397edo
| ← 396edo | 397edo | 398edo → |
397 equal divisions of the octave (abbreviated 397edo or 397ed2), also called 397-tone equal temperament (397tet) or 397 equal temperament (397et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 397 equal parts of about 3.02 ¢ each. Each step represents a frequency ratio of 21/397, or the 397th root of 2.
Theory
397edo is only consistent to the 5-odd-limit, with three mappings possible for the 7-limit:
- ⟨397 629 922 1115] (patent val)
- ⟨397 629 922 1114] (397d val)
- ⟨397 629 921 1114] (397cd val)
Using the patent val, it tempers out 129140163/128000000 and [55 -1 -23⟩ in the 5-limit; 6144/6125, 16875/16807 and 129140163/128000000 in the 7-limit; supporting grendel.
Using the 397d val, it tempers out 129140163/128000000 and [55 -1 -23⟩ in the 5-limit; 3136/3125, 420175/419904 and 33756345/33554432 in the 7-limit; supporting sengagen.
Using the 397cd val, it tempers out 390625000/387420489 and [-55 23 8⟩ in the 5-limit; 2401/2400, 390625/387072 and 14348907/14336000 in the 7-limit; supporting cotritone.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | -0.70 | +0.59 | +1.45 | -1.19 | -0.23 | +0.84 | -1.29 | +0.44 | +1.15 | +0.56 |
| relative (%) | +0 | -23 | +19 | +48 | -39 | -7 | +28 | -43 | +15 | +38 | +18 | |
| Steps (reduced) |
397 (0) |
629 (232) |
922 (128) |
1115 (321) |
1373 (182) |
1469 (278) |
1623 (35) |
1686 (98) |
1796 (208) |
1929 (341) |
1967 (379) | |
Subsets and supersets
397edo is the 78th prime edo. 1588edo, which quadruples it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-629 397⟩ | [⟨397 629]] | 0.2194 | 0.2195 | 7.26 |
| 2.3.5 | 129140163/128000000, [55 -1 -23⟩ | [⟨397 629 922]] | 0.0618 | 0.2859 | 9.46 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 8\397 | 24.18 | 686/675 | Sengagen (397d) |
| 1 | 37\397 | 111.84 | 16/15 | Vavoom |
| 1 | 128\397 | 386.90 | 5/4 | Grendel (397) |
| 1 | 171\397 | 516.88 | 27/20 | Gravity |
| 1 | 193\397 | 583.38 | 7/5 | Cotritone (397cd) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct