225edo
| ← 224edo | 225edo | 226edo → |
225 equal divisions of the octave (abbreviated 225edo or 225ed2), also called 225-tone equal temperament (225tet) or 225 equal temperament (225et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 225 equal parts of about 5.33 ¢ each. Each step represents a frequency ratio of 21/225, or the 225th root of 2.
Theory
225edo is inconsistent to the 5-odd-limit and higher limits, with rather large errors in harmonics 3, 5, 7, 11, and 13. It has three mappings possible for the 7-limit:
- ⟨225 357 522 632] (patent val),
- ⟨225 356 522 631] (225bd),
- ⟨225 357 523 632] (225c).
Using the patent val, it tempers out 20000/19683 and 2109375/2097152 in the 5-limit; 3125/3087, 10976/10935, and 589824/588245 in the 7-limit.
Using the 225bd val, it tempers out 78732/78125 (sensipent comma) and [-52 27 4⟩ in the 5-limit; 225/224, 177147/175000, and 40353607/40000000 in the 7-limit.
Using the 225c val, it tempers out 131072000/129140163 (rodan comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 2401/2400, 4375/4374, and 2097152/2066715 in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +2.04 | -2.31 | +1.84 | -1.24 | -1.98 | +2.14 | -0.27 | +1.71 | +1.15 | -1.45 | +1.06 |
| relative (%) | +38 | -43 | +35 | -23 | -37 | +40 | -5 | +32 | +22 | -27 | +20 | |
| Steps (reduced) |
357 (132) |
522 (72) |
632 (182) |
713 (38) |
778 (103) |
833 (158) |
879 (204) |
920 (20) |
956 (56) |
988 (88) |
1018 (118) | |