User:Contribution/Collection of tunings: Difference between revisions

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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 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 ===
=== Notable Local Maxima of the Riemann Zeta Function ===
{| class="wikitable sortable"
{|class="wikitable sortable"
|+ style="font-size: 105%;" | Notable Local Maxima of the Riemann Zeta Function
|+ style="font-size: 105%;" |
|- style="white-space: nowrap;"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Integer limit
|-
|-
!ZPI
!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)
!Height
!Height
!Integral
!Gap
!EDO
!EDO
!Octave
!Octave (cents)
!Consistent
!Consistent
!Distinct
!Distinct
|-
|-
|[[34zpi]]
|[[15zpi (σ = 1)]]
|12.0231830072926
|6.95688550773
|99.8071807833375
|172.490980147
|5.193290
|2.55384
|1.269599
|[[7edo]]
|15.899282
|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]]
|[[12edo]]
|1197.68616940005
|1197.79916124
|10
|10
|6
|6
|-
|-
|[[42zpi]]
|[[42zpi (σ = 1)]]
|13.9002525327005
|13.9020220557
|86.3293668353859
|86.3183783764
|4.592177
|2.50514
|0.984037
|14.097244
|[[14edo]]
|[[14edo]]
|1208.61113569540
|1208.45729727
|7
|7
|5
|5
|-
|-
|[[47zpi]]
|[[47zpi (σ = 1)]]
|15.0534898676781
|15.0534708836
|79.7157343943591
|79.7158349246
|5.050324
|2.69313
|1.104057
|14.918297
|[[15edo]]
|[[15edo]]
|1195.73601591539
|1195.73752387
|8
|8
|7
|7
|-
|-
|[[56zpi]]
|[[56zpi (σ = 1)]]
|17.0445886606675
|17.0432556931
|70.4035764012981
|70.4090827252
|5.056957
|2.65741
|1.032175
|14.269437
|[[17edo]]
|[[17edo]]
|1196.86079882207
|1196.95440633
|4
|4
|4
|4
|-
|-
|[[65zpi]]
|[[65zpi (σ = 1)]]
|18.9480867166984
|18.9489976130
|63.3309324546460
|63.3278880767
|5.980169
|3.02387
|1.313799
|16.699651
|[[19edo]]
|[[19edo]]
|1203.28771663827
|1203.22987346
|10
|10
|7
|7
|-
|-
|[[80zpi]]
|[[80zpi (σ = 1)]]
|22.0251467420146
|22.0251749360
|54.4831784348982
|54.4831086920
|6.062600
|2.99601
|1.258178
|16.213941
|[[22edo]]
|[[22edo]]
|1198.62992556776
|1198.62839122
|12
|12
|8
|8
|-
|-
|[[90zpi]]
|[[90zpi (σ = 1)]]
|24.0057421830853
|24.0053572889
|49.9880399800983
|49.9888414723
|5.721613
|2.82476
|1.092055
|14.821136
|[[24edo]]
|[[24edo]]
|1199.71295952236
|1199.73219533
|6
|6
|6
|6
|-
|-
|[[100zpi]]
|[[100zpi (σ = 1)]]
|25.9356996537225
|25.9356337472
|46.2682717652372
|46.2683893402
|5.545073
|2.71167
|1.031155
|14.793013
|[[26edo]]
|[[26edo]]
|1202.97506589617
|1202.97812285
|14
|14
|9
|9
|-
|-
|[[106zpi]]
|[[106zpi (σ = 1)]]
|27.0866140827635
|27.0853383248
|44.3023257293579
|44.3044124320
|6.069233
|2.90524
|1.185939
|16.215619
|[[27edo]]
|[[27edo]]
|1196.16279469266
|1196.21913566
|10
|10
|8
|8
|-
|-
|[[116zpi]]
|[[116zpi (σ = 1)]]
|28.9399661541990
|28.9431579907
|41.4651487014917
|41.4605759463
|5.566209
|2.68561
|1.000619
|14.904418
|[[29edo]]
|[[29edo]]
|1202.48931234326
|1202.35670244
|8
|8
|7
|7
|-
|-
|[[127zpi]]
|[[127zpi (σ = 1)]]
|30.9783816349790
|30.9779815456
|38.7366910944446
|38.7371913897
|7.003472
|3.23190
|1.403777
|17.739476
|[[31edo]]
|[[31edo]]
|1200.83742392778
|1200.85293308
|12
|12
|9
|9
|-
|-
|[[144zpi]]
|[[144zpi (σ = 1)]]
|34.0448410043159
|34.0437506778
|35.2476312005063
|35.2487600839
|6.685147
|3.07414
|1.241437
|16.236989
|[[34edo]]
|[[34edo]]
|1198.41946081721
|1198.45784285
|6
|6
|6
|6
|-
|-
|[[155zpi]]
|[[155zpi (σ = 1)]]
|35.9823877000425
|35.9827898689
|33.3496490006021
|33.3492762616
|6.027497
|2.80355
|1.028887
|14.706508
|[[36edo]]
|[[36edo]]
|1200.58736402167
|1200.57394542
|8
|8
|8
|8
|-
|-
|[[184zpi]]
|[[184zpi (σ = 1)]]
|40.9880783925993
|40.9880790756
|29.2768055263764
|29.2768050385
|7.570230
|3.32966
|1.423937
|17.722623
|[[41edo]]
|[[41edo]]
|1200.34902658143
|1200.34900658
|16
|16
|10
|10
|-
|-
|[[214zpi]]
|[[196zpi (σ = 1)]]
|46.0089748051542
|43.0234004818
|26.0818678330031
|27.8917981043
|7.495674
|2.78019
|1.356067
|[[43edo]]
|17.747832
|1199.34731849
|8
|8
|-
|[[214zpi (σ = 1)]]
|46.0106419996
|26.0809227572
|3.25119
|[[46edo]]
|[[46edo]]
|1199.76592031814
|1199.72244683
|14
|14
|11
|11
|-
|-
|[[238zpi]]
|[[238zpi (σ = 1)]]
|49.9385162652878
|49.9382924730
|24.0295485277387
|24.0296562132
|6.655352
|2.90274
|1.111229
|15.942083
|[[50edo]]
|[[50edo]]
|1201.47742638693
|1201.48281066
|10
|10
|9
|9
|-
|-
|[[257zpi]]
|[[257zpi (σ = 1)]]
