393edo

← 392edo

393edo

394edo →

Prime factorization	3 × 131
Step size	3.05344¢
Fifth	230\393 (702.29¢)
Semitones (A1:m2)	38:29 (116¢ : 88.55¢)
Consistency limit	5
Distinct consistency limit	5

393 equal divisions of the octave (abbreviated 393edo or 393ed2), also called 393-tone equal temperament (393tet) or 393 equal temperament (393et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 393 equal parts of about 3.05 ¢ each. Each step represents a frequency ratio of 2^1/393, or the 393rd root of 2.

Theory

393et is only consistent to the 5-limit, with three mappings possible for the 7-limit:

• ⟨393 623 913 1103] (patent val),
• ⟨393 623 912 1103] (393c),
• ⟨393 623 913 1104] (393d).

Using the patent val, it tempers out 393216/390625 and [-46 51 -15⟩ in the 5-limit; 10976/10935, 393216/390625 and 5250987/5242880 in the 7-limit.

Using the 393c val, it tempers out 2109375/2097152 and [32 -48 19⟩ in the 5-limit; 2401/2400, 1071875/1062882 and 2109375/2097152 in the 7-limit.

Using the 393d val, it tempers out 393216/390625 and [-46 51 -15⟩ in the 5-limit; 250047/250000, 2460375/2458624 and 2097152/2083725 in the 7-limit.

Odd harmonics

Approximation of odd harmonics in 393edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.34	+1.47	-0.89	+0.67	+1.35	-0.83	-1.25	-1.14	-1.33	-0.55	+0.73
	Relative (%)	+11.0	+48.2	-29.0	+21.9	+44.3	-27.3	-40.8	-37.3	-43.6	-18.1	+24.0
Steps (reduced)		623 (230)	913 (127)	1103 (317)	1246 (67)	1360 (181)	1454 (275)	1535 (356)	1606 (34)	1669 (97)	1726 (154)	1778 (206)

Subsets and supersets

393 factors into 3 × 131, with 3edo and 131edo as its subset edos. 786edo, which doubles it, gives a good correction to the harmonic 7.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[623 -393⟩	[⟨393 623]]	-0.1057	0.1057	3.43
2.3.5	393216/390625, [-46 51 -15⟩	-0.2819	0.2636	8.63

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	127\393	387.79	5/4	Würschmidt