243edo: Difference between revisions
+prime error table; +links; +category |
+RTT table and rank-2 temperaments, +links, +categories |
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The '''243 equal division''' divides the octave into 243 equal parts of 4.938 | 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]] | |||