|52.9968291550147
|52.9969882711
|22.6428640945673
|22.6427961125
|8.249774
|3.46399
|1.486620
|18.069918
|[[53edo]]
|[[53edo]]
|1200.07179701207
|1200.06819396
|10
|10
|10
|10
|-
|-
|[[289zpi]]
|[[289zpi (σ = 1)]]
|58.0667185533159
|58.0645692462
|20.6658827964969
|20.6666477609
|7.814035
|3.25823
|1.358357
|18.056292
|[[58edo]]
|[[58edo]]
|1198.62120219682
|1198.66557013
|16
|16
|12
|12
|-
|-
|[[301zpi]]
|[[301zpi (σ = 1)]]
|59.9201656607655
|59.9223835273
|20.0266469020418
|20.0259056693
|7.046396
|2.98826
|1.131000
|15.932359
|[[60edo]]
|[[60edo]]
|1201.59881412251
|1201.55434016
|10
|10
|10
|10
|-
|-
|[[334zpi]]
|[[321zpi (σ = 1)]]
|65.0158450885860
|63.0197888699
|18.4570391781413
|19.0416378969
|7.813349
|2.87513
|1.269821
|[[63edo]]
|16.514861
|1199.62318750
|8
|8
|-
|[[334zpi (σ = 1)]]
|65.0145858034
|18.4573966776
|3.23462
|[[65edo]]
|[[65edo]]
|1199.70754657919
|1199.73078404
|6
|6
|6
|6
|-
|-
|[[354zpi]]
|[[354zpi (σ = 1)]]
|68.0493056282519
|68.0496579343
|17.6342725163943
|17.6341812204
|7.666604
|3.14200
|1.254592
|17.034505
|[[68edo]]
|[[68edo]]
|1199.13053111481
|1199.12432299
|10
|10
|10
|10
|-
|-
|[[380zpi]]
|[[380zpi (σ = 1)]]
|71.9506065993786
|71.9512656175
|16.6781081733140
|16.6779554147
|9.157547
|3.61665
|1.625363
|19.964746
|[[72edo]]
|[[72edo]]
|1200.82378847861
|1200.81278986
|18
|18
|13
|13
|-
|-
|[[414zpi]]
|[[414zpi (σ = 1)]]
|76.9918536925042
|76.9924672555
|15.5860645308353
|15.5859403235
|8.194847
|3.28825
|1.311364
|17.029289
|[[77edo]]
|[[77edo]]
|1200.12696887432
|1200.11740491
|10
|10
|10
|10
|-
|-
|[[435zpi]]
|[[435zpi (σ = 1)]]
|80.0731374302484
|80.0733926855
|14.9862992572924
|14.9862514845
|7.873146
|3.14833
|1.247325
|17.087322
|[[80edo]]
|[[80edo]]
|1198.90394058339
|1198.90011876
|12
|12
|12
|12
|-
|-
|[[462zpi]]
|[[462zpi (σ = 1)]]
|83.9972142607288
|83.9950884037
|14.2861880666087
|14.2865496400
|8.020965
|3.19687
|1.241945
|16.733121
|[[84edo]]
|[[84edo]]
|1200.03979759513
|1200.07016976
|10
|10
|10
|10
|-
|-
|[[483zpi]]
|[[483zpi (σ = 1)]]
|87.0139255957575
|87.0139579095
|13.7908960178956
|13.7908908965
|8.869041
|3.44872
|1.439474
|18.061741
|[[87edo]]
|[[87edo]]
|1199.80795355692
|1199.80750799
|16
|16
|14
|14
|-
|-
|[[532zpi]]
|[[497zpi (σ = 1)]]
|93.9836761074943
|89.0215260329
|12.7681747480009
|13.4798857476
|8.806201
|3.02681
|1.394050
|[[89edo]]
|17.832744
|1199.70983154
|12
|12
|-
|[[532zpi (σ = 1)]]
|93.9843698073
|12.7680805059
|3.39762
|[[94edo]]
|[[94edo]]
|1200.20842631208
|1200.19956756
|24
|24
|15
|15
|-
|-
|[[568zpi]]
|[[568zpi (σ = 1)]]
|99.0473345956631
|99.0456175574
|12.1154194093028
|12.1156294402
|9.406495
|3.56676
|1.510412
|18.536483
|[[99edo]]
|[[99edo]]
|1199.42652152097
|1199.44731458
|12
|12
|12
|12
|-
|-
|[[596zpi]]
|[[596zpi (σ = 1)]]
|102.936629522070
|102.936325452
|11.6576577800491
|11.6576922163
|8.543510
|3.25007
|1.340775
|18.270998
|[[103edo]]
|[[103edo]]
|1200.73875134506
|1200.74229828
|15
|15
|15
|15
|-
|-
|[[655zpi]]
|[[655zpi (σ = 1)]]
|111.059577998833
|111.058159333
|10.8050113427643
|10.8051493669
|9.038544
|3.39509
|1.394739
|18.041165
|[[111edo]]
|[[111edo]]
|1199.35625904684
|1199.37157972
|22
|22
|16
|16
|-
|-
|[[706zpi]]
|[[706zpi (σ = 1)]]
|117.969513574257
|117.971388652
|10.1721195895637
|10.1719579104
|9.850823
|3.62695
|1.544280
|18.861062
|[[118edo]]
|[[118edo]]
|1200.31011156852
|1200.29103343
|12
|12
|12
|12
|-
|-
|[[796zpi]]
|[[796zpi (σ = 1)]]
|130.003910460506
|130.004267285
|9.23049157328654
|9.23046623824
|10.355108
|3.72487
|1.634018
|19.594551
|[[130edo]]
|[[130edo]]
|1199.96390452725
|1199.96061097
|16
|16
|16
|16
|-
|-
|[[872zpi]]
|[[872zpi (σ = 1)]]
|139.990541024216
|139.992781938
|8.57200773152536
|8.57187051639
|10.076688
|3.60746
|1.548424
|19.514765
|[[140edo]]
|[[140edo]]
|1200.08108241355
|1200.06187229
|10
|10
|10
|10
|-
|-
|[[965zpi]]
|[[965zpi (σ = 1)]]
|152.052848107925
|152.050659206
|7.89199291517551
|7.89210652729
|10.468420
|3.68901
|1.593855
|19.487224
|[[152edo]]
|[[152edo]]
|1199.58292310668
|1199.60019215
|15
|15
|15
|15
|-
|-
|[[1114zpi]]
|[[1114zpi (σ = 1)]]
|170.995891689006
|170.995049914
|7.01771246166817
|7.01774700849
|11.076998
|3.82285
|1.652856
|19.091741
|[[171edo]]
|[[171edo]]
|1200.02883094526
|1200.03473845
|14
|14
|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
|}
|}


Line 426: Line 550:


{|class="wikitable sortable"
{|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="3"|Tuning
!colspan="3"|Strength
!colspan="1"|Strength
!colspan="2"|Closest EDT
!colspan="2"|Closest EDT
!colspan="2"|No-2 Integer limit
!colspan="2"|No-2 Integer limit
|-
|-
!No-2 ZPI analog
!No-2 ZPI (σ = 1)
!Steps per octave
!Steps per octave
!Cents
!Cents
!Height
!Height
!Integral
!Gap
!EDT
!EDT
!Tritave
!Tritave
Line 442: Line 565:
!Distinct
!Distinct
|-
|-
|[[no-2 93zpi analog]]
|[[no-2 93zpi (σ = 1)]]
|24.5738316304204
|24.5747239922
|48.8324335434323
|48.8306603314
|4.665720
|2.12985
|0.766618
|13.261693
|[[39edt]]
|[[39edt]]
|1904.46490819386
|1904.39575293
|15
|15
|15
|15
|-
|-
|[[no-2 151zpi analog]]
|[[no-2 151zpi (σ = 1)]]
|35.3059427335609
|35.3061077059
|33.9886123153798
|33.9884534992
|4.738265
|2.08576
|0.709543
|13.081926
|[[56edt]]
|[[56edt]]
|1903.36228966127
|1903.35339595
|15
|15
|15
|15
|-
|-
|[[no-2 199zpi analog]]
|[[no-2 207zpi (σ = 1)]]
|43.5176229677494
|44.8164999984
|27.5750355411028
|26.7758526445
|4.824506
|2.10342
|0.678480
|12.871286
|[[69edt]]
|1902.67745233609
|9
|9
|-
|[[no-2 207zpi analog]]
|44.8152489207676
|26.7766001282638
|4.819120
|0.732965
|14.719415
|[[71edt]]
|[[71edt]]
|1901.13860910673
|1901.08553776
|17
|17
|17
|17
|-
|-
|[[no-2 222zpi analog]]
|[[no-2 222zpi (σ = 1)]]
|47.3521317910583
|47.3516876312
|25.3420480686067
|25.3422857776
|5.059485
|2.11876
|0.721113
|13.412098
|[[75edt]]
|[[75edt]]
|1900.65360514550
|1900.67143332
|15
|15
|15
|15
|-
|-
|[[no-2 233zpi analog]]
|[[no-2 233zpi (σ = 1)]]
|49.1685275266548
|49.1657210129
|24.4058559481869
|24.4072491012
|4.790248
|2.07714
|0.736865
|15.624024
|[[78edt]]
|[[78edt]]
|1903.65676395858
|1903.76542989
|21
|21
|21
|21
|-
|-
|[[no-2 249zpi analog]]
|[[no-2 273zpi (σ = 1)]]
|51.6860577447882
|55.5359583782
|23.2170928168922
|21.6076220712
|4.848916
|2.19450
|0.664134
|13.043858
|[[82edt]]
|1903.80161098516
|17
|17
|-
|[[no-2 273zpi analog]]
|55.5353711835277
|21.6078505360910
|5.441186
|0.771944
|14.061502
|[[88edt]]
|[[88edt]]
|1901.49084717601
|1901.47074227
|11
|11
|11
|11
|-
|-
|[[no-2 317zpi analog]]
|[[no-2 363zpi (σ = 1)]]
|62.4092182976906
|69.4191721809
|19.2279287055965
|17.2862908372
|5.154539
|2.08043
|0.705887
|[[110edt]]
|14.235540
|1901.49199210
|[[99edt]]
|23
|1903.56494185405
|25
|23
|23
|-
|-
|[[no-2 326zpi analog]]
|[[no-2 380zpi (σ = 1)]]
|63.7619933650274
|71.9200195089
|18.8199887843874
|16.6852012582
|4.961196
|2.07565
|0.662970
|[[114edt]]
|13.437518
|1902.11294344
|[[101edt]]
|1900.81886722313
|9
|9
|-
|[[no-2 342zpi analog]]
|66.2581615380500
|18.1109764011620
|5.073625
|0.677884
|13.529076
|[[105edt]]
|1901.65252212201
|17
|17
|17
|17
|-
|-
|[[no-2 363zpi analog]]
|[[no-2 453zpi (σ = 1)]]
|69.4221749409126
|82.6700405439
|17.2855431426825
|14.5155366092
|5.247825
|2.38406
|0.705262
|14.276498
|[[110edt]]
|1901.40974569508
|23
|23
|-
|[[no-2 453zpi analog]]
|82.6705208991009
|14.5154522670130
|6.410342
|0.925687
|16.646686
|[[131edt]]
|[[131edt]]
|1901.52424697870
|1901.53529581
|27
|27
|27
|27
|-
|-
|[[no-2 492zpi analog]]
|[[no-2 492zpi (σ = 1)]]
|88.3242305963095
|88.3238806401
|13.5863057271867
|13.5863595587
|5.480169
|2.12238
|0.696272
|13.636687
|[[140edt]]
|[[140edt]]
|1902.08280180614
|1902.09033822
|9
|9
|9
|9
|-
|-
|[[no-2 510zpi analog]]
|[[no-2 510zpi (σ = 1)]]
|90.8297848520406
|90.8334979880
|13.2115252937654
|13.2109852266
|5.712975
|2.23067
|0.810755
|16.378662
|[[144edt]]
|[[144edt]]
|1902.45964230221
|1902.38187263
|39
|39
|27
|27
|-
|-
|[[no-2 550zpi analog]]
|[[no-2 550zpi (σ = 1)]]
|96.5193707902430
|96.5187261015
|12.4327374927449
|12.4328205362
|6.047703
|2.24293
|0.795582
|14.790729
|[[153edt]]
|[[153edt]]
|1902.20883638997
|1902.22154203
|15
|15
|15
|15
|-
|-
|[[no-2 627zpi analog]]
|[[no-2 627zpi (σ = 1)]]
|107.244707551072
|107.244021785
|11.1893633485693
|11.1894348983
|6.217266
|2.29774
|0.828658
|15.375247
|[[170edt]]
|[[170edt]]
|1902.19176925679
|1902.20393272
|15
|15
|15
|15
|-
|-
|[[no-2 687zpi analog]]
|[[no-2 687zpi (σ = 1)]]
|115.410497106759
|115.412802617
|10.3976677172610
|10.3974600113
|5.985004
|2.18983
|0.754232
|14.631506
|[[183edt]]
|[[183edt]]
|1902.77319225877
|1902.73518207
|15
|15
|15
|15
|-
|-
|[[no-2 697zpi analog]]
|[[no-2 697zpi (σ = 1)]]
|116.733331758968
|116.734850378
|10.2798402300191
|10.2797064983
|5.835644
|2.15793
|0.746180
|15.041001
|[[185edt]]
|[[185edt]]
|1901.77044255353
|1901.74570218
|29
|29
|29
|29
|-
|-
|[[no-2 777zpi analog]]
|[[no-2 777zpi (σ = 1)]]
|127.487421022497
|127.486291223
|9.41269334947362
|9.41277676594
|6.134922
|2.21095
|0.758067
|14.474624
|[[202edt]]
|[[202edt]]
|1901.36405659367
|1901.38090672
|17
|17
|17
|17
|-
|-
|[[no-2 810zpi analog]]
|[[no-2 810zpi (σ = 1)]]
|131.820548689719
