# 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

897 equal divisions of the octave (abbreviated 897edo or 897ed2), also called 897-tone equal temperament (897tet) or 897 equal temperament (897et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 897 equal parts of about 1.34 ¢ each. Each step represents a frequency ratio of 21/897, or the 897th root of 2.

## Theory

897edo shares the same approximation to harmonic 3 with 299edo. It has sharp approximations to 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, 42592/42525, 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.

### Odd 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)

### Subsets and supersets

Since 897 factors into 3 × 13 × 23, 897edo has subset edos 3, 13, 23, 39, 69, and 299.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 [33 -34 9, [70 -9 -24 [897 ​1422​ 2083​]] -0.1255 0.0996 7.44
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, 102487/102400, 28824005/28697814 [897 ​1422​ 2083 ​2518 ​3103]] (897) -0.0480 0.1234 9.22
2.3.5.7.11.13 676/675, 1001/1000, 3025/3024, 4096/4095, 28824005/28697814 [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\897 560.54 864/625 Whoosh
1 440\897 588.63 128/91 Countritonic (897df)
3 103\897 137.79 13/12 Avicenna (897)

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

## Music

Francium
• 13|3|23 (2023) – 3-, 13- and 23edo polymicrotonal