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]] | |||