476edo

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← 475edo476edo477edo →
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 476-tone equal temperament (476tet), 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 of 476edo 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 -24 -30 +12 +31 -41 +32 +37 -1 +26 -22
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