677edo

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677edo

678edo →

Prime factorization	677 (prime)
Step size	1.77253¢
Fifth	396\677 (701.92¢)
Semitones (A1:m2)	64:51 (113.4¢ : 90.4¢)
Consistency limit	11
Distinct consistency limit	11

677 equal divisions of the octave (677edo), or 677-tone equal temperament (677tet), 677 equal temperament (677et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 677 equal parts of about 1.77 ¢ each.

Theory

While it does well as a 2.3.5.11 tuning system, it's notable for its high accuracy among EDOs of about its size and lower with the first several metallic ratios. Among those, it tunes acoustic phi (the golden ratio) and the acoustic copper ratio each with less than 1% relative error. The first nine metallic ratios are all tuned within 20% of an edostep.

Prime harmonics

Approximation of prime harmonics in 677edo

Harmonic		2	3	5	7	11	13	17	19	23
Error	absolute (¢)	+0.00	-0.03	+0.10	+0.75	-0.06	-0.35	-0.38	+0.27	-0.80
	relative (%)	+0	-2	+5	+42	-4	-20	-21	+15	-45
Steps (reduced)		677 (0)	1073 (396)	1572 (218)	1901 (547)	2342 (311)	2505 (474)	2767 (59)	2876 (168)	3062 (354)

Subsets and supersets

677edo is the 123rd prime EDO.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[-1073 677⟩	[⟨677 1073]]	+0.0110	0.0110	0.62
2.3.5	[38 -2 -15⟩, [-31 43 -16⟩	[⟨677 1073 1572]]	-0.0066	0.0264	1.49
2.3.5.7	703125/702464, 589824/588245, 14348907/14336000	[⟨677 1073 1572 1901]]	-0.0714	0.1145	6.46
2.3.5.7.11	3025/3024, 24057/24010, 131072/130977, 759375/758912	[⟨677 1073 1572 1901 2342]]	-0.0535	0.1084	6.12

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (reduced)*	Cents (reduced)*	Associated Ratio*	Temperaments
1	109\677	193.21	262144/234375	Luna
1	125\677	221.57	8388608/7381125	Fortune
1	281\677	498.08	4/3	Counterschismic