449edo
| ← 448edo | 449edo | 450edo → |
449 equal divisions of the octave (abbreviated 449edo or 449ed2), also called 449-tone equal temperament (449tet) or 449 equal temperament (449et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 449 equal parts of about 2.673 ¢ each. Each step represents a frequency ratio of 21/449, or the 449th root of 2.
Theory
449edo is consistent to the 7-odd-limit, but the errors of harmonics 3, 5, and 7 are all quite large, giving us the option of treating it as a full 7-limit temperament, or a 2.9.15.21.11.13 subgroup temperament.
Using the patent val, the equal temperament tempers out 4375/4374 and 26873856/26796875 in the 7-limit; 41503/41472, 160083/160000, 539055/537824, 805255/802816, and 825000/823543 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.94 | +1.21 | +1.33 | -0.79 | -0.76 | -1.33 | -0.52 | -0.72 | -0.85 | -0.40 | -0.21 |
| relative (%) | +35 | +45 | +50 | -30 | -28 | -50 | -19 | -27 | -32 | -15 | -8 | |
| Steps (reduced) |
712 (263) |
1043 (145) |
1261 (363) |
1423 (76) |
1553 (206) |
1661 (314) |
1754 (407) |
1835 (39) |
1907 (111) |
1972 (176) |
2031 (235) | |
Subsets and supersets
449edo is the 87th prime edo. 898edo, which doubles it, gives a good correction to the harmonic 3, 5 and 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1423 449⟩ | [⟨449 1423]] | 0.1249 | 0.1249 | 4.67 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 127\449 | 339.421 | 243\200 | Amity (7-limit) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Music
- Little Victorious Dance (2023)