400edo

← 399edo

400edo

401edo →

Prime factorization	24 × 52
Step size	3 ¢
Fifth	234\400 (702 ¢) (→ 117\200)
Semitones (A1:m2)	38:30 (114 ¢ : 90 ¢)
Consistency limit	21
Distinct consistency limit	21

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.

400 factors into 2⁴ × 5², with subset edos 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, and 200. Notably, 200edo 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, and the corresponding temperament is 97 & 400, with comma list 2432/2431, 2601/2600, 2926/2925, 6175/6174, 17689/17680, and 22477/22440.

Prime harmonics

Approximation of prime harmonics in 400edo

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)

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	85/57, 94/63
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

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
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	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]