209edo: Difference between revisions

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== 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 ~~and~~ the 13-limit [[spectacle]] temperament.

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 ===

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=== 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.

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~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

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| {{monzo| -331 209 }}

| {{mapping| 209 331 }}

| 0.4658

| +0.4658

| 0.4660

| 8.12

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| {{monzo| -13 17 -6 }}, {{monzo| -27 -2 13 }}

| {{mapping| 209 331 485 }}

| 0.5439

| +0.5439

| 0.3962

| 6.90

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=== Rank-2 temperaments ===

{| 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 per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| [[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

<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]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|209}}
+{{ED intro}}
 == Theory ==
-edo 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.
+edo 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 ===
@@ Line 9: / Line 9: @@
 === 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.
+Since 209 factors into {{nowrap|11 &times; 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|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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 simple badness|Relative]] (%)
@@ Line 25: / Line 26: @@
 | {{monzo| -331 209 }}
 | {{mapping| 209 331 }}
-| 0.4658
+| +0.4658
 | 0.4660
 | 8.12
@@ Line 32: / Line 33: @@
 | {{monzo| -13 17 -6 }}, {{monzo| -27 -2 13 }}
 | {{mapping| 209 331 485 }}
-| 0.5439
+| +0.5439
 | 0.3962
 | 6.90
@@ Line 39: / Line 40: @@
 === Rank-2 temperaments ===
 {| 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
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 64: / Line 66: @@
 | [[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
+<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]]