Table of zeta-stretched edos: Difference between revisions
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=== Calculation instructions === | === Calculation instructions === | ||
How to calculate the third column using the free version of Wolfram Cloud: | How to calculate the third column using the free version of Wolfram Cloud: | ||
#Copy-paste | # Copy-paste <code>Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9, 12.1}]</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'''". | # 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 | # Ensure that cell is still selected | ||
#In the menu select Evaluation > Evaluate Cells | # In the menu select Evaluation > Evaluate Cells | ||
=== Table === | === 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. | |||
{| class="wikitable sortable" | {| class="wikitable sortable" | ||
!Tuning | |+ style="font-size: 105%;" | Zeta-optimal tunings for selected EDOs | ||
!Associated edo | ! Tuning !! Associated edo !! No. of steps per 1200 cents !! Step size (cents) !! Tuning of 2/1 (cents) !! Gram point index | ||
!No. of steps per 1200 cents | |||
!Step size (cents) | |||
!Tuning of 2/1 (cents) | |||
!Gram point index | |||
|- | |- | ||
|[[1zpi]] | | [[1zpi]] || 1edo || 1.127 || 1065.177 || 1065.177 || −1 | ||
|1edo | |||
|1.127 | |||
|1065.177 | |||
|1065.177 | |||
| | |||
|- | |- | ||
|[[2zpi]] | | [[2zpi]] || 2edo || 1.973 || 608.283 || 1216.565 || 0 | ||
|2edo | |||
|1.973 | |||
|608.283 | |||
|1216.565 | |||
|0 | |||
|- | |- | ||
|[[4zpi]] | | [[4zpi]] || 3edo || 3.060 || 392.187 || 1176.562 || 2 | ||
|3edo | |||
|3.060 | |||
|392.187 | |||
|1176.562 | |||
|2 | |||
|- | |- | ||
|[[6zpi]] | | [[6zpi]] || 4edo || 3.904 || 307.342 || 1229.367 || 4 | ||
|4edo | |||
|3.904 | |||
|307.342 | |||
|1229.367 | |||
|4 | |||
|- | |- | ||
|[[9zpi]] | | [[9zpi]] || 5edo || 5.034 || 238.357 || 1191.783 || 7 | ||
|5edo | |||
|5.034 | |||
|238.357 | |||
|1191.783 | |||
|7 | |||
|- | |- | ||
|[[12zpi]] | | [[12zpi]] || 6edo || 6.035 || 198.843 || 1193.056 || 10 | ||
|6edo | |||
|6.035 | |||
|198.843 | |||
|1193.056 | |||
|10 | |||
|- | |- | ||
|[[15zpi]] | | [[15zpi]] || 7edo || 6.957 || 172.496 || 1207.471 || 13 | ||
|7edo | |||
|6.957 | |||
|172.496 | |||
|1207.471 | |||
|13 | |||
|- | |- | ||
|[[19zpi]] | | [[19zpi]] || 8edo || 8.137 || 147.467 || 1179.734 || 17 | ||
|8edo | |||
|8.137 | |||
|147.467 | |||
|1179.734 | |||
|17 | |||
|- | |- | ||
|[[22zpi]] | | [[22zpi]] || 9edo || 8.950 || 134.078 || 1206.705 || 20 | ||
|9edo | |||
|8.950 | |||
|134.078 | |||
|1206.705 | |||
|20 | |||
|- | |- | ||
