209edo: Difference between revisions
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{{ED intro}} | |||
[[ | == 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]] | |||
Latest revision as of 13:31, 13 March 2026
| ← 208edo | 209edo | 210edo → |
209 equal divisions of the octave (abbreviated 209edo or 209ed2), also called 209-tone equal temperament (209tet) or 209 equal temperament (209et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 209 equal parts of about 5.74 ¢ each. Each step represents a frequency ratio of 21/209, or the 209th root of 2.
Theory
209edo is only consistent to the 5-odd-limit. The equal temperament 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 supports the 13-limit spectacle temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.48 | -1.62 | +1.51 | +2.79 | -0.12 | -2.25 | +2.64 | -1.61 | +1.05 | +0.03 | -2.44 |
| Relative (%) | -25.7 | -28.3 | +26.3 | +48.6 | -2.1 | -39.2 | +46.0 | -28.0 | +18.3 | +0.6 | -42.4 | |
| Steps (reduced) |
331 (122) |
485 (67) |
587 (169) |
663 (36) |
723 (96) |
773 (146) |
817 (190) |
854 (18) |
888 (52) |
918 (82) |
945 (109) | |
Subsets and supersets
Since 209 factors into 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
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-331 209⟩ | [⟨209 331]] | +0.4658 | 0.4660 | 8.12 |
| 2.3.5 | [-13 17 -6⟩, [-27 -2 13⟩ | [⟨209 331 485]] | +0.5439 | 0.3962 | 6.90 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated 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 (1\209) |
700.48 (5.74) |
3/2 (225/224) |
Enneadecal (209d) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct