Table of zeta-stretched edos: Difference between revisions

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Added introduction
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'''Explanation of what a 'zeta stretched edo' 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''':
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]].
 
=== Calculation instructions ===
How to calculate the third column using the free version of Wolfram Cloud:
#Copy-paste '''Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9,12.1}]''' into a cell.
#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'''".
#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'''".
Line 7: Line 10:
#In the menu select Evaluation > Evaluate Cells
#In the menu select Evaluation > Evaluate Cells


=== Table ===
{| class="wikitable sortable"
{| class="wikitable sortable"
!Edo
!Edo
!Zeta peak index
!No. of steps per 1200 cents
!No. of steps per 1200 cents
!Step size (cents)
!Step size (cents)
!Octave size (cents)
!Octave size (cents)
!Zeta peak height
!Zeta peak height
!Zeta peak index
!Gram point index
!Gram point index
|-
|-
|1edo
|1edo
|[[1zpi]]
|1.127
|1.127
|1064.774
|1064.774
|1064.774
|1064.774
|1.6
|1.6
|[[1zpi]]
| -1
| -1
|-
|-
|2edo
|2edo
|[[2zpi]]
|1.972
|1.972
|608.519
|608.519
|1217.039
|1217.039
|2.3
|2.3
|[[2zpi]]
|0
|0
|-
|-
|3edo
|3edo
|[[4zpi]]
|3.06
|3.06
|392.157
|392.157
|1176.471
|1176.471
|2.8
|2.8
|[[4zpi]]
|2
|2
|-
|-
|4edo
|4edo
|[[6zpi]]
|3.904
|3.904
|307.377
|307.377
|1229.508
|1229.508
|3.0
|3.0
|[[6zpi]]
|4
|4
|-
|-
|5edo
|5edo
|[[9zpi]]
|5.034
|5.034
|238.379
|238.379
|1191.895
|1191.895
|3.7
|3.7
|[[9zpi]]
|7
|7
|-
|-
|6edo
|6edo
|[[12zpi]]
|6.035
|6.035
|198.840
|198.840
|1193.041
|1193.041
|2.9
|2.9
|[[12zpi]]
|10
|10
|-
|-
|7edo
|7edo
|[[15zpi]]
|6.957
|6.957
|172.488
|172.488
|1207.417
|1207.417
|4.2
|4.2
|[[15zpi]]
|13
|13
|-
|-
|8edo
|8edo
|[[19zpi]]
|8.137
|8.137
|147.474
|147.474
|1179.796
|1179.796
|3.6
|3.6
|[[19zpi]]
|17
|17
|-
|-
|9edo
|9edo
|[[22zpi]]
|8.95
|8.95
|134.078
|134.078
|1206.704
|1206.704
|4.0
|4.0
|[[22zpi]]
|20
|20
|-
|-
|10edo
|10edo
|[[26zpi]]
|10.008
|10.008
|119.904
|119.904
|1199.041
|1199.041
|4.5
|4.5
|[[26zpi]]
|24
|24
|-
|-
|11edo
|11edo
|[[30zpi]]
|11.037
|11.037
|108.725
|108.725
|1195.977
|1195.977
|2.7
|2.7
|[[30zpi]]
|28
|28
|-
|-
|12edo
|12edo
|[[34zpi]]
|12.023
|12.023
|99.809
|99.809
|1197.704
|1197.704
|5.2
|5.2
|[[34zpi]]
|32
|32
|-
|-
|13edo
|13edo
|[[38zpi]]
|12.969
|12.969
|92.528
|92.528
