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* '''The α–β–γ family, with an equave sliding from 3/1 down to 4/3'''
* '''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. In practice, you can make convincing music with ''any'' equal-step interval, every real-number step size repeated ad infinitum forms its own viable lattice. 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.
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.
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.


From the original Riemann zeta: 12edo, (14edo), (15edo), (17edo), 19edo, 22edo, 24edo, (26edo), 27edo, (29edo), 31edo, 34edo, 36edo, 41edo, 46edo, 50edo, 53edo, 58edo, 60edo, 65edo, 68edo, 72edo, 77edo, 80edo, 84edo, 87edo, 94edo, 99edo, 103edo, 111edo, 118edo, 130edo, 140edo, 152edo, 171edo
=== 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
|-
|[[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 ===
 
{|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
|-
|[[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
|}


From the no-2 Riemann zeta: 39edt, 56edt, 69edt, 71edt, 75edt, 78edt, 82edt, 88edt, 99edt, 101edt, 105edt, 110edt, 131edt, 140edt, 144edt, 153edt, 170edt, 183edt, 185edt, 202edt, 209edt, 213edt, 215edt, 219edt, 245edt
=== Notable Local Maxima of the Riemann Zeta Function after removing the prime 2 from the zeta product ===


From the no-3 Riemann zeta: 16edo, 21edo, 25edo, (28edo), 35edo, 37edo, 43edo, 47edo, (52edo), 56edo, (66edo), 74edo, 78edo, 93edo, 109edo, 124edo
{|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
|-
|[[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
|}


From the no-2 no-3 Riemann zeta:
=== Notable Local Maxima of the Riemann Zeta Function after removing the primes 2 and 3 from the zeta product ===


From the alpha-beta-gamma set: 5ed2/1, 7ed2/1, 12ed2/1, 7ed5/3, 9ed5/3, 16ed5/3, 9ed3/2, 11ed3/2, 20ed3/2, 11ed7/5, 13ed7/5, 24ed7/5, 13ed4/3, 15ed4/3, 28ed4/3
{|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
|-
!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 ===
{| class="wikitable sortable"
{| class="wikitable sortable"
|+ style="font-size: 105%;" | α–β–γ family
|+ style="font-size: 105%;" | α–β–γ family
|- style="white-space: nowrap;"
|- style="white-space: nowrap;"
! colspan="4" |Optimization
! colspan="4" |Optimization
! colspan="3" |Equal division of a ratio
! rowspan="2" |Equal division of a ratio
|- style="white-space: nowrap;"
|- style="white-space: nowrap;"
!Proposed name
!Proposed name
Line 34: Line 983:
!Cents
!Cents
!Optimization method
!Optimization method
!EDR
!Steps per octave
!Cents
|-
|-
|[[Alpha 3/1]]
|[[Alpha 3/1]]
Line 43: Line 989:
|Dave Benson
|Dave Benson
|[[3edt|3ed3/1]]
|[[3edt|3ed3/1]]
|1.89278926071437
|633.985000288463
|-
|-
|[[Beta 3/1]]
|[[Beta 3/1]]
Line 51: Line 995:
|Dave Benson
|Dave Benson
|[[5edt|5ed3/1]]
