407edo

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← 406edo407edo408edo →
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