400edo
| ← 399edo | 400edo | 401edo → |
400 equal divisions of the octave (abbreviated 400edo or 400ed2), also called 400-tone equal temperament (400tet) or 400 equal temperament (400et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 400 equal parts of exactly 3 ¢ each. Each step represents a frequency ratio of 21/400, or the 400th root of 2.
Theory
400edo is a strong 17- and 19-limit system, distinctly and purely consistent to the 21-odd-limit. It shares its excellent harmonic 3 with 200edo, which is a semiconvergent, while correcting the higher harmonics to near-just qualities.
As an equal temperament, it tempers out the unidecma, [-7 22 -12⟩, and the quintosec 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.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.04 | +0.69 | +0.17 | +0.68 | -0.53 | +0.04 | -0.51 | -1.27 | -0.58 | +0.96 | +0.66 | -0.06 |
| Relative (%) | +0.0 | +1.5 | +22.9 | +5.8 | +22.7 | -17.6 | +1.5 | -17.1 | -42.5 | -19.2 | +32.1 | +21.9 | -2.1 | |
| Steps (reduced) |
400 (0) |
634 (234) |
929 (129) |
1123 (323) |
1384 (184) |
1480 (280) |
1635 (35) |
1699 (99) |
1809 (209) |
1943 (343) |
1982 (382) |
2084 (84) |
2143 (143) | |
| Harmonic | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | 97 | 101 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.48 | +0.49 | -0.50 | -0.17 | -0.88 | -1.31 | +0.30 | +0.21 | +1.46 | -0.05 | -0.88 | +0.10 | -0.85 |
| Relative (%) | +49.4 | +16.4 | -16.8 | -5.7 | -29.5 | -43.6 | +10.1 | +7.0 | +48.8 | -1.6 | -29.3 | +3.5 | -28.5 | |
| Steps (reduced) |
2171 (171) |
2222 (222) |
2291 (291) |
2353 (353) |
2372 (372) |
2426 (26) |
2460 (60) |
2476 (76) |
2522 (122) |
2550 (150) |
2590 (190) |
2640 (240) |
2663 (263) | |
Subsets and supersets
Since 400 factors into 24 × 52, 400edo has subset edos 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200.
Interval table
All intervals
See Table of 400edo intervals.
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, 55/52 |
| 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 | 525/352, 3/2 / (81/80)^(5/12) |
| 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 |
- 400et has lower absolute errors than any previous equal temperaments in the 17- and 19-limit. It is the first to beat 354 in the 17-limit, and 311 in the 19-limit; it is bettered by 422 in either subgroup.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 83\400 | 249.00 | [-26 18 -1⟩ | Monzismic |
| 1 | 33\400 | 99.00 | 18/17 | Gregorian leap day |
| 1 | 101\400 | 303.00 | 25/21 | Quinmite |
| 1 | 153\400 | 459.00 | 125/96 | Majvamic |
| 1 | 169\400 | 507.00 | 525/352 | Gregorian leap week |
| 2 | 61\400 | 183.00 | 10/9 | Unidecmic |
| 5 | 123\400 (37\400) |
369.00 (111.00) |
1024/891 (16/15) |
Quintosec |
| 10 | 83\400 (3\400) |
249.00 (9.00) |
15/13 (176/175) |
Decoid |
| 80 | 166\400 (1\400) |
498.00 (3.00) |
4/3 (245/243) |
Octogintic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
- Huntington7
- Huntington10
- Huntington17
- Monzismic[29]
- GregorianLeapWeek[71]
- ISOWeek[71]
- GregorianLeapDay[97]
Music
- Etude in Monzismic (2023)
- thank you all (2023)