328edo: Difference between revisions
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+rank-2 temperaments |
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=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 53\328 | |||
| 193.90 | |||
| 28/25 | |||
| [[Hemiwürschmidt]] | |||
|- | |||
| 1 | |||
| 117\328 | |||
| 428.05 | |||
| 2800/2187 | |||
| [[Osiris]] | |||
|- | |||
| 2 | |||
| 17\328 | |||
| 62.20 | |||
| 28/27 | |||
| [[Eagle]] | |||
|- | |||
| 2 | |||
| 111\328<br>(53\328) | |||
| 406.10<br>(193.90) | |||
| 495/392<br>(28/25) | |||
| [[Semihemiwürschmidt]] | |||
|- | |||
| 8 | |||
| 136\328<br>(13\328) | |||
| 497.56<br>(47.56) | |||
| 4/3<br>(36/35) | |||
| [[Twilight]] | |||
|- | |||
| 41 | |||
| 49\328<br>(1\328) | |||
| 179.27<br>(3.66) | |||
| 567/512<br>(352/351) | |||
| [[Hemicounterpyth]] | |||
|} | |} | ||
Revision as of 15:58, 29 December 2021
The 328 equal divisions of the octave (328edo), or the 328(-tone) equal temperament (328tet, 328et) when viewed from a regular temperament perspective, divides the octave into 328 equal parts of 3.659 cents each.
Theory
328edo is enfactored in the 5-limit, with the same tuning as 164edo. It tempers out 2401/2400, 3136/3125, and 6144/6125 in the 7-limit, 9801/9800, 16384/16335 and 19712/19683 in the 11-limit, 676/675, 1001/1000, 1716/1715 and 2080/2079 in the 13-limit, 936/935, 1156/1155 and 2601/2600 in the 17-limit, so that it supports würschmidt and hemiwürschmidt, and provides the optimal patent val for 7-limit hemiwürschmidt, 11- and 13-limit semihemiwür, and 13-limit semiporwell.
328 factors into 23 × 41, with subset edos 2, 4, 8, 41, 82, and 164.
Prime harmonics
Script error: No such module "primes_in_edo".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 2401/2400, 3136/3125, 589824/588245 | [⟨328 520 762 921]] | -0.298 | 0.229 | 6.27 |
| 2.3.5.7.11 | 2401/2400, 3136/3125, 9801/9800, 19712/19683 | [⟨328 520 762 921 1135]] | -0.303 | 0.205 | 5.61 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647 | [⟨328 520 762 921 1135 1214]] | -0.295 | 0.188 | 5.15 |
| 2.3.5.7.11.13.17 | 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125 | [⟨328 520 762 921 1135 1214 1341]] | -0.293 | 0.174 | 4.77 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 53\328 | 193.90 | 28/25 | Hemiwürschmidt |
| 1 | 117\328 | 428.05 | 2800/2187 | Osiris |
| 2 | 17\328 | 62.20 | 28/27 | Eagle |
| 2 | 111\328 (53\328) |
406.10 (193.90) |
495/392 (28/25) |
Semihemiwürschmidt |
| 8 | 136\328 (13\328) |
497.56 (47.56) |
4/3 (36/35) |
Twilight |
| 41 | 49\328 (1\328) |
179.27 (3.66) |
567/512 (352/351) |
Hemicounterpyth |