239edo: Difference between revisions
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== Theory == | == Theory == | ||
239edo has a sharp tendency, with [[prime harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[5120/5103]], and 29360128/29296875 in the 7-limit, [[support]]ing the [[hemififths]] temperament and providing an excellent tuning. It also supports and provides a good tuning for [[quasiorwell]] and [[alphaquarter]]. In the 11-limit, it tempers out [[3025/3024]], [[4000/3993]], [[5632/5625]], and 12005/11979. | 239edo has a sharp tendency, with [[prime harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[5120/5103]], and 29360128/29296875 in the 7-limit, [[support]]ing the [[hemififths]] temperament and providing an excellent tuning. It also supports and provides a good tuning for [[quasiorwell]] and [[alphaquarter]]. In the 11-limit, it tempers out [[3025/3024]], [[4000/3993]], [[5632/5625]], and 12005/11979. | ||
It also encompasses a large variety of higher primes, specifically commendable in the 2.3.5.7.11.17.29.31.37.43.53.59 subgroup. | |||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 04:27, 26 March 2025
| ← 238edo | 239edo | 240edo → |
239 equal divisions of the octave (abbreviated 239edo or 239ed2), also called 239-tone equal temperament (239tet) or 239 equal temperament (239et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 239 equal parts of about 5.02 ¢ each. Each step represents a frequency ratio of 21/239, or the 239th root of 2.
Theory
239edo has a sharp tendency, with prime harmonics 3 through 11 all tuned sharp. The equal temperament tempers out 2401/2400, 5120/5103, and 29360128/29296875 in the 7-limit, supporting the hemififths temperament and providing an excellent tuning. It also supports and provides a good tuning for quasiorwell and alphaquarter. In the 11-limit, it tempers out 3025/3024, 4000/3993, 5632/5625, and 12005/11979.
It also encompasses a large variety of higher primes, specifically commendable in the 2.3.5.7.11.17.29.31.37.43.53.59 subgroup.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.97 | +0.30 | +0.21 | +0.98 | -2.03 | +0.48 | -1.28 | -0.66 | -0.29 | -0.27 |
| Relative (%) | +0.0 | +19.4 | +5.9 | +4.2 | +19.6 | -40.5 | +9.6 | -25.5 | -13.1 | -5.7 | -5.3 | |
| Steps (reduced) |
239 (0) |
379 (140) |
555 (77) |
671 (193) |
827 (110) |
884 (167) |
977 (21) |
1015 (59) |
1081 (125) |
1161 (205) |
1184 (228) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.30 | -2.28 | +0.62 | +2.28 | +0.14 | +0.24 | -2.24 | +1.03 | +1.06 | -1.85 | +1.99 |
| Relative (%) | -5.9 | -45.5 | +12.3 | +45.3 | +2.7 | +4.8 | -44.6 | +20.5 | +21.0 | -36.8 | +39.6 | |
| Steps (reduced) |
1245 (50) |
1280 (85) |
1297 (102) |
1328 (133) |
1369 (174) |
1406 (211) |
1417 (222) |
1450 (16) |
1470 (36) |
1479 (45) |
1507 (73) | |
Subsets and supersets
239edo is the 52nd prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [379 -239⟩ | [⟨239 379]] | −0.307 | 0.307 | 6.12 |
| 2.3.5 | [3 -18 11⟩, [32 -7 -9⟩ | [⟨239 379 555]] | −0.247 | 0.265 | 5.27 |
| 2.3.5.7 | 2401/2400, 5120/5103, 29360128/29296875 | [⟨239 379 555 671]] | −0.204 | 0.241 | 4.80 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 4000/3993, 5120/5103 | [⟨239 379 555 671 827]] | −0.220 | 0.218 | 4.34 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 3\239 | 15.06 | 121/120 | Yarman I (239) |
| 1 | 7\239 | 35.15 | 1990656/1953125 | Gammic (5-limit) |
| 1 | 9\239 | 45.19 | 250/243 | Quartonic (5-limit) |
| 1 | 11\239 | 55.23 | 33/32 | Escapade / alphaquarter |
| 1 | 35\239 | 175.73 | 72/65 | Quadrafifths (239f) |
| 1 | 54\239 | 271.13 | 90/77 | Quasiorwell (239) |
| 1 | 70\239 | 351.46 | 49/40 | Hemififths (7-limit) |
| 1 | 79\239 | 396.65 | 44/35 | Squarschmidt |
| 1 | 83\239 | 416.74 | 14/11 | Unthirds (239f) |
| 1 | 116\239 | 582.43 | 7/5 | Neptune (7-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct