243edo

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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

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 2^1/243, 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

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)

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

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 distinct