692edo

692 equal divisions of the octave (abbreviated 692edo or 692ed2), also called 692-tone equal temperament (692tet) or 692 equal temperament (692et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 692 equal parts of about 1.73 ¢ each. Each step represents a frequency ratio of 2^1/692, or the 692nd root of 2.

← 691edo

692edo

693edo →

Prime factorization	22 × 173
Step size	1.7341 ¢
Fifth	405\692 (702.312 ¢)
Semitones (A1:m2)	67:51 (116.2 ¢ : 88.44 ¢)
Consistency limit	19
Distinct consistency limit	19

Theory

692edo is consistent to the 19-odd-limit and almost the 21-odd-limit; the only inconsistently mapped intervals in the 21-odd-limit are 21/16 and its octave complement. It has a sharp tendency, with odd harmonics 3 through 19 all tuned sharp.

As an equal temperament, it tempers out [23 6 -14⟩ (vishnuzma) in the 5-limit; 4375/4374 (ragisma) and 29360128/29296875 (quasiorwellisma) in the 7-limit; 3025/3024, 5632/5625, 9801/9800, and 19712/19683 in the 11-limit; 1716/1715, 2080/2079, and 6656/6655 in the 13-limit; 2431/2430, 2500/2499, and 4914/4913 in the 17-limit; 1216/1215, 1331/1330, 1540/1539, 2432/2431, 2926/2925, 3136/3135, and 3250/3249 in the 19-limit. It supports and provides a good tuning for vishnu and its 13-limit extension acyuta.

Prime harmonics

Approximation of prime harmonics in 692edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.357	+0.391	+0.538	+0.127	+0.513	+0.825	+0.753	-0.529	+0.481	-0.527
	Relative (%)	+0.0	+20.6	+22.6	+31.0	+7.3	+29.6	+47.6	+43.4	-30.5	+27.7	-30.4
Steps (reduced)		692 (0)	1097 (405)	1607 (223)	1943 (559)	2394 (318)	2561 (485)	2829 (61)	2940 (172)	3130 (362)	3362 (594)	3428 (660)

Subsets and supersets

Since 692 factors into primes as 2² × 173, 692edo has subset edos 2, 4, 173, and 346.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[1097 -692⟩	[⟨692 1097]]	−0.1127	0.1127	6.50
2.3.5	[23 6 -14⟩, [80 -49 1⟩	[⟨692 1097 1607]]	−0.1313	0.0957	5.52
2.3.5.7	4375/4374, 29360128/29296875, 283115520/282475249	[⟨692 1097 1607 1943]]	−0.1464	0.0869	5.01
2.3.5.7.11	3025/3024, 4375/4374, 5632/5625, 283115520/282475249	[⟨692 1097 1607 1943 2394]]	−0.1245	0.0892	5.15
2.3.5.7.11.13	1716/1715, 2080/2079, 3025/3024, 5632/5625, 200000/199927	[⟨692 1097 1607 1943 2394 2561]]	−0.1268	0.0816	4.71
2.3.5.7.11.13.17	1716/1715, 2080/2079, 2431/2430, 2500/2499, 4914/4913, 5632/5625	[⟨692 1097 1607 1943 2394 2561 2829]]	−0.1375	0.0800	4.61
2.3.5.7.11.13.17.19	1216/1215, 1331/1330, 1540/1539, 1716/1715, 2431/2430, 2500/2499, 4914/4913	[⟨692 1097 1607 1943 2394 2561 2829 2940]]	−0.1425	0.0760	4.38

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
2	41\692	71.10	25/24	Vishnu / acyuta

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