168edo

Prime factorization

2³ × 3 × 7

Step size

7.14286 ¢

Fifth

98\168 (700 ¢) (→ 7\12)

Semitones (A1:m2)

14:14 (100 ¢ : 100 ¢)

Consistency limit

5

Distinct consistency limit

5

168 equal divisions of the octave (abbreviated 168edo or 168ed2), also called 168-tone equal temperament (168tet) or 168 equal temperament (168et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 168 equal parts of about 7.14 ¢ each. Each step represents a frequency ratio of 2^1/168, or the 168th root of 2.

Theory

168edo is closely related to 84edo, but the patent vals differ on the mapping for 11 and 17. It is contorted in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit.

Stacking alternating steps of 43 and 53 produces an optimal Whitewood[14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. Dotcom is also supported.

Odd harmonics

Approximation of odd harmonics in 168edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-1.96	-0.60	+2.60	+3.23	-1.32	+2.33	-2.55	+2.19	+2.49	+0.65	+0.30
Error	Relative (%)	-27.4	-8.4	+36.4	+45.3	-18.5	+32.6	-35.8	+30.6	+34.8	+9.1	+4.2
Steps (reduced)		266 (98)	390 (54)	472 (136)	533 (29)	581 (77)	622 (118)	656 (152)	687 (15)	714 (42)	738 (66)	760 (88)

Subsets and supersets

Since 168 factors into 2³ × 3 × 7, 168edo has subset edos 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, and 84.

Notation

168edo can be notated using ups and downs notation, in which Helmholtz–Ellis arrow accidentals can be used in combination with Stein–Zimmerman quarter tone accidentals: