468edo

468 equal divisions of the octave (abbreviated 468edo or 468ed2), also called 468-tone equal temperament (468tet) or 468 equal temperament (468et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 468 equal parts of about 2.56 ¢ each. Each step represents a frequency ratio of 2^1/468, or the 468th root of 2.

← 467edo

468edo

469edo →

Prime factorization	22 × 32 × 13
Step size	2.5641 ¢
Fifth	274\468 (702.564 ¢) (→ 137\234)
Semitones (A1:m2)	46:34 (117.9 ¢ : 87.18 ¢)
Consistency limit	13
Distinct consistency limit	13

Theory

468edo is consistent to the 13-odd-limit with a sharp tuning tendency and a 2.3.7.13 subgroup inherited from 234edo.

As an equal temperament, it tempers out the undim comma and the ennealimma in the 5-limit; 2401/2400 and 4375/4374 in the 7-limit, supporting ennealimmal; 3025/3024 and 9801/9800 in the 11-limit, supporting hemiennealimmal; and 676/675, 1001/1000, 1716/1715, 2080/2079 in the 13-limit, supporting 13-limit hemiennealimmal. It also provides the optimal patent val for 13-limit unlit.

Odd harmonics

Approximation of odd harmonics in 468edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.61	+0.87	+0.40	+1.22	-0.04	+0.50	-1.09	+0.17	-0.08	+1.01	-0.07
	Relative (%)	+23.8	+33.8	+15.8	+47.5	-1.4	+19.4	-42.5	+6.7	-3.0	+39.5	-2.7
Steps (reduced)		742 (274)	1087 (151)	1314 (378)	1484 (80)	1619 (215)	1732 (328)	1828 (424)	1913 (41)	1988 (116)	2056 (184)	2117 (245)

Subsets and supersets

Since 468 factors into primes as 2² × 3² × 13, 468edo has subset edos 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, and 234.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	[41 -20 -4⟩, [1 -27 18⟩	[⟨468 742 1087]]	−0.2524	0.1785	6.96
2.3.5.7	2401/2400, 4375/4374, [34 -14 -10 3⟩	[⟨468 742 1087 1314]]	−0.2253	0.1615	6.30
2.3.5.7.11	2401/2400, 3025/3024, 4375/4374, 5767168/5740875	[⟨468 742 1087 1314 1619]]	−0.1782	0.1725	6.73
2.3.5.7.11.13	676/675, 1001/1000, 1716/1715, 3025/3024, 495616/494325	[⟨468 742 1087 1314 1619 1732]]	−0.1709	0.1583	6.17
2.3.5.7.11.13.17	676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 7616/7605	[⟨468 742 1087 1314 1619 1732 1913]]	−0.1526	0.1533	5.98

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	179\468	458.97	125/96	Majvam
4	194\468 (40\468)	497.44 (102.56)	4/3 (35/33)	Unlit
9	123\468 (19\468)	315.38 (48.72)	6/5 (36/35)	Ennealimmal
18	97\468 (7\468)	248.72 (17.95)	231/200 (99/98)	Hemiennealimmal

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