[[462zpi (σ = 1)]]
|29
|83.9950884037
|41.3793103448276
|14.2865496400
|116zpi
|3.19687
|28.9399661541990
|[[84edo]]
|41.4651487014917
|1200.07016976
|EDO ≤ 29
|10
|10
|-
|-
|31edo
|[[483zpi (σ = 1)]]
|31
|87.0139579095
|38.7096774193548
|13.7908908965
|127zpi
|3.44872
|30.9783816349790
|[[87edo]]
|38.7366910944446
|1199.80750799
|Has a strong zeta peak
|16
|14
|-
|-
|13ed4/3
|[[497zpi (σ = 1)]]
|31.3224709154917
|89.0215260329
|38.3111537795856
|13.4798857476
|
|3.02681
|31.3266790320926
|[[89edo]]
|38.3060074376432
|1199.70983154
|Alpha 4/3 analogue
|12
|12
|-
|-
|34edo
|[[532zpi (σ = 1)]]
|34
|93.9843698073
|35.2941176470588
|12.7680805059
|144zpi
|3.39762
|34.0448410043159
|[[94edo]]
|35.2476312005063
|1200.19956756
|Has a strong zeta peak
|24
|15
|-
|-
|20ed3/2
|[[568zpi (σ = 1)]]
|34.1902258270291
|99.0456175574
|35.0977500432694
|12.1156294402
|
|3.56676
|34.1894540921914
|[[99edo]]
|35.0985422804417
|1199.44731458
|Carlos Gamma 3/2
|12
|12
|-
|-
|56edt
|[[596zpi (σ = 1)]]
|35.3320662000016
|102.936325452
|33.9634821583105
|11.6576922163
|151zpi no-2 analogue
|3.25007
|35.3059427335609
|[[103edo]]
|33.9886123153798
|1200.74229828
|Has a strong no-2 zeta peak
|15
|15
|-
|-
|36edo
|[[655zpi (σ = 1)]]
|36
|111.058159333
|33.3333333333333
|10.8051493669
|155zpi no-5 analogue
|3.39509
|35.9775957344990
|[[111edo]]
|33.3540909419168
|1199.37157972
|Has a strong no-5 zeta peak
|22
|16
|-
|-
|15ed4/3
|[[706zpi (σ = 1)]]
|36.1413125947981
|117.971388652
|33.2029999423075
|10.1719579104
|
|3.62695
|36.1372975038827
|[[118edo]]
|33.2066890135066
|1200.29103343
|Beta 4/3 analogue
|12
|12
|-
|-
|37edo
|[[796zpi (σ = 1)]]
|37
|130.004267285
|32.4324324324324
|9.23046623824
|161zpi no-3 analogue
|3.72487
|37.0117501336435
|[[130edo]]
|32.4221360964286
|1199.96061097
|Has a strong no-3 zeta peak
|16
|16
|-
|-
|41edo
|[[872zpi (σ = 1)]]
|41
|139.992781938
|29.2682926829268
|8.57187051639
|184zpi
|3.60746
|40.9880783925993
|[[140edo]]
|29.2768055263764
|1200.06187229
|Has a strong zeta peak
|10
|10
|-
|-
|96ed5
|[[965zpi (σ = 1)]]
|41.3449495750457
|152.050659206
|29.0241011860920
|7.89210652729
|186zpi no-2 no-3 analogue
|3.68901
|41.3477989230936
|[[152edo]]
|29.0221010852836
|1199.60019215
|Has a strong no-2 no-3 zeta peak
|15
|15
|-
|-
|66edt
|[[1114zpi (σ = 1)]]
|41.6413637357162
|170.995049914
|28.8175000131119
|7.01774700849
|188zpi no-2 no-5 analogue
|3.82285
|41.6281274155763
|[[171edo]]
|28.8266629920756
|1200.03473845
|Has a strong no-2 no-5 zeta peak
|14
|14
|-
|-
|46edo
|[[1210zpi (σ = 1)]]
|46
|183.000273182
|26.0869565217391
|6.55736726036
|214zpi
|3.76064
|46.0089748051542
|[[183edo]]
|26.0818678330031
|1199.99820865
|Has a strong zeta peak
