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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="3" | EDRs !! colspan="3" | Optimization !! Comments
* '''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'''
|- style="white-space: nowrap;"
 
! EDR !! Steps per octave !! Cents !! Optimization !! Steps per octave !! Cents !! Why it matters
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)]]
|9.49840814199707
|35.9827898689
|126.336958999921
|33.3492762616
|
|2.80355
|9.50583353877785
|[[36edo]]
|126.238272015258
|1200.57394542
|Alpha 5/3 analogue
|8
|8
|-
|-
|10edo
|[[184zpi (σ = 1)]]
|40.9880790756
|29.2768050385
|3.32966
|[[41edo]]
|1200.34900658
|16
|10
|10
|120.
| 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
|109.090909090909
| 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
|100.
|34zpi
|12.0231830072926
|99.8071807833375
|EDO ≤ 29, Has a strong zeta peak
|-
|-
|9ed5/3
|[[301zpi (σ = 1)]]
|12.2122390397105
|59.9223835273
|98.2620792221608
|20.0259056693
|
|2.98826
|12.2053823008782
|[[60edo]]
|98.3172808862904
|1201.55434016
|Beta 5/3 analogue
|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
|92.3076923076923
| 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
|85.7142857142857
|42zpi
|13.9002525327005
|86.3293668353859
|EDO ≤ 29
|-
|-
|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
|80.
|47zpi
|15.0534898676781
|79.7157343943591
|EDO ≤ 29
|-
|-
|9ed3/2
|[[655zpi (σ = 1)]]
|15.3856016221631
|111.058159333
|77.9950000961542
|10.8051493669
|
|3.39509
|15.3915238996928
|[[111edo]]
|77.9649895501219
|1199.37157972
|Carlos Alpha 3/2
|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
|75.
|16
| 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
|70.5882352941176
|10
|56zpi
|17.0445886606675
|70.4035764012981
|EDO ≤ 29
|-
|-
|18edo
|[[no-3 95zpi (σ = 1)]]
|18
|24.9617781085
|66.6666666666667
|48.0734984016
| colspan="3" |None
|2.64675
|
|[[25edo]]
|1201.83746004
|14
|11
|-
|[[no-3 127zpi (σ = 1)]]
|31.0146799866
|38.6913552073
|2.60405
|[[31edo]]
|1199.43201143
|11
|11
|-
|-
|11ed3/2
|[[no-3 161zpi (σ = 1)]]
|18.8046242048660
|37.0135086000
|63.8140909877625
|32.4205957606
|
|2.92705
|18.7990736394111
|[[37edo]]
|63.8329325698408
|1199.56204314
|Carlos Beta 3/2
|22
|16
|-
|-
|19edo
|[[no-3 196zpi (σ = 1)]]
|43.0494972034
|27.8748900209
|2.71380
|[[43edo]]
|1198.62027090
|22
|19
|19
|63.1578947368421
|65zpi
|18.9480867166984
|63.3309324546460
|EDO ≤ 29, Has a strong zeta peak
|-
|-
|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
|60.
|19
| 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
|57.1428571428571
| colspan="3" |None
|
|-
|-
|16ed5/3
|[[no-2 273zpi (σ = 1)]]
|21.7106471817076
|55.5359583782
|55.2724195624655
|21.6076220712
|
|2.19450
|21.7094399215509
|[[88edt]]
|55.2754932571412
|1901.47074227
|Gamma 5/3 analogue
|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
|-
|-
|22edo
|[[no-2 810zpi (σ = 1)]]
|22
|131.822840677
|54.5454545454545
|9.10312654342
|80zpi
|2.25360
|22.0251467420146
|[[209edt]]
|54.4831784348982
|1902.55344758
|EDO ≤ 29, Has a strong zeta peak
|21
|21
|-
|-
|11ed7/5
|[[no-2 829zpi (σ = 1)]]
|22.6604698881676
|134.373782790
|52.9556538731173
|8.93031345169
|
|2.13475
|22.6653911133366
|[[213edt]]
|52.9441558718088
|1902.15676521
|Alpha 7/5 analogue
|29
|29
|-
|-
|23edo
|[[no-2 839zpi (σ = 1)]]
|23
|135.657892938
|52.1739130434783
|8.84578091263
| colspan="3" |None
|2.11125
|
|[[215edt]]
|1901.84289622
|15
|15
|-
|-
|24edo
|[[no-2 858zpi (σ = 1)]]
|24
|138.196070465
|50.
|8.68331491602
|90zpi
|2.20051
|24.0057421830853
|[[219edt]]
|49.9880399800983
|1901.64596661
|EDO ≤ 29
|11
|11
|-
|-
|39edt
|[[no-2 902zpi (σ = 1)]]
|24.6062603892868
|143.873905513
|48.7680769452663
|8.34063686336
|93zpi no-2 analogue
|2.09948
|24.5738316304204
|[[228edt]]
|48.8324335434323
|1901.66520485
|Has a strong no-2 zeta peak
|11
|11
|-
|-
|25edo
|[[no-2 965zpi (σ = 1)]]
|25
|152.075713777
|48.