|[[5edt|5ed3/1]]
|3.15464876785729
|380.391000173077
|-
|-
|[[Alpha 2/1]]
|[[Alpha 2/1]]
Line 59: Line 1,001:
|Dave Benson
|Dave Benson
|[[5edo|5ed2/1]]
|[[5edo|5ed2/1]]
|5.00000000000000
|240.000000000000
|-
|-
|[[Gamma 3/1]]
|[[Gamma 3/1]]
Line 67: Line 1,007:
|Dave Benson
|Dave Benson
|[[8edt|8ed3/1]]
|[[8edt|8ed3/1]]
|5.04743802857166
|237.744375108173
|-
|-
|[[Beta 2/1]]
|[[Beta 2/1]]
Line 75: Line 1,013:
|Dave Benson
|Dave Benson
|[[7edo|7ed2/1]]
|[[7edo|7ed2/1]]
|7.00000000000000
|171.428571428571
|-
|-
|[[Alpha 5/3]]
|[[Alpha 5/3]]
Line 83: Line 1,019:
|Dave Benson
|Dave Benson
|[[7ed5/3]]
|[[7ed5/3]]
|9.49840814199707
|126.336958999921
|-
|-
|[[Gamma 2/1]]
|[[Gamma 2/1]]
Line 91: Line 1,025:
|Dave Benson
|Dave Benson
|[[12edo|12ed2/1]]
|[[12edo|12ed2/1]]
|12.0000000000000
|100.000000000000
|-
|-
|[[Beta 5/3]]
|[[Beta 5/3]]
Line 99: Line 1,031:
|Dave Benson
|Dave Benson
|[[9ed5/3]]
|[[9ed5/3]]
|12.2122390397105
|98.2620792221608
|-
|-
|[[Carlos Alpha|Alpha 3/2]]
|[[Carlos Alpha|Alpha 3/2]]
Line 107: Line 1,037:
|Dave Benson
|Dave Benson
|[[9edf|9ed3/2]]
|[[9edf|9ed3/2]]
|15.3856016221631
|77.9950000961542
|-
|-
|[[Carlos Beta|Beta 3/2]]
|[[Carlos Beta|Beta 3/2]]
Line 115: Line 1,043:
|Dave Benson
|Dave Benson
|[[11edf|11ed3/2]]
|[[11edf|11ed3/2]]
|18.8046242048660
|63.8140909877625
|-
|-
|[[Gamma 5/3]]
|[[Gamma 5/3]]
Line 123: Line 1,049:
|Dave Benson
|Dave Benson
|[[16ed5/3]]
|[[16ed5/3]]
|21.7106471817076
|55.2724195624655
|-
|-
|[[Alpha 7/5]]
|[[Alpha 7/5]]
Line 131: Line 1,055:
|Dave Benson
|Dave Benson
|[[11ed7/5]]
|[[11ed7/5]]
|22.6604698881676
|52.9556538731173
|-
|-
|[[Beta 7/5]]
|[[Beta 7/5]]
Line 139: Line 1,061:
|Dave Benson
|Dave Benson
|[[13ed7/5]]
|[[13ed7/5]]
|26.7805553223799
|44.8086302003300
|-
|-
|[[Alpha 4/3]]
|[[Alpha 4/3]]
Line 147: Line 1,067:
|Dave Benson
|Dave Benson
|[[13ed4/3]]
|[[13ed4/3]]
|31.3224709154917
|38.3111537795856
|-
|-
|[[Carlos Gamma|Gamma 3/2]]
|[[Carlos Gamma|Gamma 3/2]]
Line 155: Line 1,073:
|Dave Benson
|Dave Benson
|[[20edf|20ed3/2]]
|[[20edf|20ed3/2]]
|34.1902258270291
|35.0977500432694
|-
|-
|[[Beta 4/3]]
|[[Beta 4/3]]
Line 163: Line 1,079:
|Dave Benson
|Dave Benson
|[[15ed4/3]]
|[[15ed4/3]]
|36.1413125947981
|33.2029999423075
|-
|-
|[[Gamma 7/5]]
|[[Gamma 7/5]]
Line 171: Line 1,085:
|Dave Benson
|Dave Benson
|[[24ed7/5]]
|[[24ed7/5]]
|49.4410252105475
|24.2713413585121
|-
|-
|[[Gamma 4/3]]
|[[Gamma 4/3]]
Line 179: Line 1,091:
|Dave Benson
|Dave Benson
|[[28ed4/3]]
|[[28ed4/3]]
|67.4637835102899
|17.7873213976647
|}
=== Equal divisions of a ratio & optimization ===
{| class="wikitable sortable"
|+ style="font-size: 105%;" | EDRs collection & optimization
|- 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]]
|9.49840814199707
|126.336958999921
|[[Benson Alpha 5/3]]
|9.50583353877785
|126.238272015258
|Alpha 5/3
|-
|[[10edo]]
|10
|120.
