897edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|897}} | {{EDO intro|897}} | ||
==Theory== | |||
== Theory == | |||
897edo has sharp approximations to harmonics [[3/1|3]] and [[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]], 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. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|897}} | {{Harmonics in equal|897}} | ||
==Regular temperament properties== | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo| 474 -299}} | | {{monzo| 474 -299 }} | ||
|{{ | | {{mapping| 897 1422 }} | ||
| -0.1218 | | -0.1218 | ||
| 0.1218 | | 0.1218 | ||
| 9.10 | | 9.10 | ||
|- | |- | ||
|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 | | -0.1255 | ||
| 0.010 | | 0.010 | ||
| 0.75 | | 0.75 | ||
|- | |- | ||
|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 | | -0.0706 | ||
| 0.1283 | | 0.1283 | ||
| 9.59 | | 9.59 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|3025/3024, 5632/5625, 160083/160000, 28824005/28697814 | | 3025/3024, 5632/5625, 160083/160000, 28824005/28697814 | ||
|{{ | | {{mapping| 897 1422 2083 2518 3103 }} (897) | ||
| -0.0480 | | -0.0480 | ||
| 0.1234 | | 0.1234 | ||
| 9.22 | | 9.22 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|2080/2079, 1001/1000, 3025/3024, 4096/4095, 2058857500/2051893701 | | 2080/2079, 1001/1000, 3025/3024, 4096/4095, 2058857500/2051893701 | ||
|{{ | | {{mapping| 897 1422 2083 2518 3103 3319 }} (897) | ||
| -0.0222 | | -0.0222 | ||
| 0.1265 | | 0.1265 | ||
| Line 55: | Line 63: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|419\897 | | 419\897 | ||
|560.535 | | 560.535 | ||
|864/625 | | 864/625 | ||
|[[Whoosh]] | | [[Whoosh]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
==Music== | == Music == | ||
* [https://www.youtube.com/watch?v=opR71xbJ-PQ 13|3|23] | ; [[User:Francium|Francium]] | ||
* [https://www.youtube.com/watch?v=opR71xbJ-PQ ''13|3|23''] – 3-, 13- and 23edo polymicrotonal | |||