|131.822840677
|9.10328482112888
|9.10312654342
|6.140639
|2.25360
|0.820704
|16.484428
|[[209edt]]
|[[209edt]]
|1902.58652761594
|1902.55344758
|21
|21
|21
|21
|-
|-
|[[no-2 829zpi analog]]
|[[no-2 829zpi (σ = 1)]]
|134.375301622234
|134.373782790
|8.93021251311149
|8.93031345169
|5.870928
|2.13475
|0.707721
|14.252150
|[[213edt]]
|[[213edt]]
|1902.13526529275
|1902.15676521
|29
|29
|29
|29
|-
|-
|[[no-2 839zpi analog]]
|[[no-2 839zpi (σ = 1)]]
|135.657235331861
|135.657892938
|8.84582379306507
|8.84578091263
|5.733350
|2.11125
|0.672634
|13.637550
|[[215edt]]
|[[215edt]]
|1901.85211550899
|1901.84289622
|15
|15
|15
|15
|-
|-
|[[no-2 858zpi analog]]
|[[no-2 858zpi (σ = 1)]]
|138.196733558228
|138.196070465
|8.68327325185579
|8.68331491602
|5.998270
|2.20051
|0.762777
|15.383590
|[[219edt]]
|[[219edt]]
|1901.63684215642
|1901.64596661
|11
|11
|-
|[[no-2 902zpi (σ = 1)]]
|143.873905513
|8.34063686336
|2.09948
|[[228edt]]
|1901.66520485
|11
|11
|11
|11
|-
|-
|[[no-2 985zpi analog]]
|[[no-2 965zpi (σ = 1)]]
|154.604938100947
|152.075713777
|7.76171844664157
|7.89080629768
|7.104335
|2.10893
|0.924588
|[[241edt]]
|16.674411
|1901.68431774
|15
|15
|-
|[[no-2 985zpi (σ = 1)]]
|154.604034485
|7.76176381166
|2.40811
|[[245edt]]
|[[245edt]]
|1901.62101942718
|1901.63213386
|21
|21
|21
|21
|}
=== Notable Local Maxima of the Riemann Zeta Function after removing the prime 3 from the zeta product ===
{|class="wikitable sortable"
!colspan="3"|Tuning
!colspan="3"|Strength
!colspan="2"|Closest EDO
!colspan="2"|No-3 Integer limit
|-
|-
!No-3 ZPI analog
|[[no-2 1029zpi (σ = 1)]]
!Steps per octave
|160.260260060
!Cents
|7.48782012177
!Height
|2.17192
!Integral
|[[254edt]]
!Gap
|1901.90631093
!EDO
|9
!Octave
|9
!Consistent
!Distinct
|-
|-
|[[no-3 51zpi analog]]
|[[no-2 1049zpi (σ = 1)]]
|15.9698898591818
|162.750022676
|75.1414073973756
|7.37327086209
|5.367776
|2.14738
|0.953376
|[[258edt]]
|13.070433
|1902.30388242
|[[16edo]]
|17
|1202.26251835801
|17
|26
|8
|-
|-
|[[no-3 75zpi analog]]
|[[no-2 1069zpi (σ = 1)]]
|21.0437746046821
|165.332187903
|57.0239903507143
|7.25811480039
|5.752828
|2.19607
|0.956754
|[[262edt]]
|12.853639
|1901.62607770
|[[21edo]]
|17
|1197.50379736500
|17
|17
|10
|-
|-
|[[no-3 95zpi analog]]
|[[no-2 1134zpi (σ = 1)]]
|24.9596545948521
|173.506549648
|48.0775883912872
|6.91616542681
|6.060198
|2.26764
|0.954994
|[[275edt]]
|12.605015
|1901.94549237
|[[25edo]]
|29
|1201.93970978218
|29
|14
|11
|-
|[[no-3 111zpi analog]]
|28.0369867749215
|42.8006051304121
|5.701943
|0.838390
|11.937782
|[[28edo]]
|1198.41694365154
|16
|8
|-
|-
|[[no-3 149zpi analog]]
|[[no-2 1159zpi (σ = 1)]]
|34.9357059709719
|176.625850825
|34.3488121006365
|6.79402247404
|6.001080
|2.14379
|0.875916
|[[280edt]]
|12.775820
|1902.32629273
|[[35edo]]
|1202.20842352228
|14
|11
|-
|[[no-3 161zpi analog]]
|37.0117501336435
|32.4221360964286
|7.215934
|1.160421
|15.095854
|[[37edo]]
|1199.61903556786
|22
|16
|-
|[[no-3 196zpi analog]]
|43.0546167485686
|27.8715754690789
|6.495142
|1.018487
|15.545919
|[[43edo]]
|1198.47774517039
|22
|19
|-
|[[no-3 220zpi analog]]
|47.0058691719873
|25.5287269683150
|6.758393
|0.939366
|13.012654
|[[47edo]]
|1199.85016751081
|10
|10
|-
|[[no-3 251zpi analog]]
|52.0433965143593
|23.0576803277801
|6.442846
|0.856289
|12.619985
|[[52edo]]
|1198.99937704456
|11
|11
|11
|11
|-
|-
|[[no-3 276zpi analog]]
|[[no-2 1179zpi (σ = 1)]]
|55.9872265526305
|179.167803205
|21.4334603424577
|6.69763193238
|6.932381
|2.29964
|1.003267
|[[284edt]]
|14.804703
|1902.12746880
|[[56edo]]
|15
|1200.27377917763
|15
|20
|19
|-
|-
|[[no-3 340zpi analog]]
|[[no-2 1245zpi (σ = 1)]]
|65.9172827630736
|187.354933401
|18.2046338941664
|6.40495544056
|7.029648
|2.28021
|0.948492
|[[297edt]]
|13.998526
|1902.27176585
|[[66edo]]
|21
|1201.50583701498
|21
|16
|16
|-
|-
|[[no-3 394zpi analog]]
|[[no-2 1266zpi (σ = 1)]]
|74.0597618189548
|189.909845446
|16.2031306950932
|6.31878772364
|7.464214
|2.17116
|1.007842
|[[301edt]]
|14.386154
|1901.95510482
|[[74edo]]
|17
|1199.03167143690
|16
|16
|-
|[[no-3 421zpi analog]]
|78.0110209886063
|15.3824419267024
|7.592394
|1.008960
|14.204322
|[[78edo]]
|1199.83047028279
|17
|17
|16
|-
|-
|[[no-3 525zpi analog]]
|[[no-2 1297zpi (σ = 1)]]
|93.0076810773635
|193.736743714
|12.9021601882735
|6.19397217583
|8.466134
|2.12380
|1.133255
|[[307edt]]
|15.018535
|1901.54945798
|[[93edo]]
|21
|1199.90089750944
|21
|35
|19
|-
|-
|[[no-3 640zpi analog]]
|[[no-2 1343zpi (σ = 1)]]
|108.976082315502
|199.415414525
|11.0115905665045
|6.01758897555
|8.633826
|2.36503
|1.182085
|[[316edt]]
|16.319873
|1901.55811627
|[[109edo]]
|39
|1200.26337174899
|39
|16
|16
|-
|[[no-3 751zpi analog]]
|124.014367753602
|9.67629817203298
|9.498846
|1.276085
|16.564895
|[[124edo]]
|1199.86097333209
|28
|26
|}
|}


Line 917: Line 866:


{|class="wikitable sortable"
{|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="3"|Tuning
!colspan="3"|Strength
!colspan="1"|Strength
!colspan="2"|Closest ED5
!colspan="2"|Closest ED5
!colspan="2"|No-2 No-3 Integer limit
!colspan="2"|No-2 No-3 Integer limit
Line 926: Line 876:
!Cents
!Cents
!Height
!Height
!Integral
!Gap
!ED5
!ED5
!Pentave
!Pentave
Line 933: Line 881:
!Distinct
!Distinct
|-
|-
|[[no-2 no-3 55zpi analog]]
|[[no-2 no-3 186zpi (σ = 1)]]
|16.7630030425585
|41.3464998527
|71.5862185882446
|29.0230129340
|3.480299
|1.75534
|0.477759
|9.649416
|[[39ed5]]
|2791.86252494154
|13
|13
|-
|[[no-2 no-3 125zpi analog]]
|30.5974484926723
|39.2189564527704
|3.769318
|0.448541
|9.828199
|[[71ed5]]
|2784.54590814670
|19
|19
|-
|[[no-2 no-3 176zpi analog]]
|39.5828667040955
|30.3161468564337
|3.603524
|0.421674
|10.452207
|[[92ed5]]
|2789.08551079190
|11
|11
|-
|[[no-2 no-3 186zpi analog]]
|41.3477989230936
|29.0221010852836
|4.469823
|0.556068
|11.567493
|[[96ed5]]
|[[96ed5]]
|2786.12170418722
|2786.20924167
|35
|35
|23
|23
|-
|-
|[[no-2 no-3 212zpi analog]]
|[[no-2 no-3 565zpi (σ = 1)]]
|45.6783815054539
|98.6253027359
|26.2706330752267
|12.1672630320
|3.818225
|1.74188
|0.433470
|10.611042
|[[106ed5]]
|2784.68710597403
|13
|13
|-
|[[no-2 no-3 235zpi analog]]
|49.4631517377883
|24.2604839732289
|3.853032
|0.428042
|10.508697
|[[115ed5]]
|2789.95565692132
|25
|25
|-
|[[no-2 no-3 284zpi analog]]
|57.2705618247184
|20.9531731794898
|3.913350
|0.465932
|11.922515
|[[133ed5]]
|2786.77203287214
|17
|17
|-
|[[no-2 no-3 298zpi analog]]
|59.4923782274424
|20.1706510271339
|4.083075
|0.465782
|11.463643
|[[138ed5]]
|2783.54984174448
|23
|23
|-
|[[no-2 no-3 312zpi analog]]
|61.6047959566046
|19.4790029147292
|4.416896
|0.501431
|11.339301
|[[143ed5]]
|2785.49741680628
|25
|23
|-
|[[no-2 no-3 340zpi analog]]
|65.8904943328257
|18.2120351676004
|4.092923
|0.526694
|13.998526
|[[153ed5]]
|2786.44138064287
|13
|13
|-
|[[no-2 no-3 368zpi analog]]
|70.2158409653819
|17.0901606176251
|4.382540
|0.518334
|12.481351
|[[163ed5]]
|2785.69618067290
|19
|19
|-
|[[no-2 no-3 423zpi analog]]
|78.3601842342727
|15.3138996765548
|4.270381
|0.502072
|12.963711
|[[182ed5]]
|2787.12974113297
|19
|19
|-
|[[no-2 no-3 438zpi analog]]
|80.4944089071946
|14.9078677176639
|4.243838
|0.450422
|11.371118
|[[187ed5]]
|2787.77126320314
|7
|7
|-
|[[no-2 no-3 465zpi analog]]
|84.4075187897342
|14.2167429774745
|4.301350
|0.486089
|12.332303
|[[196ed5]]
|2786.48162358500
|17
|17
|-
|[[no-2 no-3 477zpi analog]]
|86.1814871554687
|13.9241041157161
|4.459348
|0.505570
|12.446285
|[[200ed5]]
|2784.82082314323
|25
|25
|-
|[[no-2 no-3 565zpi analog]]
|98.6257548378926
|12.1672072570942
|4.883729
|0.545550
|12.639964
|[[229ed5]]
|[[229ed5]]
|2786.29046187457
|2786.30323433
|29
|29
|29
|29
|-
|-
|[[no-2 no-3 581zpi analog]]
|[[no-2 no-3 671zpi (σ = 1)]]
|100.797128599965
|113.258011095
|11.9051010347969
|10.5952769998
|4.579796
|1.77217
|0.536282
|13.693791
|[[234ed5]]
|2785.79364214247
|25
|25
|-
|[[no-2 no-3 671zpi analog]]
|113.256639862217
|10.5954052800778
|5.104294
|0.563708
|12.937931
|[[263ed5]]
|[[263ed5]]
|2786.59158866045
|2786.55785095
|19
|19
|19
|19
|-
|-
|[[no-2 no-3 764zpi analog]]
|[[no-2 no-3 764zpi (σ = 1)]]
|125.745930952370
|125.745000550
|9.54305233506547
|9.54312294522
|5.001815
|1.75634
|0.548008
|12.976730
|[[292ed5]]
|[[292ed5]]
|2786.57128183912
|2786.59190001
|37
|37
|37
|37
|-
|-
|[[no-2 no-3 905zpi analog]]
|[[no-2 no-3 905zpi (σ = 1)]]
|144.300058486204
|144.297529480
|8.31600494545005
|8.31615069448
|5.030210
|1.73926
|0.539592
|13.254432
|[[335ed5]]
|[[335ed5]]
|2785.86165672577
|2785.91048265
|43
|43
|41
|41
|-
|-
|[[no-2 no-3 938zpi analog]]
|[[no-2 no-3 938zpi (σ = 1)]]
|148.561761173834
|148.562870929
|8.07744866861039
|8.07738833059
|5.510552
|1.79949
|0.600083
|13.846076
|[[345ed5]]
|[[345ed5]]
|2786.71979067058
|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
|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
|}
|}