|[[26zpi]] | | [[26zpi]] || 10edo || 10.008 || 119.899 || 1198.986 || 24 | ||
|10edo | |||
|10.008 | |||
|119.899 | |||
|1198.986 | |||
|24 | |||
|- | |- | ||
|[[30zpi]] | | [[30zpi]] || 11edo || 11.037 || 108.722 || 1195.938 || 28 | ||
|11edo | |||
|11.037 | |||
|108.722 | |||
|1195.938 | |||
|28 | |||
|- | |- | ||
|[[34zpi]] | | [[34zpi]] || 12edo || 12.023 || 99.807 || 1197.686 || 32 | ||
|12edo | |||
|12.023 | |||
|99.807 | |||
|1197.686 | |||
|32 | |||
|- | |- | ||
|[[38zpi]] | | [[38zpi]] || 13edo || 12.969 || 92.531 || 1202.900 || 36 | ||
|13edo | |||
|12.969 | |||
|92.531 | |||
|1202.900 | |||
|36 | |||
|- | |- | ||
|[[42zpi]] | | [[42zpi]] || 14edo || 13.900 || 86.329 || 1208.611 || 40 | ||
|14edo | |||
|13.900 | |||
|86.329 | |||
|1208.611 | |||
|40 | |||
|- | |- | ||
|[[47zpi]] | | [[47zpi]] || 15edo || 15.053 || 79.716 || 1195.736 || 45 | ||
|15edo | |||
|15.053 | |||
|79.716 | |||
|1195.736 | |||
|45 | |||
|- | |- | ||
|[[51zpi]] | | [[51zpi]] || 16edo || 15.944 || 75.262 || 1204.187 || 49 | ||
|16edo | |||
|15.944 | |||
|75.262 | |||
|1204.187 | |||
|49 | |||
|- | |- | ||
|[[56zpi]] | | [[56zpi]] || 17edo || 17.045 || 70.404 || 1196.861 || 54 | ||
|17edo | |||
|17.045 | |||
|70.404 | |||
|1196.861 | |||
|54 | |||
|- | |- | ||
|[[61zpi]] | | [[61zpi]] || 18edo || 18.119 || 66.228 || 1192.113 || 59 | ||
|18edo | |||
|18.119 | |||
|66.228 | |||
|1192.113 | |||
|59 | |||
|- | |- | ||
|[[65zpi]] | | [[65zpi]] || 19edo || 18.948 || 63.331 || 1203.288 || 63 | ||
|19edo | |||
|18.948 | |||
|63.331 | |||
|1203.288 | |||
|63 | |||
|- | |- | ||
|[[70zpi]] | | [[70zpi]] || 20edo || 19.982 || 60.054 || 1201.087 || 68 | ||
|20edo | |||
|19.982 | |||
|60.054 | |||
|1201.087 | |||
|68 | |||
|- | |- | ||
|[[75zpi]] | | [[75zpi]] || 21edo || 21.028 || 57.067 || 1198.406 || 73 | ||
|21edo | |||
|21.028 | |||
|57.067 | |||
|1198.406 | |||
|73 | |||
|- | |- | ||
|[[80zpi]] | | [[80zpi]] || 22edo || 22.025 || 54.483 || 1198.630 || 78 | ||
|22edo | |||
|22.025 | |||
|54.483 | |||
|1198.630 | |||
|78 | |||
|- | |- | ||
|[[84zpi]] | | [[84zpi]] || 23edo || 22.807 || 52.615 || 1210.148 || 82 | ||
|23edo | |||
|22.807 | |||
|52.615 | |||
|1210.148 | |||
|82 | |||
|- | |- | ||
|[[90zpi]] | | [[90zpi]] || 24edo || 24.006 || 49.988 || 1199.713 || 88 | ||
|24edo | |||
|24.006 | |||
|49.988 | |||
|1199.713 | |||
|88 | |||
|- | |- | ||
|[[95zpi]] | | [[95zpi]] || 25edo || 24.965 || 48.067 || 1201.678 || 93 | ||
|25edo | |||
|24.965 | |||
|48.067 | |||
|1201.678 | |||
|93 | |||
|- | |- | ||
|[[100zpi]] | | [[100zpi]] || 26edo || 25.936 || 46.268 || 1202.975 || 98 | ||
|26edo | |||
|25.936 | |||
|46.268 | |||
|1202.975 | |||