| colspan="3" |None
|EDO ≤ 29
|-
|[[11edo]]
|11
|109.090909090909
| colspan="3" |None
|EDO ≤ 29
|-
|[[12edo]]
|12
|100.
|[[34zpi]]
|12.0231830072926
|99.8071807833375
|EDO ≤ 29, strong zeta peak
|-
|[[9ed5/3]]
|12.2122390397105
|98.2620792221608
|[[Benson Beta 5/3]]
|12.2053823008782
|98.3172808862904
|Beta 5/3
|-
|[[13edo]]
|13
|92.3076923076923
| colspan="3" |None
|EDO ≤ 29
|-
|[[14edo]]
|14
|85.7142857142857
|[[42zpi]]
|13.9002525327005
|86.3293668353859
|EDO ≤ 29, medium zeta peak
|-
|[[15edo]]
|15
|80.
|[[47zpi]]
|15.0534898676781
|79.7157343943591
|EDO ≤ 29, medium zeta peak
|-
|[[9edf|9ed3/2]]
|15.3856016221631
|77.9950000961542
|[[Benson Alpha 3/2]]
|15.3915238996928
|77.9649895501219
|Alpha 3/2
|-
|[[16edo]]
|16
|75.
| colspan="3" |None
|EDO ≤ 29
|-
|[[17edo]]
|17
|70.5882352941176
|[[56zpi]]
|17.0445886606675
|70.4035764012981
|EDO ≤ 29, medium zeta peak
|-
|[[18edo]]
|18
|66.6666666666667
| colspan="3" |None
|EDO ≤ 29
|-
|[[11edf|11ed3/2]]
|18.8046242048660
|63.8140909877625
|[[Benson Beta 3/2]]
|18.7990736394111
|63.8329325698408
|Beta 3/2
|-
|[[19edo]]
|19
|63.1578947368421
|[[65zpi]]
|18.9480867166984
|63.3309324546460
|EDO ≤ 29, strong zeta peak
|-
|[[20edo]]
|20
|60.
| colspan="3" |None
|EDO ≤ 29
|-
|[[21edo]]
|21
|57.1428571428571
| colspan="3" |None
|EDO ≤ 29
|-
|[[16ed5/3]]
|21.7106471817076
|55.2724195624655
|[[Benson Gamma 5/3]]
|21.7094399215509
|55.2754932571412
|Gamma 5/3
|-
|[[22edo]]
|22
|54.5454545454545
|[[80zpi]]
|22.0251467420146
|54.4831784348982
|EDO ≤ 29, strong zeta peak
|-
|[[11ed7/5]]
|22.6604698881676
|52.9556538731173
|[[Benson Alpha 7/5]]
|22.6653911133366
|52.9441558718088
|Alpha 7/5
|-
|[[23edo]]
|23
|52.1739130434783
| colspan="3" |None
|EDO ≤ 29
|-
|[[24edo]]
|24
|50.
|[[90zpi]]
|24.0057421830853
|49.9880399800983
|EDO ≤ 29, medium zeta peak
|-
|[[39edt]]
|24.6062603892868
|48.7680769452663
|[[93zpi no-2 analogue]]
|24.5738316304204
|48.8324335434323
|strong no-2 zeta peak
|-
|[[25edo]]
|25
|48.
| colspan="3" |None
|EDO ≤ 29
|-
|[[26edo]]
|26
|46.1538461538462
|[[100zpi]]
|25.9356996537225
|46.2682717652372
|EDO ≤ 29, medium zeta peak
|-
|[[13ed7/5]]
|26.7805553223799
|44.8086302003300
|[[Benson Beta 7/5]]
|26.7758951088566
|44.8164289231577
|Beta 7/5
|-
|[[27edo]]
|27
|44.4444444444444
|[[106zpi]]
|27.0866140827635
|44.3023257293579
|EDO ≤ 29, strong zeta peak
|-
|[[28edo]]
|28
|42.8571428571429
| colspan="3" |None
|EDO ≤ 29
|-
|[[29edo]]
|29
|41.3793103448276
|[[116zpi]]
|28.9399661541990
|41.4651487014917
|EDO ≤ 29, medium zeta peak
|-
|[[31edo]]
|31
|38.7096774193548
|[[127zpi]]
|30.9783816349790
|38.7366910944446
|strong zeta peak
|-
|[[13ed4/3]]
|31.3224709154917
|38.3111537795856
|[[Benson Alpha 4/3]]
|31.3266790320926
|38.3060074376432
|Alpha 4/3
|-
|[[34edo]]
|34
|35.2941176470588
|[[144zpi]]
|34.0448410043159
|35.2476312005063
|strong zeta peak
|-
|[[20edf|20ed3/2]]
|34.1902258270291
|35.0977500432694
|[[Benson Gamma 3/2]]
|34.1894540921914
|35.0985422804417
|Gamma 3/2
|-
|[[56edt]]
|35.3320662000016
|33.9634821583105
|[[151zpi no-2 analogue]]
|35.3059427335609
|33.9886123153798
|strong no-2 zeta peak
|-
|[[36edo]]
|36
|33.3333333333333
|[[155zpi no-5 analogue]]
|35.9775957344990
|33.3540909419168
|strong no-5 zeta peak
|-
|[[15ed4/3]]
|36.1413125947981
|33.2029999423075
|[[Benson Beta 4/3]]
|36.1372975038827
|33.2066890135066
|Beta 4/3
|-
|[[37edo]]
|37
|32.4324324324324
|[[161zpi no-3 analogue]]
|37.0117501336435
|32.4221360964286
|strong no-3 zeta peak
|-
|[[41edo]]
|41
|29.2682926829268
|[[184zpi]]
|40.9880783925993
|29.2768055263764
|strong zeta peak
|-
|[[96ed5]]
|41.3449495750457
|29.0241011860920
|[[186zpi no-2 no-3 analogue]]
|41.3477989230936
|29.0221010852836
|strong no-2 no-3 zeta peak
|-
|[[66edt]]
|41.6413637357162
|28.8175000131119
|[[188zpi no-2 no-5 analogue]]
|41.6281274155763
|28.8266629920756
|strong no-2 no-5 zeta peak
|-
|[[46edo]]
|46
|26.0869565217391
|[[214zpi]]
|46.0089748051542
|26.0818678330031
|strong zeta peak
|-
|[[24ed7/5]]
|49.4410252105475
|24.2713413585121
|[[Benson Gamma 7/5]]
|49.4404896216012
|24.2716042900130
|Gamma 7/5
|-
|[[50edo]]
|50
|24.0
|[[238zpi]]
|49.9385162652878
|24.0295485277387
|medium zeta peak
|-
|[[53edo]]
|53
|22.6415094339623
|[[257zpi]]
|52.9968291550147
|22.6428640945673
|strong zeta peak
|-
|[[57edo]]
|57
|21.0526315789474
|[[282zpi no-3 no-5 analogue]]
|56.9949885079207
|21.0544827083040
|strong no-3 no-5 zeta peak
|-
|[[58edo]]
|58
|20.6896551724138
|[[289zpi]]
|58.0667185533159
|20.6658827964969
|strong zeta peak
|-
|[[60edo]]
|60
|20.