|1202.868
|1202.868
|3.1
|3.1
|[[38zpi]]
|36
|36
|-
|-
|14edo
|14edo
|[[42zpi]]
|13.9
|13.9
|86.331
|86.331
|1208.633
|1208.633
|4.6
|4.6
|[[42zpi]]
|40
|40
|-
|-
|15edo
|15edo
|[[47zpi]]
|15.053
|15.053
|79.718
|79.718
|1195.775
|1195.775
|5.1
|5.1
|[[47zpi]]
|45
|45
|-
|-
|16edo
|16edo
|[[51zpi]]
|15.945
|15.945
|75.259
|75.259
|1204.139
|1204.139
|4.2
|4.2
|[[51zpi]]
|49
|49
|-
|-
|17edo
|17edo
|[[56zpi]]
|17.045
|17.045
|70.402
|70.402
|1196.832
|1196.832
|5.1
|5.1
|[[56zpi]]
|54
|54
|-
|-
|18edo
|18edo
|[[61zpi]]
|18.119
|18.119
|66.229
|66.229
|1192.119
|1192.119
|3.5
|3.5
|[[61zpi]]
|59
|59
|-
|-
|19edo
|19edo
|[[65zpi]]
|18.948
|18.948
|63.331
|63.331
|1203.293
|1203.293
|6.0
|6.0
|[[65zpi]]
|63
|63
|-
|-
|20edo
|20edo
|[[70zpi]]
|19.982
|19.982
|60.054
|60.054
|1201.081
|1201.081
|3.4
|3.4
|[[70zpi]]
|68
|68
|-
|-
|21edo
|21edo
|[[75zpi]]
|21.028
|21.028
|57.067
|57.067
|1198.402
|1198.402
|4.1
|4.1
|[[75zpi]]
|73
|73
|-
|-
|22edo
|22edo
|[[80zpi]]
|22.025
|22.025
|54.484
|54.484
|1198.638
|1198.638
|6.1
|6.1
|[[80zpi]]
|78
|78
|-
|-
|23edo (1st peak)
|23edo (1st peak)
|[[84zpi]]
|22.807
|22.807
|52.615
|52.615
|1210.148
|1210.148
|3.7
|3.7
|[[84zpi]]
|82
|82
|-
|-
|23edo (2nd peak)
|23edo (2nd peak)
|[[85zpi]]
|23.026
|23.026
|52.115
|52.115
|1198.645
|1198.645
|2.2
|2.2
|[[85zpi]]
|83
|83
|-
|-
|24edo
|24edo
|[[90zpi]]
|24.006
|24.006
|49.988
|49.988
|1199.700
|1199.700
|5.7
|5.7
|[[90zpi]]
|88
|88
|-
|-
|25edo
|25edo
|[[95zpi]]
|24.965
|24.965
|48.067
|48.067
|1201.682
|1201.682
|3.9
|3.9
|[[95zpi]]
|93
|93
|-
|-
|26edo
|26edo
|[[100zpi]]
|25.936
|25.936
|46.268
|46.268
|1202.961
|1202.961
|5.6
|5.6
|[[100zpi]]
|98
|98
|-
|-
|27edo
|27edo
|[[106zpi]]
|27.087
|27.087
|44.302
|44.302
|1196.146
|1196.146
|6.1
|6.1
|[[106zpi]]
|104
|104
|-
|-
|28edo
|28edo
|[[111zpi]]
|28.032
|28.032
|42.808
|42.808
|1198.630
|1198.630
|3.7
|3.7
|[[111zpi]]
|109
|109
|-
|-
|29edo
|29edo
|[[116zpi]]
|28.94
|28.94
|41.465
|41.465
|1202.488
|1202.488
|5.6
|5.6
|[[116zpi]]
|114
|114
|-
|-
|30edo
|30edo
|[[122zpi]]
|30.062
|30.062
|39.918
|39.918
|1197.525
|1197.525
|3.3
|3.3
|[[122zpi]]
|120
|120
|-
|-
|31edo
|31edo
|[[127zpi]]
|30.978
|30.978
|38.737
|38.737
|1200.852
|1200.852
|7.0
|7.0
|[[127zpi]]
|125
|125
|-
|-
|32edo
|32edo
|[[133zpi]]
|32.07
|32.07
|37.418
|37.418
|1197.381
|1197.381
|4.5
|4.5
|[[133zpi]]
|131
|131
|-
|-
|33edo
|33edo
|[[138zpi]]
|32.972
|32.972
|36.395
|36.395
|1201.019
|1201.019
|3.3
|3.3
|[[138zpi]]
|136
|136
|-
|-
|34edo
|34edo
