415edo
| ← 414edo | 415edo | 416edo → |
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
| 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 | +40 | -5 | +48 | +34 | +32 | -36 | -30 | +11 | +19 | -28 | |
| 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.