|7.89080629768
| colspan="3" |None
|2.10893
|
|[[241edt]]
|1901.68431774
|15
|15
|-
|-
|26edo
|[[no-2 985zpi (σ = 1)]]
|26
|154.604034485
|46.1538461538462
|7.76176381166
|100zpi
|2.40811
|25.9356996537225
|[[245edt]]
|46.2682717652372
|1901.63213386
|EDO ≤ 29
|21
|21
|-
|-
|13ed7/5
|[[no-2 1029zpi (σ = 1)]]
|26.7805553223799
|160.260260060
|44.8086302003300
|7.48782012177
|
|2.17192
|26.7758951088566
|[[254edt]]
|44.8164289231577
|1901.90631093
|Beta 7/5 analogue
|9
|9
|-
|-
|27edo
|[[no-2 1049zpi (σ = 1)]]
|27
|162.750022676
|44.4444444444444
|7.37327086209
|106zpi
|2.14738
|27.0866140827635
|[[258edt]]
|44.3023257293579
|1902.30388242
|EDO ≤ 29, Has a strong zeta peak
|17
|17
|-
|-
|28edo
|[[no-2 1069zpi (σ = 1)]]
|28
|165.332187903
|42.8571428571429
|7.25811480039
| colspan="3" |None
|2.19607
|
|[[262edt]]
|1901.62607770
|17
|17
|-
|-
|29edo
|[[no-2 1134zpi (σ = 1)]]
|173.506549648
|6.91616542681
|2.26764
|[[275edt]]
|1901.94549237
|29
|29
|29
|41.3793103448276
|116zpi
|28.9399661541990
|41.4651487014917
|EDO ≤ 29
|-
|-
|31edo
|[[no-2 1159zpi (σ = 1)]]
|31
|176.625850825
|38.7096774193548
|6.79402247404
|127zpi
|2.14379
|30.9783816349790
|[[280edt]]
|38.7366910944446
|1902.32629273
|Has a strong zeta peak
|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
|-
|-
|13ed4/3
|[[no-2 1266zpi (σ = 1)]]
|31.3224709154917
|189.909845446
|38.3111537795856
|6.31878772364
|
|2.17116
|31.3266790320926
|[[301edt]]
|38.3060074376432
|1901.95510482
|Alpha 4/3 analogue
|17
|17
|-
|[[no-2 1297zpi (σ = 1)]]
|193.736743714
|6.19397217583
|2.12380
|[[307edt]]
|1901.54945798
|21
|21
|-
|-
|34edo
|[[no-2 1343zpi (σ = 1)]]
|34
|199.415414525
|35.2941176470588
|6.01758897555
|144zpi
|2.36503
|34.0448410043159
|[[316edt]]
|35.2476312005063
|1901.55811627
|Has a strong zeta peak
|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
|34.1902258270291
!Steps per octave
|35.0977500432694
!Cents
|
!Height
|34.1894540921914
!ED5
|35.0985422804417
!Pentave
|Carlos Gamma 3/2
!Consistent
!Distinct
|-
|-
|56edt
|[[no-2 no-3 186zpi (σ = 1)]]
|35.3320662000016
|41.3464998527
|33.9634821583105
|29.0230129340
|151zpi no-2 analogue
|1.75534
|35.3059427335609
|[[96ed5]]
|33.9886123153798
|2786.20924167
|Has a strong no-2 zeta peak
|35
|23
|-
|-
|36edo
|[[no-2 no-3 565zpi (σ = 1)]]
|36
|98.6253027359
|33.3333333333333
|12.1672630320
|155zpi no-5 analogue
|1.74188
|35.9775957344990
|[[229ed5]]
|33.3540909419168
|2786.30323433
|Has a strong no-5 zeta peak
|29
|29
|-
|-
|15ed4/3
|[[no-2 no-3 671zpi (σ = 1)]]
|36.1413125947981
|113.258011095
|33.2029999423075
|10.5952769998
|
|1.77217
|36.1372975038827
|[[263ed5]]
|33.2066890135066
|2786.55785095
|Beta 4/3 analogue
|19
|19
|-
|-
|37edo
|[[no-2 no-3 764zpi (σ = 1)]]
|125.745000550
|9.54312294522
|1.75634
|[[292ed5]]
|2786.59190001
|37
|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
|-
|-
|66edt
|[[no-2 no-3 1046zpi (σ = 1)]]
|41.6413637357162
|162.414291729
|28.8175000131119
|7.38851234841
|188zpi no-2 no-5 analogue
|1.73251
|41.6281274155763
|[[377ed5]]
|28.8266629920756
|2785.46915535
|Has a strong no-2 no-5 zeta peak
|23
|23
|-
|-
|46edo
|[[no-2 no-3 1145zpi (σ = 1)]]
|46
|174.880594782
|26.0869565217391
|6.86182478678
|214zpi
|1.74084
|46.0089748051542
|[[406ed5]]
|26.0818678330031
|2785.90086343
|Has a strong zeta peak
|25
|25
|-
|[[no-2 no-3 1196zpi (σ = 1)]]
|181.292147244
|6.61915046096
|1.77770
|[[421ed5]]
|2786.66234406
|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
|
|
|}
|}
Retrieved from "https://en.xen.wiki/w/User:Contribution/Collection_of_tunings"