468edo

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468edo

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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

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.

Theory

468et tempers out 78125000/78121827, 4375/4374, 250047/250000, 2401/2400, 420175/419904, 200120949/200000000 and 40353607/40310784 in the 7-limit; 21437500/21434787, 151263/151250, 117649/117612, 514714375/514434888, 47265625/47258883, 9801/9800, 3025/3024, 1890625/1889568, 160083/160000, 41503/41472, 3294225/3294172, 43923/43904, 102487/102400 and 1771561/1769472 in the 11-limit. It provides the optimal patent val for 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

468 factors into 2² × 3² × 13, with 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	[371 -234⟩	[⟨468 742]]	−0.1922	0.1921	7.49
2.3.5	2199023255552/2179240250625, 7629394531250/7625597484987	[⟨468 742 1087]]	−0.2524	0.1785	6.96
2.3.5.7	4375/4374, 2401/2400, 47141561040896/46708681640625	[⟨468 742 1087 1314]]	−0.2253	0.1615	6.30
2.3.5.7.11	3025/3024, 4375/4374, 2401/2400, 5767168/5740875	[⟨468 742 1087 1314 1619]]	−0.1782	0.1725	6.73
2.3.5.7.11.13	2080/2079, 1001/1000, 3025/3024, 1716/1715, 1982464/1974375	[⟨468 742 1087 1314 1619 1732]]	−0.1709	0.1583	6.17
2.3.5.7.11.13.17	2080/2079, 1001/1000, 3025/3024, 1716/1715, 17920/17901, 30464/30375	[⟨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)	Undim
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 it is distinct