405edo

 ← 404edo 405edo 406edo →
Prime factorization 34 × 5
Step size 2.96296¢
Fifth 237\405 (702.222¢) (→79\135)
Semitones (A1:m2) 39:30 (115.6¢ : 88.89¢)
Consistency limit 7
Distinct consistency limit 7

405 equal divisions of the octave (abbreviated 405edo or 405ed2), also called 405-tone equal temperament (405tet) or 405 equal temperament (405et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 405 equal parts of about 2.96 ¢ each. Each step represents a frequency ratio of 21/405, or the 405th root of 2.

Theory

405edo is enfactored in the 3-limit, with the same tuning as 135edo. Like 135edo, it is consistent to the 7-odd-limit with a poor approximation to the harmonic 5. Using the patent val, the equal temperament tempers out 15625/15552 in the 5-limit; 2100875/2097152, and 2460375/2458624 in the 7-limit; 1375/1372, 4000/3993, 19712/19683, and 41503/41472 in the 11-limit. It supports marthirds, novemkleismic and kleirtismic. It provides the optimal patent val for 7- and 11-limit novemkleismic.

Prime harmonics

Approximation of prime harmonics in 405edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.27 -1.13 +0.06 -0.21 +0.95 -1.25 -1.22 -0.13 -1.43 -1.33
Relative (%) +0.0 +9.0 -38.1 +2.1 -7.0 +32.2 -42.2 -41.1 -4.3 -48.2 -45.0
Steps
(reduced)
405
(0)
642
(237)
940
(130)
1137
(327)
1401
(186)
1499
(284)
1655
(35)
1720
(100)
1832
(212)
1967
(347)
2006
(386)

Subsets and supersets

Since 405 factors into 34 × 5, 405edo has subset edos 3, 5, 9, 15, 27, 45, 81, and 135.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5 15625/15552, [110 -65 -3 [405 642 940]] +0.1058 0.2776 9.37
2.3.5.7 15625/15552, 2100875/2097152, 2460375/2458624 [405 642 940 1137]] +0.0737 0.2467 8.33
2.3.5.7.11 1375/1372, 4000/3993, 19712/19683, 41503/41472 [405 642 940 1137 1401]] +0.0709 0.2207 7.45

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 56\405 165.93 11/10 Satin
1 107\405 317.04 6/5 Hanson
9 107\405
(17\405)
317.04
(50.37)
6/5
(36/35)
Novemkleismic

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct