400edo

The 400 equal divisions of the octave (400edo) 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 Sym454 calendar scale with 231\400 as the generator, which is close to 5/12 syntonic comma meantone. The leap week scale offers an interest in that 1/7th of its generator, 33\400, 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.

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

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	83\270	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]^{[clarification needed]}
• LeapDay[97]^{[clarification needed]}