|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
|-
|-
|24ed7/5
!No-3 ZPI analog
|49.4410252105475
!Steps per octave
|24.2713413585121
!Cents
|
!Height
|49.4404896216012
!EDO
|24.2716042900130
!Octave
|Gamma 7/5 analogue
!Consistent
!Distinct
|-
|-
|53edo
|[[no-3 51zpi (σ = 1)]]
|53
|15.9687074547
|22.6415094339623
|75.1469712502
|257zpi
|2.56677
|52.9968291550147
|[[16edo]]
|22.6428640945673
|1202.35154000
|Has a strong zeta peak
|26
|8
|-
|-
|57edo
|[[no-3 75zpi (σ = 1)]]
|57
|21.0417134383
|21.0526315789474
|57.0295762045
|282zpi no-3 no-5 analogue
|2.60042
|56.9949885079207
|[[21edo]]
|21.0544827083040
|1197.62110029
|Has a strong no-3 no-5 zeta peak
|17
|10
|-
|-
|58edo
|[[no-3 95zpi (σ = 1)]]
|58
|24.9617781085
|20.6896551724138
|48.0734984016
|289zpi
|2.64675
|58.0667185533159
|[[25edo]]
|20.6658827964969
|1201.83746004
|Has a strong zeta peak
|14
|11
|-
|-
|60edo
|[[no-3 127zpi (σ = 1)]]
|60
|31.0146799866
|20.0000000000000
|38.6913552073
|301zpi
|2.60405
|59.9201656607655
|[[31edo]]
|20.0266469020418
|1199.43201143
|Has a strong zeta peak
|11
|11
|-
|-
|65edo
|[[no-3 161zpi (σ = 1)]]
|65
|37.0135086000
|18.4615384615385
|32.4205957606
|334zpi
|2.92705
|65.0158450885860
|[[37edo]]
|18.4570391781413
|1199.56204314
|Has a strong zeta peak
|22
|16
|-
|-
|28ed4/3
|[[no-3 196zpi (σ = 1)]]
|67.4637835102899
|43.0494972034
|17.7873213976647
|27.8748900209
|
|2.71380
|67.4633901646646
|[[43edo]]
|17.7874251067289
|1198.62027090
|Gamma 4/3 analogue
|22
|19
|-
|-
|68edo
|[[no-3 220zpi (σ = 1)]]
|68
|47.0043385196
|17.6470588235294
|25.5295582875
|354zpi
|2.69328
|68.0493056282519
|[[47edo]]
|17.6342725163943
|1199.88923951
|Has a strong zeta peak
|10
|10
|-
|-
|72edo
|[[no-3 276zpi (σ = 1)]]
|72
|55.9891415481
|16.6666666666667
|21.4327272543
|380zpi
|2.76321
|71.9506065993786
|[[56edo]]
|16.6781081733140
|1200.23272624
|Has a strong zeta peak
|20
|19
|-
|-
|77edo
|[[no-3 340zpi (σ = 1)]]
|77
|65.9204029312
|15.5844155844156
|18.2037722259
|414zpi
|2.65263
|76.9918536925042
|[[66edo]]
|15.5860645308353
|1201.44896691
|Has a strong zeta peak
|16
|16
|-
|-
|80edo
|[[no-3 354zpi (σ = 1)]]
|80
|68.0229453080
|15.0000000000000
|17.6411061674
|435zpi
|2.76285
|80.0731374302484
|[[68edo]]
|14.9862992572924
|1199.59521939
|Has a strong zeta peak
|11
|11
|-
|-
|83edo
|[[no-3 394zpi (σ = 1)]]
|83
|74.0566473758
|14.4578313253012
|16.2038121158
|455zpi no-3 no-5 analogue
|2.76672
|82.9585473728587
|[[74edo]]
|14.4650555970632
|1199.08209657
|Has a strong no-3 no-5 zeta peak
|16
|16
|-
|-
|84edo
|[[no-3 421zpi (σ = 1)]]
|84
|78.0097604150
|14.2857142857143
|15.3826904943
|462zpi
|2.81219
|83.9972142607288
|[[78edo]]
|14.2861880666087
|1199.84985856
|Has a strong zeta peak
|17
|16
|-
|-
|87edo
|[[no-3 525zpi (σ = 1)]]
|87
|93.0066513531
|13.7931034482759
|12.9023030347
|483zpi
|2.97919
|87.0139255957575
|[[93edo]]
|13.7908960178956
|1199.91418223
|Has a strong zeta peak
|35
|19
|-
|-
|94edo
|[[no-3 751zpi (σ = 1)]]
|94
|124.013627761
|12.7659574468085
|9.67635591079
|532zpi
|3.13747
|93.9836761074943
|[[124edo]]
|12.7681747480009
|1199.86813294
|Has a strong zeta peak
|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
|-
|-
|99edo
!No-2 ZPI (σ = 1)
|99
!Steps per octave
|12.1212121212121
!Cents
|568zpi
!Height
|99.0473345956631
!EDT
|12.1154194093028
!Tritave
|Has a strong zeta peak
!Consistent
!Distinct
|-
|-
|327ed7
|[[no-2 93zpi (σ = 1)]]
|116.479750184323
|24.5747239922
|10.3022198974591
|48.8306603314
|695zpi no-2 no-3 no-5 analogue
|2.12985
|116.481879086492
|[[39edt]]
|10.3020316070705
|1904.39575293
|Has a strong no-2 no-3 no-5 zeta peak
|15
|15
|-
|-
|171edo
|[[no-2 151zpi (σ = 1)]]
|171
|35.3061077059
|7.01754385964912
|33.9884534992
|1114zpi
|2.08576
|170.995891689006
|[[56edt]]
|7.01771246166817
|1903.35339595
|Exceptionally strong zeta peak
|15
|15
|-
|-
|270edo
|[[no-2 207zpi (σ = 1)]]
|270
|44.8164999984
|4.44444444444444
|26.7758526445
|1936zpi
|2.10342
|270.017794631965
|[[71edt]]
|4.44415154799558
|1901.08553776
|Exceptionally strong zeta peak
|17
|17
|-
|-
|311edo
|[[no-2 222zpi (σ = 1)]]
|311
|47.3516876312
|3.85852090032154
|25.3422857776
|2293zpi
|2.11876
|311.004029926555
|[[75edt]]
|3.85847090239759
|1900.67143332
|Exceptionally strong zeta peak
|15
|15
|-
|-
|342edo
|[[no-2 233zpi (σ = 1)]]
|342
|49.1657210129
|3.50877192982456
|24.4072491012
| colspan="3" |None
|2.07714
|
|[[78edt]]
|1903.76542989
|21
|21
|-
|-
|684edo
|[[no-2 273zpi (σ = 1)]]
|684
|55.5359583782
|1.75438596491228
|21.6076220712
| colspan="3" |None
|2.19450
|
|[[88edt]]
|}
|1901.47074227
{| class="wikitable sortable"
|11
|+ style="font-size: 105%;" | EDRs collection & optimization
|11
|- style="white-space: nowrap;"
! colspan="3" | EDRs !! colspan="3" | Optimization !! Comments
|
|- style="white-space: nowrap;"
! EDR !! Steps per octave !! Cents !! Optimization !! Steps per octave !! Cents !! Why it matters
|-
|-
|7ed5/3
|[[no-2 363zpi (σ = 1)]]
|9.49840814199707
|69.4191721809
|126.336958999921
|17.2862908372
|
|2.08043
|9.50583353877785
|[[110edt]]
|126.238272015258
|1901.49199210
|Alpha 5/3 analogue
|23
|23
|-
|-
|10edo
|[[no-2 380zpi (σ = 1)]]
|10
|71.9200195089
|120.
|16.6852012582
| colspan="3" |None
|2.07565
|
|[[114edt]]
|1902.11294344
|17
|17
|-
|-
|11edo
|[[no-2 453zpi (σ = 1)]]
|11
|82.6700405439
|109.090909090909
|14.5155366092
| colspan="3" |None
|2.38406
|
|[[131edt]]
|1901.53529581
|27
|27
|-
|-
|12edo
|[[no-2 492zpi (σ = 1)]]
|12
|88.3238806401
|100.
|13.5863595587
|34zpi
|2.12238
|12.0231830072926
|[[140edt]]
|99.8071807833375
|1902.09033822
|EDO ≤ 29, Has a strong zeta peak
|9
|9
|-
|-
|9ed5/3
|[[no-2 510zpi (σ = 1)]]
|12.2122390397105
|90.8334979880
|98.2620792221608
|13.2109852266
|
|2.23067
|12.2053823008782
|[[144edt]]
|98.3172808862904
|1902.38187263
|Beta 5/3 analogue
|39
|27
|-
|-
|13edo
|[[no-2 550zpi (σ = 1)]]
|13
|96.5187261015
|92.3076923076923
|12.4328205362
| colspan="3" |None
|2.24293
|
|[[153edt]]
|1902.22154203
|15
|15
|-
|-
|14edo
|[[no-2 627zpi (σ = 1)]]
|14
|107.244021785
|85.7142857142857
|11.1894348983
|42zpi
|2.29774
|13.9002525327005
|[[170edt]]
|86.3293668353859
|1902.20393272
|EDO ≤ 29
|15
|15
|-
|-
|15edo
|[[no-2 687zpi (σ = 1)]]
|115.412802617
|10.3974600113
|2.18983
|[[183edt]]
|1902.73518207
|15
|15
|15
|80.
|47zpi
|15.0534898676781
|79.7157343943591
|EDO ≤ 29
|-
|-
|9ed3/2
|[[no-2 697zpi (σ = 1)]]
|15.3856016221631
|116.734850378
|77.9950000961542
|10.2797064983
|
|2.15793
|15.3915238996928
|[[185edt]]
|77.9649895501219
|1901.74570218
|Carlos Alpha 3/2
|29
|29
|-
|-
|16edo
|[[no-2 777zpi (σ = 1)]]
|16
|127.486291223
|75.
|9.41277676594
| colspan="3" |None
|2.21095
|
|[[202edt]]
|-
|1901.38090672
|17edo
|17
|17
|17
|70.5882352941176
|56zpi
|17.0445886606675
|70.4035764012981
|EDO ≤ 29
|-
|-
|18edo
|[[no-2 810zpi (σ = 1)]]
|18
|131.822840677
|66.6666666666667
|9.10312654342
| colspan="3" |None
|2.25360
|
|[[209edt]]
|1902.55344758
|21
|21
|-
|-
|11ed3/2
|[[no-2 829zpi (σ = 1)]]
|18.8046242048660
|134.373782790
|63.8140909877625
|8.93031345169
|
|2.13475
|18.7990736394111
|[[213edt]]
|63.8329325698408
|1902.15676521
|Carlos Beta 3/2
|29
|29
|-
|-
|19edo
|[[no-2 839zpi (σ = 1)]]
|19
|135.657892938
|63.1578947368421
|8.84578091263
|65zpi
|2.11125
|18.9480867166984
|[[215edt]]
|63.3309324546460
|1901.84289622
|EDO ≤ 29, Has a strong zeta peak
|15
|15
|-
|-
|20edo
|[[no-2 858zpi (σ = 1)]]
|20
|138.196070465
|60.
|8.68331491602
| colspan="3" |None
|2.20051
|
|[[219edt]]
|1901.64596661
|11
|11
|-
|-
|21edo
|[[no-2 902zpi (σ = 1)]]
|21
|143.873905513
|57.1428571428571
|8.34063686336
| colspan="3" |None
|2.09948
|
|[[228edt]]
|1901.66520485
|11
|11
|-
|-
|16ed5/3
|[[no-2 965zpi (σ = 1)]]
|21.7106471817076
|152.075713777
|55.2724195624655
|7.89080629768
|
|2.10893
|21.7094399215509
|[[241edt]]
|55.2754932571412
|1901.68431774
|Gamma 5/3 analogue
|15
|15
|-
|-
|22edo
|[[no-2 985zpi (σ = 1)]]
|22
|154.604034485
|54.5454545454545
|7.76176381166
|80zpi
|2.40811
|22.0251467420146
|[[245edt]]
|54.4831784348982
|1901.63213386
|EDO ≤ 29, Has a strong zeta peak
|21
|21
|-
|-
|11ed7/5
|[[no-2 1029zpi (σ = 1)]]
|22.6604698881676
|160.260260060
|52.9556538731173
|7.48782012177
|
|2.17192
|22.6653911133366
|[[254edt]]
|52.9441558718088
|1901.90631093
|Alpha 7/5 analogue
|9
|9
|-
|-
|23edo
|[[no-2 1049zpi (σ = 1)]]
|23
|162.750022676
|52.1739130434783
|7.37327086209
| colspan="3" |None
|2.14738
|
|[[258edt]]
|1902.30388242
|17
|17
|-
|-
|24edo
|[[no-2 1069zpi (σ = 1)]]
|24
|165.332187903
|50.