|[[144zpi]]
|34.045
|34.045
|35.247
|35.247
|1198.414
|1198.414
|6.7
|6.7
|[[144zpi]]
|142
|142
|-
|-
|35edo
|35edo
|[[149zpi]]
|34.926
|34.926
|34.358
|34.358
|1202.543
|1202.543
|4.2
|4.2
|[[149zpi]]
|147
|147
|-
|-
|36edo
|36edo
|[[155zpi]]
|35.982
|35.982
|33.350
|33.350
|1200.600
|1200.600
|6.0
|6.0
|[[155zpi]]
|153
|153
|-
|-
|37edo
|37edo
|[[161zpi]]
|37.027
|37.027
|32.409
|32.409
|1199.125
|1199.125
|5.3
|5.3
|[[161zpi]]
|159
|159
|-
|-
|38edo (1st peak)
|38edo (1st peak)
|[[166zpi]]
|37.89
|37.89
|31.671
|31.671
|1203.484
|1203.484
|5.8
|5.8
|[[166zpi]]
|164
|164
|-
|-
|38edo (2nd peak)
|38edo (2nd peak)
|[[167zpi]]
|38.103
|38.103
|31.494
|31.494
|1196.756
|1196.756
|2.5
|2.5
|[[167zpi]]
|165
|165
|-
|-
|39edo (1st peak)
|39edo (1st peak)
|[[172zpi]]
|38.916
|38.916
|30.836
|30.836
|1202.590
|1202.590
|2.6
|2.6
|[[172zpi]]
|170
|170
|-
|-
|39edo (2nd peak)
|39edo (2nd peak)
|[[173zpi]]
|39.124
|39.124
|30.672
|30.672
|1196.197
|1196.197
|5.6
|5.6
|[[173zpi]]
|171
|171
|-
|-
|40edo
|40edo
|[[178zpi]]
|39.968
|39.968
|30.024
|30.024
|1200.961
|1200.961
|4.0
|4.0
|[[178zpi]]
|176
|176
|-
|-
|41edo
|41edo
|[[184zpi]]
|40.988
|40.988
|29.277
|29.277
|1200.351
|1200.351
|7.6
|7.6
|[[184zpi]]
|182
|182
|-
|-
|42edo
|42edo
|[[190zpi]]
|41.999
|41.999
|28.572
|28.572
|1200.029
|1200.029
|2.7
|2.7
|[[190zpi]]
|188
|188
|-
|-
|43edo
|43edo
|[[196zpi]]
|43.027
|43.027
|27.889
|27.889
|1199.247
|1199.247
|6.2
|6.2
|[[196zpi]]
|194
|194
|-
|-
|44edo
|44edo
|[[202zpi]]
|44.015
|44.015
|27.263
|27.263
|1199.591
|1199.591
|4.6
|4.6
|[[202zpi]]
|200
|200
|-
|-
|45edo (1st peak)
|45edo (1st peak)
|[[207zpi]]
|44.84
|44.84
|26.762
|26.762
|1204.282
|1204.282
|5.3
|5.3
|[[207zpi]]
|205
|205
|-
|-
|45edo (2nd peak)
|45edo (2nd peak)
|[[208zpi]]
|45.035
|45.035
|26.646
|26.646
|1199.067
|1199.067
|2.1
|2.1
|[[208zpi]]
|206
|206
|-
|-
|46edo
|46edo
|[[214zpi]]
|46.009
|46.009
|26.082
|26.082
|1199.765
|1199.765
|7.5
|7.5
|[[214zpi]]
|212
|212
|-
|-
|47edo
|47edo
|[[220zpi]]
|47.006
|47.006
|25.529
|25.529
|1199.847
|1199.847
|4.3
|4.3
|[[220zpi]]
|218
|218
|-
|-
|48edo
|48edo
|[[226zpi]]
|47.988
|47.988
|25.006
|25.006
|1200.300
|1200.300
|5.8
|5.8
|[[226zpi]]
|224
|224
|-
|-
|49edo (1st peak)
|49edo (1st peak)
|[[232zpi]]
|48.945
|48.945
|24.517
|24.517
|1201.348
|1201.348
|2.2
|2.2
|[[232zpi]]
|230
|230
|-
|-
|49edo (2nd peak)
|49edo (2nd peak)
|[[233zpi]]
|49.141
|49.141
|24.420
|24.420
|1196.557
|1196.557
|5.7
|5.7
|[[233zpi]]
|231
|231
|-
|-
|50edo
|50edo
|[[238zpi]]
|49.939
|49.939
|24.029
|24.029
|1201.466
|1201.466
|6.7
|6.7
|[[238zpi]]
|236
|236
|-
|-
|51edo
|51edo
|[[245zpi]]
|51.079
|51.079
|23.493
|23.493
|1198.144
|1198.144
|4.8
|4.8
|[[245zpi]]
|243
|243
|-
|-
|52edo
|52edo
|[[251zpi]]
|52.043
|52.043
|23.058
|23.058
|1199.009
|1199.009
|4.1
|4.1
|[[251zpi]]
|249
|249
|-
|-
|53edo
|53edo
|[[257zpi]]
|52.997
|52.997
|22.643
|22.643
|1200.068
|1200.068
|8.2
|8.2
|[[257zpi]]
|255
|255
|-
|-
|54edo (1st peak)
|54edo (1st peak)
|[[263zpi]]
|53.949
|53.949
|22.243
|22.243
|1201.134
|1201.134
|2.0
|2.0
|[[263zpi]]
|261
|261
|-
|-
|54edo (2nd peak)
|54edo (2nd peak)
|[[264zpi]]
|54.116
|54.116
|22.175
|22.175
|1197.428
|1197.428
|3.5
|3.5
|[[264zpi]]
|262
|262
|-
|-
|55edo
|55edo
|[[269zpi]]
|54.894
|54.894
|21.860
|21.860
|1202.317
|1202.317
|5.3
|5.3
|[[269zpi]]
|267
|267
|-
|-
|56edo
|56edo
|[[276zpi]]
|56.008
|56.008
|21.426
|21.426
|1199.829
|1199.829
|6.1
|6.1
|[[276zpi]]
|274
|274
|-
|-
|57edo
|57edo
|[[282zpi]]
|56.968
|56.968
|21.064
|21.064
|1200.674
|1200.674
|4.9
|4.9
|[[282zpi]]
|280
|280
|-
|-
|58edo
|58edo
|[[289zpi]]
|58.067
|58.067
|20.666
|20.666
|1198.615
|1198.615
|7.8
|7.8
|[[289zpi]]
|287
|287
|-
|-
|59edo
|59edo
|[[295zpi]]
|58.992
|58.992
|20.342
|20.342
|1200.163
|1200.163
|4.0
|4.0
|[[295zpi]]
|293
|293
|-
|-
|60edo
|60edo
|[[301zpi]]
|59.92
|59.92
|20.027
|20.027
|1201.602
|1201.602
|7.1
|7.1
|[[301zpi]]
|299
|299
|-
|-
|61edo
|61edo
|[[308zpi]]
|61.003
|61.003
|19.671
|19.671
|1199.941
|1199.941
|3.7
|3.7
|[[308zpi]]
|306
|306
|-
|-
|62edo
|62edo
|[[314zpi]]
|61.938
|61.938
|19.374
|19.374
|1201.201
|1201.201
|6.3
|6.3
|[[314zpi]]
|312
|312
|-
|-
|63edo
|63edo
|[[321zpi]]
|63.019
|63.019
|19.042
|19.042
|1199.638
|1199.638
|6.8
|6.8
|[[321zpi]]
|319
|319
|-
|-
|64edo
|64edo
|[[328zpi]]
|64.099
|64.099
|18.721
|18.721
|1198.147
|1198.147
|3.6
|3.6
|[[328zpi]]
|326
|326
|-
|-
|65edo
|65edo
|[[334zpi]]
|65.016
|65.016
|18.457
|18.457
|1199.705
|1199.705
|7.8
|7.8
|[[334zpi]]
|332
|332
|-
|-
|66edo
|66edo
|[[340zpi]]
|65.916
|65.916
|18.205
|18.205
|1201.529
|1201.529
|4.5
|4.5
|[[340zpi]]
|338
|338
|-
|-
|67edo
|67edo
|[[347zpi]]
|66.998
|66.998
|17.911
|17.911
|1200.036
|1200.036
|5.3
|5.3
|[[347zpi]]
|345
|345
|-
|-
|68edo
|68edo
|[[354zpi]]
|68.049
|68.049
|17.634
|17.634
|1199.136
|1199.136
|7.7
|7.7
|[[354zpi]]
|352
|352
|-
|-
|69edo
|69edo
|[[360zpi]]
|68.96
|68.96
|17.401
|17.401
|1200.696
|1200.696
|4.1
|4.1
|[[360zpi]]
|358
|358
|-
|-
|70edo
|70edo
