227edo: Difference between revisions
Note on 13-limit countercata tuning |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|227}} | {{EDO intro|227}} | ||
== Theory == | |||
The equal temperament tempers out 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 61 -37 -1 }} in the 5-limit; [[5120/5103]], [[65625/65536]], and 117649/116640 in the 7-limit, so that it [[support]]s [[countercata]]. In the 11-limit, it tempers out [[385/384]], 2200/2187, 3388/3375, and 12005/11979, so that it provides the [[optimal patent val]] for 11-limit countercata. In the 13-limit, it tempers out [[325/324]], [[352/351]], [[625/624]], [[676/675]], and [[847/845]], and again supplies a good tuning for 13-limit countercata, although [[140edo]] tunes it better in this case. | The equal temperament tempers out 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 61 -37 -1 }} in the 5-limit; [[5120/5103]], [[65625/65536]], and 117649/116640 in the 7-limit, so that it [[support]]s [[countercata]]. In the 11-limit, it tempers out [[385/384]], 2200/2187, 3388/3375, and 12005/11979, so that it provides the [[optimal patent val]] for 11-limit countercata. In the 13-limit, it tempers out [[325/324]], [[352/351]], [[625/624]], [[676/675]], and [[847/845]], and again supplies a good tuning for 13-limit countercata, although [[140edo]] tunes it better in this case. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
227edo is the 49th [[prime edo]]. | 227edo is the 49th [[prime edo]]. | ||
==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|360 -227}} | |||
|{{val|227 360}} | |||
| -0.3561 | |||
| 0.3560 | |||
| 6.73 | |||
|- | |||
|2.3.5 | |||
|15625/15552, {{monzo|61 -37 -1}} | |||
|{{val|227 360 527}} | |||
| -0.1785 | |||
| 0.3842 | |||
| 7.27 | |||
|- | |||
|2.3.5.7 | |||
|5120/5103, 15625/15552, 65625/65536 | |||
|{{val|227 360 527 637}} | |||
| -0.0071 | |||
| 0.4461 | |||
| 8.44 | |||
|- | |||
|2.3.5.7.11 | |||
|385/384, 2200/2187, 3388/3375, 5120/5103 | |||
|{{val|227 360 527 637 785}} | |||
| +0.0832 | |||
| 0.4380 | |||
| 8.29 | |||
|- | |||
|2.3.5.7.11.13 | |||
|325/324, 352/351, 385/384, 625/624, 847/845 | |||
|{{val|227 360 527 637 785 840}} | |||
| +0.0693 | |||
| 0.4010 | |||
| 7.59 | |||
|- | |||
|2.3.5.7.11.13.17 | |||
|352/351, 595/594, 715/714, 847/845, 1001/1000, 3185/3179 | |||
|{{val|227 360 527 637 785 840 928}} | |||
| +0.0324 | |||
| 0.3821 | |||
| 7.23 | |||
|} | |||
=== 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 | |||
|25\227 | |||
|132.16 | |||
|{{monzo|-38 5 13}} | |||
|[[Astro]] / [[kastro]] | |||
|- | |||
|1 | |||
|60\227 | |||
|317.18 | |||
|6/5 | |||
|[[Hanson]] / [[countercata]] | |||
|- | |||
|1 | |||
|94\227 | |||
|496.92 | |||
|4/3 | |||
|[[Undecental]] | |||
|} | |||
[[Category:Countercata]] | [[Category:Countercata]] | ||