|7.25811480039
|90zpi
|2.19607
|24.0057421830853
|[[262edt]]
|49.9880399800983
|1901.62607770
|EDO ≤ 29
|17
|17
|-
|-
|39edt
|[[no-2 1134zpi (σ = 1)]]
|24.6062603892868
|173.506549648
|48.7680769452663
|6.91616542681
|93zpi no-2 analogue
|2.26764
|24.5738316304204
|[[275edt]]
|48.8324335434323
|1901.94549237
|Has a strong no-2 zeta peak
|29
|29
|-
|-
|25edo
|[[no-2 1159zpi (σ = 1)]]
|25
|176.625850825
|48.
|6.79402247404
| colspan="3" |None
|2.14379
|
|[[280edt]]
|1902.32629273
|11
|11
|-
|-
|26edo
|[[no-2 1179zpi (σ = 1)]]
|26
|179.167803205
|46.1538461538462
|6.69763193238
|100zpi
|2.29964
|25.9356996537225
|[[284edt]]
|46.2682717652372
|1902.12746880
|EDO ≤ 29
|15
|15
|-
|-
|13ed7/5
|[[no-2 1245zpi (σ = 1)]]
|26.7805553223799
|187.354933401
|44.8086302003300
|6.40495544056
|
|2.28021
|26.7758951088566
|[[297edt]]
|44.8164289231577
|1902.27176585
|Beta 7/5 analogue
|21
|21
|-
|-
|27edo
|[[no-2 1266zpi (σ = 1)]]
|27
|189.909845446
|44.4444444444444
|6.31878772364
|106zpi
|2.17116
|27.0866140827635
|[[301edt]]
|44.3023257293579
|1901.95510482
|EDO ≤ 29, Has a strong zeta peak
|17
|17
|-
|-
|28edo
|[[no-2 1297zpi (σ = 1)]]
|28
|193.736743714
|42.8571428571429
|6.19397217583
| colspan="3" |None
|2.12380
|
|[[307edt]]
|1901.54945798
|21
|21
|-
|-
|29edo
|[[no-2 1343zpi (σ = 1)]]
|29
|199.415414525
|41.3793103448276
|6.01758897555
|116zpi
|2.36503
|28.9399661541990
|[[316edt]]
|41.4651487014917
|1901.55811627
|EDO ≤ 29
|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
|-
|-
|31edo
!No-2 No-3 ZPI analog
|31
!Steps per octave
|38.7096774193548
!Cents
|127zpi
!Height
|30.9783816349790
!ED5
|38.7366910944446
!Pentave
|Has a strong zeta peak
!Consistent
!Distinct
|-
|-
|13ed4/3
|[[no-2 no-3 186zpi (σ = 1)]]
|31.3224709154917
|41.3464998527
|38.3111537795856
|29.0230129340
|
|1.75534
|31.3266790320926
|[[96ed5]]
|38.3060074376432
|2786.20924167
|Alpha 4/3 analogue
|35
|23
|-
|-
|34edo
|[[no-2 no-3 565zpi (σ = 1)]]
|34
|98.6253027359
|35.2941176470588
|12.1672630320
|144zpi
|1.74188
|34.0448410043159
|[[229ed5]]
|35.2476312005063
|2786.30323433
|Has a strong zeta peak
|29
|29
|-
|-
|20ed3/2
|[[no-2 no-3 671zpi (σ = 1)]]
|34.1902258270291
|113.258011095
|35.0977500432694
|10.5952769998
|
|1.77217
|34.1894540921914
|[[263ed5]]
|35.0985422804417
|2786.55785095
|Carlos Gamma 3/2
|19
|19
|-
|-
|56edt
|[[no-2 no-3 764zpi (σ = 1)]]
|35.3320662000016
|125.745000550
|33.9634821583105
|9.54312294522
|151zpi no-2 analogue
|1.75634
|35.3059427335609
|[[292ed5]]
|33.9886123153798
|2786.59190001
|Has a strong no-2 zeta peak
|37
|-
|36edo
|36
|33.3333333333333
|155zpi no-5 analogue
|35.9775957344990
|33.3540909419168
|Has a strong no-5 zeta peak
