Table of zeta-stretched edos: Difference between revisions

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<noinclude>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]].
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]].


No such table could possibly be complete (as there are so many possible edos), so please add tunings of interest as you see fit.
No such table could possibly be complete (as there are so many possible edos), so please add tunings of interest as you see fit.
Line 5: Line 5:
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]].
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 ===
== Table ==
How to calculate the third column using the free version of Wolfram Cloud:
This is a list of zeta peak-based octave tunings, which includes all EDOs up to 100 and certain noteworthy ones above 100.
# Copy-paste <code>Plot{{!(}}Abs{{!(}}RiemannSiegelZ{{!(}}9.06472028x{{))!}}, {{(}}x, 11.9, 12.1{{)}}{{)!}}</code> into a cell.
{| class="wikitable sortable"
# 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
 
=== Table ===
This is a complete list of all zeta peak-based octave tunings, which includes all EDOs up to 100 and certain noteworthy ones above 100.
</noinclude>{| class="wikitable sortable"
|+ style="font-size: 105%;" | Zeta-optimal tunings for selected EDOs
|+ style="font-size: 105%;" | Zeta-optimal tunings for selected EDOs
|-
! Tuning !! Associated edo !! No. of steps per 1200 cents !! Step size (cents) !! Tuning of 2/1 (cents) !! Gram point index
! Tuning !! Associated edo !! No. of steps per 1200 cents !! Step size (cents) !! Tuning of 2/1 (cents) !! Gram point index
|-
|-
Line 227: Line 221:
|-
|-
| [[5818zpi]] || 684edo || 683.939 || 1.755 || 1200.107 || 5816
| [[5818zpi]] || 684edo || 683.939 || 1.755 || 1200.107 || 5816
|}<noinclude>
|}
 
== Calculation instructions ==
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 ==
== See also ==
* [[Zeta peak index]]
* [[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]]
* [[The Riemann zeta function and tuning]]


[[Category:Lists of scales]]
[[Category:Lists of scales]]
[[Category:Zeta peak indexes]]
[[Category:Zeta peak indexes]]
</noinclude>

Latest revision as of 17:20, 28 August 2025

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.

No such table could possibly be complete (as there are so many possible edos), so please add tunings of interest as you see fit.

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.

