243edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 242edo243edo244edo →
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 21/243, or the 243rd root of 2.

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)

Subsets and supersets

Since 243 factors into 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 [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