User:Contribution/Collection of tunings: Difference between revisions

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{{todo|use sigma 1.0|inline=1|comment=instead of sigma 1/2}}
{{todo|use sigma 1.0|inline=1|comment=instead of sigma 1/2}}
<s>
 
=== 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%;" | Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.5 and cents ≥ 40.0) or (height ≥ 2.8 and cents ≥ 12.0)
|- style="white-space: nowrap;"
|-
! colspan="3" |Tuning
!colspan="3"|Tuning
! colspan="3" |Strength
!colspan="1"|Strength
! colspan="2" |Closest EDO
!colspan="2"|Closest EDO
! colspan="2" |Integer limit
!colspan="2"|Integer limit
|-
|-
!ZPI
!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]]
|[[214zpi (σ = 1)]]
|46.0089748051542
|46.0106419996
|26.0818678330031
|26.0809227572
|7.495674
|3.25119
|1.356067
|17.747832
|[[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
|[[94edo]]
|1200.20842631208
|24
|15
|-
|[[568zpi]]
|99.0473345956631
|12.1154194093028
|9.406495
|1.510412
|18.536483
|[[99edo]]
|1199.42652152097
|12
|12
|12
|12
|-
|-
|[[596zpi]]
|[[532zpi (σ = 1)]]
|102.936629522070
|93.9843698073
|11.6576577800491
|12.7680805059
|8.543510
|3.39762
|1.340775
|[[94edo]]
|18.270998
|1200.19956756
|[[103edo]]
|24
|1200.73875134506
|15
|15
|15
|-
|-
|[[655zpi]]
|[[546zpi (σ = 1)]]
|111.059577998833
|95.9558568688
|10.8050113427643
|12.5057504477
|9.038544
|2.93099
|1.394739
|[[96edo]]
|18.041165
|1200.55204298
|[[111edo]]
|6
|1199.35625904684
|6
|22
|16
|-
|-
|[[706zpi]]
|[[568zpi (σ = 1)]]
|117.969513574257
|99.0456175574
|10.1721195895637
|12.1156294402
|9.850823
|3.56676
|1.544280
|[[99edo]]
|18.861062
|1199.44731458
|[[118edo]]
|1200.31011156852
|12
|12
|12
|12
|-
|[[796zpi]]
|130.003910460506
|9.23049157328654
|10.355108
|1.634018
|19.594551
|[[130edo]]
|1199.96390452725
|16
|16
|-
|[[872zpi]]
|139.990541024216
|8.57200773152536
|10.076688
|1.548424
|19.514765
|[[140edo]]
|1200.08108241355
|10
|10
|-
|[[965zpi]]
|152.052848107925
|7.89199291517551
|10.468420
|1.593855
|19.487224
|[[152edo]]
|1199.58292310668
|15
|15
|-
|[[1114zpi]]
|170.995891689006
|7.01771246166817
|11.076998
|1.652856
|19.091741
|[[171edo]]
|1200.02883094526
|14
|14
|}
|}


Line 1,163: Line 1,073:
|25
|25
|}
|}
</s>
 
=== The α–β–γ family ===
=== The α–β–γ family ===
{| class="wikitable sortable"
{| class="wikitable sortable"

Revision as of 18:12, 28 August 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.

Todo: use sigma 1.0

instead of sigma 1/2

Notable Local Maxima of the Riemann Zeta Function

Zeta Peak Indexes at sigma = 1, filtered with (height ≥ 2.5 and cents ≥ 40.0) or (height ≥ 2.8 and cents ≥ 12.0)
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
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
546zpi (σ = 1) 95.9558568688 12.5057504477 2.93099 96edo 1200.55204298 6 6
568zpi (σ = 1) 99.0456175574 12.1156294402 3.56676 99edo 1199.44731458 12 12

