1000edo: Difference between revisions
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== Theory == | == Theory == | ||
1000edo is related to [[200edo]], but the [[patent val]]s differ on the mapping for [[5/1|5]] and [[7/1|7]]. In the [[5-limit]], it [[tempers out]] {{monzo| 38 -2 -15 }} (luna comma) and {{monzo| -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. | 1000edo is related to [[200edo]], but the [[patent val]]s differ on the mapping for [[5/1|5]] and [[7/1|7]]. In the [[5-limit]], it [[tempering out|tempers out]] {{monzo| 38 -2 -15 }} ([[luna comma]]) and {{monzo| -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 === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 1000 factors into | Since 1000 factors into 2<sup>3</sup> × 5<sup>3</sup>, 1000edo has subset edos {{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, [[11/1|11]], [[13/1|13]], [[17/1|17]], and [[23/1|23]]. | [[2000edo]], which doubles 1000edo, is consistent in the 29-odd-limit and thus provides good corrections for harmonics 7, [[11/1|11]], [[13/1|13]], [[17/1|17]], and [[23/1|23]]. | ||
Revision as of 16:41, 17 January 2025
| ← 999edo | 1000edo | 1001edo → |
1000edo's step size is called a millioctave when used as an interval size unit.
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 (%) | +0.0 | +3.7 | +7.2 | -35.5 | -43.2 | -44.0 | -46.3 | +7.2 | +43.8 | +1.9 | -19.6 | |
| 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
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 distinct