129edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 128edo129edo130edo →
Prime factorization 3 × 43
Step size 9.30233¢ 
Fifth 75\129 (697.674¢) (→25\43)
Semitones (A1:m2) 9:12 (83.72¢ : 111.6¢)
Dual sharp fifth 76\129 (706.977¢)
Dual flat fifth 75\129 (697.674¢) (→25\43)
Dual major 2nd 22\129 (204.651¢)
Consistency limit 3
Distinct consistency limit 3

129 equal divisions of the octave (abbreviated 129edo or 129ed2), also called 129-tone equal temperament (129tet) or 129 equal temperament (129et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 129 equal parts of about 9.3 ¢ each. Each step represents a frequency ratio of 21/129, or the 129th root of 2.

129edo is inconsistent to the 5-odd-limit and both harmonics 3 and 5 are about halfway between its steps. The patent val is enfactored in the 5-limit, with the same tuning as 43edo. It is the last patent val that tempers out 81/80 so as to support meantone and its higher-limit expansions. It also tempers out 1029/1024 and 1728/1715 in the 7-limit; 176/175 and 540/539 in the 11-limit; 507/500, 676/675 and 847/845 in the 13-limit; 221/220 in the 17-limit; 171/170 and 286/285 in the 19-limit. It provides the optimal patent val for the 11-limit rank-3 clio temperament.

Odd harmonics

Approximation of odd harmonics in 129edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -4.28 +4.38 -1.38 +0.74 -2.48 -3.32 +0.10 -2.63 +0.16 +3.64 +4.28
Relative (%) -46.0 +47.1 -14.9 +8.0 -26.7 -35.7 +1.1 -28.3 +1.7 +39.1 +46.1
Steps
(reduced)
204
(75)
300
(42)
362
(104)
409
(22)
446
(59)
477
(90)
504
(117)
527
(11)
548
(32)
567
(51)
584
(68)

Subsets and supersets

Since 129 factors into 3 × 43, 129edo contains 3edo and 43edo as its subsets.