3ed11/9

3 equal divisions of 11/9 (abbreviated 3ed11/9) is a nonoctave tuning system that divides the interval of 11/9 into 3 equal parts of about 116 ¢ each. Each step represents a frequency ratio of (11/9)^1/3, or the cube root of 11/9.

← 2ed11/9

3ed11/9

4ed11/9 →

Prime factorization	3 (prime)
Step size	115.803 ¢
Octave	10\3ed11/9 (1158.03 ¢) (semiconvergent)
Twelfth	16\3ed11/9 (1852.84 ¢) (semiconvergent)
Consistency limit	3
Distinct consistency limit	3

11 steps of this temperament is an extremely close approximation of 9²:13², having only 0.5% relative error. 6 steps is exactly 9²:11² (since 3 steps is 9:11), so 9²:11²:13² (81:121:169) is well approximated, which represents the approximate 2:3:4 created by overtones of chimes.

9²:11²:13²:17²:23² is also very well approximated, as is 15²:19²:21².

Approximation of odd square harmonics relative to 9²

Todo: formatting

ratio | steps | relative error | absolute error

1²:9² | -66 | -30.4% | -35.2¢

3²:9² | -33 | -15.2% | -17.6¢

5²:9² | -18 | -42.5% | -49.3¢

7²:9² | -8 | -48.6% | -56.3¢

9²:9² | 0 | 0% | 0¢

11²:9² | 6 | 0% | 0¢

13²:9² | 11 | -0.51% | -0.59¢

15²:9² | 15 | -27.4% | -32.1¢

17²:9² | 19 | -1.6% | -1.8¢

19²:9² | 22 | -34.2% | -39.5¢

21²:9² | 25 | -33.4% | -38.7¢

23²:9² | 28 | -5.4% | -6.3¢

9ed11/9 is a possible correction for 15, 19, and 21.