243edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Eliora (talk | contribs)
autopatrolled
3,351 edits
Cleanup
Line 1: Line 1:
{{Infobox ET}}
{{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 ==
== 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.  
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 ==
== 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|Comma List]]
! rowspan="2" | [[Mapping]]
! 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]] (¢)
! [[TE error|Absolute]] (¢)
Line 25: Line 25:
| 2.3
| 2.3
| {{monzo| -385 243 }}
| {{monzo| -385 243 }}
| [{{val| 243 385 }}]
| {{mapping| 243 385 }}
| +0.227
| +0.227
| 0.227
| 0.227
Line 32: Line 32:
| 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
Line 39: Line 39:
| 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| 241 385 564 682 }}
| +0.318
| +0.318
| 0.192
| 0.192
| 3.90
| 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 ===
=== Rank-2 temperaments ===
Line 51: Line 50:
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
Line 92: Line 91:
| [[Ennealimmal]]
| [[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:Ennealimmal]]
[[Category:Jove]]
[[Category:Jove]]

Revision as of 12:50, 24 March 2024

← 242edo 243edo 244edo →
Prime factorization 35
Step size 4.93827 ¢ 
Fifth 142\243 (701.235 ¢)
Semitones (A1:m2) 22:19 (108.6 ¢ : 93.83 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

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 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 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 ⟨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

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 [⟨241 385 564 682]] +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 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 354et in terms of absolute error and 935et in terms of relative error.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* 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

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Retrieved from "https://en.xen.wiki/index.php?title=243edo&oldid=140059"