User:Contribution/Impressions on EDOs sizes: Difference between revisions
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Special Category: 130, 140, 152, 171, 183, 270 | Special Category: 130, 140, 152, 171, 183, 270 | ||
== | == Size-aware heuristic ranking of EDOs == | ||
EDOs can be ranked using the heuristic score <math>S = H^p/x</math>, where <math>H</math> is the height of the highest [[ZPI|zeta peak]] associated with the EDO, and <math>x</math> is that peak's unrounded steps-per-octave value. Each peak is assigned to the nearest EDO by rounding <math>x</math> to the nearest integer. | EDOs can be ranked using the heuristic score <math>S = H^p/x</math>, where <math>H</math> is the height of the highest [[ZPI|zeta peak]] associated with the EDO, and <math>x</math> is that peak's unrounded steps-per-octave value. Each peak is assigned to the nearest EDO by rounding <math>x</math> to the nearest integer. | ||
Revision as of 00:48, 10 November 2026
EDOs of choice: 12, 14, 15, 16, 17, 19, 22, 24, 26, 27, 29, 31, 34, 36, 37, 41, 46, 49, 50, 53, 72, 87, 171, 270.
| 12 | 14 | 15 | 16 | 17 | 19 | ||||
| 22 | 24 | 26 | 27 | 29 | |||||
| 31 | 34 | 36 | 37 | ||||||
| 41 | 43 | 46 | 49 | ||||||
| 50 | 53 | 58 | |||||||
| 60 | 65 | 68 | |||||||
| 72 | 77 | ||||||||
| 80 | 84 | 87 | |||||||
| 94 | 99 |
| 118 | 130 | 140 | 152 | 171 | 183 | 270 |
Small EDOs: 12, 14, 15, 16, 17
Medium-small EDOs: 19, 22, 24, 26, 27, 29, 31, 34
Medium-large EDOs: 36, 37, 41, 43, 46, 49, 50, 53
Large EDOs: 58, 60, 65, 68, 72, 80, 84, 87
Very large EDOs: 94, 99, 118
Special Category: 130, 140, 152, 171, 183, 270
Size-aware heuristic ranking of EDOs
EDOs can be ranked using the heuristic score [math]\displaystyle{ S = H^p/x }[/math], where [math]\displaystyle{ H }[/math] is the height of the highest zeta peak associated with the EDO, and [math]\displaystyle{ x }[/math] is that peak's unrounded steps-per-octave value. Each peak is assigned to the nearest EDO by rounding [math]\displaystyle{ x }[/math] to the nearest integer.
The exponent [math]\displaystyle{ p }[/math] controls the balance between peak height and the number of steps per octave. Here, we use [math]\displaystyle{ p = 3 }[/math], an arbitrary choice that appears to give a useful balance for this ranking.
Using the first 10,000 zeta peaks and setting [math]\displaystyle{ p = 3 }[/math] in the heuristic score [math]\displaystyle{ S = H^p/x }[/math], the EDOs whose selected peaks satisfy [math]\displaystyle{ x \ge 12 }[/math] and [math]\displaystyle{ S \ge 5.36 }[/math] are, in descending order: 12, 19, 31, 72, 53, 41, 22, 46, 270, 34, 15, 130, 99, 27, 58, 118, 87, 171, 24, 17, 152, 65, 183, 140, 94, 224, 311, 342, 77, 14, 111, 68, 217, 26, 494, 84, 80, 36, 243, 103, 190, 159, 29, 50, 60, 422, 441, 193, 198, 121, 43, 113.
