# 415edo

 ← 414edo 415edo 416edo →
Prime factorization 5 × 83
Step size 2.89157¢
Fifth 243\415 (702.651¢)
Semitones (A1:m2) 41:30 (118.6¢ : 86.75¢)
Consistency limit 7
Distinct consistency limit 7

415 equal divisions of the octave (abbreviated 415edo or 415ed2), also called 415-tone equal temperament (415tet) or 415 equal temperament (415et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 415 equal parts of about 2.89 ¢ each. Each step represents a frequency ratio of 21/415, or the 415th root of 2.

If harmonic 5 is used, 415edo tends very sharp. In the 5-limit the equal temperament tempers out the parakleisma, [8 14 -13; in the 7-limit 3136/3125 and 4375/4374, so that it supports parakleismic, the 99 & 316 temperament, and provides the optimal patent val. In the 11-limit it tempers out 12005/11979, 16384/16335, and 41503/41472; and in the 13-limit, 676/675, 1001/1000, 2080/2079, 3584/3575, and 10648/10647.

### Odd harmonics

Approximation of odd harmonics in 415edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.70 +1.16 -0.15 +1.39 +0.97 +0.92 -1.04 -0.86 +0.32 +0.54 -0.80
Relative (%) +24.1 +40.0 -5.2 +48.1 +33.6 +31.8 -36.0 -29.7 +11.0 +18.8 -27.8
Steps
(reduced)
658
(243)
964
(134)
1165
(335)
1316
(71)
1436
(191)
1536
(291)
1621
(376)
1696
(36)
1763
(103)
1823
(163)
1877
(217)

### Subsets and supersets

Since 415 factors into 5 × 83, 415edo contains 5edo and 83edo as subsets.