|[[367zpi]]
|70.004
|70.004
|17.142
|17.142
|1199.931
|1199.931
|5.7
|5.7
|[[367zpi]]
|365
|365
|-
|-
|71edo
|71edo
|[[374zpi]]
|71.059
|71.059
|16.887
|16.887
|1199.004
|1199.004
|3.8
|3.8
|[[374zpi]]
|372
|372
|-
|-
|72edo
|72edo
|[[380zpi]]
|71.951
|71.951
|16.678
|16.678
|1200.817
|1200.817
|9.2
|9.2
|[[380zpi]]
|378
|378
|-
|-
|73edo
|73edo
|[[387zpi]]
|72.984
|72.984
|16.442
|16.442
|1200.263
|1200.263
|3.4
|3.4
|[[387zpi]]
|385
|385
|-
|-
|74edo
|74edo
|[[394zpi]]
|74.052
|74.052
|16.205
|16.205
|1199.157
|1199.157
|5.1
|5.1
|[[394zpi]]
|392
|392
|-
|-
|75edo
|75edo
|[[401zpi]]
|75.091
|75.091
|15.981
|15.981
|1198.546
|1198.546
|6.0
|6.0
|[[401zpi]]
|399
|399
|-
|-
|76edo
|76edo
|[[407zpi]]
|75.968
|75.968
|15.796
|15.796
|1200.505
|1200.505
|2.6
|2.6
|[[407zpi]]
|405
|405
|-
|-
|77edo
|77edo
|[[414zpi]]
|76.992
|76.992
|15.586
|15.586
|1200.125
|1200.125
|8.2
|8.2
|[[414zpi]]
|412
|412
|-
|-
|78edo
|78edo
|[[420zpi]]
|78.021
|78.021
|15.380
|15.380
|1199.677
|1199.677
|5.4
|5.4
|[[420zpi]]
|418
|418
|-
|-
|79edo
|79edo
|[[427zpi]]
|78.892
|78.892
|15.211
|15.211
|1201.643
|1201.643
|5.8
|5.8
|[[427zpi]]
|425
|425
|-
|-
|80edo
|80edo
|[[435zpi]]
|80.073
|80.073
|14.986
|14.986
|1198.906
|1198.906
|7.9
|7.9
|[[435zpi]]
|433
|433
|-
|-
|81edo
|81edo
|[[441zpi]]
|80.947
|80.947
|14.825
|14.825
|1200.786
|1200.786
|5.2
|5.2
|[[441zpi]]
|439
|439
|-
|-
|82edo
|82edo
|[[448zpi]]
|81.954
|81.954
|14.642
|14.642
|1200.674
|1200.674
|6.7
|6.7
|[[448zpi]]
|446
|446
|-
|-
|83edo
|83edo
|[[455zpi]]
|82.967
|82.967
|14.464
|14.464
|1200.477
|1200.477
|3.9
|3.9
|[[455zpi]]
|453
|453
|-
|-
|84edo
|84edo
|[[462zpi]]
|83.997
|83.997
|14.286
|14.286
|1200.043
|1200.043
|8.0
|8.0
|[[462zpi]]
|460
|460
|-
|-
|85edo
|85edo
|[[469zpi]]
|84.991
|84.991
|14.119
|14.119
|1200.127
|1200.127
|3.0
|3.0
|[[469zpi]]
|467
|467
|-
|-
|86edo
|86edo
|[[476zpi]]
|86.019
|86.019
|13.950
|13.950
|1199.735
|1199.735
|2.4
|2.4
|[[476zpi]]
|474
|474
|-
|-
|87edo
|87edo
|[[483zpi]]
|87.014
|87.014
|13.791
|13.791
|1199.807
|1199.807
|8.9
|8.9
|[[483zpi]]
|481
|481
|-
|-
|88edo
|88edo
|[[490zpi]]
|88.027
|88.027
|13.632
|13.632
|1199.632
|1199.632
|2.6
|2.6
|[[490zpi]]
|488
|488
|-
|-
|89edo
|89edo
|[[497zpi]]
|89.023
|89.023
|13.480
|13.480
|1199.690
|1199.690
|7.6
|7.6
|[[497zpi]]
|495
|495
|-
|-
|90edo
|90edo
|[[504zpi]]
|90.006
|90.006
|13.332
|13.332
|1199.920
|1199.920
|4.8
|4.8
|[[504zpi]]
|502
|502
|-
|-
|91edo (1st peak)
|91edo (1st peak)
|[[510zpi]]
|90.852
|90.852