|-
|15ed4/3
|36.1413125947981
|33.2029999423075
|
|36.1372975038827
|33.2066890135066
|Beta 4/3 analogue
|-
|37edo
|37
|37
|32.4324324324324
|161zpi no-3 analogue
|37.0117501336435
|32.4221360964286
|Has a strong no-3 zeta peak
|-
|-
|41edo
|[[no-2 no-3 905zpi (σ = 1)]]
|144.297529480
|8.31615069448
|1.73926
|[[335ed5]]
|2785.91048265
|43
|41
|41
|29.2682926829268
|184zpi
|40.9880783925993
|29.2768055263764
|Has a strong zeta peak
|-
|-
|96ed5
|[[no-2 no-3 938zpi (σ = 1)]]
|41.3449495750457
|148.562870929
|29.0241011860920
|8.07738833059
|186zpi no-2 no-3 analogue
|1.79949
|41.3477989230936
|[[345ed5]]
|29.0221010852836
|2786.69897405
|Has a strong no-2 no-3 zeta peak
|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)]]
|41.6413637357162
|174.880594782
|28.8175000131119
|6.86182478678
|188zpi no-2 no-5 analogue
|1.74084
|41.6281274155763
|[[406ed5]]
|28.8266629920756
|2785.90086343
|Has a strong no-2 no-5 zeta peak
|25
|25
|-
|-
|46edo
|[[no-2 no-3 1196zpi (σ = 1)]]
|46
|181.292147244
|26.0869565217391
|6.61915046096
|214zpi
|1.77770
|46.0089748051542
|[[421ed5]]
|26.0818678330031
|2786.66234406
|Has a strong zeta peak
|35
|35
|-
|-
|24ed7/5
|[[no-2 no-3 1280zpi (σ = 1)]]
|49.4410252105475
|191.632570168
|24.2713413585121
|6.26198353937
|
|1.75036
|49.4404896216012
|[[445ed5]]
|24.2716042900130
|2786.58267502
|Gamma 7/5 analogue
|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
|22.6415094339623
|629.130000247254
|257zpi
|Dave Benson
|52.9968291550147
|[[3edt|3ed3/1]]
|22.6428640945673
|Has a strong zeta peak
|-
|-
|57edo
|[[Beta 3/1]]
|57
|3.14186231690763
|21.0526315789474
|381.939079106782
|282zpi no-3 no-5 analogue
|Dave Benson
|56.9949885079207
|[[5edt|5ed3/1]]
|21.0544827083040
|Has a strong no-3 no-5 zeta peak
|-
|-
|58edo
|[[Alpha 2/1]]
|58
|5.00991270509077
|20.6896551724138
|239.525131601721
|289zpi
|Dave Benson
|58.0667185533159
|[[5edo|5ed2/1]]
|20.6658827964969
|Has a strong zeta peak
|-
|-
|60edo
|[[Gamma 3/1]]
|60
|5.04255621376059
|20.0000000000000
|237.974540913462
|301zpi
|Dave Benson
|59.9201656607655
|[[8edt|8ed3/1]]
|20.0266469020418
|Has a strong zeta peak
|-
|-
|65edo
|[[Beta 2/1]]
|65
|6.99104980248710
|18.4615384615385
|171.648040552235
|334zpi
|Dave Benson
|65.0158450885860
|[[7edo|7ed2/1]]
|18.4570391781413
|Has a strong zeta peak
|-
|-
|28ed4/3
|[[Alpha 5/3]]
|67.4637835102899
|9.50583353877785
|17.7873213976647
|126.238272015258
|
|Dave Benson
|67.4633901646646
|[[7ed5/3]]
|17.7874251067289
|Gamma 4/3 analogue
|-
|-
|68edo
|[[Gamma 2/1]]
|68
|11.9978480914311
|17.6470588235294
|100.017935787756
|354zpi
|Dave Benson
|68.0493056282519
|[[12edo|12ed2/1]]
|17.6342725163943
