# 476edo

 ← 475edo 476edo 477edo →
Prime factorization 22 × 7 × 17
Step size 2.52101¢
Fifth 278\476 (700.84¢) (→139\238)
Semitones (A1:m2) 42:38 (105.9¢ : 95.8¢)
Dual sharp fifth 279\476 (703.361¢)
Dual flat fifth 278\476 (700.84¢) (→139\238)
Dual major 2nd 81\476 (204.202¢)
Consistency limit 7
Distinct consistency limit 7

476 equal divisions of the octave (abbreviated 476edo or 476ed2), also called 476-tone equal temperament (476tet) or 476 equal temperament (476et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 476 equal parts of about 2.52 ¢ each. Each step represents a frequency ratio of 21/476, or the 476th root of 2.

## Theory

476edo is consistent to the 7-odd-limit, but the harmonic 3 is about halfway its steps, while its 5 and 7 are both tuned flat. To start with, consider the 2.3.5.7 patent val, as well as 2.9.15.21 and 2.9.5.7 subgroups.

Using the patent val, the equal temperament tempers out 2401/2400 and 19683/19600 in the 7-limit, supporting harry. The 11-limit 476e val tempers out 3025/3024 and 41503/41472, whereas the patent val tempers out 243/242, 441/440, 540/539, 4000/3993, 8019/8000, and 9801/9800, supporting 11-limit harry.

### Odd harmonics

Approximation of odd harmonics in 476edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.11 -0.60 -0.76 +0.29 +0.78 -1.03 +0.81 +0.93 -0.03 +0.65 -0.54
Relative (%) -44.2 -23.8 -30.1 +11.6 +31.1 -40.9 +32.0 +36.8 -1.3 +25.7 -21.5
Steps
(reduced)
754
(278)
1105
(153)
1336
(384)
1509
(81)
1647
(219)
1761
(333)
1860
(432)
1946
(42)
2022
(118)
2091
(187)
2153
(249)

### Subsets and supersets

476 factors into 22 × 7 × 17, with subset edos 2, 4, 7, 14, 17, 28, 34, 68, 119, and 238. 952edo, which doubles it, gives a good correction to the harmonic 3, but unfortunately it is inconsistent in the 5-odd-limit.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [1509 -476 [476 1509]] -0.0460 0.0460 1.82
2.9.5 [33 -17 9, [-65 0 28 [476 1509 1105]] +0.0554 0.1482 5.88
2.9.5.7 703125/702464, 4802000/4782969, [25 3 -3 8 [476 1509 1105 1336]] +0.1091 0.1586 6.29

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 205\476 516.81 27/20 Larry (476)
2 205\476
(33\476)
516.81
(83.19)
27/20
(21/20)
Harry (11-limit, 476)
28 197\476
(6\476)
496.64
(15.13)
4/3
(105/104)
Oquatonic (5-limit)

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