|[[301zpi]]
|59.9201656607655
|20.0266469020418
|medium zeta peak
|-
|[[65edo]]
|65
|18.4615384615385
|[[334zpi]]
|65.0158450885860
|18.4570391781413
|strong zeta peak
|-
|[[28ed4/3]]
|67.4637835102899
|17.7873213976647
|[[Benson Gamma 4/3]]
|67.4633901646646
|17.7874251067289
|Gamma 4/3
|-
|[[68edo]]
|68
|17.6470588235294
|[[354zpi]]
|68.0493056282519
|17.6342725163943
|strong zeta peak
|-
|[[72edo]]
|72
|16.6666666666667
|[[380zpi]]
|71.9506065993786
|16.6781081733140
|strong zeta peak
|-
|[[77edo]]
|77
|15.5844155844156
|[[414zpi]]
|76.9918536925042
|15.5860645308353
|strong zeta peak
|-
|[[80edo]]
|80
|15.
|[[435zpi]]
|80.0731374302484
|14.9862992572924
|medium zeta peak
|-
|[[131edt]]
|82.6517977178609
|14.5187404646213
|[[453zpi no-2 analogue]]
|82.6705208991009
|14.5154522670130
|strong no-2 zeta peak
|-
|[[83edo]]
|83
|14.4578313253012
|[[455zpi no-3 no-5 analogue]]
|82.9585473728587
|14.4650555970632
|strong no-3 no-5 zeta peak
|-
|[[84edo]]
|84
|14.2857142857143
|[[462zpi]]
|83.9972142607288
|14.2861880666087
|medium zeta peak
|-
|[[87edo]]
|87
|13.7931034482759
|[[483zpi]]
|87.0139255957575
|13.7908960178956
|strong zeta peak
|-
|[[94edo]]
|94
|12.7659574468085
|[[532zpi]]
|93.9836761074943
|12.7681747480009
|strong zeta peak
|-
|[[99edo]]
|99
|12.1212121212121
|[[568zpi]]
|99.0473345956631
|12.1154194093028
|strong zeta peak
|-
|[[103edo]]
|103
|11.6504854368932
|[[596zpi]]
|102.936629522070
|11.6576577800491
|medium zeta peak
|-
|[[111edo]]
|111
|10.8108108108108
|[[655zpi]]
|111.059577998833
|10.8050113427643
|medium zeta peak
|-
|[[327ed7]]
|116.479750184323
|10.3022198974591
|[[695zpi no-2 no-3 no-5 analogue]]
|116.481879086492
|10.3020316070705
|strong no-2 no-3 no-5 zeta peak
|-
|[[118edo]]
|118
|10.1694915254237
|[[706zpi]]
|117.969513574257
|10.1721195895637
|strong zeta peak
|-
|[[130edo]]
|130
|9.23076923076923
|[[796zpi]]
|130.003910460506
|9.23049157328654
|strong zeta peak
|-
|[[140edo]]
|140
|8.57142857142857
|[[872zpi]]
|139.990541024216
|8.57200773152536
|strong zeta peak
|-
|[[152edo]]
|152
|7.89473684210526
|[[965zpi]]
|152.052848107925
|7.89199291517551
|strong zeta peak
|-
|[[171edo]]
|171
|7.01754385964912
|[[1114zpi]]
|170.995891689006
|7.01771246166817
|exceptionally strong zeta peak
|-
|[[270edo]]
|270
|4.44444444444444
|[[1936zpi]]
|270.017794631965
|4.44415154799558
|exceptionally strong zeta peak
|-
|[[311edo]]
|311
|3.85852090032154
|[[2293zpi]]
|311.004029926555
|3.85847090239759
|exceptionally strong zeta peak
|-
|[[342edo]]
|342
|3.50877192982456
| colspan="3" |None
|171*2^n family
|-
|[[684edo]]
|684
|1.75438596491228
| colspan="3" |None
|171*2^n family
|}
|}


Retrieved from "https://en.xen.wiki/w/User:Contribution/Collection_of_tunings"