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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
|-
|-
|5edo
|[[26zpi (σ = 1)]]
|10.0089746115
|119.892401228
|2.57426
|[[10edo]]
|1198.92401228
|8
|5
|5
|
|EDO ≤ 29, Alpha 2/1 analogue
|-
|-
|6edo
|[[34zpi (σ = 1)]]
|12.0220488259
|99.8165967700
|2.85866
|[[12edo]]
|1197.79916124
|10
|6
|6
|
|EDO ≤ 29
|-
|-
|7edo
|[[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
|7
|
|EDO ≤ 29, Beta 2/1 analogue
|-
|-
|8edo
|[[80zpi (σ = 1)]]
|22.0251749360
|54.4831086920
|2.99601
|[[22edo]]
|1198.62839122
|12
|8
|8
|
|EDO ≤ 29
|-
|-
|9edo
|[[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
|9
|
|EDO ≤ 29
|-
|-
|7ed5/3
|[[106zpi (σ = 1)]]
|
|27.0853383248
|
|44.3044124320
|Alpha 5/3 analogue
|2.90524
|-
|[[27edo]]
|10edo
|1196.21913566
|10
|10
|
|8
|EDO ≤ 29
|-
|-
|26zpi
|[[116zpi (σ = 1)]]
|10.0084563372591
|28.9431579907
|119.898609691954
|41.4605759463
|
|2.68561
|-
|[[29edo]]
|11edo
|1202.35670244
|11
|8
|
|7
|EDO ≤ 29
|-
|-
|12edo
|[[127zpi (σ = 1)]]
|30.9779815456
|38.7371913897
|3.23190
|[[31edo]]
|1200.85293308
|12
|12
|
|9
|EDO ≤ 29, Has a strong zeta peak, Gamma 2/1 analogue
|-
|-
|34zpi
|[[144zpi (σ = 1)]]
|12.0231830072926
|34.0437506778
|99.8071807833375
|35.2487600839
|
|3.07414
|[[34edo]]
|1198.45784285
|6
|6
|-
|-
|9ed5/3
|[[155zpi (σ = 1)]]
|
|35.9827898689
|
|33.3492762616
|Beta 5/3 analogue
|2.80355
|[[36edo]]
|1200.57394542
|8
|8
|-
|-
|13edo
|[[184zpi (σ = 1)]]
|13
|40.9880790756
|
|29.2768050385
|EDO ≤ 29
|3.32966
|[[41edo]]
|1200.34900658
|16
|10
|-
|-
|42zpi
|[[196zpi (σ = 1)]]
|13.9002525327005
|43.0234004818
|86.3293668353859
|27.8917981043
|
|2.78019
|[[43edo]]
|1199.34731849
|8
|8
|-
|-
|14edo
|[[214zpi (σ = 1)]]
|46.0106419996
|26.0809227572
|3.25119
|[[46edo]]
|1199.72244683
|14
|14
|
|11
|EDO ≤ 29
|-
|-
|15edo
|[[238zpi (σ = 1)]]
|15
|49.9382924730
|
|24.0296562132
|EDO ≤ 29
|2.90274
|[[50edo]]
|1201.48281066
|10
|9
|-
|-
|47zpi
|[[257zpi (σ = 1)]]
|15.0534898676781
|52.9969882711
|79.7157343943591
|22.6427961125
|
|3.46399
|[[53edo]]
|1200.06819396
|10
|10
|-
|-
|9ed3/2
|[[289zpi (σ = 1)]]
|
|58.0645692462
|
|20.6666477609
|Carlos Alpha 3/2
|3.25823
|[[58edo]]
|1198.66557013
|16
|12
|-
|-
|51zpi
|[[301zpi (σ = 1)]]
|15.9443732426877
|59.9223835273
|75.2616601314409
|20.0259056693
|
|2.98826
|[[60edo]]
|1201.55434016
|10
|10
|-
|-
|16edo
|[[321zpi (σ = 1)]]
|16
|63.0197888699
|
|19.0416378969
|EDO ≤ 29
|2.87513
|[[63edo]]
|1199.62318750
|8
|8
|-
|-
|17edo
|[[334zpi (σ = 1)]]
|17
|65.0145858034
|
|18.4573966776
|EDO ≤ 29
