243edo: Difference between revisions

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The '''243 equal division of the octave''' ('''243edo''') divides the [[octave]] into 243 [[equal]] parts of 4.938 [[cent]]s each. It tempers out the [[semicomma]] (5-limit orwell comma) 2109375/2097152 in the 5-limit, and [[2401/2400]] and [[4375/4374]] in the 7-limit. In the ~~11-limit~~ it tempers out [[243/242]] ~~and~~ [[441/440]], and provides the [[optimal patent val]] for the [[Ragismic microtemperaments #Ennealimmal|ennealimnic]] temperament. In the 13-limit it tempers out [[364/363]] ~~and~~ [[625/624]], and provides the optimal temperament for 13-limit ennealimnic and the rank-3 [[Breed family #Jovial|jovial]] temperament, and in the 17-limit it tempers out 375/374 and 595/594 and provides the optimal patent val for 17-limit ennealimnic.

The '''243 equal division of the octave''' ('''243edo''') divides the [[octave]] into 243 [[equal]] parts of 4.938 [[cent]]s each.

== Theory ==

243et tempers out the [[semicomma]] (i.e. the 5-limit orwell comma) 2109375/2097152 in the 5-limit, and [[2401/2400]] and [[4375/4374]] in the 7-limit.

Using the [[patent val]], it tempers out [[243/242]], [[441/440]], and [[540/539]] in the 11-limit, and provides the [[optimal patent val]] for the [[Ragismic microtemperaments #Ennealimmal|ennealimnic]] temperament. In the 13-limit it tempers out [[364/363]], [[625/624]], [[729/728]], and [[2080/2079]], and provides the optimal temperament for 13-limit ennealimnic and the rank-3 [[Breed family #Jovial|jovial]] temperament, and in the 17-limit it tempers out 375/374 and 595/594 and provides the optimal patent val for 17-limit ennealimnic.

Using the alternative val 243e {{val| 241 385 564 682 '''840''' }}, with an lower error, it tempers out [[385/384]], 1375/1372, [[8019/8000]], and [[14641/14580]], and in the 13-limit, 625/624, 729/728, [[847/845]], [[1001/1000]], and [[1716/1715]]. It provides a good tuning for [[fibo]].

=== Prime harmonics ===

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| 2.3.5.7

| 2401/2400, 4375/4374, 2109375/2097152

| [{{val| 241 ~~382 560 677~~ }}]

| [{{val| 241 385 564 682 }}]

| +0.318

| 0.192

@@ Line 1: / Line 1: @@
-The '''243 equal division of the octave''' ('''243edo''') divides the [[octave]] into 243 [[equal]] parts of 4.938 [[cent]]s each. It tempers out the [[semicomma]] (5-limit orwell comma) 2109375/2097152 in the 5-limit, and [[2401/2400]] and [[4375/4374]] in the 7-limit. In the 11-limit it tempers out [[243/242]] and [[441/440]], and provides the [[optimal patent val]] for the [[Ragismic microtemperaments #Ennealimmal|ennealimnic]] temperament. In the 13-limit it tempers out [[364/363]] and [[625/624]], and provides the optimal temperament for 13-limit ennealimnic and the rank-3 [[Breed family #Jovial|jovial]] temperament, and in the 17-limit it tempers out 375/374 and 595/594 and provides the optimal patent val for 17-limit ennealimnic.
+The '''243 equal division of the octave''' ('''243edo''') divides the [[octave]] into 243 [[equal]] parts of 4.938 [[cent]]s each.
+== Theory ==
+et tempers out the [[semicomma]] (i.e. the 5-limit orwell comma) 2109375/2097152 in the 5-limit, and [[2401/2400]] and [[4375/4374]] in the 7-limit.
+Using the [[patent val]], it tempers out [[243/242]], [[441/440]], and [[540/539]] in the 11-limit, and provides the [[optimal patent val]] for the [[Ragismic microtemperaments #Ennealimmal|ennealimnic]] temperament. In the 13-limit it tempers out [[364/363]], [[625/624]], [[729/728]], and [[2080/2079]], and provides the optimal temperament for 13-limit ennealimnic and the rank-3 [[Breed family #Jovial|jovial]] temperament, and in the 17-limit it tempers out 375/374 and 595/594 and provides the optimal patent val for 17-limit ennealimnic.
+Using the alternative val 243e {{val| 241 385 564 682 '''840''' }}, with an lower error, it tempers out [[385/384]], 1375/1372, [[8019/8000]], and [[14641/14580]], and in the 13-limit, 625/624, 729/728, [[847/845]], [[1001/1000]], and [[1716/1715]]. It provides a good tuning for [[fibo]].
 === Prime harmonics ===
@@ Line 31: / Line 38: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, 2109375/2097152
-| [{{val| 241 382 560 677 }}]
+| [{{val| 241 385 564 682 }}]
 | +0.318
 | 0.192