| Tuning | Strength | Closest EDO | Integer limit | |||||
|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per octave | Cents | Height | [math]\displaystyle{ H^p/N }[/math] | EDO | Octave | Consistent | Distinct |
| 34zpi | 12.0231830072926 | 99.8071807833375 | 5.193 | 11.650 | 12edo | 1197.68616940005 | 10 | 6 |
| 65zpi | 18.9480867166984 | 63.3309324546460 | 5.980 | 11.287 | 19edo | 1203.28771663827 | 10 | 7 |
| 127zpi | 30.9783816349790 | 38.7366910944446 | 7.003 | 11.089 | 31edo | 1200.83742392778 | 12 | 9 |
| 380zpi | 71.9506065993786 | 16.6781081733140 | 9.158 | 10.673 | 72edo | 1200.82378847861 | 18 | 13 |
| 257zpi | 52.9968291550147 | 22.6428640945673 | 8.250 | 10.594 | 53edo | 1200.07179701207 | 10 | 10 |
| 184zpi | 40.9880783925993 | 29.2768055263764 | 7.570 | 10.584 | 41edo | 1200.34902658143 | 16 | 10 |
| 15zpi | 6.95668765658792 | 172.495885863671 | 4.167 | 10.400 | 7edo | 1207.47120104570 | 6 | 5 |
| 80zpi | 22.0251467420146 | 54.4831784348982 | 6.063 | 10.117 | 22edo | 1198.62992556776 | 12 | 8 |
| 9zpi | 5.03447559860338 | 238.356503373041 | 3.665 | 9.777 | 5edo | 1191.78251686521 | 10 | 4 |
| 214zpi | 46.0089748051542 | 26.0818678330031 | 7.496 | 9.154 | 46edo | 1199.76592031814 | 14 | 11 |
| 26zpi | 10.0084563372591 | 119.898609691954 | 4.477 | 8.967 | 10edo | 1198.98609691954 | 8 | 5 |
| 1936zpi | 270.017794631965 | 4.44415154799558 | 13.371 | 8.853 | 270edo | 1199.92091795881 | 22 | 22 |
| 144zpi | 34.0448410043159 | 35.2476312005063 | 6.685 | 8.776 | 34edo | 1198.41946081721 | 6 | 6 |
| 47zpi | 15.0534898676781 | 79.7157343943591 | 5.050 | 8.557 | 15edo | 1195.73601591539 | 8 | 7 |
| 796zpi | 130.003910460506 | 9.23049157328654 | 10.355 | 8.541 | 130edo | 1199.96390452725 | 16 | 16 |
| 568zpi | 99.0473345956631 | 12.1154194093028 | 9.406 | 8.403 | 99edo | 1199.42652152097 | 12 | 12 |
| 106zpi | 27.0866140827635 | 44.3023257293579 | 6.069 | 8.254 | 27edo | 1196.16279469266 | 10 | 8 |
| 289zpi | 58.0667185533159 | 20.6658827964969 | 7.814 | 8.217 | 58edo | 1198.62120219682 | 16 | 12 |
| 706zpi | 117.969513574257 | 10.1721195895637 | 9.851 | 8.103 | 118edo | 1200.31011156852 | 12 | 12 |
| 483zpi | 87.0139255957575 | 13.7908960178956 | 8.869 | 8.018 | 87edo | 1199.80795355692 | 16 | 14 |
| 1114zpi | 170.995891689006 | 7.01771246166817 | 11.077 | 7.948 | 171edo | 1200.02883094526 | 14 | 14 |
| 90zpi | 24.0057421830853 | 49.9880399800983 | 5.722 | 7.803 | 24edo | 1199.71295952236 | 6 | 6 |
| 56zpi | 17.0445886606675 | 70.4035764012981 | 5.057 | 7.587 | 17edo | 1196.86079882207 | 4 | 4 |
| 4zpi | 3.05976162780486 | 392.187413913321 | 2.847 | 7.546 | 3edo | 1176.56224173996 | 6 | 4 |
| 965zpi | 152.052848107925 | 7.89199291517551 | 10.468 | 7.545 | 152edo | 1199.58292310668 | 15 | 15 |
| 334zpi | 65.0158450885860 | 18.4570391781413 | 7.813 | 7.337 | 65edo | 1199.70754657919 | 6 | 6 |
| 1210zpi | 182.999728466810 | 6.55738677895164 | 11.021 | 7.315 | 183edo | 1200.00178054815 | 18 | 18 |
| 872zpi | 139.990541024216 | 8.57200773152536 | 10.077 | 7.309 | 140edo | 1200.08108241355 | 10 | 10 |
| 532zpi | 93.9836761074943 | 12.7681747480009 | 8.806 | 7.266 | 94edo | 1200.20842631208 | 24 | 15 |
