317edo

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317edo

318edo →

Prime factorization	317 (prime)
Step size	3.78549 ¢
Fifth	185\317 (700.315 ¢)
Semitones (A1:m2)	27:26 (102.2 ¢ : 98.42 ¢)
Dual sharp fifth	186\317 (704.101 ¢)
Dual flat fifth	185\317 (700.315 ¢)
Dual major 2nd	54\317 (204.416 ¢)
Consistency limit	5
Distinct consistency limit	5

Theory

317et tempers out 589824/588245, 16875/16807, 65625/65536 and 49009212/48828125 in the 7-limit; 1835008/1830125, 14700/14641, 2097152/2096325, 4000/3993, 2734375/2725888, 1953125/1951488, 172032/171875, 441/440, 5767168/5764801, 825000/823543, 537109375/536870912, 134775333/134217728, 160083/160000, 16808715/16777216, 539055/537824 and 3294225/3294172 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 317edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-1.64	-0.19	+0.26	+0.51	+1.36	-0.15	-1.83	+1.04	+1.54	-1.38	+0.12
	Relative (%)	-43.3	-5.1	+6.8	+13.4	+36.0	-3.9	-48.4	+27.4	+40.7	-36.5	+3.1
Steps (reduced)		502 (185)	736 (102)	890 (256)	1005 (54)	1097 (146)	1173 (222)	1238 (287)	1296 (28)	1347 (79)	1392 (124)	1434 (166)

Subsets and supersets

317edo is the 66th prime edo. 634edo, 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	[1005 -317⟩	[⟨317 1005]]	-0.0799	0.0799	2.11
2.9.5	[-53 5 16⟩, [33 -17 9⟩	[⟨317 1005 736]]	-0.0254	0.1009	2.67
2.9.5.7	420175/419904, 703125/702464, 33554432/33480783	[⟨317 1005 736 890]]	-0.0422	0.0921	2.43
2.9.5.7.11	6250/6237, 12005/11979, 46656/46585, 151263/151250	[⟨317 1005 736 890 1097]]	-0.1126	0.1631	4.31
2.9.5.7.11.13	1575/1573, 4459/4455, 6250/6237, 67392/67375, 190125/189728	[⟨317 1005 736 890 1097 1173]]	-0.0871	0.1594	4.21
2.9.5.7.11.13.17	936/935, 1225/1224, 1575/1573, 12376/12375, 17920/17901, 34000/33957	[⟨317 1005 736 890 1097 1173 1296]]	-0.1109	0.1587	4.19

Music

Francium

• "Narrator Of Myths" from Mysteries (2023) Spotify | Bandcamp | YouTube