Table of zeta-stretched edos
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:
- 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
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 |