|3.23462
|[[65edo]]
|1199.73078404
|6
|6
|-
|-
|<s>56zpi no-5 analogue</s>
|[[354zpi (σ = 1)]]
|<s>17.0347927675362037047409871190907115584257359751871828691074626558313</s>
|68.0496579343
|
|17.6341812204
|
|3.14200
|[[68edo]]
|1199.12432299
|10
|10
|-
|-
|56zpi
|[[380zpi (σ = 1)]]
|17.0445886606675
|71.9512656175
|70.4035764012981
|16.6779554147
|
|3.61665
|[[72edo]]
|1200.81278986
|18
|13
|-
|-
|18edo
|[[414zpi (σ = 1)]]
|18
|76.9924672555
|
|15.5859403235
|EDO ≤ 29
|3.28825
|[[77edo]]
|1200.11740491
|10
|10
|-
|-
|11ed3/2
|[[435zpi (σ = 1)]]
|
|80.0733926855
|
|14.9862514845
|Carlos Beta 3/2
|3.14833
|[[80edo]]
|1198.90011876
|12
|12
|-
|-
|65zpi
|[[462zpi (σ = 1)]]
|18.9480867166984
|83.9950884037
|63.3309324546460
|14.2865496400
|
|3.19687
|[[84edo]]
|1200.07016976
|10
|10
|-
|-
|19edo
|[[483zpi (σ = 1)]]
|19
|87.0139579095
|
|13.7908908965
|EDO ≤ 29, Has a strong zeta peak
|3.44872
|[[87edo]]
|1199.80750799
|16
|14
|-
|-
|20edo
|[[497zpi (σ = 1)]]
|20
|89.0215260329
|
|13.4798857476
|EDO ≤ 29
|3.02681
|[[89edo]]
|1199.70983154
|12
|12
|-
|-
|21edo
|[[532zpi (σ = 1)]]
|21
|93.9843698073
|
|12.7680805059
|EDO ≤ 29
|3.39762
|[[94edo]]
|1200.19956756
|24
|15
|-
|-
|75zpi
|[[568zpi (σ = 1)]]
|21.0279383259393
|99.0456175574
|57.0669354931351
|12.1156294402
|
|3.56676
|[[99edo]]
|1199.44731458
|12
|12
|-
|-
|16ed5/3
|[[596zpi (σ = 1)]]
|
|102.936325452
|
|11.6576922163
|Gamma 5/3 analogue
|3.25007
|[[103edo]]
|1200.74229828
|15
|15
|-
|-
|22edo
|[[655zpi (σ = 1)]]
|111.058159333
|10.8051493669
|3.39509
|[[111edo]]
|1199.37157972
|22
|22
|
|16
|EDO ≤ 29, Has a strong zeta peak
|-
|[[706zpi (σ = 1)]]
|117.971388652
|10.1719579104
|3.62695
|[[118edo]]
|1200.29103343
|12
|12
|-
|-
|80zpi
|[[796zpi (σ = 1)]]
|22.0251467420146
|130.004267285
|54.4831784348982
|9.23046623824
|
|3.72487
|[[130edo]]
|1199.96061097
|16
|16
|-
|-
|11ed7/5
|[[872zpi (σ = 1)]]
|
|139.992781938
|
|8.57187051639
|Alpha 7/5 analogue
|3.60746
|[[140edo]]
|1200.06187229
|10
|10
|-
|-
|23edo
|[[965zpi (σ = 1)]]
|23
|152.050659206
|
|7.89210652729
|EDO ≤ 29
|3.68901
|[[152edo]]
|1199.60019215
|15
|15
|-
|-
|24edo
|[[1114zpi (σ = 1)]]
|24
|170.995049914
|
|7.01774700849
|EDO ≤ 29
|3.82285
|[[171edo]]
|1200.03473845
|14
|14
|-
|-
|90zpi
|[[1210zpi (σ = 1)]]
|24.0057421830853
|183.000273182
|49.9880399800983
|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
|-
|-
|93zpi no-2 analogue
!No-3 ZPI analog
|24.5738316304204445883184323365600165414701853056787276394517489970293
!Steps per octave
|48.8324335434322607337830293873763777285246843568212813459755275268185837463