|98 | |||
|- | |- | ||
|[[106zpi]] | | [[106zpi]] || 27edo || 27.087 || 44.302 || 1196.163 || 104 | ||
|27edo | |||
|27.087 | |||
|44.302 | |||
|1196.163 | |||
|104 | |||
|- | |- | ||
|[[111zpi]] | | [[111zpi]] || 28edo || 28.032 || 42.808 || 1198.629 || 109 | ||
|28edo | |||
|28.032 | |||
|42.808 | |||
|1198.629 | |||
|109 | |||
|- | |- | ||
|[[116zpi]] | | [[116zpi]] || 29edo || 28.940 || 41.465 || 1202.489 || 114 | ||
|29edo | |||
|28.940 | |||
|41.465 | |||
|1202.489 | |||
|114 | |||
|- | |- | ||
|[[122zpi]] | | [[122zpi]] || 30edo || 30.061 || 39.918 || 1197.555 || 120 | ||
|30edo | |||
|30.061 | |||
|39.918 | |||
|1197.555 | |||
|120 | |||
|- | |- | ||
|[[127zpi]] | | [[127zpi]] || 31edo || 30.978 || 38.737 || 1200.837 || 125 | ||
|31edo | |||
|30.978 | |||
|38.737 | |||
|1200.837 | |||
|125 | |||
|- | |- | ||
|[[133zpi]] | | [[133zpi]] || 32edo || 32.070 || 37.418 || 1197.375 || 131 | ||
|32edo | |||
|32.070 | |||
|37.418 | |||
|1197.375 | |||
|131 | |||
|- | |- | ||
|[[138zpi]] | | [[138zpi]] || 33edo || 32.972 || 36.394 || 1201.009 || 136 | ||
|33edo | |||
|32.972 | |||
|36.394 | |||
|1201.009 | |||
|136 | |||
|- | |- | ||
|[[144zpi]] | | [[144zpi]] || 34edo || 34.045 || 35.248 || 1198.419 || 142 | ||
|34edo | |||
|34.045 | |||
|35.248 | |||
|1198.419 | |||
|142 | |||
|- | |- | ||
|[[149zpi]] | | [[149zpi]] || 35edo || 34.925 || 34.359 || 1202.564 || 147 | ||
|35edo | |||
|34.925 | |||
|34.359 | |||
|1202.564 | |||
|147 | |||
|- | |- | ||
|[[155zpi]] | | [[155zpi]] || 36edo || 35.982 || 33.350 || 1200.587 || 153 | ||
|36edo | |||
|35.982 | |||
|33.350 | |||
|1200.587 | |||
|153 | |||
|- | |- | ||
|[[161zpi]] | | [[161zpi]] || 37edo || 37.028 || 32.408 || 1199.108 || 159 | ||
|37edo | |||
|37.028 | |||
|32.408 | |||
|1199.108 | |||
|159 | |||
|- | |- | ||
|[[166zpi]] | | [[166zpi]] || 38edo || 37.890 || 31.671 || 1203.480 || 164 | ||
|38edo | |||
|37.890 | |||
|31.671 | |||
|1203.480 | |||
|164 | |||
|- | |- | ||
|[[173zpi]] | | [[173zpi]] || 39edo || 39.124 || 30.672 || 1196.204 || 171 | ||
|39edo | |||
|39.124 | |||
|30.672 | |||
|1196.204 | |||
|171 | |||
|- | |- | ||
|[[178zpi]] | | [[178zpi]] || 40edo || 39.968 || 30.024 || 1200.965 || 176 | ||
|40edo | |||
|39.968 | |||
|30.024 | |||
|1200.965 | |||
|176 | |||
|- | |- | ||
|[[184zpi]] | | [[184zpi]] || 41edo || 40.988 || 29.277 || 1200.349 || 182 | ||
|41edo | |||
|40.988 | |||
|29.277 | |||
|1200.349 | |||
|182 | |||
|- | |- | ||
|[[190zpi]] | | [[190zpi]] || 42edo || 41.999 || 28.572 || 1200.032 || 188 | ||
|42edo | |||
|41.999 | |||
|28.572 | |||
|1200.032 | |||
|188 | |||
|- | |- | ||
|[[196zpi]] | | [[196zpi]] || 43edo || 43.026 || 27.890 || 1199.261 || 194 | ||
|43edo | |||
|43.026 | |||
|27.890 | |||
|1199.261 | |||
|194 | |||
|- | |- | ||
|[[202zpi]] | | [[202zpi]] || 44edo || 44.015 || 27.263 || 1199.579 || 200 | ||
|44edo | |||
|44.015 | |||
|27.263 | |||
|1199.579 | |||
|200 | |||
|- | |- | ||
|[[207zpi]] | | [[207zpi]] || 45edo || 44.840 || 26.762 || 1204.289 || 205 | ||
|45edo | |||
|44.840 | |||
|26.762 | |||
|1204.289 | |||
|205 | |||
|- | |- | ||
|[[214zpi]] | | [[214zpi]] || 46edo || 46.009 || 26.082 || 1199.766 || 212 | ||
|46edo | |||
|46.009 | |||
|26.082 | |||
|1199.766 | |||
|212 | |||
|- | |- | ||
|[[220zpi]] | | [[220zpi]] || 47edo || 47.006 || 25.529 || 1199.846 || 218 | ||
|47edo | |||
|47.006 | |||
|25.529 | |||
|1199.846 | |||
|218 | |||
|- | |- | ||
|[[226zpi]] | | [[226zpi]] || 48edo || 47.988 || 25.006 || 1200.292 || 224 | ||
|48edo | |||
|47.988 | |||
|25.006 | |||
|1200.292 | |||
|224 | |||
|- | |- | ||
|[[233zpi]] | | [[233zpi]] || 49edo || 49.141 || 24.419 || 1196.552 || 231 | ||
|49edo | |||
|49.141 | |||
|24.419 | |||
|1196.552 | |||
|231 | |||
|- | |- | ||
|[[238zpi]] | | [[238zpi]] || 50edo || 49.939 || 24.030 || 1201.477 || 236 | ||
|50edo | |||
|49.939 | |||
|24.030 | |||
|1201.477 | |||
|236 | |||
|- | |- | ||
|[[245zpi]] | | [[245zpi]] || 51edo || 51.080 || 23.493 || 1198.128 || 243 | ||
|51edo | |||
|51.080 | |||
|23.493 | |||
|1198.128 | |||
|243 | |||
|- | |- | ||
|[[251zpi]] | | [[251zpi]] || 52edo || 52.043 || 23.058 || 1199.018 || 249 | ||
|52edo | |||
|52.043 | |||
|23.058 | |||
|1199.018 | |||
|249 | |||
|- | |- | ||
|[[257zpi]] | | [[257zpi]] || 53edo || 52.997 || 22.643 || 1200.072 || 255 | ||
|53edo | |||
|52.997 | |||
|22.643 | |||
|1200.072 | |||
|255 | |||
|- | |- | ||
|[[264zpi]] | | [[264zpi]] || 54edo || 54.116 || 22.175 || 1197.430 || 262 | ||
|54edo | |||
|54.116 | |||
|22.175 | |||
|1197.430 | |||
|262 | |||
|- | |- | ||
|[[269zpi]] | | [[269zpi]] || 55edo || 54.894 || 21.860 || 1202.325 || 267 | ||
|55edo | |||
|54.894 | |||
|21.860 | |||
|1202.325 | |||
|267 | |||
|- | |- | ||
|[[276zpi]] | | [[276zpi]] || 56edo || 56.008 || 21.425 || 1199.821 || 274 | ||
|56edo | |||
|56.008 | |||
|21.425 | |||
|1199.821 | |||
|274 | |||
|- | |- | ||
|[[282zpi]] | | [[282zpi]] || 57edo || 56.968 || 21.064 || 1200.668 || 280 | ||
|57edo | |||
|56.968 | |||
|21.064 | |||
|1200.668 | |||
|280 | |||
|- | |- | ||
|[[289zpi]] | | [[289zpi]] || 58edo || 58.067 || 20.666 || 1198.621 || 287 | ||
|58edo | |||
|58.067 | |||
|20.666 | |||
|1198.621 | |||
|287 | |||
|- | |- | ||
|[[295zpi]] | | [[295zpi]] || 59edo || 58.992 || 20.342 || 1200.157 || 293 | ||
