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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
400edo is a strong 17- and 19-limit system, [[distinctly consistent]] to the [[21-odd-limit]] | 400edo is a strong 17- and 19-limit system, [[consistency|distinctly and purely consistent]] to the [[21-odd-limit]]. It shares its excellent [[harmonic]] [[3/1|3]] with [[200edo]], which is a semiconvergent, while correcting the higher harmonics to near-just qualities. | ||
As an equal temperament, it [[tempering out|tempers out]] the unidecma, {{monzo| -7 22 -12 }}, and the quintosec comma, {{monzo| 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]], [[support]]ing 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 === | === Prime harmonics === | ||
{{Harmonics in equal|400|columns=13}} | {{Harmonics in equal|400|columns=13}} | ||
{{Harmonics in equal|400 | {{Harmonics in equal|400|columns=13|start=14|collapsed=true|title=Approximation of prime harmonics in 400edo (continued)}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 400 factors into | Since 400 factors into 2<sup>4</sup> × 5<sup>2</sup>, 400edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200 }}. | ||
Of edos that are a multiple of 400, {{EDOs| 1600 and 2000}} are notable for their high consistency limits, as [[Interval size measure|interval size measures]], and perhaps as ways of tuning various temperaments. | |||
== Interval table == | == Interval table == | ||
=== All intervals === | === All intervals === | ||
See [[Table of 400edo intervals]]. | |||
=== Selected intervals === | === Selected intervals === | ||
| Line 24: | Line 24: | ||
|- | |- | ||
! Step | ! Step | ||
! Eliora's | ! Eliora's naming system | ||
! Associated | ! Associated ratio | ||
|- | |- | ||
| 0 | | 0 | ||
Latest revision as of 12:33, 14 August 2025
| ← 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.
Of edos that are a multiple of 400, 1600 and 2000 are notable for their high consistency limits, as interval size measures, and perhaps as ways of tuning various temperaments.
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)