243edo: Difference between revisions

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The '''243 equal division''' divides the octave into 243 equal parts of 4.938 cents 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. 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.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|243}}
{{Harmonics in equal|243}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -385 243 }}
| [{{val| 243 385 }}]
| +0.227
| 0.227
| 4.60
|-
| 2.3.5
| 2109375/2097152, {{monzo| 1 -27 18 }}
| [{{val| 243 385 564 }}]
| +0.314
| 0.222
| 4.50
|-
| 2.3.5.7
| 2401/2400, 4375/4374, 2109375/2097152
| [{{val| 241 382 560 677 }}]
| +0.318
| 0.192
| 3.90
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 11\243
| 54.32
| 405/392
| [[Quinwell]]
|-
| 1
| 47\243
| 232.10
| 8/7
| [[Quadrawell]]
|-
| 1
| 55\243
| 271.60
| 75/64
| [[Sabric]]
|-
| 1
| 64\243
| 316.05
| 6/5
| [[Counterkleismic]]
|-
| 1
| 92\243
| 454.32
| 13/10
| [[Fibo]]
|-
| 9
| 64\243<br>(10\243)
| 316.05<br>(49.38)
| 6/5<br>(36/35)
| [[Ennealimmal]]
|}


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

Revision as of 21:18, 16 March 2022

The 243 equal division of the octave (243edo) divides the octave into 243 equal parts of 4.938 cents 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 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 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.

Prime harmonics

Approximation of prime harmonics in 243edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.72 -1.13 -0.92 +1.77 -1.02 -1.25 -1.22 -1.11 -2.42 +0.64
Relative (%) +0.0 -14.6 -22.9 -18.7 +35.8 -20.7 -25.3 -24.6 -22.6 -48.9 +13.0
Steps
(reduced) 		243
(0) 		385
(142) 		564
(78) 		682
(196) 		841
(112) 		899
(170) 		993
(21) 		1032
(60) 		1099
(127) 		1180
(208) 		1204
(232)

Regular temperament properties

Subgroup Comma list Mapping Optimal 8ve
stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-385 243⟩ [⟨243 385]] +0.227 0.227 4.60
2.3.5 2109375/2097152, [1 -27 18⟩ [⟨243 385 564]] +0.314 0.222 4.50
2.3.5.7 2401/2400, 4375/4374, 2109375/2097152 [⟨241 382 560 677]] +0.318 0.192 3.90

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 11\243 54.32 405/392 Quinwell
1 47\243 232.10 8/7 Quadrawell
1 55\243 271.60 75/64 Sabric
1 64\243 316.05 6/5 Counterkleismic
1 92\243 454.32 13/10 Fibo
9 64\243
(10\243) 		316.05
(49.38) 		6/5
(36/35) 		Ennealimmal
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