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.
In fact, 225edo sets the sequential record for being worst for integer harmonics as a whole, when evaluated using how close are its harmonic errors to [math]\displaystyle{ 1 - \phi }[/math] = 38.1966%, the golden ratio, the number most difficult to approximate with rationals. While other edos with poor approximation usually end up approximating the square of that harmonic well, 225edo performs worst due to it setting a record of how close its harmonic errors are to 38.1966%.
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 (%) | +38.3 | -43.4 | +34.5 | -23.3 | -37.2 | +40.1 | -5.0 | +32.1 | +21.6 | -27.1 | +19.9 | |
| Steps (reduced) |
357 (132) |
522 (72) |
632 (182) |
713 (38) |
778 (103) |
833 (158) |
879 (204) |
920 (20) |
956 (56) |
988 (88) |
1018 (118) | |