400edo
| ← 399edo | 400edo | 401edo → |
Theory
400edo is a strong 17- and 19-limit system, distinctly 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.
The equal temperament 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) | |
| Harmonic | 103 | 107 | 109 | 113 | 127 | 131 | 137 | 139 | 149 | 151 | 157 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.20 | +1.24 | -0.82 | -0.21 | -1.42 | -1.11 | -0.64 | +1.27 | +1.00 | -1.09 | +0.46 |
| Relative (%) | +40.0 | +41.3 | -27.4 | -7.2 | -47.4 | -36.9 | -21.3 | +42.4 | +33.3 | -36.2 | +15.2 | |
| Steps (reduced) |
2675 (275) |
2697 (297) |
2707 (307) |
2728 (328) |
2795 (395) |
2813 (13) |
2839 (39) |
2848 (48) |
2888 (88) |
2895 (95) |
2918 (118) | |
| Harmonic | 163 | 167 | 173 | 179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 | 227 | 229 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.47 | -1.45 | +0.45 | +1.42 | +0.18 | +0.09 | +0.05 | +0.54 | +1.05 | -1.32 | -1.08 | +1.14 | +0.96 |
| Relative (%) | -49.1 | -48.2 | +14.9 | +47.4 | +6.2 | +2.8 | +1.7 | +17.9 | +35.0 | -44.0 | -36.0 | +38.1 | +31.8 | |
| Steps (reduced) |
2939 (139) |
2953 (153) |
2974 (174) |
2994 (194) |
3000 (200) |
3031 (231) |
3037 (237) |
3049 (249) |
3055 (255) |
3088 (288) |
3120 (320) |
3131 (331) |
3136 (336) | |
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
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
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf
Scales
- Huntington7
- Huntington10
- Huntington17
- Monzismic[29]
- GregorianLeapWeek[71]
- ISOWeek[71]
- GregorianLeapDay[97]
Music
- Etude in Monzismic (2023)
- thank you all (2023)