|59edo | |||
|58.992 | |||
|20.342 | |||
|1200.157 | |||
|293 | |||
|- | |- | ||
|[[301zpi]] | | [[301zpi]] || 60edo || 59.920 || 20.027 || 1201.599 || 299 | ||
|60edo | |||
|59.920 | |||
|20.027 | |||
|1201.599 | |||
|299 | |||
|- | |- | ||
|[[308zpi]] | | [[308zpi]] || 61edo || 61.003 || 19.671 || 1199.937 || 306 | ||
|61edo | |||
|61.003 | |||
|19.671 | |||
|1199.937 | |||
|306 | |||
|- | |- | ||
|[[314zpi]] | | [[314zpi]] || 62edo || 61.938 || 19.374 || 1201.200 || 312 | ||
|62edo | |||
|61.938 | |||
|19.374 | |||
|1201.200 | |||
|312 | |||
|- | |- | ||
|[[321zpi]] | | [[321zpi]] || 63edo || 63.019 || 19.042 || 1199.633 || 319 | ||
|63edo | |||
|63.019 | |||
|19.042 | |||
|1199.633 | |||
|319 | |||
|- | |- | ||
|[[328zpi]] | | [[328zpi]] || 64edo || 64.099 || 18.721 || 1198.140 || 326 | ||
|64edo | |||
|64.099 | |||
|18.721 | |||
|1198.140 | |||
|326 | |||
|- | |- | ||
|[[334zpi]] | | [[334zpi]] || 65edo || 65.016 || 18.457 || 1199.708 || 332 | ||
|65edo | |||
|65.016 | |||
|18.457 | |||
|1199.708 | |||
|332 | |||
|- | |- | ||
|[[340zpi]] | | [[340zpi]] || 66edo || 65.916 || 18.205 || 1201.533 || 338 | ||
|66edo | |||
|65.916 | |||
|18.205 | |||
|1201.533 | |||
|338 | |||
|- | |- | ||
|[[347zpi]] | | [[347zpi]] || 67edo || 66.998 || 17.911 || 1200.029 || 345 | ||
|67edo | |||
|66.998 | |||
|17.911 | |||
|1200.029 | |||
|345 | |||
|- | |- | ||
|[[354zpi]] | | [[354zpi]] || 68edo || 68.049 || 17.634 || 1199.131 || 352 | ||
|68edo | |||
|68.049 | |||
|17.634 | |||
|1199.131 | |||
|352 | |||
|- | |- | ||
|[[360zpi]] | | [[360zpi]] || 69edo || 68.960 || 17.401 || 1200.696 || 358 | ||
|69edo | |||
|68.960 | |||
|17.401 | |||
|1200.696 | |||
|358 | |||
|- | |- | ||
|[[367zpi]] | | [[367zpi]] || 70edo || 70.004 || 17.142 || 1199.931 || 365 | ||
|70edo | |||
|70.004 | |||
|17.142 | |||
|1199.931 | |||
|365 | |||
|- | |- | ||
|[[374zpi]] | | [[374zpi]] || 71edo || 71.059 || 16.887 || 1198.998 || 372 | ||
|71edo | |||
|71.059 | |||
|16.887 | |||
|1198.998 | |||
|372 | |||
|- | |- | ||
|[[380zpi]] | | [[380zpi]] || 72edo || 71.951 || 16.678 || 1200.824 || 378 | ||
|72edo | |||
|71.951 | |||
|16.678 | |||
|1200.824 | |||
|378 | |||
|- | |- | ||
|[[387zpi]] | | [[387zpi]] || 73edo || 72.983 || 16.442 || 1200.273 || 385 | ||
|73edo | |||
|72.983 | |||
|16.442 | |||
|1200.273 | |||
|385 | |||
|- | |- | ||
|[[394zpi]] | | [[394zpi]] || 74edo || 74.052 || 16.205 || 1199.155 || 392 | ||
|74edo | |||
|74.052 | |||
|16.205 | |||
|1199.155 | |||
|392 | |||
|- | |- | ||
|[[401zpi]] | | [[401zpi]] || 75edo || 75.091 || 15.981 || 1198.544 || 399 | ||
|75edo | |||