|Has a strong zeta peak
|-
|-
|72edo
|[[Beta 5/3]]
|72
|12.2053823008782
|16.6666666666667
|98.3172808862904
|380zpi
|Dave Benson
|71.9506065993786
|[[9ed5/3]]
|16.6781081733140
|Has a strong zeta peak
|-
|-
|77edo
|[[Carlos Alpha|Alpha 3/2]]
|77
|15.3915238996928
|15.5844155844156
|77.9649895501219
|414zpi
|Dave Benson
|76.9918536925042
|[[9edf|9ed3/2]]
|15.5860645308353
|Has a strong zeta peak
|-
|-
|80edo
|[[Carlos Beta|Beta 3/2]]
|80
|18.7990736394111
|15.0000000000000
|63.8329325698408
|435zpi
|Dave Benson
|80.0731374302484
|[[11edf|11ed3/2]]
|14.9862992572924
|Has a strong zeta peak
|-
|-
|83edo
|[[Gamma 5/3]]
|83
|21.7094399215509
|14.4578313253012
|55.2754932571412
|455zpi no-3 no-5 analogue
|Dave Benson
|82.9585473728587
|[[16ed5/3]]
|14.4650555970632
|Has a strong no-3 no-5 zeta peak
|-
|-
|84edo
|[[Alpha 7/5]]
|84
|22.6653911133366
|14.2857142857143
|52.9441558718088
|462zpi
|Dave Benson
|83.9972142607288
|[[11ed7/5]]
|14.2861880666087
|Has a strong zeta peak
|-
|-
|87edo
|[[Beta 7/5]]
|87
|26.7758951088566
|13.7931034482759
|44.8164289231577
|483zpi
|Dave Benson
|87.0139255957575
|[[13ed7/5]]
|13.7908960178956
|Has a strong zeta peak
|-
|-
|94edo
|[[Alpha 4/3]]
|94
|31.3266790320926
|12.7659574468085
|38.3060074376432
|532zpi
|Dave Benson
|93.9836761074943
|[[13ed4/3]]
|12.7681747480009
|Has a strong zeta peak
|-
|-
|99edo
|[[Carlos Gamma|Gamma 3/2]]
|99
|34.1894540921914
|12.1212121212121
|35.0985422804417
|568zpi
|Dave Benson
|99.0473345956631
|[[20edf|20ed3/2]]
|12.1154194093028
|Has a strong zeta peak
|-
|-
|327ed7
|[[Beta 4/3]]
|116.479750184323
|36.1372975038827
|10.3022198974591
|33.2066890135065
|695zpi no-2 no-3 no-5 analogue
|Dave Benson
|116.481879086492
|[[15ed4/3]]
|10.3020316070705
|Has a strong no-2 no-3 no-5 zeta peak
|-
|-
|171edo
|[[Gamma 7/5]]
|171
|49.4404896216012
|7.01754385964912
|24.2716042900130
|1114zpi
|Dave Benson
|170.995891689006
|[[24ed7/5]]
|7.01771246166817
|Exceptionally strong zeta peak
|-
|-
|270edo
|[[Gamma 4/3]]
|270
|67.4633901646646
|4.44444444444444
|17.7874251067289
|1936zpi
|Dave Benson
|270.017794631965
|[[28ed4/3]]
|4.44415154799558
|}
|Exceptionally strong zeta peak
 
== 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>
|3.85852090032154
|16 11 16 12 16 11 12
|2293zpi
|
|311.004029926555
|3.85847090239759
|Exceptionally strong zeta peak
|-
|-
|342edo
|[[36edo|833 Cent Acoustic Golden Scale [11]]]
|342
|25\36<2/1>
|3.50877192982456
|3 1 3 3 1 3 1 3 3 1 3
| colspan="3" |None
|
|
|-
|-
|684edo
|833 Cent Logarithmic Golden Scale [8]
|684
|ϕ
|1.75438596491228
|ϕ 1 ϕ ϕ 1 ϕ 1 ϕ
| colspan="3" |None
|
|
|}
|}
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