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Table of zeta-stretched edos: Difference between revisions

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'''Explanation of what this is''': [[The Riemann Zeta Function and Tuning#Optimal Octave Stretch]].
This table lists tuning instructions for [[EDO|equal divisions of the octave]] which have been stretched or compressed using optimal octave stretch based on zeta peaks, as described here: [[the Riemann zeta function and tuning]].


'''Instructions on how to calculate the second column using the free version of Wolfram Cloud''':
No such table could possibly be complete (as there are so many possible edos), so please add tunings of interest as you see fit.
#Copy-paste '''Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9,12.1}]''' into a cell.
#Change "'''11.9'''" and "'''12.1'''" to whatever values you want, e.g. to view the curve around 15edo you might use the values "'''14.9'''" and "'''15.1'''".
#Ensure that cell is still selected
#In the menu select Evaluation > Evaluate Cells


All of the tunings listed on this page are [[zeta peak index]] tunings, e.g. 1zpi, 2zpi, 3zpi... However, not all zeta peak index tunings are listed here - only those with intervals close to the octave. For a more complete table see: [[zeta peak index]].
== Table ==
This is a list of zeta peak-based octave tunings, which includes all EDOs up to 100 and certain noteworthy ones above 100.
{| class="wikitable sortable"
{| class="wikitable sortable"
|'''Edo'''
|+ style="font-size: 105%;" | Zeta-optimal tunings for selected EDOs
|'''No. of steps per 1200 [[cents]]'''
|'''Step size (cents)'''
|'''[[Octave]] size (cents)'''
|-
|-
|'''[[1edo]]'''
! Tuning !! Associated edo !! No. of steps per 1200 cents !! Step size (cents) !! Tuning of 2/1 (cents) !! Gram point index
|'''1.127'''
|'''1064.774'''
|'''1064.774'''
|-
|-
|'''[[2edo]]'''
| [[1zpi]] || 1edo || 1.127 || 1065.177 || 1065.177 || −1
|'''1.972'''
|'''608.519'''
|'''1217.039'''
|-
|-
|'''[[3edo]]'''
| [[2zpi]] || 2edo || 1.973 || 608.283 || 1216.565 || 0
|'''3.06'''
|'''392.157'''
|'''1176.471'''
|-
|-
|'''[[4edo]]'''
| [[4zpi]] || 3edo || 3.060 || 392.187 || 1176.562 || 2
|'''3.904'''
|'''307.377'''
|'''1229.508'''
|-
|-
|'''[[5edo]]'''
| [[6zpi]] || 4edo || 3.904 || 307.342 || 1229.367 || 4
|'''5.034'''
|'''238.379'''
|'''1191.895'''
|-
|-
|[[6edo]]
| [[9zpi]] || 5edo || 5.034 || 238.357 || 1191.783 || 7
|6.035
|198.840
|1193.041
|-
|-
|'''[[7edo]]'''
| [[12zpi]] || 6edo || 6.035 || 198.843 || 1193.056 || 10
|'''6.957'''
|'''172.488'''
|'''1207.417'''
|-
|-
|[[8edo]]
| [[15zpi]] || 7edo || 6.957 || 172.496 || 1207.471 || 13
|8.137
|147.474
|1179.796
|-
|-
|[[9edo]]
| [[19zpi]] || 8edo || 8.137 || 147.467 || 1179.734 || 17
|8.95
|134.078
|1206.704
|-
|-
|'''[[10edo]]'''
| [[22zpi]] || 9edo || 8.950 || 134.078 || 1206.705 || 20
|'''10.008'''
|'''119.904'''
|'''1199.041'''
|-
|-
|[[11edo]]
| [[26zpi]] || 10edo || 10.008 || 119.899 || 1198.986 || 24
