289edo: Difference between revisions

Line 3:

== Theory ==

289edo is the [[optimal patent val]] for [[13-limit]] [[History (temperament)|history]] temperament, which tempers out [[364/363]], [[441/440]] and [[676/675]], and provides a good tuning for the 11-limit version also, and is also the optimal patent val for [[sextilififths]] in both the 11- and 13-limit~~. It is uniquely~~ [[consistent]] ~~in the 9-odd-limit, and~~ 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 has decent 11- and 13-limit interpretations despite not being [[consistent]]. It 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.

~~289~~ is ~~17 squared. In light of containing~~ [[~~17edo~~]] ~~as a subset, 289edo~~ [[~~support~~]]s the [[~~chlorine~~]] temperament, which tempers out the [[~~septendecima~~]] ~~{{monzo|~~-52 -~~17 34}}~~ and the ~~ragisma 4375/4374~~.

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.

=== Prime harmonics ===

=== Divisors ===

289 is 17 squared. In light of containing [[17edo]] as a subset, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo|-52 -17 34}} and the ragisma 4375/4374.

== Regular temperament properties ==

~~289edo has decent 11- and 13-limit interpretations despite not being consistent.~~

{| class="wikitable center-4 center-5 center-6"

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

Line 124:

Line 125:

| 93\289<br>(8\289)

| 386.16<br>(33.22)

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

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

| [[Chlorine]]

|}

[[Category:Equal divisions of the octave|###]]

[[Category:Gentle]]

[[Category:History (temperament)]]

[[Category:Sextilififths]]

[[Category:Quintaschis]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is the [[optimal patent val]] for [[13-limit]] [[History (temperament)|history]] temperament, which tempers out [[364/363]], [[441/440]] and [[676/675]], and provides a good tuning for the 11-limit version also, and is also the optimal patent val for [[sextilififths]] in both the 11- and 13-limit. It is uniquely [[consistent]] in the 9-odd-limit, and 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 has decent 11- and 13-limit interpretations despite not being [[consistent]]. It 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.
-is 17 squared. In light of containing [[17edo]] as a subset, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo|-52 -17 34}} and the ragisma 4375/4374.
+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.
 === Prime harmonics ===
 {{Harmonics in equal|289}}
+=== Divisors ===
+is 17 squared. In light of containing [[17edo]] as a subset, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo|-52 -17 34}} and the ragisma 4375/4374.
 == Regular temperament properties ==
-edo has decent 11- and 13-limit interpretations despite not being consistent.
 {| class="wikitable center-4 center-5 center-6"
 ! rowspan="2" | [[Subgroup]]
@@ Line 124: / Line 125: @@
 | 93\289<br>(8\289)
 | 386.16<br>(33.22)
-| {{monzo|-23 5 9 -2}}<br>(100352/98415)
+| {{monzo| -23 5 9 -2 }}<br>(100352/98415)
 | [[Chlorine]]
 |}
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+[[Category:Gentle]]
 [[Category:History (temperament)]]
 [[Category:Sextilififths]]
+[[Category:Quintaschis]]