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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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. | |||
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 === | === Subsets and supersets === | ||
Since 243 factors into | 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 == | ||
| Line 28: | Line 33: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| -385 243 }} | ||
| {{ | | {{Mapping| 243 385 }} | ||
| +0.227 | | +0.227 | ||
| 0.227 | | 0.227 | ||
| Line 36: | Line 41: | ||
| 2.3.5 | | 2.3.5 | ||
| 2109375/2097152, {{monzo| 1 -27 18 }} | | 2109375/2097152, {{monzo| 1 -27 18 }} | ||
| {{ | | {{Mapping| 243 385 564 }} | ||
| +0.314 | | +0.314 | ||
| 0.222 | | 0.222 | ||
| Line 43: | Line 48: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 4375/4374, 2109375/2097152 | | 2401/2400, 4375/4374, 2109375/2097152 | ||
| {{ | | {{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 | * 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]]. | ||
* 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 === | ||
| Line 54: | Line 102: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 91: | Line 139: | ||
|- | |- | ||
| 9 | | 9 | ||
| 64\243<br | | 64\243<br>(10\243) | ||
| 316.05<br | | 316.05<br>(49.38) | ||
| 6/5<br | | 6/5<br>(36/35) | ||
| [[Ennealimmal]] | | [[Ennealimmal]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[ | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct | ||
[[Category:Ennealimmal]] | [[Category:Ennealimmal]] | ||
[[Category:Jove]] | [[Category:Jove]] | ||
Latest revision as of 20:01, 12 August 2025
| ← 242edo | 243edo | 244edo → |
243 equal divisions of the octave (abbreviated 243edo or 243ed2), also called 243-tone equal temperament (243tet) or 243 equal temperament (243et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 243 equal parts of about 4.94 ¢ each. Each step represents a frequency ratio of 21/243, or the 243rd root of 2.
Theory
243edo is a strong higher-limit system, especially if we skip prime 11. It is consistent to the no-11 29-odd-limit tending flat, with the 3, 5, 7, 13, 17, 19, 23, and 29 all tuned flat.
As an equal temperament, it 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 supports ennealimmal, quadrawell, and sabric.
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
| 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) | |
Octave stretch
243edo can benefit from slightly 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 35, 243edo has subset edos 3, 9, 27, and 81.
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 | [⟨243 385 564 682]] | +0.318 | 0.192 | 3.90 |
| 2.3.5.7.13 | 625/624, 729/728, 2401/2400, 10985/10976 | [⟨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 | [⟨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 | [⟨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 | [⟨243 385 564 682 899 993 1032 1099]] | +0.298 | 0.138 | 2.80 |
| 2.3.5.7.11 | 385/384, 1375/1372, 4375/4374, 14641/14580 | [⟨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 | [⟨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 217 and is only bettered by 270et.
- 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 354et in terms of absolute error and 935et in terms of relative error, and in the no-11 23-limit by 422 in terms of absolute error and 2460 in terms of relative error.
Rank-2 temperaments
| 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 distinct