200edo
| ← 199edo | 200edo | 201edo → |
(semiconvergent)
200 equal divisions of the octave (abbreviated 200edo or 200ed2), also called 200-tone equal temperament (200tet) or 200 equal temperament (200et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 200 equal parts of exactly 6 ¢ each. Each step represents a frequency ratio of 21/200, or the 200th root of 2.
Theory
200edo contains a perfect fifth of exactly 702 cents and a perfect fourth of exactly 498 cents, which is accurate due to 200 being the denominator of a continued fraction convergent to log2(3/2). Only about 0.045 cents sharp, it is the next best fifth in absolute error after 53edo's. In light of having its perfect fifth precise and the step divisible by 9, it is essentially a perfect edo for Carlos Alpha, even up many octaves (the difference between 13 steps of 200edo and 1 step of Carlos Alpha is only 0.03501 cents).
It tempers out the schisma (32805/32768) and the quartemka, [2 -32 21⟩ in the 5-limit, and the gamelisma, 1029/1024, in the 7-limit, so that it supports the guiron temperament.
One step of 200edo is close to 289/288. Unfortunately, it is not compatible with any obvious 2.3.17 subgroup mappings of 200edo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.04 | -2.31 | -2.83 | +0.68 | -0.53 | -2.96 | +2.49 | +1.73 | +2.42 | +0.96 |
| Relative (%) | +0.0 | +0.7 | -38.6 | -47.1 | +11.4 | -8.8 | -49.3 | +41.4 | +28.8 | +40.4 | +16.1 | |
| Steps (reduced) |
200 (0) |
317 (117) |
464 (64) |
561 (161) |
692 (92) |
740 (140) |
817 (17) |
850 (50) |
905 (105) |
972 (172) |
991 (191) | |
Subsets and supersets
200 factorizes as 23 × 52, and has subset edos 2, 4, 5, 8, 10, 20, 25, 40, 50, 100.
400edo, which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [317 -200⟩ | [⟨200 317]] | −0.0142 | 0.0142 | 0.24 |
| 2.3.5 | 32805/32768, [2 -32 21⟩ | [⟨200 317 464]] | +0.3226 | 0.4767 | 7.95 |
| 2.3.5.7 | 1029/1024, 10976/10935, 390625/387072 | [⟨200 317 464 561]] | +0.4937 | 0.5082 | 8.47 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 23\200 | 138.00 | 27/25 | Quartemka |
| 1 | 39\200 | 234.00 | 8/7 | Guiron |
| 1 | 83\200 | 498.00 | 4/3 | Helmholtz |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
- 34 34 15 34 34 34 15 = Pythagorean tuning
- 32 32 20 32 32 32 20 = Meantone tuning in the same way of 50edo
- 27 27 27 27 27 27 27 11 = Porcupine tuning
- 26 26 26 9 26 26 26 26 9 = Superdiatonic tuning
- 24 24 24 16 24 24 24 24 16 = Superdiatonic tuning in the same way of 25edo
- 22 22 8 22 22 22 8 22 22 22 8 = Sensi
- 16 16 16 8 16 16 16 16 8 16 16 16 16 8 = Ketradektriatoh tuning
Music
- Fugue on Elgar’s Enigma Theme – YouTube | soonlabel archive[dead link] | play[dead link]