209edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
==Theory== | |||
== Theory == | |||
===Odd harmonics=== | 209edo is only [[consistent]] to the [[5-odd-limit]]. The equal temperament [[tempering out|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 13-limit [[marvo]] temperament. It also [[support]]s the 13-limit [[spectacle]] temperament. | ||
=== Odd harmonics === | |||
{{Harmonics in equal|209}} | {{Harmonics in equal|209}} | ||
===Subsets and supersets=== | |||
209 factors into | === Subsets and supersets === | ||
==Regular temperament properties== | Since 209 factors into {{nowrap|11 × 19}}, 209edo contains [[11edo]] and [[19edo]] as its subsets. [[627edo]], which triples it, gives a good correction to the harmonic 7. | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|-331 209}} | ! rowspan="2" | [[Comma list]] | ||
|{{ | ! rowspan="2" | [[Mapping]] | ||
| 0.4658 | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -331 209 }} | |||
| {{mapping| 209 331 }} | |||
| +0.4658 | |||
| 0.4660 | | 0.4660 | ||
| 8.12 | | 8.12 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|-13 17 -6}}, {{monzo|-27 -2 13}} | | {{monzo| -13 17 -6 }}, {{monzo| -27 -2 13 }} | ||
|{{ | | {{mapping| 209 331 485 }} | ||
| 0.5439 | | +0.5439 | ||
| 0.3962 | | 0.3962 | ||
| 6.90 | | 6.90 | ||
|} | |} | ||
=== 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 | ||
|71\209 | | 71\209 | ||
|407.66 | | 407.66 | ||
|15625/12288 | | 15625/12288 | ||
|[[Ditonic]] | | [[Ditonic]] | ||
|- | |- | ||
|1 | | 1 | ||
|90\209 | | 90\209 | ||
|516.75 | | 516.75 | ||
|27/20 | | 27/20 | ||
|[[ | | [[Larry]] / [[marvo]] (209) / [[zarvo]] (209d) | ||
|- | |- | ||
|19 | | 19 | ||
|122\209<br>(1\209) | | 122\209<br>(1\209) | ||
|700.48<br>(5.74) | | 700.48<br>(5.74) | ||
|3/2<br>(225/224) | | 3/2<br>(225/224) | ||
|[[Enneadecal]] (209d) | | [[Enneadecal]] (209d) | ||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
[[Category: | [[Category:Marvo]] | ||