|13.208
|13.208
|1201.955
|1201.955
|6.7
|6.7
|[[510zpi]]
|508
|508
|-
|-
|91edo (2nd peak)
|91edo (2nd peak)
|[[511zpi]]
|91.033
|91.033
|13.182
|13.182
|1199.565
|1199.565
|3.4
|3.4
|[[511zpi]]
|509
|509
|-
|-
|92edo
|92edo
|[[518zpi]]
|91.993
|91.993
|13.044
|13.044
|1200.091
|1200.091
|4.5
|4.5
|[[518zpi]]
|516
|516
|-
|-
|93edo
|93edo
|[[525zpi]]
|93.002
|93.002
|12.903
|12.903
|1199.974
|1199.974
|5.6
|5.6
|[[525zpi]]
|523
|523
|-
|-
|94edo
|94edo
|[[532zpi]]
|93.984
|93.984
|12.768
|12.768
|1200.204
|1200.204
|8.8
|8.8
|[[532zpi]]
|530
|530
|-
|-
|95edo (1st peak)
|95edo (1st peak)
|[[539zpi]]
|94.962
|94.962
|12.637
|12.637
|1200.480
|1200.480
|0.9
|0.9
|[[539zpi]]
|537
|537
|-
|-
|95edo (2nd peak)
|95edo (2nd peak)
|[[540zpi]]
|95.117
|95.117
|12.616
|12.616
|1198.524
|1198.524
|5.3
|5.3
|[[540zpi]]
|538
|538
|-
|-
|96edo
|96edo
|[[546zpi]]
|95.954
|95.954
|12.506
|12.506
|1200.575
|1200.575
|7.3
|7.3
|[[546zpi]]
|544
|544
|-
|-
|97edo
|97edo
|[[553zpi]]
|96.925
|96.925
|12.381
|12.381
|1200.929
|1200.929
|4.2
|4.2
|[[553zpi]]
|551
|551
|-
|-
|98edo
|98edo
|[[560zpi]]
|97.923
|97.923
|12.255
|12.255
|1200.944
|1200.944
|4.3
|4.3
|[[560zpi]]
|558
|558
|-
|-
|99edo
|99edo
|[[568zpi]]
|99.047
|99.047
|12.115
|12.115
|1199.431
|1199.431
|9.4
|9.4
|[[568zpi]]
|566
|566
|-
|-
|100edo
|100edo
|[[575zpi]]
|100.024
|100.024
|11.997
|11.997
|1199.712
|1199.712
|4.3
|4.3
|[[575zpi]]
|573
|573
|-
|-
|270edo
|270edo
|[[1936zpi]]
|270.018
|270.018
|4.444
|4.444
|1199.920
|1199.920
|13.4
|13.4
|[[1936zpi]]
|1934
|1934
|-
|-
|311edo
|311edo
|[[2293zpi]]
|311.004
|311.004
|3.858
|3.858
|1199.985
|1199.985
|13.1
|13.1
|[[2293zpi]]
|2291
|2291
|-
|-
|342edo
|342edo
|[[2568zpi]]
|341.975
|341.975
|3.509
|3.509
|1200.088
|1200.088
|13.5
|13.5
|[[2568zpi]]
|2566
|2566
|-
|-
|494edo
|494edo
|[[3971zpi]]
|494.014
|494.014
|2.429
|2.429
|1199.966
|1199.966
|14.7
|14.7
|[[3971zpi]]
|3969
|3969
|-
|-
|684edo
|684edo
|[[5818zpi]]
|683.939
|683.939
|1.755
|1.755
|1200.107
|1200.107
|14.3
|14.3
|[[5818zpi]]
|5816
|5816
|}
|}

Revision as of 05:54, 30 March 2024

This table lists tuning instructions for 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.

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.

Calculation instructions

How to calculate the third column using the free version of Wolfram Cloud:

  1. Copy-paste Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9,12.1}] into a cell.
  2. 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".
