897edo
| ← 896edo | 897edo | 898edo → |
897 equal divisions of the octave (abbreviated 897edo or 897ed2), also called 897-tone equal temperament (897tet) or 897 equal temperament (897et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 897 equal parts of about 1.34 ¢ each. Each step represents a frequency ratio of 21/897, or the 897th root of 2.
Theory
897edo shares the same approximation to harmonic 3 with 299edo. It has sharp approximations to 5 and flat ones starting with 7. In the 13-limit at least, we want to consider the patent val ⟨897 1422 2083 2518 3103 3319] as well as the 897def val ⟨897 1422 2083 2519 3104 3320].
The patent val tempers out 250047/250000 in the 7-limit; 3025/3024, 5632/5625, 42592/42525, 102487/102400, 131072/130977, 160083/160000, and 172032/171875 in the 11-limit; 676/675, 1001/1000, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit.
The 897def val tempers out 4375/4374 in the 7-limit; 4000/3993, 46656/46585, and 172032/171875 in the 11-limit; 1716/1715, and 10648/10647 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.386 | +0.308 | -0.264 | -0.566 | -0.147 | -0.394 | -0.643 | -0.608 | -0.523 | +0.122 | +0.488 |
| Relative (%) | +28.9 | +23.0 | -19.7 | -42.3 | -11.0 | -29.4 | -48.1 | -45.4 | -39.1 | +9.1 | +36.5 | |
| Steps (reduced) |
1422 (525) |
2083 (289) |
2518 (724) |
2843 (152) |
3103 (412) |
3319 (628) |
3504 (813) |
3666 (78) |
3810 (222) |
3940 (352) |
4058 (470) | |
Subsets and supersets
Since 897 factors into 3 × 13 × 23, 897edo has subset edos 3, 13, 23, 39, 69, and 299.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [33 -34 9⟩, [70 -9 -24⟩ | [⟨897 1422 2083]] | −0.1255 | 0.0996 | 7.44 |
| 2.3.5.7 | 250047/250000, 67108864/66976875, 28824005/28697814 | [⟨897 1422 2083 2518]] (897) | −0.0706 | 0.1283 | 9.59 |
| 2.3.5.7.11 | 3025/3024, 5632/5625, 102487/102400, 28824005/28697814 | [⟨897 1422 2083 2518 3103]] (897) | −0.0480 | 0.1234 | 9.22 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 3025/3024, 4096/4095, 28824005/28697814 | [⟨897 1422 2083 2518 3103 3319]] (897) | −0.0222 | 0.1265 | 9.46 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 419\897 | 560.54 | 864/625 | Whoosh |
| 1 | 440\897 | 588.63 | 128/91 | Countritonic (897df) |
| 3 | 103\897 | 137.79 | 13/12 | Avicenna (897) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Music
- 13|3|23 (2023) – 3-, 13-, and 23edo polymicrotonal