243edo: Difference between revisions

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The '''243 equal divisions of the octave''' ('''243edo'''), or the '''243(-tone) equal temperament''' ('''243tet''', '''243et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 243 [[equal]] parts of about 4.934 [[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.

243et [[tempering out|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.

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== Regular temperament properties ==

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

! rowspan="2" | Subgroup

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

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

! rowspan="2" | [[Comma list|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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| 2.3

| {{monzo| -385 243 }}

| [{{~~val~~| 243 385 }}]

| {{mapping| 243 385 }}

| +0.227

| 0.227

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| 2.3.5

| 2109375/2097152, {{monzo| 1 -27 18 }}

| [{{~~val~~| 243 385 564 }}]

| {{mapping| 243 385 564 }}

| +0.314

| 0.222

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| 2.3.5.7

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

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

| {{mapping| 241 385 564 682 }}

| +0.318

| 0.192

| 3.90

|}

* 243et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, despite inconsistency in the corresponding odd limit. The same subgroup is only better tuned by [[270edo|270et]]. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is [[354edo|354et]] in terms of absolute error and [[935edo|935et]] in terms of relative error.

243et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, ~~even though it is inconsistent~~. The same subgroup is only better tuned by [[270edo|270et]]. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is [[354edo|354et]] in terms of absolute error and [[935edo|935et]] in terms of relative error.

=== Rank-2 temperaments ===

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

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| [[Ennealimmal]]

|}

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

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

[[Category:Ennealimmal]]

[[Category:Jove]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''243 equal divisions of the octave''' ('''243edo'''), or the '''243(-tone) equal temperament''' ('''243tet''', '''243et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 243 [[equal]] parts of about 4.934 [[cent]]s each.
+{{EDO intro}}
 == 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.
+et [[tempering out|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.
@@ Line 14: / Line 14: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal 8ve <br>stretch (¢)
+! rowspan="2" | Optimal 8ve <br>Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 25: / Line 25: @@
 | 2.3
 | {{monzo| -385 243 }}
-| [{{val| 243 385 }}]
+| {{mapping| 243 385 }}
 | +0.227
 | 0.227
@@ Line 32: / Line 32: @@
 | 2.3.5
 | 2109375/2097152, {{monzo| 1 -27 18 }}
-| [{{val| 243 385 564 }}]
+| {{mapping| 243 385 564 }}
 | +0.314
 | 0.222
@@ Line 39: / Line 39: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, 2109375/2097152
-| [{{val| 241 385 564 682 }}]
+| {{mapping| 241 385 564 682 }}
 | +0.318
 | 0.192
 | 3.90
 |}
+* 243et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, despite inconsistency in the corresponding odd limit. The same subgroup is only better tuned by [[270edo|270et]]. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is [[354edo|354et]] in terms of absolute error and [[935edo|935et]] in terms of relative error.
-et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, even though it is inconsistent. The same subgroup is only better tuned by [[270edo|270et]]. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is [[354edo|354et]] in terms of absolute error and [[935edo|935et]] in terms of relative error.
 === Rank-2 temperaments ===
@@ Line 51: / Line 50: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 92: / Line 91: @@
 | [[Ennealimmal]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Ennealimmal]]
 [[Category:Jove]]