  3. Ensure that cell is still selected
  4. In the menu select Evaluation > Evaluate Cells

Table

Edo Zeta peak index No. of steps per 1200 cents Step size (cents) Octave size (cents) Zeta peak height Gram point index
1edo 1zpi 1.127 1064.774 1064.774 1.6 -1
2edo 2zpi 1.972 608.519 1217.039 2.3 0
3edo 4zpi 3.06 392.157 1176.471 2.8 2
4edo 6zpi 3.904 307.377 1229.508 3.0 4
5edo 9zpi 5.034 238.379 1191.895 3.7 7
6edo 12zpi 6.035 198.840 1193.041 2.9 10
7edo 15zpi 6.957 172.488 1207.417 4.2 13
8edo 19zpi 8.137 147.474 1179.796 3.6 17
9edo 22zpi 8.95 134.078 1206.704 4.0 20
10edo 26zpi 10.008 119.904 1199.041 4.5 24
11edo 30zpi 11.037 108.725 1195.977 2.7 28
12edo 34zpi 12.023 99.809 1197.704 5.2 32
13edo 38zpi 12.969 92.528 1202.868 3.1 36
14edo 42zpi 13.9 86.331 1208.633 4.6 40
15edo 47zpi 15.053 79.718 1195.775 5.1 45
16edo 51zpi 15.945 75.259 1204.139 4.2 49
17edo 56zpi 17.045 70.402 1196.832 5.1 54
18edo 61zpi 18.119 66.229 1192.119 3.5 59
19edo 65zpi 18.948 63.331 1203.293 6.0 63
20edo 70zpi 19.982 60.054 1201.081 3.4 68
21edo 75zpi 21.028 57.067 1198.402 4.1 73
22edo 80zpi 22.025 54.484 1198.638 6.1 78
23edo (1st peak) 84zpi 22.807 52.615 1210.148 3.7 82
23edo (2nd peak) 85zpi 23.026 52.115 1198.645 2.2 83
24edo 90zpi 24.006 49.988 1199.700 5.7 88
25edo 95zpi 24.965 48.067 1201.682 3.9 93
26edo 100zpi 25.936 46.268 1202.961 5.6 98
27edo 106zpi 27.087 44.302 1196.146 6.1 104
28edo 111zpi 28.032 42.808 1198.630 3.7 109
29edo 116zpi 28.94 41.465 1202.488 5.6 114
30edo 122zpi 30.062 39.918 1197.525 3.3 120
31edo 127zpi 30.978 38.737 1200.852 7.0 125
32edo 133zpi 32.07 37.418 1197.381 4.5 131
33edo 138zpi 32.972 36.395 1201.019 3.3 136
34edo 144zpi 34.045 35.247 1198.414 6.7 142
35edo 149zpi 34.926 34.358 1202.543 4.2 147
36edo 155zpi 35.982 33.350 1200.600 6.0 153
37edo 161zpi 37.027 32.409 1199.125 5.3 159
38edo (1st peak) 166zpi 37.89 31.671 1203.484 5.8 164
38edo (2nd peak) 167zpi 38.103 31.494 1196.756 2.5 165
39edo (1st peak) 172zpi 38.916 30.836 1202.590 2.6 170
39edo (2nd peak) 173zpi 39.124 30.672 1196.197 5.6 171
40edo 178zpi 39.968 30.024 1200.961 4.0 176
41edo 184zpi 40.988 29.277 1200.351 7.6 182
42edo 190zpi 41.999 28.572 1200.029 2.7 188
43edo 196zpi 43.027 27.889 1199.247 6.2 194
44edo 202zpi 44.015 27.263 1199.591 4.6 200
45edo (1st peak) 207zpi 44.84 26.762 1204.282 5.3 205
45edo (2nd peak) 208zpi 45.035 26.646 1199.067 2.1 206
46edo 214zpi 46.009 26.082 1199.765 7.5 212
47edo 220zpi 47.006 25.529 1199.847 4.3 218
48edo 226zpi 47.988 25.006 1200.300 5.8 224
49edo (1st peak) 232zpi 48.945 24.517 1201.348 2.2 230
49edo (2nd peak) 233zpi 49.141 24.420 1196.557 5.7 231
50edo 238zpi 49.939 24.029 1201.466 6.7 236
51edo 245zpi 51.079 23.493 1198.144 4.8 243