|75.091 | |||
|15.981 | |||
|1198.544 | |||
|399 | |||
|- | |- | ||
|[[407zpi]] | | [[407zpi]] || 76edo || 75.968 || 15.796 || 1200.503 || 405 | ||
|76edo | |||
|75.968 | |||
|15.796 | |||
|1200.503 | |||
|405 | |||
|- | |- | ||
|[[414zpi]] | | [[414zpi]] || 77edo || 76.992 || 15.586 || 1200.127 || 412 | ||
|77edo | |||
|76.992 | |||
|15.586 | |||
|1200.127 | |||
|412 | |||
|- | |- | ||
|[[420zpi]] | | [[420zpi]] || 78edo || 77.851 || 15.414 || 1202.292 || 418 | ||
|78edo | |||
|77.851 | |||
|15.414 | |||
|1202.292 | |||
|418 | |||
|- | |- | ||
|[[427zpi]] | | [[427zpi]] || 79edo || 78.892 || 15.211 || 1201.637 || 425 | ||
|79edo | |||
|78.892 | |||
|15.211 | |||
|1201.637 | |||
|425 | |||
|- | |- | ||
|[[435zpi]] | | [[435zpi]] || 80edo || 80.073 || 14.986 || 1198.904 || 433 | ||
|80edo | |||
|80.073 | |||
|14.986 | |||
|1198.904 | |||
|433 | |||
|- | |- | ||
|[[441zpi]] | | [[441zpi]] || 81edo || 80.948 || 14.824 || 1200.777 || 439 | ||
|81edo | |||
|80.948 | |||
|14.824 | |||
|1200.777 | |||
|439 | |||
|- | |- | ||
|[[448zpi]] | | [[448zpi]] || 82edo || 81.954 || 14.642 || 1200.671 || 446 | ||
|82edo | |||
|81.954 | |||
|14.642 | |||
|1200.671 | |||
|446 | |||
|- | |- | ||
|[[455zpi]] | | [[455zpi]] || 83edo || 82.967 || 14.464 || 1200.484 || 453 | ||
|83edo | |||
|82.967 | |||
|14.464 | |||
|1200.484 | |||
|453 | |||
|- | |- | ||
|[[462zpi]] | | [[462zpi]] || 84edo || 83.997 || 14.286 || 1200.040 || 460 | ||
|84edo | |||
|83.997 | |||
|14.286 | |||
|1200.040 | |||
|460 | |||
|- | |- | ||
|[[469zpi]] | | [[469zpi]] || 85edo || 84.991 || 14.119 || 1200.131 || 467 | ||
|85edo | |||
|84.991 | |||
|14.119 | |||
|1200.131 | |||
|467 | |||
|- | |- | ||
|[[476zpi]] | | [[476zpi]] || 86edo || 86.019 || 13.950 || 1199.741 || 474 | ||
|86edo | |||
|86.019 | |||
|13.950 | |||
|1199.741 | |||
|474 | |||
|- | |- | ||
|[[483zpi]] | | [[483zpi]] || 87edo || 87.014 || 13.791 || 1199.808 || 481 | ||
|87edo | |||
|87.014 | |||
|13.791 | |||
|1199.808 | |||
|481 | |||
|- | |- | ||
|[[490zpi]] | | [[490zpi]] || 88edo || 88.027 || 13.632 || 1199.635 || 488 | ||
|88edo | |||
|88.027 | |||
|13.632 | |||
|1199.635 | |||
|488 | |||
|- | |- | ||
|[[497zpi]] | | [[497zpi]] || 89edo || 89.023 || 13.480 || 1199.691 || 495 | ||
|89edo | |||
|89.023 | |||
|13.480 | |||
|1199.691 | |||
|495 | |||
|- | |- | ||
|[[504zpi]] | | [[504zpi]] || 90edo || 90.006 || 13.332 || 1199.917 || 502 | ||
|90edo | |||
|90.006 | |||
|13.332 | |||
|1199.917 | |||
|502 | |||
|- | |- | ||
|[[510zpi]] | | [[510zpi]] || 91edo || 90.852 || 13.208 || 1201.956 || 508 | ||
|91edo | |||
|90.852 | |||
|13.208 | |||
|1201.956 | |||
