769edo

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769edo

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Prime factorization	769 (prime)
Step size	1.56047¢
Fifth	450\769 (702.211¢)
Semitones (A1:m2)	74:57 (115.5¢ : 88.95¢)
Consistency limit	9
Distinct consistency limit	9

769 equal divisions of the octave (abbreviated 769edo or 769ed2), also called 769-tone equal temperament (769tet) or 769 equal temperament (769et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 769 equal parts of about 1.56 ¢ each. Each step represents a frequency ratio of 2^1/769, or the 769th root of 2.

Theory

769edo is consistent to the 9-odd-limit, despite of a large error in its harmonic 5. As an equal temperament, it tempers out 2401/2400, 359661568/358722675 and [8 14 -13 0⟩ in the 7-limit.

Odd harmonics

Approximation of odd harmonics in 769edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.256	+0.682	+0.225	+0.511	-0.473	+0.565	-0.622	-0.404	+0.536	+0.480	+0.594
	Relative (%)	+16.4	+43.7	+14.4	+32.8	-30.3	+36.2	-39.9	-25.9	+34.4	+30.8	+38.1
Steps (reduced)		1219 (450)	1786 (248)	2159 (621)	2438 (131)	2660 (353)	2846 (539)	3004 (697)	3143 (67)	3267 (191)	3378 (302)	3479 (403)

Subsets and supersets

769edo is the 136th prime edo. 1538edo, which doubles it, gives a good correction to the harmonic 5.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[1219 -769⟩	[⟨769 1219]]	-0.0807	0.0806	5.17
2.3.5	[8 14 -13⟩, [103 -43 -15⟩	[⟨769 1219 1786]]	-0.1517	0.1202	7.70
2.3.5.7	2401/2400, 359661568/358722675, 1224440064/1220703125	[⟨769 1219 1786 2159]]	-0.1338	0.1086	6.96

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	202\769	315.215	6/5	Parakleismic
1	220\769	343.303	8000/6561	Raider
1	225\769	351.105	49/40	Newt
1	334\769	521.196	875/648	Maviloid

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