52edo 251zpi 52.043 23.058 1199.009 4.1 249
53edo 257zpi 52.997 22.643 1200.068 8.2 255
54edo (1st peak) 263zpi 53.949 22.243 1201.134 2.0 261
54edo (2nd peak) 264zpi 54.116 22.175 1197.428 3.5 262
55edo 269zpi 54.894 21.860 1202.317 5.3 267
56edo 276zpi 56.008 21.426 1199.829 6.1 274
57edo 282zpi 56.968 21.064 1200.674 4.9 280
58edo 289zpi 58.067 20.666 1198.615 7.8 287
59edo 295zpi 58.992 20.342 1200.163 4.0 293
60edo 301zpi 59.92 20.027 1201.602 7.1 299
61edo 308zpi 61.003 19.671 1199.941 3.7 306
62edo 314zpi 61.938 19.374 1201.201 6.3 312
63edo 321zpi 63.019 19.042 1199.638 6.8 319
64edo 328zpi 64.099 18.721 1198.147 3.6 326
65edo 334zpi 65.016 18.457 1199.705 7.8 332
66edo 340zpi 65.916 18.205 1201.529 4.5 338
67edo 347zpi 66.998 17.911 1200.036 5.3 345
68edo 354zpi 68.049 17.634 1199.136 7.7 352
69edo 360zpi 68.96 17.401 1200.696 4.1 358
70edo 367zpi 70.004 17.142 1199.931 5.7 365
71edo 374zpi 71.059 16.887 1199.004 3.8 372
72edo 380zpi 71.951 16.678 1200.817 9.2 378
73edo 387zpi 72.984 16.442 1200.263 3.4 385
74edo 394zpi 74.052 16.205 1199.157 5.1 392
75edo 401zpi 75.091 15.981 1198.546 6.0 399
76edo 407zpi 75.968 15.796 1200.505 2.6 405
77edo 414zpi 76.992 15.586 1200.125 8.2 412
78edo 420zpi 78.021 15.380 1199.677 5.4 418
79edo 427zpi 78.892 15.211 1201.643 5.8 425
80edo 435zpi 80.073 14.986 1198.906 7.9 433
81edo 441zpi 80.947 14.825 1200.786 5.2 439
82edo 448zpi 81.954 14.642 1200.674 6.7 446
83edo 455zpi 82.967 14.464 1200.477 3.9 453
84edo 462zpi 83.997 14.286 1200.043 8.0 460
85edo 469zpi 84.991 14.119 1200.127 3.0 467
86edo 476zpi 86.019 13.950 1199.735 2.4 474
87edo 483zpi 87.014 13.791 1199.807 8.9 481
88edo 490zpi 88.027 13.632 1199.632 2.6 488
89edo 497zpi 89.023 13.480 1199.690 7.6 495
90edo 504zpi 90.006 13.332 1199.920 4.8 502
91edo (1st peak) 510zpi 90.852 13.208 1201.955 6.7 508
91edo (2nd peak) 511zpi 91.033 13.182 1199.565 3.4 509
92edo 518zpi 91.993 13.044 1200.091 4.5 516
93edo 525zpi 93.002 12.903 1199.974 5.6 523
94edo 532zpi 93.984 12.768 1200.204 8.8 530
95edo (1st peak) 539zpi 94.962 12.637 1200.480 0.9 537
95edo (2nd peak) 540zpi 95.117 12.616 1198.524 5.3 538
96edo 546zpi 95.954 12.506 1200.575 7.3 544
97edo 553zpi 96.925 12.381 1200.929 4.2 551
98edo 560zpi 97.923 12.255 1200.944 4.3 558
99edo 568zpi 99.047 12.115 1199.431 9.4 566
100edo 575zpi 100.024 11.997 1199.712 4.3 573
270edo 1936zpi 270.018 4.444 1199.920 13.4 1934
311edo 2293zpi 311.004 3.858 1199.985 13.1 2291
342edo 2568zpi 341.975 3.509 1200.088 13.5 2566
494edo 3971zpi 494.014 2.429 1199.966 14.7 3969
684edo 5818zpi 683.939 1.755 1200.107 14.3 5816

See also

Retrieved from "https://en.xen.wiki/index.php?title=Table_of_zeta-stretched_edos&oldid=140399"