!Cents
|
!Height
!EDO
!Octave
!Consistent
!Distinct
|-
|-
|<s>93zpi</s>
|[[no-3 51zpi (σ = 1)]]
|<s>24.5782550666850</s>
|15.9687074547
|<s>48.8236449961234</s>
|75.1469712502
|
|2.56677
|[[16edo]]
|1202.35154000
|26
|8
|-
|-
|39edt
|[[no-3 75zpi (σ = 1)]]
|24.6062603892868400468815574593676733176838399651433366869555428097087228538
|21.0417134383
|
|57.0295762045
|Has a strong no-2 zeta peak
|2.60042
|[[21edo]]
|1197.62110029
|17
|10
|-
|-
|25edo
|[[no-3 95zpi (σ = 1)]]
|25
|24.9617781085
|
|48.0734984016
|EDO ≤ 29
|2.64675
|[[25edo]]
|1201.83746004
|14
|11
|-
|-
|100zpi
|[[no-3 127zpi (σ = 1)]]
|25.9356996537225
|31.0146799866
|46.2682717652372
|38.6913552073
|
|2.60405
|[[31edo]]
|1199.43201143
|11
|11
|-
|-
|26edo
|[[no-3 161zpi (σ = 1)]]
|26
|37.0135086000
|
|32.4205957606
|EDO ≤ 29
|2.92705
|[[37edo]]
|1199.56204314
|22
|16
|-
|-
|13ed7/5
|[[no-3 196zpi (σ = 1)]]
|
|43.0494972034
|
|27.8748900209
|Beta 7/5 analogue
|2.71380
|[[43edo]]
|1198.62027090
|22
|19
|-
|-
|27edo
|[[no-3 220zpi (σ = 1)]]
|27
|47.0043385196
|
|25.5295582875
|EDO ≤ 29, Has a strong zeta peak
|2.69328
|[[47edo]]
|1199.88923951
|10
|10
|-
|-
|106zpi
|[[no-3 276zpi (σ = 1)]]
|27.0866140827635
|55.9891415481
|44.3023257293579
|21.4327272543
|
|2.76321
|[[56edo]]
|1200.23272624
|20
|19
|-
|-
|28edo
|[[no-3 340zpi (σ = 1)]]
|28
|65.9204029312
|
|18.2037722259
|EDO ≤ 29
|2.65263
|[[66edo]]
|1201.44896691
|16
|16
|-
|-
|116zpi
|[[no-3 354zpi (σ = 1)]]
|28.9399661541990
|68.0229453080
|41.4651487014917
|17.6411061674
|
|2.76285
|[[68edo]]
|1199.59521939
|11
|11
|-
|-
|29edo
|[[no-3 394zpi (σ = 1)]]
|29
|74.0566473758
|
|16.2038121158
|EDO ≤ 29
|2.76672
|[[74edo]]
|1199.08209657
|16
|16
|-
|-
|127zpi
|[[no-3 421zpi (σ = 1)]]
|30.9783816349790
|78.0097604150
|38.7366910944446
|15.3826904943
|
|2.81219
|[[78edo]]
|1199.84985856
|17
|16
|-
|-
|31edo
|[[no-3 525zpi (σ = 1)]]
|31
|93.0066513531
|
|12.9023030347
|Has a strong zeta peak
|2.97919
|[[93edo]]
|1199.91418223
|35
|19
|-
|-
|13ed4/3
|[[no-3 751zpi (σ = 1)]]
|
|124.013627761
|
|9.67635591079
|Alpha 4/3 analogue
|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
|-
|-
|34edo
!No-2 ZPI (σ = 1)
|34
!Steps per octave
|
!Cents
|Has a strong zeta peak
!Height
!EDT
!Tritave
!Consistent
!Distinct
|-
|-
|144zpi
|[[no-2 93zpi (σ = 1)]]
|34.0448410043159
|24.5747239922
|35.2476312005063
|48.8306603314
|
|2.12985
|[[39edt]]
|1904.39575293
|15
|15
|-
|-
|20ed3/2
|[[no-2 151zpi (σ = 1)]]
|
|35.3061077059
|
|33.9884534992
|Carlos Gamma 3/2
|2.08576
|[[56edt]]
|1903.35339595
|15
|15
|-
|-
|<s>151zpi</s>
|[[no-2 207zpi (σ = 1)]]
|<s>35.2948974160593</s>
|44.8164999984
|<s>33.9992488391253</s>
|26.7758526445
|
|2.10342
|[[71edt]]
|1901.08553776
|17
|17
|-
|-
|151zpi no-2 analogue
|[[no-2 222zpi (σ = 1)]]
|35.3059427335608633586867709728239574896988978536248407971925774931920
|47.3516876312
|33.9886123153797795726938859938695575674205028551304634432771826217692714955
|25.3422857776
|
|2.11876
|[[75edt]]
|1900.67143332
|15
|15
|-
|-
|56edt
|[[no-2 233zpi (σ = 1)]]
|35.3320662000016164775735184031946078407767958473853039607566768549663712773
|49.1657210129
|
|24.4072491012
|Has a strong no-2 zeta peak
|2.07714
|[[78edt]]
|1903.76542989
|21
|21
|-
|-
|155zpi no-5 analogue
|[[no-2 273zpi (σ = 1)]]
|35.9775957344990354876843659181629374042162799645238247644819739175425
|55.5359583782
|33.3540909419168338960282298282173036675588854165862895775989035929190051321
|21.6076220712
|
|2.19450
|[[88edt]]
|1901.47074227
|11
|11
|-
|-
|<s>155zpi</s>
|[[no-2 363zpi (σ = 1)]]
|<s>35.9823877000425</s>
|69.4191721809
|<s>33.3496490006021</s>
|17.2862908372
|
|2.08043
|[[110edt]]
|1901.49199210
|23
|23
|-
|-
|36edo
|[[no-2 380zpi (σ = 1)]]
|36
|71.9200195089
|
|16.6852012582
|Has a strong no-5 zeta peak
|2.07565
|[[114edt]]
|1902.11294344
|17
|17
|-
|-
|15ed4/3
|[[no-2 453zpi (σ = 1)]]
|
|82.6700405439
|
|14.5155366092
|Beta 4/3 analogue
|2.38406
|[[131edt]]
|1901.53529581
|27
|27
|-
|-
|37edo
|[[no-2 492zpi (σ = 1)]]
|37
|88.3238806401
|
|13.5863595587
|Has a strong no-3 zeta peak
|2.12238
|[[140edt]]
|1902.09033822
|9
|9
|-
|-
|161zpi no-3 analogue
|[[no-2 510zpi (σ = 1)]]
|37.0117501336435252522939269985227920601261578745487306336979972897294
|90.8334979880
|32.4221360964286053986540281462323756320027885683144327873809896041665053646
|13.2109852266
|
|2.23067
|[[144edt]]
|1902.38187263
|39
|27
|-
|-
|<s>161zpi</s>
|[[no-2 550zpi (σ = 1)]]
|<s>37.0275229191254</s>
|96.5187261015
|<s>32.4083250889078</s>
|12.4328205362
|
|2.24293
|[[153edt]]
|1902.22154203
|15
|15
|-
|-
|184zpi
|[[no-2 627zpi (σ = 1)]]
|40.9880783925993
|107.244021785
|29.2768055263764
|11.1894348983
|
|2.29774
|[[170edt]]
|1902.20393272
|15
|15
|-
|-
|41edo
|[[no-2 687zpi (σ = 1)]]
|41
|115.412802617
|
|10.3974600113
|Has a strong zeta peak
|2.18983
|[[183edt]]
|1902.73518207
|15
|15
|-
|-
|<s>186zpi</s>
|[[no-2 697zpi (σ = 1)]]
|<s>41.3438354846780</s>
|116.734850378
|<s>29.0248832971658</s>
|10.2797064983
|
|2.15793
|[[185edt]]
|1901.74570218
|29
|29
|-
|-
|96ed5
|[[no-2 777zpi (σ = 1)]]
|41.3449495750457328643302306013407006787000254796570031349096996442235599236
|127.486291223
|
|9.41277676594
|Has a strong no-2 no-3 zeta peak
|2.21095
|[[202edt]]
|1901.38090672
|17
|17
|-
|-
|186zpi no-2 no-3 analogue
|[[no-2 810zpi (σ = 1)]]
|41.3477989230936
|131.822840677
|29.0221010852836
|9.10312654342
|
|2.25360
|[[209edt]]
|1902.55344758
|21
|21
|-
|-
|188zpi no-2 no-5 analogue
|[[no-2 829zpi (σ = 1)]]
|41.6281274155763001275416027845619755345480144939422820248677372190977
|134.373782790
|28.8266629920755754571831740158108063867663530357929200798818480023767470361
|8.93031345169
|
|2.13475
|[[213edt]]
|1902.15676521
|29
|29
|-
|-
|66edt
|[[no-2 839zpi (σ = 1)]]
|41.6413637357161908485687895466222163837726522487041082394632262933532232911
|135.657892938
|
|8.84578091263
|Has a strong no-2 no-5 zeta peak
|2.11125
|[[215edt]]
|1901.84289622
|15
|15
|-
|-
|<s>188zpi</s>
|[[no-2 858zpi (σ = 1)]]
|<s>41.6477381898475</s>
|138.196070465
|<s>28.8130893094339</s>
|8.68331491602
|
|2.20051
|[[219edt]]
|1901.64596661
|11
|11
|-
|-
|46edo
|[[no-2 902zpi (σ = 1)]]
|46
|143.873905513
|
|8.34063686336
|Has a strong zeta peak
|2.09948
|[[228edt]]
|1901.66520485
|11
|11
|-
|-
|214zpi
|[[no-2 965zpi (σ = 1)]]
|46.0089748051542
|152.075713777
|26.0818678330031
|7.89080629768
|
|2.10893
|[[241edt]]
|1901.68431774
|15
|15
|-
|-
|24ed7/5
|[[no-2 985zpi (σ = 1)]]
|
|154.604034485
|
|7.76176381166
|Gamma 7/5 analogue
|2.40811
|[[245edt]]
|1901.63213386
|21
|21
|-
|-
|257zpi
|[[no-2 1029zpi (σ = 1)]]
|52.9968291550147
|160.260260060
|22.6428640945673
|7.48782012177
|
|2.17192
|[[254edt]]
|1901.90631093
|9
|9
|-
|-
|53edo
|[[no-2 1049zpi (σ = 1)]]
|53
|162.750022676
|
|7.37327086209
|Has a strong zeta peak
|2.14738
|[[258edt]]
|1902.30388242
|17
|17
|-
|-
|<s>282zpi</s>
|[[no-2 1069zpi (σ = 1)]]
|<s>56.9682909142655</s>
|165.332187903
|<s>21.0643496713977</s>
|7.25811480039
|
|2.19607
|[[262edt]]
|1901.62607770
|17
|17
|-
|-
|282zpi no-3 no-5 analogue
|[[no-2 1134zpi (σ = 1)]]
|56.9949885079206769176514037038198357725273287855611008976484058072516
|173.506549648
|21.0544827083039808806917479490481104480956786904618314090632684997207904034
|6.91616542681
|
|2.26764
|[[275edt]]
|1901.94549237
|29
|29
|-
|-
|57edo
|[[no-2 1159zpi (σ = 1)]]
|57
|176.625850825
|
|6.79402247404
|Has a strong no-3 no-5 zeta peak
|2.14379
|[[280edt]]
|1902.32629273
|11
|11
|-
|-
|58edo
|[[no-2 1179zpi (σ = 1)]]
|58
|179.167803205
|
|6.69763193238
|Has a strong zeta peak
|2.29964
|[[284edt]]
|1902.12746880
|15
|15
|-
|-
|289zpi
|[[no-2 1245zpi (σ = 1)]]
|58.0667185533159
