684edo

This page presents a novelty topic. It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks.

This page is a stub. You can help the Xenharmonic Wiki by expanding it.

← 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

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 as 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

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (Reduced)	Cents (Reduced)	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