416edo

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416edo

417edo →

Prime factorization	25 × 13
Step size	2.88462 ¢
Fifth	243\416 (700.962 ¢)
Semitones (A1:m2)	37:33 (106.7 ¢ : 95.19 ¢)
Dual sharp fifth	244\416 (703.846 ¢) (→ 61\104)
Dual flat fifth	243\416 (700.962 ¢)
Dual major 2nd	71\416 (204.808 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

416et is consistent to the 7-odd-limit and the harmonic 3 is about halfway its steps. It is suitable for the 2.9.5.7.11.19.23.29.31.37 subgroup, tempering out 1540/1539, 5632/5625, 9801/9800, 10241/10240, 1045/1044, 26125/26082, 46000/45927, 17600/17577 and 1036/1035. It supports tridecatonic, fermionic, embankment and polder.

Odd harmonics

Approximation of odd harmonics in 416edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-0.99	+0.22	+0.40	+0.90	-0.36	-1.10	-0.77	-1.11	-0.40	-0.59	+0.57
	Relative (%)	-34.4	+7.8	+14.0	+31.1	-12.4	-38.3	-26.6	-38.5	-13.8	-20.4	+19.8
Steps (reduced)		659 (243)	966 (134)	1168 (336)	1319 (71)	1439 (191)	1539 (291)	1625 (377)	1700 (36)	1767 (103)	1827 (163)	1882 (218)

Subsets and supersets

416 factors into 2⁵ × 13, with subset edos 2, 4, 8, 13, 16, 26, 32, 52, 104, and 208. 832edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.9	[1319 -416⟩	[⟨416 1319]]	-0.1416	0.1416	4.91
2.9.5 [56 -14 -5⟩, [-5 -16 24⟩	[⟨416 1319 966]]	-0.1267	0.1175	4.07
2.9.5.7	420175/419904, 102760448/102515625, 1280000000/1275989841	[⟨416 1319 966 1168]]	-0.1310	0.1021	3.54
2.9.5.7.11	5632/5625, 9801/9800, 41503/41472, 774400000/771895089	[⟨416 1319 966 1168 1439]]	-0.0842	0.1308	4.53