897edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
==Theory== | |||
== Theory == | |||
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 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 === | |||
{{Harmonics in equal|897}} | {{Harmonics in equal|897}} | ||
==Regular temperament properties== | |||
=== Subsets and supersets === | |||
Since 897 factors into {{factorization|897}}, 897edo has subset edos {{EDOs| 3, 13, 23, 39, 69, and 299 }}. | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
![[ | ! rowspan="2" | [[Subgroup]] | ||
![[ | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |- | ||
| | ! [[TE error|Absolute]] (¢) | ||
| | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo| 33 -34 9}}, {{monzo| 70 -9 -24}} | | {{monzo| 33 -34 9 }}, {{monzo| 70 -9 -24 }} | ||
|{{ | | {{mapping| 897 1422 2083 }} | ||
| | | −0.1255 | ||
| 0. | | 0.0996 | ||
| | | 7.44 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|250047/250000, 67108864/66976875, 28824005/28697814 | | 250047/250000, 67108864/66976875, 28824005/28697814 | ||
|{{ | | {{mapping| 897 1422 2083 2518 }} (897) | ||
| | | −0.0706 | ||
| 0.1283 | | 0.1283 | ||
| 9.59 | | 9.59 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|3025/3024, 5632/5625, | | 3025/3024, 5632/5625, 102487/102400, 28824005/28697814 | ||
|{{ | | {{mapping| 897 1422 2083 2518 3103 }} (897) | ||
| | | −0.0480 | ||
| 0.1234 | | 0.1234 | ||
| 9.22 | | 9.22 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| | | 676/675, 1001/1000, 3025/3024, 4096/4095, 28824005/28697814 | ||
|{{ | | {{mapping| 897 1422 2083 2518 3103 3319 }} (897) | ||
| | | −0.0222 | ||
| 0.1265 | | 0.1265 | ||
| 9.46 | | 9.46 | ||
| Line 53: | Line 58: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|419\897 | | 419\897 | ||
|560. | | 560.54 | ||
|864/625 | | 864/625 | ||
|[[Whoosh]] | | [[Whoosh]] | ||
|- | |||
| 1 | |||
| 440\897 | |||
| 588.63 | |||
| 128/91 | |||
| [[Countritonic]] (897df) | |||
|- | |||
| 3 | |||
| 103\897 | |||
| 137.79 | |||
| 13/12 | |||
| [[Avicenna (temperament)|Avicenna]] (897) | |||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Music == | |||
; [[User:Francium|Francium]] | |||
* [https://www.youtube.com/watch?v=opR71xbJ-PQ ''13|3|23''] (2023) – 3-, 13-, and 23edo polymicrotonal | |||
[[Category:Listen]] | |||
Latest revision as of 06:30, 21 February 2025
| ← 896edo | 897edo | 898edo → |
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
| 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 distinct
Music
- 13|3|23 (2023) – 3-, 13-, and 23edo polymicrotonal