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 4914/4913 and 24576/24565 in the 17-limit, and 1729/1728 with 93347/93312 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 can be treated as 5/12 syntonic comma meantone, which is the first meantone in the continued fraction that offers good precision. Other items like 1/3 and 2/5 eventually become inconsistent with the edo.

The leap week scale offers an interest in that 1/7th of its generator, 33\400, is associated to 18/17, making it an interpretation of 18/17s equal temperament. Since it tempers out the 93347/93312, a stack of three 18/17s is equated with 19/16.

Table of intervals

Step	Name	Associated ratio	Notes
0	unison	1/1 exact
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, twelfth root of 800000/6561
234	perfect fifth	3/2
323	harmonic seventh	7/4
372	5/12-meantone seventh	12500/6561
400	octave	2/1 exact

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]}