764edo

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764edo

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Prime factorization	22 × 191
Step size	1.57068 ¢
Fifth	447\764 (702.094 ¢)
Semitones (A1:m2)	73:57 (114.7 ¢ : 89.53 ¢)
Consistency limit	17
Distinct consistency limit	17

Theory

764edo is a very strong 17-limit system distinctly consistent to the 17-odd-limit, and is the fourteenth zeta integral edo. In the 5-limit it tempers out the hemithirds comma, [38 -2 -15⟩; in the 7-limit 4375/4374; in the 11-limit 3025/3024 and 9801/9800; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647; and in the 17-limit 2431/2430, 2500/2499, 4914/4913 and 5832/5831. It provides the optimal patent val for the abigail temperament in the 11-limit.

Prime harmonics

Approximation of prime harmonics in 764edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.139	+0.074	+0.284	-0.009	-0.214	+0.280	-0.654	-0.002	-0.781	-0.009
	Relative (%)	+0.0	+8.9	+4.7	+18.1	-0.6	-13.6	+17.8	-41.7	-0.1	-49.7	-0.6
Steps (reduced)		764 (0)	1211 (447)	1774 (246)	2145 (617)	2643 (351)	2827 (535)	3123 (67)	3245 (189)	3456 (400)	3711 (655)	3785 (729)

Subsets and supersets

Since 764 factors into 2² × 191, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 jinns.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[1211 -764⟩	[⟨764 1211]]	-0.0439	0.0439	2.80
2.3.5	[38 -2 -15⟩, [25 -48 22⟩	[⟨764 1211 1774]]	-0.0399	0.0363	2.31
2.3.5.7	4375/4374, 52734375/52706752, [31 -6 -2 -6⟩	[⟨764 1211 1774 2145]]	-0.0552	0.0412	2.62
2.3.5.7.11	3025/3024, 4375/4374, 131072/130977, 35156250/35153041	[⟨764 1211 1774 2145 2643]]	-0.0436	0.0435	2.77
2.3.5.7.11.13	1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875	[⟨764 1211 1774 2145 2643 2827]]	-0.0267	0.0548	3.49
2.3.5.7.11.13.17	1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913	[⟨764 1211 1774 2145 2643 2827 3123]]	-0.0327	0.0528	3.36

• 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats 684 and is only bettered by 935. In the 17-limit it beats 742 and is only bettered by 814.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	123\764	193.19	262144/234375	Lunatic (7-limit)
1	277\764	435.08	9/7	Supermajor
2	133\764	208.90	44/39	Abigail
2	277\764 (105\764)	435.08 (164.92)	9/7 (11/10)	Semisupermajor