|11.037
|108.725
|1195.977
|-
|-
|'''[[12edo]]'''
| [[30zpi]] || 11edo || 11.037 || 108.722 || 1195.938 || 28
|'''12.023'''
|'''99.809'''
|'''1197.704'''
|-
|-
|[[13edo]]
| [[34zpi]] || 12edo || 12.023 || 99.807 || 1197.686 || 32
|12.969
|92.528
|1202.868
|-
|-
|[[14edo]]
| [[38zpi]] || 13edo || 12.969 || 92.531 || 1202.900 || 36
|13.9
|86.331
|1208.633
|-
|-
|[[15edo]]
| [[42zpi]] || 14edo || 13.900 || 86.329 || 1208.611 || 40
|15.053
|79.718
|1195.775
|-
|-
|[[16edo]]
| [[47zpi]] || 15edo || 15.053 || 79.716 || 1195.736 || 45
|15.945
|75.259
|1204.139
|-
|-
|[[17edo]]
| [[51zpi]] || 16edo || 15.944 || 75.262 || 1204.187 || 49
|17.045
|70.402
|1196.832
|-
|-
|[[18edo]]
| [[56zpi]] || 17edo || 17.045 || 70.404 || 1196.861 || 54
|18.119
|66.229
|1192.119
|-
|-
|'''[[19edo]]'''
| [[61zpi]] || 18edo || 18.119 || 66.228 || 1192.113 || 59
|'''18.948'''
|'''63.331'''
|'''1203.293'''
|-
|-
|[[20edo]]
| [[65zpi]] || 19edo || 18.948 || 63.331 || 1203.288 || 63
|19.982
|60.054
|1201.081
|-
|-
|[[21edo]]
| [[70zpi]] || 20edo || 19.982 || 60.054 || 1201.087 || 68
|21.028
|57.067
|1198.402
|-
|-
|'''[[22edo]]'''
| [[75zpi]] || 21edo || 21.028 || 57.067 || 1198.406 || 73
|'''22.025'''
|'''54.484'''
|'''1198.638'''
|-
|-
|[[23edo]]
| [[80zpi]] || 22edo || 22.025 || 54.483 || 1198.630 || 78
|23.026
|52.115
|1198.645
|-
|-
|[[24edo]]
| [[84zpi]] || 23edo || 22.807 || 52.615 || 1210.148 || 82
|24.006
|49.988
|1199.700
|-
|-
|[[25edo]]
| [[90zpi]] || 24edo || 24.006 || 49.988 || 1199.713 || 88
|24.965
|48.067
|1201.682
|-
|-
|[[26edo]]
| [[95zpi]] || 25edo || 24.965 || 48.067 || 1201.678 || 93
|25.936
|46.268
|1202.961
|-
|-
|'''[[27edo]]'''
| [[100zpi]] || 26edo || 25.936 || 46.268 || 1202.975 || 98
|'''27.087'''
|'''44.302'''
|'''1196.146'''
|-
|-
|[[28edo]]
| [[106zpi]] || 27edo || 27.087 || 44.302 || 1196.163 || 104
|28.032
|42.808
|1198.630
|-
|-
|[[29edo]]
| [[111zpi]] || 28edo || 28.032 || 42.808 || 1198.629 || 109
|28.94
|41.465
|1202.488
|-
|-
|[[30edo]]
| [[116zpi]] || 29edo || 28.940 || 41.465 || 1202.489 || 114
|30.062
|39.918
|1197.525
|-
|-
|'''[[31edo]]'''
| [[122zpi]] || 30edo || 30.061 || 39.918 || 1197.555 || 120
|'''30.978'''
|'''38.737'''
|'''1200.852'''
|-
|-
|[[32edo]]
| [[127zpi]] || 31edo || 30.978 || 38.737 || 1200.837 || 125
|32.07
|37.418
|1197.381
|-
|-
|[[33edo]]
| [[133zpi]] || 32edo || 32.070 || 37.418 || 1197.375 || 131
|32.972
|36.395
|1201.019
|-
|-
|[[34edo]]
| [[138zpi]] || 33edo || 32.972 || 36.394 || 1201.009 || 136
|34.045
|35.247
|1198.414
|-
|-
|[[35edo]]
| [[144zpi]] || 34edo || 34.045 || 35.248 || 1198.419 || 142
|34.926
|34.358
|1202.543
|-
|-
|[[36edo]]
| [[149zpi]] || 35edo || 34.925 || 34.359 || 1202.564 || 147
|35.982
|33.350
|1200.600
|-
|-
|[[37edo]]
| [[155zpi]] || 36edo || 35.982 || 33.350 || 1200.587 || 153
|37.027
|32.409
|1199.125
|-
|-
|[[38edo]] (1st peak)
| [[161zpi]] || 37edo || 37.028 || 32.408 || 1199.108 || 159
|37.89
|31.671
|1203.484
|-
|-
|[[38edo]] (2nd peak)
