400edo
| ← 399edo | 400edo | 401edo → |
The 400 equal divisions of the octave (400edo), or the 400(-tone) equal temperament (400tet, 400et) when viewed from a regular temperament perspective, is the equal division of the octave into 400 parts of exact 3 cents each.
Theory
400edo is consistent in the 21-odd-limit. It tempers out the unidecma, [-7 22 -12⟩, and the qintosec comma, [47 -15 -10⟩, in the 5-limit; 2401/2400, 1959552/1953125, and 14348907/14336000 in the 7-limit; 5632/5625, 9801/9800, 117649/117612, and 131072/130977 in the 11-limit; 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095, 4225/4224 and 39366/39325 in the 13-limit, supporting the decoid temperament and the quinmite temperament. It tempers out 936/935, 1156/1155, 2058/2057, 2601/2600, 4914/4913 and 24576/24565 in the 17-limit, and 969/968, 1216/1215, 1521/1520, and 1729/1728 in the 19-limit.
400edo doubles 200edo, which holds a record for the best 3/2 fifth approximation.
400 is also the number of years in the Gregorian calendar's leap cycle. 400edo supports the LeapWeek[71] scale with 231\400 as the generator, which is close to 5/12 syntonic comma meantone. Likewise, 400edo contains LeapDay[97] scale, which is a maximal evenness version of the leap rule currently in use in the world today. The scale has a 33\400 generator which is associated to 18/17, making it an approximation of 18/17 equal-step tuning. Since it tempers out the 93347/93312, a stack of three 18/17's is equated with 19/16.
Prime harmonics
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Selected intervals
| Step | Eliora's Naming System | Associated ratio |
|---|---|---|
| 0 | unison | 1/1 |
| 28 | 5/12-meantone semitone | 6561/6250 |
| 33 | small septendecimal semitone | 18/17 |
| 35 | septendecimal semitone | 17/16 |
| 37 | diatonic semitone | 16/15 |
| 99 | undevicesimal minor third | 19/16 |
| 100 | symmetric minor third | |
| 200 | symmetric tritone | 99/70, 140/99 |
| 231 | Gregorian leap week fifth | 118/79 |
| 234 | perfect fifth | 3/2 |
| 323 | harmonic seventh | 7/4 |
| 372 | 5/12-meantone seventh | 12500/6561 |
| 400 | octave | 2/1 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [-7 22 -12⟩, [47 -15 -10⟩ | [⟨400 634 929]] | -0.1080 | 0.1331 | 4.44 |
| 2.3.5.7 | 2401/2400, 1959552/1953125, 14348907/14336000 | [⟨400 634 929 1123]] | -0.0965 | 0.1170 | 3.90 |
| 2.3.5.7.11 | 2401/2400, 5632/5625, 9801/9800, 46656/46585 | [⟨400 634 929 1123 1384]] | -0.1166 | 0.1121 | 3.74 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 1716/1715, 4096/4095, 39366/39325 | [⟨400 634 929 1123 1384 1480]] | -0.0734 | 0.1407 | 4.69 |
| 2.3.5.7.11.13.17 | 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 4096/4095 | [⟨400 634 929 1123 1384 1480 1635]] | -0.0645 | 0.1321 | 4.40 |
| 2.3.5.7.11.13.17.19 | 676/675, 936/935, 969/968, 1001/1000, 1156/1155, 1216/1215, 1716/1715 | [⟨400 634 929 1123 1384 1480 1635 1699]] | -0.0413 | 0.1380 | 4.60 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 83\400 | 249.00 | [-26 18 -1⟩ | Monzismic |
| 1 | 101\400 | 303.00 | 25/21 | Quinmite |
| 1 | 153\400 | 459.00 | 125/96 | Majvam |
| 2 | 61\400 | 183.00 | 10/9 | Unidecmic |
| 5 | 123\400 (37\400) |
369.00 (111.00) |
10125/8192 (16/15) |
Qintosec (5-limit) |
| 10 | 83\400 (3\400) |
249.00 (9.00) |
15/13 (176/175) |
Decoid |
Scales
- Huntington7
- Huntington10
- Huntington17
- LeapWeek[71]
- LeapDay[97]