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243edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3<sup>5</sup>
{{ED intro}}
| Step size = 4.93827¢
| Fifth = 142\243 (701.23¢)
| Semitones = 22:19 (108.64¢ : 93.83)
| Consistency = 9
}}
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 ==
== 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.  
243edo is a strong higher-limit system, especially if we skip [[prime harmonic|prime]] [[11/1|11]]. It is [[consistent]] to the no-11 [[29-odd-limit]] tending flat, with the [[3/1|3]], [[5/1|5]], [[7/1|7]], [[13/1|13]], [[17/1|17]], [[19/1|19]], [[23/1|23]], and [[29/1|29]] all tuned flat.  


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.  
As an equal temperament, it [[tempering out|tempers out]] the [[semicomma]] (2109375/2097152, the 5-limit orwell comma) and the [[ennealimma]] in the 5-limit, and [[2401/2400]] and [[4375/4374]] in the 7-limit. It [[support]]s [[ennealimmal]], [[quadrawell]], and [[sabric]].  


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]].  
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 ===
=== Prime harmonics ===
{{Harmonics in equal|243}}
{{Harmonics in equal|243}}
=== Octave stretch ===
243edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[385edt]] or [[628ed6]]. This improves most of the approximated harmonics, including the 11 if we use the 243e val.
=== Subsets and supersets ===
Since 243 factors into primes as 3<sup>5</sup>, 243edo has subset edos {{EDOs| 3, 9, 27, and 81 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| -385 243 }}
| {{Monzo| -385 243 }}
| [{{val| 243 385 }}]
| {{Mapping| 243 385 }}
| +0.227
| +0.227
| 0.227
| 0.227
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| 2.3.5
| 2.3.5
| 2109375/2097152, {{monzo| 1 -27 18 }}
| 2109375/2097152, {{monzo| 1 -27 18 }}
| [{{val| 243 385 564 }}]
| {{Mapping| 243 385 564 }}
| +0.314
| +0.314
| 0.222
| 0.222
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 4375/4374, 2109375/2097152
| 2401/2400, 4375/4374, 2109375/2097152
| [{{val| 241 385 564 682 }}]
| {{Mapping| 243 385 564 682 }}
| +0.318
| +0.318
| 0.192
| 0.192
| 3.90
| 3.90
|-
| 2.3.5.7.13
| 625/624, 729/728, 2401/2400, 10985/10976
| {{Mapping| 243 385 564 682 899 }}
| +0.309
| 0.173
| 3.50
|-
| 2.3.5.7.13.17
| 625/624, 729/728, 833/832, 1225/1224, 10985/10976
| {{Mapping| 243 385 564 682 899 993 }}
| +0.309
| 0.158
| 3.20
|-
| 2.3.5.7.13.17.19
| 513/512, 625/624, 729/728, 833/832, 1225/1224, 1445/1444
| {{Mapping| 243 385 564 682 899 993 1032 }}
| +0.306
| 0.146
| 2.96
|-
| 2.3.5.7.13.17.19.23
| 513/512, 625/624, 729/728, 833/832, 875/874, 897/896, 1105/1104
| {{Mapping| 243 385 564 682 899 993 1032 1099 }}
| +0.298
| 0.138
| 2.80
|- style="border-top: double;"
| 2.3.5.7.11
| 385/384, 1375/1372, 4375/4374, 14641/14580
| {{Mapping| 243 385 564 682 840 }} (243e)
| +0.437
| 0.295
| 5.97
|-
| 2.3.5.7.11.13
| 385/384, 625/624, 729/728, 847/845, 1716/1715
| {{Mapping| 243 385 564 682 840 899 }} (243e)
| +0.410
| 0.276
| 5.59
|}
|}
 
* 243et (243e val) has lower absolute errors than any previous equal temperaments in the 19-, 23-limit, and somewhat beyond, despite inconsistency in the corresponding odd limits. In both the 19- and 23-limit, it beats [[217edo|217]] and is only bettered by [[270edo|270et]].  
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.  
* It is much stronger in the no-11 subgroups of the limits above, holding the record of lowest relative errors until being bettered in the no-11 19-limit by [[354edo|354et]] in terms of absolute error and [[935edo|935et]] in terms of relative error, and in the no-11 23-limit by [[422edo|422]] in terms of absolute error and [[2460edo|2460]] in terms of relative error.  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Ennealimmal]]
| [[Ennealimmal]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave]]
[[Category:Ennealimmal]]
[[Category:Ennealimmal]]
[[Category:Jove]]
[[Category:Jove]]
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