|187.354933401
|20.6658827964969
|6.40495544056
|
|2.28021
|[[297edt]]
|1902.27176585
|21
|21
|-
|-
|301zpi
|[[no-2 1266zpi (σ = 1)]]
|59.9201656607655
|189.909845446
|20.0266469020418
|6.31878772364
|
|2.17116
|[[301edt]]
|1901.95510482
|17
|17
|-
|-
|60edo
|[[no-2 1297zpi (σ = 1)]]
|60
|193.736743714
|
|6.19397217583
|Has a strong zeta peak
|2.12380
|[[307edt]]
|1901.54945798
|21
|21
|-
|-
|65edo
|[[no-2 1343zpi (σ = 1)]]
|65
|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
|-
|-
|334zpi
!No-2 No-3 ZPI analog
|65.0158450885860
!Steps per octave
|18.4570391781413
!Cents
|
!Height
!ED5
!Pentave
!Consistent
!Distinct
|-
|-
|28ed4/3
|[[no-2 no-3 186zpi (σ = 1)]]
|
|41.3464998527
|
|29.0230129340
|Gamma 4/3 analogue
|1.75534
|[[96ed5]]
|2786.20924167
|35
|23
|-
|-
|68edo
|[[no-2 no-3 565zpi (σ = 1)]]
|68
|98.6253027359
|
|12.1672630320
|Has a strong zeta peak
|1.74188
|[[229ed5]]
|2786.30323433
|29
|29
|-
|-
|354zpi
|[[no-2 no-3 671zpi (σ = 1)]]
|68.0493056282519
|113.258011095
|17.6342725163943
|10.5952769998
|
|1.77217
|[[263ed5]]
|2786.55785095
|19
|19
|-
|-
|380zpi
|[[no-2 no-3 764zpi (σ = 1)]]
|71.9506065993786
|125.745000550
|16.6781081733140
|9.54312294522
|
|1.75634
|[[292ed5]]
|2786.59190001
|37
|37
|-
|-
|72edo
|[[no-2 no-3 905zpi (σ = 1)]]
|72
|144.297529480
|
|8.31615069448
|Has a strong zeta peak
|1.73926
|[[335ed5]]
|2785.91048265
|43
|41
|-
|-
|414zpi
|[[no-2 no-3 938zpi (σ = 1)]]
|76.9918536925042
|148.562870929
|15.5860645308353
|8.07738833059
|
|1.79949
|[[345ed5]]
|2786.69897405
|25
|25
|-
|-
|77edo
|[[no-2 no-3 1046zpi (σ = 1)]]
|77
|162.414291729
|
|7.38851234841
|Has a strong zeta peak
|1.73251
|[[377ed5]]
|2785.46915535
|23
|23
|-
|-
|80edo
|[[no-2 no-3 1145zpi (σ = 1)]]
|80
|174.880594782
|
|6.86182478678
|Has a strong zeta peak
|1.74084
|[[406ed5]]
|2785.90086343
|25
|25
|-
|-
|435zpi
|[[no-2 no-3 1196zpi (σ = 1)]]
|80.0731374302484
|181.292147244
|14.9862992572924
|6.61915046096
|
|1.77770
|[[421ed5]]
|2786.66234406
|35
|35
|-
|-
|455zpi no-3 no-5 analogue
|[[no-2 no-3 1280zpi (σ = 1)]]
|82.9585473728587401934282446836610895074185494886540503684148508037660
|191.632570168
|14.4650555970631644892614919440394905869155594072293855522604093941309631517
|6.26198353937
|
|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
|-
|-
|<s>455zpi</s>
|[[Alpha 3/1]]
|<s>82.9665108106434</s>
|1.90739592696007
|<s>14.4636671866169</s>
|629.130000247254
|
|Dave Benson
|[[3edt|3ed3/1]]
|-
|-
|83edo
|[[Beta 3/1]]
|83
|3.14186231690763
|
|381.939079106782
|Has a strong no-3 no-5 zeta peak
|Dave Benson
|[[5edt|5ed3/1]]
|-
|-
|462zpi
|[[Alpha 2/1]]
|83.9972142607288
|5.00991270509077
|14.2861880666087
|239.525131601721
|
|Dave Benson
|[[5edo|5ed2/1]]
|-
|-
|84edo
|[[Gamma 3/1]]
|84
|5.04255621376059
|
|237.974540913462
|Has a strong zeta peak
|Dave Benson
|[[8edt|8ed3/1]]
|-
|-
|87edo
|[[Beta 2/1]]
|87
|6.99104980248710
|
|171.648040552235
|Has a strong zeta peak
|Dave Benson
|[[7edo|7ed2/1]]
|-
|-
|483zpi
|[[Alpha 5/3]]
|87.0139255957575
|9.50583353877785
|13.7908960178956
|126.238272015258
|
|Dave Benson
|[[7ed5/3]]
|-
|-
|532zpi
|[[Gamma 2/1]]
|93.9836761074943
|11.9978480914311
|12.7681747480009
|100.017935787756
|
|Dave Benson
|[[12edo|12ed2/1]]
|-
|-
|94edo
|[[Beta 5/3]]
|94
|12.2053823008782
|
|98.3172808862904
|Has a strong zeta peak
|Dave Benson
|[[9ed5/3]]
|-
|-
|99edo
|[[Carlos Alpha|Alpha 3/2]]
|99
|15.3915238996928
|
|77.9649895501219
|Has a strong zeta peak
|Dave Benson
|[[9edf|9ed3/2]]
|-
|-
|568zpi
|[[Carlos Beta|Beta 3/2]]
|99.0473345956631
|18.7990736394111
|12.1154194093028
|63.8329325698408
|
|Dave Benson
|[[11edf|11ed3/2]]
|-
|-
|327ed7
|[[Gamma 5/3]]
|116.479750184323251720135904506422003080366592226710079912941501697613698777
|21.7094399215509
|
|55.2754932571412
|Has a strong no-2 no-3 no-5 zeta peak
|Dave Benson
|[[16ed5/3]]
|-
|-
|695zpi no-2 no-3 no-5 analogue
|[[Alpha 7/5]]
|116.481879086491562246584713240674074197523436163157572853694264779074
|22.6653911133366
|10.3020316070704971370763940790472253291607124811459581948058543261473166559
|52.9441558718088
|
|Dave Benson
|[[11ed7/5]]
|-
|-
|<s>695zpi</s>
|[[Beta 7/5]]
|<s>116.484048333840</s>
|26.7758951088566
|<s>10.3018397554387</s>
|44.8164289231577
|
|Dave Benson
|[[13ed7/5]]
|-
|-
|1114zpi
|[[Alpha 4/3]]
|170.995891689006
|31.3266790320926
|7.01771246166817
|38.3060074376432
|
|Dave Benson
|[[13ed4/3]]
|-
|-
|171edo
|[[Carlos Gamma|Gamma 3/2]]
|171
|34.1894540921914
|
|35.0985422804417
|Exceptionally strong zeta peak
|Dave Benson
|[[20edf|20ed3/2]]
|-
|-
|270edo
|[[Beta 4/3]]
|270
|36.1372975038827
|
|33.2066890135065
|Exceptionally strong zeta peak
|Dave Benson
|[[15ed4/3]]
|-
|-
|1936zpi
|[[Gamma 7/5]]
|270.017794631965
|49.4404896216012
|4.44415154799558
|24.2716042900130
|
|Dave Benson
|[[24ed7/5]]
|-
|-
|311edo
|[[Gamma 4/3]]
|311
|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]]
|311.004029926555
|94\93<2/1>
|3.85847090239759
|16 11 16 12 16 11 12
|
|
|-
|-
|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
|}
|}
Retrieved from "https://en.xen.wiki/w/User:Contribution/Collection_of_tunings"