209edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|209}}
{{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]].
== 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 and the 13-limit [[spectacle]] temperament.
 
=== Odd harmonics ===
{{Harmonics in equal|209}}
{{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.
=== Subsets and supersets ===
==Regular temperament properties==
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 ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|-331 209}}
| {{monzo| -331 209 }}
|{{val|209 331}}
| {{mapping| 209 331 }}
| 0.4658
| 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 }}
|{{val|209 331 485}}
| {{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
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! 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
|[[Gravity]] / [[Zarvo]] (209d)
| [[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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 10:11, 13 April 2024

← 208edo 209edo 210edo →
Prime factorization 11 × 19
Step size 5.74163 ¢ 
Fifth 122\209 (700.478 ¢)
Semitones (A1:m2) 18:17 (103.3 ¢ : 97.61 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

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 and the 13-limit spectacle temperament.

Odd harmonics

Approximation of odd harmonics in 209edo
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

Table of rank-2 temperaments by generator
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 it is distinct

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