684edo

← 683edo

684edo

685edo →

Prime factorization	22 × 32 × 19
Step size	1.75439¢
Fifth	400\684 (701.754¢) (→100\171)
Semitones (A1:m2)	64:52 (112.3¢ : 91.23¢)
Consistency limit	17
Distinct consistency limit	17

684 equal divisions of the octave (abbreviated 684edo or 684ed2), also called 684-tone equal temperament (684tet) or 684 equal temperament (684et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 684 equal parts of about 1.75 ¢ each. Each step represents a frequency ratio of 2^1/684, or the 684th root of 2.

Theory

684edo divides the steps of 171edo into four. It is consistent to the 17-odd-limit, tempering out 2401/2400, 3025/3024, 4225/4224, 4375/4374, and 32805/32768 in the 13-limit; 1089/1088, 1225/1224, 1701/1700, 2025/2023, 2058/2057, 2500/2499, 8624/8619, and 14875/14872 in the 17-limit.

Prime harmonics

Approximation of prime harmonics in 684edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.201	-0.349	-0.405	-0.441	-0.177	+0.308	+0.733	-0.204	+0.247	+0.578
	Relative (%)	+0.0	-11.4	-19.9	-23.1	-25.1	-10.1	+17.5	+41.8	-11.6	+14.1	+33.0
Steps (reduced)		684 (0)	1084 (400)	1588 (220)	1920 (552)	2366 (314)	2531 (479)	2796 (60)	2906 (170)	3094 (358)	3323 (587)	3389 (653)

Subsets and supersets

Since 684 factors into 2² × 3² × 19, 684edo has subset edos 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5.7.11.13	2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768	[⟨684 1084 1588 1920 2366 2531]]	+0.0994	0.0558	3.18
2.3.5.7.11.13.17	1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005	[⟨684 1084 1588 1920 2366 2531 2796]]	+0.0744	0.0800	4.56

• 684et is the first equal temperament past 494 with a lower 13-limit absolute error. The next equal temperament that is better tuned is 764.

Rank-2 temperaments

Note: 11-limit temperaments supported by 342et are not shown.

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
18	271\684 (5\684)	475.44 (8.77)	1053/800 (1287/1280)	Semihemiennealimmal
38	151\684 (7\684)	264.91 (12.28)	500/429 (144/143)	Semihemienneadecal

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct