196edo

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196edo

197edo →

Prime factorization	22 × 72
Step size	6.12245¢
Fifth	115\196 (704.082¢)
Semitones (A1:m2)	21:13 (128.6¢ : 79.59¢)
Dual sharp fifth	115\196 (704.082¢)
Dual flat fifth	114\196 (697.959¢) (→57\98)
Dual major 2nd	33\196 (202.041¢)
Consistency limit	5
Distinct consistency limit	5

196 equal divisions of the octave (abbreviated 196edo or 196ed2), also called 196-tone equal temperament (196tet) or 196 equal temperament (196et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 196 equal parts of about 6.12 ¢ each. Each step represents a frequency ratio of 2^1/196, or the 196th root of 2.

Theory

The equal temperament tempers out 9765625/9565938 (fifive comma) and [32 -7 -9⟩ (escapade comma) in the 5-limit. Using the patent val, it tempers out 245/243, 65625/65536, and 235298/234375 in the 7-limit; 385/384, 896/891, 3388/3375, and 117649/117128 in the 11-limit; 352/351, 364/363, 625/624, 1001/1000, and 9295/9261 in the 13-limit; 289/288, 442/441, 715/714, and 1156/1155 in the 17-limit.

196edo can also treated as a 2.9.5.7.11.13.17 subgroup temperament (with the patent 9), providing a distinct flat tendency for harmonics 5, 7, 9, 11, 13, and 17. With the patent 9, it tempers out 321489/320000, 420175/419904, and 703125/702464 in the 2.9.5.7 subgroup; 441/440, 8019/8000, 41503/41472, and 9453125/9437184 in the 2.9.5.7.11 subgroup; 729/728, 1001/1000, 1575/1573, 6656/6655, and 10985/10976 in the 2.9.5.7.11.13 subgroup; 833/832, 936/935, 1089/1088, 1225/1224, 2025/2023, and 14365/14336 in the 2.9.5.7.11.13.17 subgroup.

Since it is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates peppermint temperament.

Odd harmonics

Approximation of odd harmonics in 196edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+2.13	-0.60	-1.48	-1.87	-0.30	-1.75	+1.53	-0.87	+2.49	+0.65	+2.34
	Relative (%)	+34.7	-9.8	-24.2	-30.5	-4.9	-28.6	+24.9	-14.3	+40.6	+10.6	+38.2
Steps (reduced)		311 (115)	455 (63)	550 (158)	621 (33)	678 (90)	725 (137)	766 (178)	801 (17)	833 (49)	861 (77)	887 (103)

Subsets and supersets

Since 196 factors into 2² × 7², 196edo has subset edos 2, 4, 7, 14, 28, 49, and 98.

Intervals

See: Table of 196edo intervals