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

Tuning Strength Closest EDT No-2 Integer limit
No-2 ZPI analog Steps per octave Cents Height Integral Gap EDT Tritave Consistent Distinct
no-2 93zpi analog 24.5738316304204 48.8324335434323 4.665720 0.766618 13.261693 39edt 1904.46490819386 15 15
no-2 151zpi analog 35.3059427335609 33.9886123153798 4.738265 0.709543 13.081926 56edt 1903.36228966127 15 15
no-2 199zpi analog 43.5176229677494 27.5750355411028 4.824506 0.678480 12.871286 69edt 1902.67745233609 9 9
no-2 207zpi analog 44.8152489207676 26.7766001282638 4.819120 0.732965 14.719415 71edt 1901.13860910673 17 17
no-2 222zpi analog 47.3521317910583 25.3420480686067 5.059485 0.721113 13.412098 75edt 1900.65360514550 15 15
no-2 233zpi analog 49.1685275266548 24.4058559481869 4.790248 0.736865 15.624024 78edt 1903.65676395858 21 21
no-2 249zpi analog 51.6860577447882 23.2170928168922 4.848916 0.664134 13.043858 82edt 1903.80161098516 17 17
no-2 273zpi analog 55.5353711835277 21.6078505360910 5.441186 0.771944 14.061502 88edt 1901.49084717601 11 11
no-2 317zpi analog 62.4092182976906 19.2279287055965 5.154539 0.705887 14.235540 99edt 1903.56494185405 25 23
no-2 326zpi analog 63.7619933650274 18.8199887843874 4.961196 0.662970 13.437518 101edt 1900.81886722313 9 9
no-2 342zpi analog 66.2581615380500 18.1109764011620 5.073625 0.677884 13.529076 105edt 1901.65252212201 17 17
no-2 363zpi analog 69.4221749409126 17.2855431426825 5.247825 0.705262 14.276498 110edt 1901.40974569508 23 23
no-2 453zpi analog 82.6705208991009 14.5154522670130 6.410342 0.925687 16.646686 131edt 1901.52424697870 27 27
no-2 492zpi analog 88.3242305963095 13.5863057271867 5.480169 0.696272 13.636687 140edt 1902.08280180614 9 9
no-2 510zpi analog 90.8297848520406 13.2115252937654 5.712975 0.810755 16.378662 144edt 1902.45964230221 39 27
no-2 550zpi analog 96.5193707902430 12.4327374927449 6.047703 0.795582 14.790729 153edt 1902.20883638997 15 15
no-2 627zpi analog 107.244707551072 11.1893633485693 6.217266 0.828658 15.375247 170edt 1902.19176925679 15 15
no-2 687zpi analog 115.410497106759 10.3976677172610 5.985004 0.754232 14.631506 183edt 1902.77319225877 15 15
no-2 697zpi analog 116.733331758968 10.2798402300191 5.835644 0.746180 15.041001 185edt 1901.77044255353 29 29
no-2 777zpi analog 127.487421022497 9.41269334947362 6.134922 0.758067 14.474624 202edt 1901.36405659367 17 17
no-2 810zpi analog 131.820548689719 9.10328482112888 6.140639 0.820704 16.484428 209edt 1902.58652761594 21 21
no-2 829zpi analog 134.375301622234 8.93021251311149 5.870928 0.707721 14.252150 213edt 1902.13526529275 29 29
no-2 839zpi analog 135.657235331861 8.84582379306507 5.733350 0.672634 13.637550 215edt 1901.85211550899 15 15
no-2 858zpi analog 138.196733558228 8.68327325185579 5.998270 0.762777 15.383590 219edt 1901.63684215642 11 11
no-2 985zpi analog 154.604938100947 7.76171844664157 7.104335 0.924588 16.674411 245edt 1901.62101942718 21 21

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

Tuning Strength Closest EDO No-3 Integer limit
No-3 ZPI analog Steps per octave Cents Height Integral Gap EDO Octave Consistent Distinct
no-3 51zpi analog 15.9698898591818 75.1414073973756 5.367776 0.953376 13.070433 16edo 1202.26251835801 26 8
no-3 75zpi analog 21.0437746046821 57.0239903507143 5.752828 0.956754 12.853639 21edo 1197.50379736500 17 10
no-3 95zpi analog 24.9596545948521 48.0775883912872 6.060198 0.954994 12.605015 25edo 1201.93970978218 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 34.9357059709719 34.3488121006365 6.001080 0.875916 12.775820 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
no-3 276zpi analog 55.9872265526305 21.4334603424577 6.932381 1.003267 14.804703 56edo 1200.27377917763 20 19
no-3 340zpi analog 65.9172827630736 18.2046338941664 7.029648 0.948492 13.998526 66edo 1201.50583701498 16 16
no-3 394zpi analog 74.0597618189548 16.2031306950932 7.464214 1.007842 14.386154 74edo 1199.03167143690 16 16
no-3 421zpi analog 78.0110209886063 15.3824419267024 7.592394 1.008960 14.204322 78edo 1199.83047028279 17 16
no-3 525zpi analog 93.0076810773635 12.9021601882735 8.466134 1.133255 15.018535 93edo 1199.90089750944 35 19
no-3 640zpi analog 108.976082315502 11.0115905665045 8.633826 1.182085 16.319873 109edo 1200.26337174899 16 16
no-3 751zpi analog 124.014367753602 9.67629817203298 9.498846 1.276085 16.564895 124edo 1199.86097333209 28 26

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

Tuning Strength Closest ED5 No-2 No-3 Integer limit
No-2 No-3 ZPI analog Steps per octave Cents Height Integral Gap ED5 Pentave Consistent Distinct
no-2 no-3 55zpi analog 16.7630030425585 71.5862185882446 3.480299 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 2786.12170418722 35 23
no-2 no-3 212zpi analog 45.6783815054539 26.2706330752267 3.818225 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 2786.29046187457 29 29
no-2 no-3 581zpi analog 100.797128599965 11.9051010347969 4.579796 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 2786.59158866045 19 19
no-2 no-3 764zpi analog 125.745930952370 9.54305233506547 5.001815 0.548008 12.976730 292ed5 2786.57128183912 37 37
no-2 no-3 905zpi analog 144.300058486204 8.31600494545005 5.030210 0.539592 13.254432 335ed5 2785.86165672577 43 41
no-2 no-3 938zpi analog 148.561761173834 8.07744866861039 5.510552 0.600083 13.846076 345ed5 2786.71979067058 25 25

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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