|508 | |||
|- | |- | ||
|[[518zpi]] | | [[518zpi]] || 92edo || 91.993 || 13.044 || 1200.089 || 516 | ||
|92edo | |||
|91.993 | |||
|13.044 | |||
|1200.089 | |||
|516 | |||
|- | |- | ||
|[[525zpi]] | | [[525zpi]] || 93edo || 93.002 || 12.903 || 1199.969 || 523 | ||
|93edo | |||
|93.002 | |||
|12.903 | |||
|1199.969 | |||
|523 | |||
|- | |- | ||
|[[532zpi]] | | [[532zpi]] || 94edo || 93.984 || 12.768 || 1200.208 || 530 | ||
|94edo | |||
|93.984 | |||
|12.768 | |||
|1200.208 | |||
|530 | |||
|- | |- | ||
|[[540zpi]] | | [[540zpi]] || 95edo || 95.117 || 12.616 || 1198.526 || 538 | ||
|95edo | |||
|95.117 | |||
|12.616 | |||
|1198.526 | |||
|538 | |||
|- | |- | ||
|[[546zpi]] | | [[546zpi]] || 96edo || 95.954 || 12.506 || 1200.570 || 544 | ||
|96edo | |||
|95.954 | |||
|12.506 | |||
|1200.570 | |||
|544 | |||
|- | |- | ||
|[[553zpi]] | | [[553zpi]] || 97edo || 96.925 || 12.381 || 1200.927 || 551 | ||
|97edo | |||
|96.925 | |||
|12.381 | |||
|1200.927 | |||
|551 | |||
|- | |- | ||
|[[560zpi]] | | [[560zpi]] || 98edo || 97.923 || 12.254 || 1200.941 || 558 | ||
|98edo | |||
|97.923 | |||
|12.254 | |||
|1200.941 | |||
|558 | |||
|- | |- | ||
|[[568zpi]] | | [[568zpi]] || 99edo || 99.047 || 12.115 || 1199.427 || 566 | ||
|99edo | |||
|99.047 | |||
|12.115 | |||
|1199.427 | |||
|566 | |||
|- | |- | ||
|[[575zpi]] | | [[575zpi]] || 100edo || 99.869 || 12.016 || 1201.577 || 573 | ||
|100edo | |||
|99.869 | |||
|12.016 | |||
|1201.577 | |||
|573 | |||
|- | |- | ||
|[[1936zpi]] | | [[1936zpi]] || 270edo || 270.018 || 4.444 || 1199.920 || 1934 | ||
|270edo | |||
|270.018 | |||
|4.444 | |||
|1199.920 | |||
|1934 | |||
|- | |- | ||
|[[2293zpi]] | | [[2293zpi]] || 311edo || 311.004 || 3.858 || 1199.985 || 2291 | ||
|311edo | |||
|311.004 | |||
|3.858 | |||
|1199.985 | |||
|2291 | |||
|- | |- | ||
|[[2568zpi]] | | [[2568zpi]] || 342edo || 341.975 || 3.509 || 1200.088 || 2566 | ||
|342edo | |||
|341.975 | |||
|3.509 | |||
|1200.088 | |||
|2566 | |||
|- | |- | ||
|[[3971zpi]] | | [[3971zpi]] || 494edo || 494.014 || 2.429 || 1199.966 || 3969 | ||
|494edo | |||
|494.014 | |||
|2.429 | |||
|1199.966 | |||
|3969 | |||
|- | |- | ||
|[[5818zpi]] | | [[5818zpi]] || 684edo || 683.939 || 1.755 || 1200.107 || 5816 | ||
|684edo | |||
|683.939 | |||
|1.755 | |||
|1200.107 | |||
|5816 | |||
|} | |} | ||
Revision as of 15:13, 26 July 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.
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.
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
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.
| 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 |