624edo

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624edo

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Prime factorization	24 × 3 × 13
Step size	1.92308 ¢
Fifth	365\624 (701.923 ¢)
Semitones (A1:m2)	59:47 (113.5 ¢ : 90.38 ¢)
Consistency limit	27
Distinct consistency limit	27

Theory

624edo is consistent to the 27-odd-limit. The equal temperament tempers out [23 6 -14⟩ (vishnuzma) and [-69 45 -1⟩ (counterschisma) in the 5-limit; 250047/250000, 2460375/2458624, and 134217728/133984375 in the 7-limit; 9801/9800, 46656/46585, 131072/130977, and 151263/151250 in the 11-limit; 1716/1715, 2080/2079, 4096/4095, 34398/34375, and 39366/39325 in the 13-limit; 936/935, 1701/1700, 2025/2023, and 2058/2057 in the 17-limit; 1521/1520, 2376/2375, 2432/2431, and 3328/3325 in the 19-limit; 2024/2023, 2025/2024, and 3888/3887 in the 23-limit.

Prime harmonics

Approximation of prime harmonics in 624edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.032	+0.225	+0.405	+0.605	-0.143	+0.814	+0.564	+0.572	-0.731	-0.805
	Relative (%)	+0.0	-1.7	+11.7	+21.1	+31.5	-7.4	+42.3	+29.3	+29.7	-38.0	-41.8
Steps (reduced)		624 (0)	989 (365)	1449 (201)	1752 (504)	2159 (287)	2309 (437)	2551 (55)	2651 (155)	2823 (327)	3031 (535)	3091 (595)

Subsets and supersets

Since 624 factors into 2⁴ × 3 × 13, 624edo has subset edos 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[-989 624⟩	[⟨624 989]]	+0.0101	0.0101	0.52
2.3.5	[23 6 -14⟩, [-69 45 -1⟩	[⟨624 989 1449]]	-0.0256	0.0510	2.65
2.3.5.7	250047/250000, 2460375/2458624, [27 0 -8 -3⟩	[⟨624 989 1449 1752]]	-0.0552	0.0678	3.52
2.3.5.7.11	9801/9800, 46656/46585, 131072/130977, 151263/151250	[⟨624 989 1449 1752 2159]]	-0.0792	0.0772	4.02
2.3.5.7.11.13	1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325	[⟨624 989 1449 1752 2159 2309]]	-0.0595	0.0831	4.32
2.3.5.7.11.13.17	936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011	[⟨624 989 1449 1752 2159 2309 2551]]	-0.0795	0.0911	4.74
2.3.5.7.11.13.17.19	936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011	[⟨624 989 1449 1752 2159 2309 2551 2651]]	-0.0861	0.0870	4.53

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	259\624	498.08	4/3	Counterschismic
1	311\624	598.08	847/600	Vydubychi
2	37\624	71.15	25/24	Vishnu (5-limit)
3	73\624	140.38	243/224	Septichrome
6	177\624 (73\624)	340.38 (140.38)	3400/2793 (425/392)	Semiseptichrome
12	259\624 (1\624)	498.08 (1.92)	4/3 (32805/32768)	Atomic
13	259\624 (19\624)	498.08 (36.54)	4/3 (?)	Aluminium
16	259\624 (14\624)	498.08 (48.077)	4/3 (?)	Sulfur
24	303\624 (17\624)	582.692 (32.692)	7/5 (?)	Chromium
26	259\624 (19\624)	498.08 (36.54)	4/3 (?)	Iron

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