214edo: Difference between revisions
m Infobox ET added |
No edit summary |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|214}} | |||
==Theory== | |||
214et is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which tempers out the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; [[6144/6125]] (porwell), 16875/16807 (mirkwai), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]] and 1375/1372 in the 11-limit; 1188/1183, [[351/350]] and [[847/845]] in the 13-limit. It can be viewed as a 2.13/5 subgroup temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning. | |||
===Prime harmonics=== | |||
{{Harmonics in equal|214}} | |||
===Subsets and supersets=== | |||
214 factors into 2 × 107, with [[2edo]] and [[107edo]] as its subset edos. | |||
==Regular temperament properties== | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |[[Subgroup]] | |||
! rowspan="2" |[[Comma list|Comma List]] | |||
! rowspan="2" |[[Mapping]] | |||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | |||
! colspan="2" |Tuning Error | |||
|- | |||
![[TE error|Absolute]] (¢) | |||
![[TE simple badness|Relative]] (%) | |||
|- | |||
|2.3 | |||
|{{monzo|-339 214}} | |||
|{{val|214 339}} | |||
| +0.3219 | |||
| 0.3220 | |||
| 5.74 | |||
|- | |||
|2.3.5 | |||
|78732/78125, {{monzo|-49 28 2}} | |||
|{{val|214 339 497}} | |||
| +0.1281 | |||
| 0.3797 | |||
| 6.77 | |||
|- | |||
|2.3.5.7 | |||
|6144/6125, 16875/16807, 78732/78125 | |||
|{{val|214 339 497 601}} | |||
| -0.0169 | |||
| 0.4137 | |||
| 7.38 | |||
|- | |||
|2.3.5.7.11 | |||
|540/539, 1375/1372, 3025/3024, 5632/5625 | |||
|{{val|214 339 497 601 740}} | |||
| +0.0897 | |||
| 0.4270 | |||
| 7.61 | |||
|- | |||
|2.3.5.7.11.13 | |||
|540/539, 847/845, 1001/1000, 1375/1372, 5632/5625 | |||
|{{val|214 339 497 601 740 792}} | |||
| +0.0480 | |||
| 0.4008 | |||
| 7.15 | |||
|- | |||
|2.3.5.7.11.13.17 | |||
|351/350, 715/714, 936/935, 1275/1274, 5544/5525, 5850/5831 | |||
|{{val|214 339 497 601 740 792 875}} | |||
| -0.0144 | |||
| 0.4012 | |||
| 7.15 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
|1 | |||
|69\214 | |||
|386.92 | |||
|5/4 | |||
|[[Grendel]] | |||
|- | |||
|1 | |||
|79\214 | |||
|442.99 | |||
|9/7 | |||
|[[Sensi]] | |||
|- | |||
|1 | |||
|105\214 | |||
|588.79 | |||
|7/5 | |||
|[[Untriton]] / [[aufo]] | |||
|- | |||
|2 | |||
|28\214 | |||
|157.01 | |||
|35/32 | |||
|[[Bison]] | |||
|- | |||
|2 | |||
|29\214 | |||
|162.62 | |||
|1125/1024 | |||
|[[Kwazy]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||