289edo: Difference between revisions

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== Theory ==

289edo is a strong 5-limit system with decent 11- and 13-limit interpretations despite in[[consistency]] in the [[13-odd-limit]]. ~~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~~]], [[quintaschis]], and [[quincy]] 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 ==

{~~{comma basis begin}}~~

{| 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

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| 0.2657

| 6.40

~~{{comma basis end}~~}

|}

* 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 ===

{{rank-2 ~~begin}}~~

{| 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

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| 83.04

| 21/20

| [[~~Sextilififths~~]]

| [[Sextilifourths]]

|-

| 1

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

| [[Chlorine]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

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

[[Category:History (temperament)]]

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[[Category:Quincy]]

[[Category:Quintaschis]]

[[Category:~~Sextilififths~~]]

[[Category:Sextilifourths]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|289}}
+{{ED intro}}
 == Theory ==
-edo is a strong 5-limit system with decent 11- and 13-limit interpretations despite in[[consistency]] in the [[13-odd-limit]]. 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]], [[quintaschis]], and [[quincy]] 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 14: / Line 14: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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
@@ Line 50: / Line 59: @@
 | 0.2657
 | 6.40
-{{comma basis end}}
+|}
 * 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 ===
-{{rank-2 begin}}
+{| 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
@@ Line 72: / Line 88: @@
 | 83.04
 | 21/20
-| [[Sextilififths]]
+| [[Sextilifourths]]
 |-
 | 1
@@ Line 115: / Line 131: @@
 | {{monzo| -23 5 9 -2 }}<br />(100352/98415)
 | [[Chlorine]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:History (temperament)]]
@@ Line 122: / Line 138: @@
 [[Category:Quincy]]
 [[Category:Quintaschis]]
-[[Category:Sextilififths]]
+[[Category:Sextilifourths]]