1000edo
| ← 999edo | 1000edo | 1001edo → |
1000 equal divisions of the octave (abbreviated 1000edo or 1000ed2), also called 1000-tone equal temperament (1000tet) or 1000 equal temperament (1000et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1000 equal parts of exactly 1.2 ¢ each. Each step represents a frequency ratio of 21/1000, or the 1000th root of 2.
1000edo is notable mostly because it is the equal division corresponding to millioctaves.
Theory
1000edo is related to 200edo, but the patent vals differ on the mapping for 5 and 7. In the 5-limit, it tempers out [38 -2 -15⟩ (luna comma) and [-17 62 -35⟩ (senior comma). In the 7-limit, it tempers out 4375/4374, 201768035/201326592, and 165288374272/164794921875, leading to the lunatic temperament and seniority temperament. It also tempers out 3025/3024, 9801/9800, and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, 4459/4455, and 10648/10647 in the 13-limit, leading to the deca temperament and donar temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | +0.045 | +0.086 | -0.426 | -0.518 | -0.528 | -0.555 | +0.087 | +0.526 | +0.023 | -0.236 |
| relative (%) | +0 | +4 | +7 | -35 | -43 | -44 | -46 | +7 | +44 | +2 | -20 | |
| Steps (reduced) |
1000 (0) |
1585 (585) |
2322 (322) |
2807 (807) |
3459 (459) |
3700 (700) |
4087 (87) |
4248 (248) |
4524 (524) |
4858 (858) |
4954 (954) | |
Subsets and supersets
1000edo carries the interval size measure millioctave. Since 1000 factors into 23 × 53, 1000edo has subset edos 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, and 500.
2000edo, which doubles 1000edo, is consistent in the 29-odd-limit and thus provides good corrections for harmonics 7, 11, 13, 17, and 23.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [317 -200⟩ | [⟨1000 1585]] | -0.0142 | 0.0142 | 1.18 |
| 2.3.5 | [38 -2 -15⟩, [55 -64 20⟩ | [⟨1000 1585 2322]] | -0.0219 | 0.0159 | 1.33 |
| 2.3.5.7 | 4375/4374, 201768035/201326592, [12 -3 -14 9⟩ | [⟨1000 1585 2322 2807]] | +0.0215 | 0.0764 | 6.37 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 391314/390625, [-32 13 1 2 1⟩ | [⟨1000 1585 2322 2807 3459]] | +0.0472 | 0.0854 | 7.12 |
| 2.3.5.7.11.13 | 1001/1000, 3025/3024, 4225/4224, 4375/4374, 708883245/708837376 | [⟨1000 1585 2322 2807 3459 3700]] | +0.0631 | 0.0857 | 7.14 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 161\1000 | 193.200 | 262144/234375 | Lunatic (7-limit) |
| 1 | 269\1000 | 322.800 | 3087/2560 | Seniority |
| 4 | 317\1000 (67\1000) |
380.400 (80.400) |
5103/4096 (22/21) |
Quasithird |
| 10 | 263\1000 (37\1000) |
315.600 (44.400) |
6/5 (15/14) |
Deca |
| 25 | 301\1000 (21\1000) |
361.200 (25.200) |
[54 13 -32⟩ (?) |
Manganese |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct