209edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|209}} | |||
==Theory== | |||
509et tempers out 129140163/128000000 (graviton) and 1220703125/1207959552 (ditonma) in the 5-limit. Using the patent val, it tempers out 225/224, 2125764/2100875, and 2500000/2470629 in the 7-limit; 243/242, 3025/3024, 4000/3993, and 16896/16807 in the 11-limit; 351/350, 625/624, 1573/1568, 1625/1617, and 15379/15360 in the 13-limit, so that it provides the [[optimal patent val]] for the 11-limit [[Marvel temperaments|marvo temperament]] and the 13-limit [[Marvel family|spectacle temperament]]. | |||
===Odd harmonics=== | |||
{{Harmonics in equal|209}} | |||
===Subsets and supersets=== | |||
209 factors into 11 × 19, with subset edos {{EDOs|11, and 19}}. [[627edo]], which triples it, gives a good correction to the harmonic 7. | |||
==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|-331 209}} | |||
|{{val|209 331}} | |||
| 0.4658 | |||
| 0.4660 | |||
| 8.12 | |||
|- | |||
|2.3.5 | |||
|{{monzo|-13 17 -6}}, {{monzo|-27 -2 13}} | |||
|{{val|209 331 485}} | |||
| 0.5439 | |||
| 0.3962 | |||
| 6.90 | |||
|} | |||
=== 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 | |||
|71\209 | |||
|407.66 | |||
|15625/12288 | |||
|[[Ditonic]] | |||
|- | |||
|1 | |||
|90\209 | |||
|516.75 | |||
|27/20 | |||
|[[Gravity]] / [[Zarvo]] (209d) | |||
|- | |||
|19 | |||
|122\209<br>(1\209) | |||
|700.48<br>(5.74) | |||
|3/2<br>(225/224) | |||
|[[Enneadecal]] (209d) | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||