388edo

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388edo

389edo →

Prime factorization	22 × 97
Step size	3.09278 ¢
Fifth	227\388 (702.062 ¢)
Semitones (A1:m2)	37:29 (114.4 ¢ : 89.69 ¢)
Consistency limit	37
Distinct consistency limit	27

The 388 equal divisions of the octave (388edo), or the 388(-tone) equal temperament (388tet, 388et) when viewed from a regular temperament perspective, divides the octave into 388 equal parts of about 3.09 cents each.

Theory

388edo is the first edo that is uniquely consistent through to the 27-odd-limit; it is also consistent through the 37-odd-limit.

388et tempers out the vishnuzma, [23 6 -14⟩, the tricot comma, [39 -29 3⟩, the minortone comma, [-16 35 -17⟩, and the raider comma, [71 -99 31⟩, in the 5-limit, and provides a tuning with less error than any previous equal temperaments. It tempers out 4375/4374 and 235298/234375 in the 7-limit, and 5632/5625, 3025/3024 and 9801/9800 in the 11-limit and 847/845, 1001/1000 and 4096/4095 in the 13-limit. It is the optimal patent val for cuthbert temperament, which tempers out cuthbert, the 847/845 comma, and for a number of other temperaments tempering out cuthbert, e.g. neusec, the 190&198 temperament. By tempering out cuthbert it supports the cuthbert triad, in addition to sinbadmic chords.

Prime harmonics

Approximation of prime harmonics in 388edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.11	+0.28	-0.78	-0.80	+0.71	+0.20	-0.61	-0.44	+0.32	-0.71
	Relative (%)	+0.0	+3.5	+9.2	-25.4	-25.9	+22.9	+6.4	-19.6	-14.2	+10.3	-22.8
Steps (reduced)		388 (0)	615 (227)	901 (125)	1089 (313)	1342 (178)	1436 (272)	1586 (34)	1648 (96)	1755 (203)	1885 (333)	1922 (370)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[615 -388⟩	[⟨388 615]]	+0.0337	0.0337	1.09
2.3.5	[23 6 -14⟩, [39 -29 3⟩	[⟨388 615 901]]	-0.0633	0.0501	1.62
2.3.5.7	4375/4374, 235298/234375, 2100875/2097152	[⟨388 615 901 1089]]	+0.0224	0.1546	5.00
2.3.5.7.11	3025/3024, 4375/4374, 5632/5625, 235298/234375	[⟨388 615 901 1089 1342]]	+0.0643	0.1617	5.23
2.3.5.7.11.13	847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374	[⟨388 615 901 1089 1342 1436]]	+0.0216	0.1758	5.68
2.3.5.7.11.13.17	833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700	[⟨388 615 901 1089 1342 1436 1586]]	+0.0116	0.1646	5.32
2.3.5.7.11.13.17.19	833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330	[⟨388 615 901 1089 1342 1436 1586 1648]]	+0.0280	0.1600	5.17

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	59\388	182.47	10/9	Mitonic
1	111\388	343.30	8000/6561	Raider
1	145\388	448.45	35/27	Semidimfourth
1	183\388	565.97	75/52	Trillium / pseudotrillium
2	23\388	71.13	25/24	Vishnu / ananta
2	49\388	151.54	12/11	Neusec
4	123\388 (26\388)	380.41 (80.41)	81/65 (22/21)	Quasithird
97	161\388 (1\388)	497.938 (3.09)	4/3 (?)	Berkelium