| [[166zpi]] || 38edo || 37.890 || 31.671 || 1203.480 || 164
|38.103
|31.494
|1196.756
|-
|-
|[[39edo]] (1st peak)
| [[173zpi]] || 39edo || 39.124 || 30.672 || 1196.204 || 171
|38.916
|30.836
|1202.590
|-
|-
|[[39edo]] (2nd peak)
| [[178zpi]] || 40edo || 39.968 || 30.024 || 1200.965 || 176
|39.124
|30.672
|1196.197
|-
|-
|[[40edo]]
| [[184zpi]] || 41edo || 40.988 || 29.277 || 1200.349 || 182
|39.968
|30.024
|1200.961
|-
|-
|'''[[41edo]]'''
| [[190zpi]] || 42edo || 41.999 || 28.572 || 1200.032 || 188
|'''40.988'''
|'''29.277'''
|'''1200.351'''
|-
|-
|[[42edo]]
| [[196zpi]] || 43edo || 43.026 || 27.890 || 1199.261 || 194
|41.999
|28.572
|1200.029
|-
|-
|[[43edo]]
| [[202zpi]] || 44edo || 44.015 || 27.263 || 1199.579 || 200
|43.027
|27.889
|1199.247
|-
|-
|[[44edo]]
| [[207zpi]] || 45edo || 44.840 || 26.762 || 1204.289 || 205
|44.015
|27.263
|1199.591
|-
|-
|[[45edo]]
| [[214zpi]] || 46edo || 46.009 || 26.082 || 1199.766 || 212
|44.84
|26.762
|1204.282
|-
|-
|[[45edo]]
| [[220zpi]] || 47edo || 47.006 || 25.529 || 1199.846 || 218
|45.035
|26.646
|1199.067
|-
|-
|[[46edo]]
| [[226zpi]] || 48edo || 47.988 || 25.006 || 1200.292 || 224
|46.009
|26.082
|1199.765
|-
|-
|[[47edo]]
| [[233zpi]] || 49edo || 49.141 || 24.419 || 1196.552 || 231
|47.006
|25.529
|1199.847
|-
|-
|[[48edo]]
| [[238zpi]] || 50edo || 49.939 || 24.030 || 1201.477 || 236
|47.988
|25.006
|1200.300
|-
|-
|[[49edo]] (1st peak)
| [[245zpi]] || 51edo || 51.080 || 23.493 || 1198.128 || 243
|48.945
|24.517
|1201.348
|-
|-
|[[49edo]] (2nd peak)
| [[251zpi]] || 52edo || 52.043 || 23.058 || 1199.018 || 249
|49.141
|24.420
|1196.557
|-
|-
|[[50edo]]
| [[257zpi]] || 53edo || 52.997 || 22.643 || 1200.072 || 255
|49.939
|24.029
|1201.466
|-
|-
|[[51edo]]
| [[264zpi]] || 54edo || 54.116 || 22.175 || 1197.430 || 262
|51.079
|23.493
|1198.144
|-
|-
|[[52edo]]
| [[269zpi]] || 55edo || 54.894 || 21.860 || 1202.325 || 267
|52.043
|23.058
|1199.009
|-
|-
|'''[[53edo]]'''
| [[276zpi]] || 56edo || 56.008 || 21.425 || 1199.821 || 274
|'''52.997'''
|'''22.643'''
|'''1200.068'''
|-
|-
|[[54edo]] (1st peak)
| [[282zpi]] || 57edo || 56.968 || 21.064 || 1200.668 || 280
|53.949
|22.243
|1201.134
|-
|-
|[[54edo]] (2nd peak)
| [[289zpi]] || 58edo || 58.067 || 20.666 || 1198.621 || 287
|54.116
|22.175
|1197.428
|-
|-
|[[55edo]]
| [[295zpi]] || 59edo || 58.992 || 20.342 || 1200.157 || 293
|54.894
|21.860
|1202.317
|-
|-
|[[56edo]]
| [[301zpi]] || 60edo || 59.920 || 20.027 || 1201.599 || 299
|56.008
|21.426
|1199.829
|-
|-
|[[57edo]]
| [[308zpi]] || 61edo || 61.003 || 19.671 || 1199.937 || 306
|56.968
|21.064
|1200.674
|-
|-
|[[58edo]]
| [[314zpi]] || 62edo || 61.938 || 19.374 || 1201.200 || 312
|58.067
|20.666
|1198.615
|-
|-
|[[59edo]]
| [[321zpi]] || 63edo || 63.019 || 19.042 || 1199.633 || 319
|58.992
|20.342
|1200.163
|-
|-
|[[60edo]]
| [[328zpi]] || 64edo || 64.099 || 18.721 || 1198.140 || 326
|59.92
|20.027
|1201.602
|-
|-
|[[61edo]]
| [[334zpi]] || 65edo || 65.016 || 18.457 || 1199.708 || 332
|61.003
