328edo: Difference between revisions
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== Theory == | == Theory == | ||
328edo is [[enfactoring|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 | 328edo is [[enfactoring|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 [[support]]s [[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 2<sup>3</sup> × 41, with subset edos 2, 4, 8, 41, 82, and 164. | 328 factors into 2<sup>3</sup> × 41, with subset edos 2, 4, 8, 41, 82, and 164. | ||
Revision as of 18:42, 25 January 2022
| ← 327edo | 328edo | 329edo → |
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 about 3.66 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 |