# 407edo

 ← 406edo 407edo 408edo →
Prime factorization 11 × 37
Step size 2.9484¢
Fifth 238\407 (701.72¢)
Semitones (A1:m2) 38:31 (112¢ : 91.4¢)
Consistency limit 7
Distinct consistency limit 7

407 equal divisions of the octave (abbreviated 407edo or 407ed2), also called 407-tone equal temperament (407tet) or 407 equal temperament (407et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 407 equal parts of about 2.95 ¢ each. Each step represents a frequency ratio of 21/407, or the 407th root of 2.

## Theory

407edo is a strong 5-limit system and 2.3.5.11.13.19.23 subgroup system. The equal temperament tempers out 32805/32768 in the 5-limit; using the patent val, 16875/16807, 4096000/4084101, and 26873856/26796875 in the 7-limit. It supports and provides the optimal patent val for the subsemifourth temperament in the 7- and 11-limit. Essentially tempered chords available in 407et include pinkanberry chords.

### Prime harmonics

Approximation of prime harmonics in 407edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.24 -0.07 +1.20 +0.03 -0.23 +1.19 +0.28 -0.26 -0.58 -1.06
Relative (%) +0.0 -8.0 -2.5 +40.7 +1.1 -7.9 +40.3 +9.4 -9.0 -19.8 -35.8
Steps
(reduced)
407
(0)
645
(238)
945
(131)
1143
(329)
1408
(187)
1506
(285)
1664
(36)
1729
(101)
1841
(213)
1977
(349)
2016
(388)

### Subsets and supersets

407 factors into 11 × 37, with 11edo and 37edo as its subset edos. 814edo, which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-645 407 [407 645]] +0.0742 0.0742 2.52
2.3.5 32805/32768, [30 47 -45 [407 645 945]] +0.0599 0.0638 2.16

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 63\407 185.75 [24 4 -13 Pirate
1 83\407 244.72 15/13 Subsemifourth (407f)
1 169\407 498.28 4/3 Helmholtz

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