897edo: Difference between revisions
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m Adopt template: Factorization; misc. cleanup |
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== Theory == | == Theory == | ||
897edo shares the same approximation to [[3/1|harmonic 3]] | 897edo shares the same approximation to [[3/1|harmonic 3]] with [[299edo]]. It has sharp approximations to [[5/1|5]] and flat ones starting with [[7/1|7]]. In the 13-limit at least, we want to consider the [[patent val]] {{val| 897 1422 2083 '''2518''' '''3103''' '''3319''' }} as well as the 897def [[val]] {{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 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. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 897 factors into | Since 897 factors into {{factorization|897}}, 897edo has subset edos {{EDOs| 3, 13, 23, 39, 69, and 299 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
Revision as of 11:33, 2 November 2023
| ← 896edo | 897edo | 898edo → |
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
| 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
| 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
- 13|3|23 (2023) – 3-, 13- and 23edo polymicrotonal