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). The error is only about 1/22 cents. 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).
The equal temperament 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 52 × 23. 200edo's subset edos are: 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 it is 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]