|19.671
|1199.941
|-
|-
|[[62edo]]
| [[340zpi]] || 66edo || 65.916 || 18.205 || 1201.533 || 338
|61.938
|19.374
|1201.201
|-
|-
|[[63edo]]
| [[347zpi]] || 67edo || 66.998 || 17.911 || 1200.029 || 345
|63.019
|19.042
|1199.638
|-
|-
|[[64edo]]
| [[354zpi]] || 68edo || 68.049 || 17.634 || 1199.131 || 352
|64.099
|18.721
|1198.147
|-
|-
|[[65edo]]
| [[360zpi]] || 69edo || 68.960 || 17.401 || 1200.696 || 358
|65.016
|18.457
|1199.705
|-
|-
|[[66edo]]
| [[367zpi]] || 70edo || 70.004 || 17.142 || 1199.931 || 365
|65.916
|18.205
|1201.529
|-
|-
|[[67edo]]
| [[374zpi]] || 71edo || 71.059 || 16.887 || 1198.998 || 372
|66.998
|17.911
|1200.036
|-
|-
|[[68edo]]
| [[380zpi]] || 72edo || 71.951 || 16.678 || 1200.824 || 378
|68.049
|17.634
|1199.136
|-
|-
|[[69edo]]
| [[387zpi]] || 73edo || 72.983 || 16.442 || 1200.273 || 385
|68.96
|17.401
|1200.696
|-
|-
|[[70edo]]
| [[394zpi]] || 74edo || 74.052 || 16.205 || 1199.155 || 392
|70.004
|17.142
|1199.931
|-
|-
|[[71edo]]
| [[401zpi]] || 75edo || 75.091 || 15.981 || 1198.544 || 399
|71.059
|16.887
|1199.004
|-
|-
|'''[[72edo]]'''
| [[407zpi]] || 76edo || 75.968 || 15.796 || 1200.503 || 405
|'''71.951'''
|'''16.678'''
|'''1200.817'''
|-
|-
|[[73edo]]
| [[414zpi]] || 77edo || 76.992 || 15.586 || 1200.127 || 412
|72.984
|16.442
|1200.263
|-
|-
|[[74edo]]
| [[420zpi]] || 78edo || 77.851 || 15.414 || 1202.292 || 418
|74.052
|16.205
|1199.157
|-
|-
|[[75edo]]
| [[427zpi]] || 79edo || 78.892 || 15.211 || 1201.637 || 425
|75.091
|15.981
|1198.546
|-
|-
|[[76edo]]
| [[435zpi]] || 80edo || 80.073 || 14.986 || 1198.904 || 433
|75.968
|15.796
|1200.505
|-
|-
|[[77edo]]
| [[441zpi]] || 81edo || 80.948 || 14.824 || 1200.777 || 439
|76.992
|15.586
|1200.125
|-
|-
|[[78edo]]
| [[448zpi]] || 82edo || 81.954 || 14.642 || 1200.671 || 446
|78.021
|15.380
|1199.677
|-
|-
|[[79edo]]
| [[455zpi]] || 83edo || 82.967 || 14.464 || 1200.484 || 453
|78.892
|15.211
|1201.643
|-
|-
|[[80edo]]
| [[462zpi]] || 84edo || 83.997 || 14.286 || 1200.040 || 460
|80.073
|14.986
|1198.906
|-
|-
|[[81edo]]
| [[469zpi]] || 85edo || 84.991 || 14.119 || 1200.131 || 467
|80.947
|14.825
|1200.786
|-
|-
|[[82edo]]
| [[476zpi]] || 86edo || 86.019 || 13.950 || 1199.741 || 474
|81.954
|14.642
|1200.674
|-
|-
|[[83edo]]
| [[483zpi]] || 87edo || 87.014 || 13.791 || 1199.808 || 481
|82.967
|14.464
|1200.477
|-
|-
|[[84edo]]
| [[490zpi]] || 88edo || 88.027 || 13.632 || 1199.635 || 488
|83.997
|14.286
|1200.043
|-
|-
|[[85edo]]
| [[497zpi]] || 89edo || 89.023 || 13.480 || 1199.691 || 495
|84.991
|14.119
|1200.127
|-
|-
|[[86edo]]
| [[504zpi]] || 90edo || 90.006 || 13.332 || 1199.917 || 502
|86.019
|13.950
|1199.735
|-
|-
|[[87edo]]
| [[510zpi]] || 91edo || 90.852 || 13.208 || 1201.956 || 508
|87.014
|13.791
|1199.807
|-
|-
|[[88edo]]
| [[518zpi]] || 92edo || 91.993 || 13.044 || 1200.089 || 516
|88.027
|13.632
|1199.632
|-
|-
|[[89edo]]
| [[525zpi]] || 93edo || 93.002 || 12.903 || 1199.969 || 523
|89.023
|13.480
|1199.690
|-
|-
|[[90edo]]