Latest revision as of 18:31, 14 December 2025

Equal-step tunings

About this list

The table that follows is not a “best-of” roster but a modest snapshot of equal-step tunings that happen to score highly under a few specific mathematical lenses. In particular, it gathers:

  • Prominent peak counts from the classic Riemann zeta function
  • Prominent peaks after removing the prime 2 from the zeta product
  • Prominent peaks after removing the prime 3
  • Prominent peaks after simultaneously removing the primes 2 and 3
  • The α–β–γ family, with an equave sliding from 3/1 down to 4/3

These tunings earn the label “optimized” only relative to the limited set of zeta-derived functions explored here. When you layer many differently pruned zeta functions in a tool such as Wolfram Mathematica, striking peaks emerge almost everywhere; the peaks simply shift as each combination of omitted primes reshapes the landscape. That ubiquity means there is no absolute “good” or “bad” equal-step tuning, only different alignments of primes that reveal different musical affordances.

Consequently, the list below is inherently biased toward a handful of functions and can only hint at the boundless diversity of xenharmonic equal-step systems. Treat it as a useful starting palette, not a definitive canon.

Notable Local Maxima of the Riemann Zeta Function

Tuning Strength Closest EDO Integer limit
ZPI (σ = 1) Steps per octave Step size (cents) Height EDO Octave (cents) Consistent Distinct
15zpi (σ = 1) 6.95688550773 172.490980147 2.55384 7edo 1207.43686103 6 5
26zpi (σ = 1) 10.0089746115 119.892401228 2.57426 10edo 1198.92401228 8 5
34zpi (σ = 1) 12.0220488259 99.8165967700 2.85866 12edo 1197.79916124 10 6
42zpi (σ = 1) 13.9020220557 86.3183783764 2.50514 14edo 1208.45729727 7 5
47zpi (σ = 1) 15.0534708836 79.7158349246 2.69313 15edo 1195.73752387 8 7
56zpi (σ = 1) 17.0432556931 70.4090827252 2.65741 17edo 1196.95440633 4 4
65zpi (σ = 1) 18.9489976130 63.3278880767 3.02387 19edo 1203.22987346 10 7
80zpi (σ = 1) 22.0251749360 54.4831086920 2.99601 22edo 1198.62839122 12 8
90zpi (σ = 1) 24.0053572889 49.9888414723 2.82476 24edo 1199.73219533 6 6
100zpi (σ = 1) 25.9356337472 46.2683893402 2.71167 26edo 1202.97812285 14 9
106zpi (σ = 1) 27.0853383248 44.3044124320 2.90524 27edo 1196.21913566 10 8
116zpi (σ = 1) 28.9431579907 41.4605759463 2.68561 29edo 1202.35670244 8 7
127zpi (σ = 1) 30.9779815456 38.7371913897 3.23190 31edo 1200.85293308 12 9
144zpi (σ = 1) 34.0437506778 35.2487600839 3.07414 34edo 1198.45784285 6 6
155zpi (σ = 1) 35.9827898689 33.3492762616 2.80355 36edo 1200.57394542 8 8
184zpi (σ = 1) 40.9880790756 29.2768050385 3.32966 41edo 1200.34900658 16 10
196zpi (σ = 1) 43.0234004818 27.8917981043 2.78019 43edo 1199.34731849 8 8
214zpi (σ = 1) 46.0106419996 26.0809227572 3.25119 46edo 1199.72244683 14 11
238zpi (σ = 1) 49.9382924730 24.0296562132 2.90274 50edo 1201.48281066 10 9
257zpi (σ = 1) 52.9969882711 22.6427961125 3.46399 53edo 1200.06819396 10 10
289zpi (σ = 1) 58.0645692462 20.6666477609 3.25823 58edo 1198.66557013 16 12
301zpi (σ = 1) 59.9223835273 20.0259056693 2.98826 60edo 1201.55434016 10 10
321zpi (σ = 1) 63.0197888699 19.0416378969 2.87513 63edo 1199.62318750 8 8
334zpi (σ = 1) 65.0145858034 18.4573966776 3.23462 65edo 1199.73078404 6 6
354zpi (σ = 1) 68.0496579343 17.6341812204 3.14200 68edo 1199.12432299 10 10
380zpi (σ = 1) 71.9512656175 16.6779554147 3.61665 72edo 1200.81278986 18 13
414zpi (σ = 1) 76.9924672555 15.5859403235 3.28825 77edo 1200.11740491 10 10
435zpi (σ = 1) 80.0733926855 14.9862514845 3.14833 80edo 1198.90011876 12 12
462zpi (σ = 1) 83.9950884037 14.2865496400 3.19687 84edo 1200.07016976 10 10
483zpi (σ = 1) 87.0139579095 13.7908908965 3.44872 87edo 1199.80750799 16 14
497zpi (σ = 1) 89.0215260329 13.4798857476 3.02681 89edo 1199.70983154 12 12
532zpi (σ = 1) 93.9843698073 12.7680805059 3.39762 94edo 1200.19956756 24 15
568zpi (σ = 1) 99.0456175574 12.1156294402 3.56676 99edo 1199.44731458 12 12
596zpi (σ = 1) 102.936325452 11.6576922163 3.25007 103edo 1200.74229828 15 15
655zpi (σ = 1) 111.058159333 10.8051493669 3.39509 111edo 1199.37157972 22 16
706zpi (σ = 1) 117.971388652 10.1719579104 3.62695 118edo 1200.29103343 12 12
796zpi (σ = 1) 130.004267285 9.23046623824 3.72487 130edo 1199.96061097 16 16
872zpi (σ = 1) 139.992781938 8.57187051639 3.60746 140edo 1200.06187229 10 10
965zpi (σ = 1) 152.050659206 7.89210652729 3.68901 152edo 1199.60019215 15 15
1114zpi (σ = 1) 170.995049914 7.01774700849 3.82285 171edo 1200.03473845 14 14
1210zpi (σ = 1) 183.000273182 6.55736726036 3.76064 183edo 1199.99820865 18 18

Notable Local Maxima of the Riemann Zeta Function after removing the prime 3 from the zeta product

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.5 and cents ≥ 40.0) or (height ≥ 2.6 and cents ≥ 15.0) or (height ≥ 2.8 and cents ≥ 12.0) or (height ≥ 3.1 and cents ≥ 6.0)
Tuning Strength Closest EDO No-3 Integer limit
No-3 ZPI analog Steps per octave Cents Height EDO Octave Consistent Distinct
no-3 51zpi (σ = 1) 15.9687074547 75.1469712502 2.56677 16edo 1202.35154000 26 8
no-3 75zpi (σ = 1) 21.0417134383 57.0295762045 2.60042 21edo 1197.62110029 17 10
no-3 95zpi (σ = 1) 24.9617781085 48.0734984016 2.64675 25edo 1201.83746004 14 11
no-3 127zpi (σ = 1) 31.0146799866 38.6913552073 2.60405 31edo 1199.43201143 11 11
no-3 161zpi (σ = 1) 37.0135086000 32.4205957606 2.92705 37edo 1199.56204314 22 16
no-3 196zpi (σ = 1) 43.0494972034 27.8748900209 2.71380 43edo 1198.62027090 22 19
no-3 220zpi (σ = 1) 47.0043385196 25.5295582875 2.69328 47edo 1199.88923951 10 10
no-3 276zpi (σ = 1) 55.9891415481 21.4327272543 2.76321 56edo 1200.23272624 20 19
no-3 340zpi (σ = 1) 65.9204029312 18.2037722259 2.65263 66edo 1201.44896691 16 16
no-3 354zpi (σ = 1) 68.0229453080 17.6411061674 2.76285 68edo 1199.59521939 11 11
no-3 394zpi (σ = 1) 74.0566473758 16.2038121158 2.76672 74edo 1199.08209657 16 16
no-3 421zpi (σ = 1) 78.0097604150 15.3826904943 2.81219 78edo 1199.84985856 17 16
no-3 525zpi (σ = 1) 93.0066513531 12.9023030347 2.97919 93edo 1199.91418223 35 19
no-3 751zpi (σ = 1) 124.013627761 9.67635591079 3.13747 124edo 1199.86813294 28 26

Notable Local Maxima of the Riemann Zeta Function after removing the prime 2 from the zeta product

