289edo: Difference between revisions

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== Theory ==

289edo ~~has~~ decent 11- and 13-limit interpretations despite ~~not being~~ [[~~consistent~~]]. ~~The~~ equal temperament [[tempering out|tempers out]] the [[schisma]], 32805/32768 in the 5-limit; [[4375/4374]] and [[65625/65536]] in the 7-limit; [[441/440]] and [[4000/3993]] in the 11-limit; and [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]] and [[2080/2079]] in the 13-limit.

289edo is a strong 5-limit system with decent [[11-limit|11-]] and [[13-limit]] interpretations despite in[[consistency]] in the [[13-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[schisma]], 32805/32768 in the [[5-limit]]; [[4375/4374]] and [[65625/65536]] in the [[7-limit]]; [[441/440]] and [[4000/3993]] in the 11-limit; and [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]] and [[2080/2079]] in the 13-limit.

It is the [[optimal patent val]] for the [[13-limit]] rank-5 temperament tempering out 364/363, and the 13-limit [[~~History~~ (temperament)|history]] temperament, which tempers out 364/363, 441/440 and 676/675. It provides a good tuning for the 11-limit version also. It is also the optimal patent val for [[~~sextilififths~~]] ~~in both the 11- and 13-limit~~, and ~~for~~ [[~~quintaschis~~]] in both the 11- and 13-limit.

It is the [[optimal patent val]] for the [[13-limit]] rank-5 temperament tempering out 364/363, and the 13-limit [[history (temperament)|history]] temperament, which tempers out 364/363, 441/440 and 676/675. It provides a good tuning for the 11-limit version also. It is also the optimal patent val for [[sextilifourths]], [[quintaschis]], and [[quincy]] in both the 11- and 13-limit.

=== Prime harmonics ===

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== 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]] (¢)

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| 6.40

|}

* 289et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[171edo|171]] and followed by [[323edo|323]].

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

! Periods per 8ve

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| 83.04

| 21/20

| [[~~Sextilififths~~]]

| [[Sextilifourths]]

|-

| 1

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|-

| 17

| 93\289 (8\289)

| 93\289 (8\289)

| 386.16 (33.22)

| 386.16 (33.22)

| {{monzo| -23 5 9 -2 }} (100352/98415)

| {{monzo| -23 5 9 -2 }} (100352/98415)

| [[Chlorine]]

|}

<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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:History (temperament)]]

[[Category:Minor minthmic]]

[[Category:~~History (temperament)]]~~

[[Category:Quincy]]

~~[[Category:Sextilififths~~]]

[[Category:Quintaschis]]

[[Category:Sextilifourths]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|289}}
+{{ED intro}}
 == Theory ==
-edo has decent 11- and 13-limit interpretations despite not being [[consistent]]. The equal temperament [[tempering out|tempers out]] the [[schisma]], 32805/32768 in the 5-limit; [[4375/4374]] and [[65625/65536]] in the 7-limit; [[441/440]] and [[4000/3993]] in the 11-limit; and [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]] and [[2080/2079]] in the 13-limit.
+edo is a strong 5-limit system with decent [[11-limit|11-]] and [[13-limit]] interpretations despite in[[consistency]] in the [[13-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[schisma]], 32805/32768 in the [[5-limit]]; [[4375/4374]] and [[65625/65536]] in the [[7-limit]]; [[441/440]] and [[4000/3993]] in the 11-limit; and [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]] and [[2080/2079]] in the 13-limit.
-It is the [[optimal patent val]] for the [[13-limit]] rank-5 temperament tempering out 364/363, and the 13-limit [[History (temperament)|history]] temperament, which tempers out 364/363, 441/440 and 676/675. It provides a good tuning for the 11-limit version also. It is also the optimal patent val for [[sextilififths]] in both the 11- and 13-limit, and for [[quintaschis]] in both the 11- and 13-limit.
+It is the [[optimal patent val]] for the [[13-limit]] rank-5 temperament tempering out 364/363, and the 13-limit [[history (temperament)|history]] temperament, which tempers out 364/363, 441/440 and 676/675. It provides a good tuning for the 11-limit version also. It is also the optimal patent val for [[sextilifourths]], [[quintaschis]], and [[quincy]] in both the 11- and 13-limit.
 === Prime harmonics ===
@@ Line 15: / Line 15: @@
 == 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]] (¢)
@@ Line 59: / Line 60: @@
 | 6.40
 |}
+* 289et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[171edo|171]] and followed by [[323edo|323]].
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-! Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 84: / Line 88: @@
 | 83.04
 | 21/20
-| [[Sextilififths]]
+| [[Sextilifourths]]
 |-
 | 1
@@ Line 123: / Line 127: @@
 |-
 | 17
-| 93\289<br>(8\289)
+| 93\289<br />(8\289)
-| 386.16<br>(33.22)
+| 386.16<br />(33.22)
-| {{monzo| -23 5 9 -2 }}<br>(100352/98415)
+| {{monzo| -23 5 9 -2 }}<br />(100352/98415)
 | [[Chlorine]]
 |}
-<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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+[[Category:History (temperament)]]
 [[Category:Minor minthmic]]
-[[Category:History (temperament)]]
+[[Category:Quincy]]
-[[Category:Sextilififths]]
 [[Category:Quintaschis]]
+[[Category:Sextilifourths]]