Zeta-optimal tunings for selected EDOs
Tuning Associated edo No. of steps per 1200 cents Step size (cents) Tuning of 2/1 (cents) Gram point index
1zpi 1edo 1.127 1065.177 1065.177 −1
2zpi 2edo 1.973 608.283 1216.565 0
4zpi 3edo 3.060 392.187 1176.562 2
6zpi 4edo 3.904 307.342 1229.367 4
9zpi 5edo 5.034 238.357 1191.783 7
12zpi 6edo 6.035 198.843 1193.056 10
15zpi 7edo 6.957 172.496 1207.471 13
19zpi 8edo 8.137 147.467 1179.734 17
22zpi 9edo 8.950 134.078 1206.705 20
26zpi 10edo 10.008 119.899 1198.986 24
30zpi 11edo 11.037 108.722 1195.938 28
34zpi 12edo 12.023 99.807 1197.686 32
38zpi 13edo 12.969 92.531 1202.900 36
42zpi 14edo 13.900 86.329 1208.611 40
47zpi 15edo 15.053 79.716 1195.736 45
51zpi 16edo 15.944 75.262 1204.187 49
56zpi 17edo 17.045 70.404 1196.861 54
61zpi 18edo 18.119 66.228 1192.113 59
65zpi 19edo 18.948 63.331 1203.288 63
70zpi 20edo 19.982 60.054 1201.087 68
75zpi 21edo 21.028 57.067 1198.406 73
80zpi 22edo 22.025 54.483 1198.630 78
84zpi 23edo 22.807 52.615 1210.148 82
90zpi 24edo 24.006 49.988 1199.713 88
95zpi 25edo 24.965 48.067 1201.678 93
100zpi 26edo 25.936 46.268 1202.975 98
106zpi 27edo 27.087 44.302 1196.163 104
111zpi 28edo 28.032 42.808 1198.629 109
116zpi 29edo 28.940 41.465 1202.489 114
122zpi 30edo 30.061 39.918 1197.555 120
127zpi 31edo 30.978 38.737 1200.837 125
133zpi 32edo 32.070 37.418 1197.375 131
138zpi 33edo 32.972 36.394 1201.009 136
144zpi 34edo 34.045 35.248 1198.419 142
149zpi 35edo 34.925 34.359 1202.564 147
155zpi 36edo 35.982 33.350 1200.587 153
161zpi 37edo 37.028 32.408 1199.108 159
166zpi 38edo 37.890 31.671 1203.480 164
173zpi 39edo 39.124 30.672 1196.204 171
178zpi 40edo 39.968 30.024 1200.965 176
184zpi 41edo 40.988 29.277 1200.349 182
190zpi 42edo 41.999 28.572 1200.032 188
196zpi 43edo 43.026 27.890 1199.261 194
202zpi 44edo 44.015 27.263 1199.579 200
207zpi 45edo 44.840 26.762 1204.289 205
214zpi 46edo 46.009 26.082 1199.766 212
220zpi 47edo 47.006 25.529 1199.846 218
226zpi 48edo 47.988 25.006 1200.292 224
233zpi 49edo 49.141 24.419 1196.552 231
238zpi 50edo 49.939 24.030 1201.477 236
245zpi 51edo 51.080 23.493 1198.128 243
251zpi 52edo 52.043 23.058 1199.018 249
257zpi 53edo 52.997 22.643 1200.072 255
264zpi 54edo 54.116 22.175 1197.430 262
269zpi 55edo 54.894 21.860 1202.325 267
276zpi 56edo 56.008 21.425 1199.821 274
282zpi 57edo 56.968 21.064 1200.668 280
289zpi 58edo 58.067 20.666 1198.621 287
295zpi 59edo 58.992 20.342 1200.157 293
301zpi 60edo 59.920 20.027 1201.599 299
308zpi 61edo 61.003 19.671 1199.937 306
314zpi 62edo 61.938 19.374 1201.200 312
321zpi 63edo 63.019 19.042 1199.633 319
328zpi 64edo 64.099 18.721 1198.140 326
334zpi 65edo 65.016 18.457 1199.708 332
340zpi 66edo 65.916 18.205 1201.533 338
347zpi 67edo 66.998 17.911 1200.029 345
354zpi 68edo 68.049 17.634 1199.131 352
360zpi 69edo 68.960 17.401 1200.696 358
367zpi 70edo 70.004 17.142 1199.931 365
374zpi 71edo 71.059 16.887 1198.998 372
380zpi 72edo 71.951 16.678 1200.824 378
387zpi 73edo 72.983 16.442 1200.273 385
394zpi 74edo 74.052 16.205 1199.155 392
401zpi 75edo 75.091 15.981 1198.544 399
407zpi 76edo 75.968 15.796 1200.503 405
414zpi 77edo 76.992 15.586 1200.127 412
420zpi 78edo 77.851 15.414 1202.292 418
427zpi 79edo 78.892 15.211 1201.637 425
435zpi 80edo 80.073 14.986 1198.904 433
441zpi 81edo 80.948 14.824 1200.777 439
448zpi 82edo 81.954 14.642 1200.671 446
455zpi 83edo 82.967 14.464 1200.484 453
462zpi 84edo 83.997 14.286 1200.040 460
469zpi 85edo 84.991 14.119 1200.131 467
476zpi 86edo 86.019 13.950 1199.741 474
483zpi 87edo 87.014 13.791 1199.808 481
490zpi 88edo 88.027 13.632 1199.635 488
497zpi 89edo 89.023 13.480 1199.691 495
504zpi 90edo 90.006 13.332 1199.917 502
510zpi 91edo 90.852 13.208 1201.956 508
518zpi 92edo 91.993 13.044 1200.089 516
525zpi 93edo 93.002 12.903 1199.969 523
532zpi 94edo 93.984 12.768 1200.208 530
540zpi 95edo 95.117 12.616 1198.526 538
546zpi 96edo 95.954 12.506 1200.570 544
553zpi 97edo 96.925 12.381 1200.927 551
560zpi 98edo 97.923 12.254 1200.941 558
568zpi 99edo 99.047 12.115 1199.427 566
575zpi 100edo 99.869 12.016 1201.577 573
1936zpi 270edo 270.018 4.444 1199.920 1934
2293zpi 311edo 311.004 3.858 1199.985 2291
2568zpi 342edo 341.975 3.509 1200.088 2566
3971zpi 494edo 494.014 2.429 1199.966 3969
5818zpi 684edo 683.939 1.755 1200.107 5816

Calculation instructions

The third column of the above table can be calculated 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

Record zeta peaks

    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

See also

Retrieved from "https://en.xen.wiki/index.php?title=Table_of_zeta-stretched_edos&oldid=208682"