209edo: Difference between revisions
Created page with "'''209edo''' is the equal division of the octave into 209 parts of 5.7416 cents each. It tempers out 129140163/128000000 (graviton) and 1220703125/1207959552 (dito..." Tags: Mobile edit Mobile web edit |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" Tags: Mobile edit Mobile web edit |
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{{Infobox ET}} | |||
{{ED intro}} | |||
[[Category: | == Theory == | ||
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}} | |||
=== Subsets and supersets === | |||
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" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[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 }} | |||
| {{mapping| 209 331 }} | |||
| +0.4658 | |||
| 0.4660 | |||
| 8.12 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| -13 17 -6 }}, {{monzo| -27 -2 13 }} | |||
| {{mapping| 209 331 485 }} | |||
| +0.5439 | |||
| 0.3962 | |||
| 6.90 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 71\209 | |||
| 407.66 | |||
| 15625/12288 | |||
| [[Ditonic]] | |||
|- | |||
| 1 | |||
| 90\209 | |||
| 516.75 | |||
| 27/20 | |||
| [[Larry]] / [[marvo]] (209) / [[zarvo]] (209d) | |||
|- | |||
| 19 | |||
| 122\209<br>(1\209) | |||
| 700.48<br>(5.74) | |||
| 3/2<br>(225/224) | |||
| [[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:Marvo]] | |||