897edo

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← 896edo897edo898edo →
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