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, consistent to the 17-odd-limit or the no-19 no-29 41-odd-limit. It 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.

In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.

Prime harmonics

Approximation of prime harmonics in 764edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Error	Absolute (¢)	+0.000	+0.139	+0.074	+0.284	-0.009	-0.214	+0.280	-0.654	-0.002	-0.781	-0.009	-0.035	-0.267	+0.524	+0.462
	Relative (%)	+0.0	+8.9	+4.7	+18.1	-0.6	-13.6	+17.8	-41.7	-0.1	-49.7	-0.6	-2.2	-17.0	+33.4	+29.4
Steps (reduced)		764 (0)	1211 (447)	1774 (246)	2145 (617)	2643 (351)	2827 (535)	3123 (67)	3245 (189)	3456 (400)	3711 (655)	3785 (729)	3980 (160)	4093 (273)	4146 (326)	4244 (424)

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 (22\16808).

Regular temperament properties

Template:Comma basis begin |- | 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 |- | 2.3.5.7.11.13.17.23 | 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095 | [⟨764 1211 1774 2145 2643 2827 3123 3456]] | −0.0286 | 0.0506 | 3.22 Template:Comma basis end

• 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.
• It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past 494 and before 1578.

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf