439edo
| ← 438edo | 439edo | 440edo → |
439 equal divisions of the octave (abbreviated 439edo or 439ed2), also called 439-tone equal temperament (439tet) or 439 equal temperament (439et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 439 equal parts of about 2.73 ¢ each. Each step represents a frequency ratio of 21/439, or the 439th root of 2.
Theory
439edo is inconsistent in the 5-odd-limit. In the 7-limit, the patent val ⟨439 696 1019 1232], and the 439cd val ⟨439 696 1020 1233] are about as viable.
The patent val tempers out 10976/10935, 15625/15552 and 703125/702464 in the 7-limit; 3025/3024, 4375/4356, 14700/14641, 59290/59049, 131072/130977, and 184877/184320 in the 11-limit.
The 439cd val tempers out 3136/3125, 2097152/2083725, and 4096000/4084101 in the 7-limit; 3136/3125, 15488/15435, 16384/16335, 19712/19683, 24057/24010, 43923/43750, 43923/43904, and 46656/46585 in the 11-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +0.55 | -0.89 | -1.17 | +0.85 | -1.35 | -1.08 | +0.44 | +0.43 | +0.95 | +0.29 |
| relative (%) | +0 | +20 | -33 | -43 | +31 | -49 | -40 | +16 | +16 | +35 | +11 | |
| Steps (reduced) |
439 (0) |
696 (257) |
1019 (141) |
1232 (354) |
1519 (202) |
1624 (307) |
1794 (38) |
1865 (109) |
1986 (230) |
2133 (377) |
2175 (419) | |
Subsets and supersets
439edo is the 85th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [696 -439⟩ | [⟨439 696]] | -0.1737 | 0.1737 | 6.36 |
| 2.3.5 | 15625/15552, [115 -74 1⟩ | [⟨439 696 1019]] (439) | +0.0122 | 0.2988 | 10.9 |
| 2.3.5.7 | 10976/10935, 15625/15552, [-25 6 -3 8⟩ | [⟨439 696 1019 1232]] (439) | +0.1135 | 0.3127 | 11.4 |
| 2.3.5 | [24 -21 4⟩, [36 11 -23⟩ | [⟨439 696 1020]] (439c) | -0.3801 | 0.3244 | 11.9 |
| 2.3.5.7 | 3136/3125, 2097152/2083725, 43046721/42875000 | [⟨439 696 1020 1233]] (439cd) | -0.4241 | 0.2911 | 10.7 |