897edo

← 896edo

897edo

898edo →

Prime factorization	3 × 13 × 23
Step size	1.33779 ¢
Fifth	525\897 (702.341 ¢) (→ 175\299)
Semitones (A1:m2)	87:66 (116.4 ¢ : 88.29 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

897edo has sharp approximations to harmonics 3 and 5 and flat ones starting with 7. In the 13-limit at least, we want to consider the patent val ⟨897 1422 2083 2518 3103 3319] as well as the 897def val ⟨897 1422 2083 2519 3104 3320].

The patent val tempers out 250047/250000 in the 7-limit; 3025/3024, 5632/5625, 102487/102400, 131072/130977, 160083/160000, and 172032/171875 in the 11-limit; 676/675, 1001/1000, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit.

The 897def val tempers out 4375/4374 in the 7-limit; 4000/3993, 46656/46585, and 172032/171875 in the 11-limit; 1716/1715, and 10648/10647 in the 13-limit.

Prime harmonics

Approximation of odd harmonics in 897edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.386	+0.308	-0.264	-0.566	-0.147	-0.394	-0.643	-0.608	-0.523	+0.122	+0.488
	Relative (%)	+28.9	+23.0	-19.7	-42.3	-11.0	-29.4	-48.1	-45.4	-39.1	+9.1	+36.5
Steps (reduced)		1422 (525)	2083 (289)	2518 (724)	2843 (152)	3103 (412)	3319 (628)	3504 (813)	3666 (78)	3810 (222)	3940 (352)	4058 (470)

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[474 -299⟩	[⟨897 1422]]	-0.1218	0.1218	9.10
2.3.5	[33 -34 9⟩, [70 -9 -24⟩	[⟨897 ​1422​ 2083​]]	-0.1255	0.010	0.75
2.3.5.7	250047/250000, 67108864/66976875, 28824005/28697814	[⟨897​ 1422 ​2083 ​2518]] (897)	-0.0706	0.1283	9.59
2.3.5.7.11	3025/3024, 5632/5625, 160083/160000, 28824005/28697814	[⟨897 ​1422​ 2083 ​2518 ​3103]] (897)	-0.0480	0.1234	9.22
2.3.5.7.11.13	2080/2079, 1001/1000, 3025/3024, 4096/4095, 2058857500/2051893701	[⟨897​ 1422 ​2083 ​2518​ 3103​ 3319]] (897)	-0.0222	0.1265	9.46

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio	Temperaments
1	419\897	560.535	864/625	Whoosh

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

Music

Francium

• 13|3|23 – 3-, 13- and 23edo polymicrotonal