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.075 and cents ≥ 6.0)
Tuning Strength Closest EDT No-2 Integer limit
No-2 ZPI (σ = 1) Steps per octave Cents Height EDT Tritave Consistent Distinct
no-2 93zpi (σ = 1) 24.5747239922 48.8306603314 2.12985 39edt 1904.39575293 15 15
no-2 151zpi (σ = 1) 35.3061077059 33.9884534992 2.08576 56edt 1903.35339595 15 15
no-2 207zpi (σ = 1) 44.8164999984 26.7758526445 2.10342 71edt 1901.08553776 17 17
no-2 222zpi (σ = 1) 47.3516876312 25.3422857776 2.11876 75edt 1900.67143332 15 15
no-2 233zpi (σ = 1) 49.1657210129 24.4072491012 2.07714 78edt 1903.76542989 21 21
no-2 273zpi (σ = 1) 55.5359583782 21.6076220712 2.19450 88edt 1901.47074227 11 11
no-2 363zpi (σ = 1) 69.4191721809 17.2862908372 2.08043 110edt 1901.49199210 23 23
no-2 380zpi (σ = 1) 71.9200195089 16.6852012582 2.07565 114edt 1902.11294344 17 17
no-2 453zpi (σ = 1) 82.6700405439 14.5155366092 2.38406 131edt 1901.53529581 27 27
no-2 492zpi (σ = 1) 88.3238806401 13.5863595587 2.12238 140edt 1902.09033822 9 9
no-2 510zpi (σ = 1) 90.8334979880 13.2109852266 2.23067 144edt 1902.38187263 39 27
no-2 550zpi (σ = 1) 96.5187261015 12.4328205362 2.24293 153edt 1902.22154203 15 15
no-2 627zpi (σ = 1) 107.244021785 11.1894348983 2.29774 170edt 1902.20393272 15 15
no-2 687zpi (σ = 1) 115.412802617 10.3974600113 2.18983 183edt 1902.73518207 15 15
no-2 697zpi (σ = 1) 116.734850378 10.2797064983 2.15793 185edt 1901.74570218 29 29
no-2 777zpi (σ = 1) 127.486291223 9.41277676594 2.21095 202edt 1901.38090672 17 17
no-2 810zpi (σ = 1) 131.822840677 9.10312654342 2.25360 209edt 1902.55344758 21 21
no-2 829zpi (σ = 1) 134.373782790 8.93031345169 2.13475 213edt 1902.15676521 29 29
no-2 839zpi (σ = 1) 135.657892938 8.84578091263 2.11125 215edt 1901.84289622 15 15
no-2 858zpi (σ = 1) 138.196070465 8.68331491602 2.20051 219edt 1901.64596661 11 11
no-2 902zpi (σ = 1) 143.873905513 8.34063686336 2.09948 228edt 1901.66520485 11 11
no-2 965zpi (σ = 1) 152.075713777 7.89080629768 2.10893 241edt 1901.68431774 15 15
no-2 985zpi (σ = 1) 154.604034485 7.76176381166 2.40811 245edt 1901.63213386 21 21
no-2 1029zpi (σ = 1) 160.260260060 7.48782012177 2.17192 254edt 1901.90631093 9 9
no-2 1049zpi (σ = 1) 162.750022676 7.37327086209 2.14738 258edt 1902.30388242 17 17
no-2 1069zpi (σ = 1) 165.332187903 7.25811480039 2.19607 262edt 1901.62607770 17 17
no-2 1134zpi (σ = 1) 173.506549648 6.91616542681 2.26764 275edt 1901.94549237 29 29
no-2 1159zpi (σ = 1) 176.625850825 6.79402247404 2.14379 280edt 1902.32629273 11 11
no-2 1179zpi (σ = 1) 179.167803205 6.69763193238 2.29964 284edt 1902.12746880 15 15
no-2 1245zpi (σ = 1) 187.354933401 6.40495544056 2.28021 297edt 1902.27176585 21 21
no-2 1266zpi (σ = 1) 189.909845446 6.31878772364 2.17116 301edt 1901.95510482 17 17
no-2 1297zpi (σ = 1) 193.736743714 6.19397217583 2.12380 307edt 1901.54945798 21 21
no-2 1343zpi (σ = 1) 199.415414525 6.01758897555 2.36503 316edt 1901.55811627 39 39

Notable Local Maxima of the Riemann Zeta Function after removing the primes 2 and 3 from the zeta product

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 1.725 and cents ≥ 6.0)
Tuning Strength Closest ED5 No-2 No-3 Integer limit
No-2 No-3 ZPI analog Steps per octave Cents Height ED5 Pentave Consistent Distinct
no-2 no-3 186zpi (σ = 1) 41.3464998527 29.0230129340 1.75534 96ed5 2786.20924167 35 23
no-2 no-3 565zpi (σ = 1) 98.6253027359 12.1672630320 1.74188 229ed5 2786.30323433 29 29
no-2 no-3 671zpi (σ = 1) 113.258011095 10.5952769998 1.77217 263ed5 2786.55785095 19 19
no-2 no-3 764zpi (σ = 1) 125.745000550 9.54312294522 1.75634 292ed5 2786.59190001 37 37
no-2 no-3 905zpi (σ = 1) 144.297529480 8.31615069448 1.73926 335ed5 2785.91048265 43 41
no-2 no-3 938zpi (σ = 1) 148.562870929 8.07738833059 1.79949 345ed5 2786.69897405 25 25
no-2 no-3 1046zpi (σ = 1) 162.414291729 7.38851234841 1.73251 377ed5 2785.46915535 23 23
no-2 no-3 1145zpi (σ = 1) 174.880594782 6.86182478678 1.74084 406ed5 2785.90086343 25 25
no-2 no-3 1196zpi (σ = 1) 181.292147244 6.61915046096 1.77770 421ed5 2786.66234406 35 35
no-2 no-3 1280zpi (σ = 1) 191.632570168 6.26198353937 1.75036 445ed5 2786.58267502 29 29

The α–β–γ family

α–β–γ family
Optimization Equal division of a ratio
Proposed name Steps per octave Cents Optimization method
Alpha 3/1 1.90739592696007 629.130000247254 Dave Benson 3ed3/1
Beta 3/1 3.14186231690763 381.939079106782 Dave Benson 5ed3/1
Alpha 2/1 5.00991270509077 239.525131601721 Dave Benson 5ed2/1
Gamma 3/1 5.04255621376059 237.974540913462 Dave Benson 8ed3/1
Beta 2/1 6.99104980248710 171.648040552235 Dave Benson 7ed2/1
Alpha 5/3 9.50583353877785 126.238272015258 Dave Benson 7ed5/3
Gamma 2/1 11.9978480914311 100.017935787756 Dave Benson 12ed2/1
Beta 5/3 12.2053823008782 98.3172808862904 Dave Benson 9ed5/3
Alpha 3/2 15.3915238996928 77.9649895501219 Dave Benson 9ed3/2
Beta 3/2 18.7990736394111 63.8329325698408 Dave Benson 11ed3/2
Gamma 5/3 21.7094399215509 55.2754932571412 Dave Benson 16ed5/3
Alpha 7/5 22.6653911133366 52.9441558718088 Dave Benson 11ed7/5
Beta 7/5 26.7758951088566 44.8164289231577 Dave Benson 13ed7/5
Alpha 4/3 31.3266790320926 38.3060074376432 Dave Benson 13ed4/3
Gamma 3/2 34.1894540921914 35.0985422804417 Dave Benson 20ed3/2
Beta 4/3 36.1372975038827 33.2066890135065 Dave Benson 15ed4/3
Gamma 7/5 49.4404896216012 24.2716042900130 Dave Benson 24ed7/5
Gamma 4/3 67.4633901646646 17.7874251067289 Dave Benson 28ed4/3

Unequal-step tunings

Unequal-step tunings from equal divisions of a ratio

Tuning Period Mode Why it matters
Stretched hemififth 94\93<2/1> 16 11 16 12 16 11 12
833 Cent Acoustic Golden Scale [11] 25\36<2/1> 3 1 3 3 1 3 1 3 3 1 3
833 Cent Logarithmic Golden Scale [8] ϕ ϕ 1 ϕ ϕ 1 ϕ 1 ϕ
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