| [[532zpi]] || 94edo || 93.984 || 12.768 || 1200.208 || 530
|90.006
|13.332
|1199.920
|-
|-
|[[91edo]] (1st peak)
| [[540zpi]] || 95edo || 95.117 || 12.616 || 1198.526 || 538
|90.852
|13.208
|1201.955
|-
|-
|[[91edo]] (2nd peak)
| [[546zpi]] || 96edo || 95.954 || 12.506 || 1200.570 || 544
|91.033
|13.182
|1199.565
|-
|-
|[[92edo]]
| [[553zpi]] || 97edo || 96.925 || 12.381 || 1200.927 || 551
|91.993
|13.044
|1200.091
|-
|-
|[[93edo]]
| [[560zpi]] || 98edo || 97.923 || 12.254 || 1200.941 || 558
|93.002
|12.903
|1199.974
|-
|-
|[[94edo]]
| [[568zpi]] || 99edo || 99.047 || 12.115 || 1199.427 || 566
|93.984
|12.768
|1200.204
|-
|-
|[[95edo]] (1st peak)
| [[575zpi]] || 100edo || 99.869 || 12.016 || 1201.577 || 573
|94.962
|12.637
|1200.480
|-
|-
|[[95edo]] (2nd peak)
| [[1936zpi]] || 270edo || 270.018 || 4.444 || 1199.920 || 1934
|95.117
|12.616
|1198.524
|-
|-
|[[96edo]]
| [[2293zpi]] || 311edo || 311.004 || 3.858 || 1199.985 || 2291
|95.954
|12.506
|1200.575
|-
|-
|[[97edo]]
| [[2568zpi]] || 342edo || 341.975 || 3.509 || 1200.088 || 2566
|96.925
|12.381
|1200.929
|-
|-
|[[98edo]]
| [[3971zpi]] || 494edo || 494.014 || 2.429 || 1199.966 || 3969
|97.923
|12.255
|1200.944
|-
|-
|'''[[99edo]]'''
| [[5818zpi]] || 684edo || 683.939 || 1.755 || 1200.107 || 5816
|'''99.047'''
|'''12.115'''
|'''1199.431'''
|-
|[[100edo]]
|100.024
|11.997
|1199.712
|}
|}
[[Category:Tables]]
 
[[Category:Zeta]]
== Calculation instructions ==
[[Category:Equal divisions of the octave]]
The third column of the above table can be calculated using the free version of Wolfram Cloud:
# Copy-paste <code><nowiki>Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9, 12.1}]</nowiki></code> into a cell.
# Change "'''11.9'''" and "'''12.1'''" to whatever values you want, e.g. to view the curve around 15edo you might use the values "'''14.9'''" and "'''15.1'''".
# Ensure that cell is still selected
# In the menu select Evaluation > Evaluate Cells
 
== Record zeta peaks ==
<pre>
    0.00000
    1.12657
    1.97277
    3.05976
    3.90445
    5.03448
    6.95669
  10.00846
  12.02318
  18.94809
  22.02515
  27.08661
  30.97838
  40.98808
  52.99683
  71.95061
  99.04733
  117.96951
  130.00391
  152.05285
  170.99589
  217.02470
  224.00255
  270.01779
  341.97485
  422.05570
  441.01827
  494.01377
  742.01093
  764.01938
  935.03297
  953.94128
1012.02423
1105.99972
1177.96567
1236.02355
1394.98350
1447.97300
1577.98315
2459.98488
2683.99168
3395.02659
5585.00172
6079.01642
7032.96529
8268.98378
8539.00834
11664.01488
14347.99444
16807.99325
28742.01019
34691.00191
36268.98775
57578.00854
58972.99326
95524.04578
102557.01877
112984.99531
148418.01630
212146.99129
241199.99851
</pre>
 
== See also ==
* [[Zeta peak index]]
* [[User:Contribution/Gallery of Zeta Peak Indexes at sigma = 1|Zeta Peak Indexes at sigma = 1]]
* [[The Riemann zeta function and tuning]]
 
[[Category:Lists of scales]]
[[Category:Zeta peak indexes]]
Retrieved from "https://en.xen.wiki/w/Table_of_zeta-stretched_edos"