| 1546zpi | 224.002551156014 | 5.35708184485908 | 11.730 | 7.206 | 224edo | 1199.98633324843 | 16 | 16 |
| 2293zpi | 311.004029926555 | 3.85847090239759 | 13.066 | 7.173 | 311edo | 1199.98445064565 | 42 | 25 |
| 2568zpi | 341.974850913987 | 3.50902996753355 | 13.479 | 7.160 | 342edo | 1200.08824889647 | 12 | 12 |
| 414zpi | 76.9918536925042 | 15.5860645308353 | 8.195 | 7.148 | 77edo | 1200.12696887432 | 10 | 10 |
| 22zpi | 8.94999197142904 | 134.078332565073 | 3.999 | 7.143 | 9edo | 1206.70499308566 | 8 | 6 |
| 42zpi | 13.9002525327005 | 86.3293668353859 | 4.592 | 6.967 | 14edo | 1208.61113569540 | 7 | 5 |
| 655zpi | 111.059577998833 | 10.8050113427643 | 9.039 | 6.649 | 111edo | 1199.35625904684 | 22 | 16 |
| 354zpi | 68.0493056282519 | 17.6342725163943 | 7.667 | 6.622 | 68edo | 1199.13053111481 | 10 | 10 |
| 1488zpi | 217.024704123598 | 5.52932443726124 | 11.271 | 6.597 | 217edo | 1199.86340288569 | 13 | 13 |
| 100zpi | 25.9356996537225 | 46.2682717652372 | 5.545 | 6.574 | 26edo | 1202.97506589617 | 14 | 9 |
| 6zpi | 3.90444812410668 | 307.341770682266 | 2.942 | 6.524 | 4edo | 1229.36708272906 | 8 | 3 |
| 3971zpi | 494.013766003673 | 2.42908210778701 | 14.755 | 6.503 | 494edo | 1199.96656124678 | 18 | 18 |
| 2zpi | 1.97276711441199 | 608.282645849800 | 2.341 | 6.499 | 2edo | 1216.56529169960 | 4 | 3 |
| 462zpi | 83.9972142607288 | 14.2861880666087 | 8.021 | 6.143 | 84edo | 1200.03979759513 | 10 | 10 |
| 435zpi | 80.0731374302484 | 14.9862992572924 | 7.873 | 6.095 | 80edo | 1198.90394058339 | 12 | 12 |
| 155zpi | 35.9823877000425 | 33.3496490006021 | 6.027 | 6.086 | 36edo | 1200.58736402167 | 8 | 8 |
| 1705zpi | 242.934231195669 | 4.93960852735271 | 11.385 | 6.074 | 243edo | 1200.32487214671 | 10 | 10 |
| 596zpi | 102.936629522070 | 11.6576577800491 | 8.544 | 6.058 | 103edo | 1200.73875134506 | 15 | 15 |
| 1266zpi | 189.933554886685 | 6.31799894818965 | 10.475 | 6.052 | 190edo | 1200.41980015603 | 16 | 16 |
| 1019zpi | 158.953545367059 | 7.54937549350618 | 9.845 | 6.003 | 159edo | 1200.35070346748 | 16 | 16 |
| 116zpi | 28.9399661541990 | 41.4651487014917 | 5.566 | 5.959 | 29edo | 1202.48931234326 | 8 | 7 |
| 19zpi | 8.13742532740116 | 147.466790995825 | 3.642 | 5.936 | 8edo | 1179.73432796660 | 7 | 4 |
| 238zpi | 49.9385162652878 | 24.0295485277387 | 6.655 | 5.903 | 50edo | 1201.47742638693 | 10 | 9 |
| 301zpi | 59.9201656607655 | 20.0266469020418 | 7.046 | 5.839 | 60edo | 1201.59881412251 | 10 | 10 |
| 3297zpi | 422.055698758402 | 2.84322662513537 | 13.497 | 5.825 | 422edo | 1199.84163580713 | 28 | 28 |
| 3473zpi | 441.018266267016 | 2.72097573226923 | 13.575 | 5.672 | 441edo | 1199.95029793073 | 18 | 18 |
| 1291zpi | 193.021683290400 | 6.21691811792258 | 10.258 | 5.592 | 193edo | 1199.86519675906 | 12 | 12 |
| 1332zpi | 198.058774452480 | 6.05880756011603 | 10.279 | 5.483 | 198edo | 1199.64389690297 | 16 | 16 |
| 729zpi | 121.077437013311 | 9.91101256849428 | 8.711 | 5.460 | 121edo | 1199.23252078781 | 15 | 15 |
| 196zpi | 43.0264994477693 | 27.8897892090130 | 6.166 | 5.449 | 43edo | 1199.26093598756 | 8 | 8 |
| 669zpi | 112.963481821010 | 10.6229020268815 | 8.464 | 5.367 | 113edo | 1200.38792903761 | 10 | 10 |