3776edo

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Prime factorization 26 × 59
Step size 0.317797¢ 
Fifth 2209\3776 (702.013¢)
Semitones (A1:m2) 359:283 (114.1¢ : 89.94¢)
Consistency limit 13
Distinct consistency limit 13

3776 equal divisions of the octave (abbreviated 3776edo or 3776ed2), also called 3776-tone equal temperament (3776tet) or 3776 equal temperament (3776et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3776 equal parts of about 0.318 ¢ each. Each step represents a frequency ratio of 21/3776, or the 3776th root of 2.

Theory

3776edo is a good 2.3.11.13.19 subgroup system. It does not tune the 15-odd-limit consistently, though a reasonable represenation exists through the 19-limit patent val. There, it provides the optimal patent val for the oganesson temperament in the 7-, 11-, 13-, 17-, and the 19-limit. It tempers out the quartisma in the 11-limit, and is a tuning for the rank-3 van gogh temperament.

In the 19-limit, and 2.3.5.17.19 subgroup, 3776edo tempers out the comma that associates 171/170 to 1 step of 118edo, hence enabling usage of this interval as microchroma. This is strengthened by 3776edo's strong and consistent approximations of 19/17 and 10/9, intervals that are one 171/170 apart.

Odd harmonics

Approximation of odd harmonics in 3776edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.058 +0.127 +0.136 +0.115 +0.059 +0.044 -0.133 -0.083 -0.055 -0.124 +0.010
Relative (%) +18.2 +40.0 +42.8 +36.3 +18.6 +14.0 -41.9 -26.0 -17.4 -39.1 +3.0
Steps
(reduced)
5985
(2209)
8768
(1216)
10601
(3049)
11970
(642)
13063
(1735)
13973
(2645)
14752
(3424)
15434
(330)
16040
(936)
16585
(1481)
17081
(1977)

The following tables show how 19-odd-limit intervals are represented in 3776edo. Prime harmonics are in bold; inconsistent intervals are in italics.

19-odd-limit intervals in 3776edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/6, 12/11 0.001 0.5
7/5, 10/7 0.009 2.8
9/5, 10/9 0.012 3.6
13/12, 24/13 0.013 4.2
13/11, 22/13 0.015 4.7
9/7, 14/9 0.021 6.5
19/17, 34/19 0.027 8.5
13/8, 16/13 0.044 14.0
15/14, 28/15 0.049 15.3
17/15, 30/17 0.051 15.9
19/16, 32/19 0.055 17.4
11/9, 18/11 0.056 17.7
3/2, 4/3 0.058 18.2
11/8, 16/11 0.059 18.6
11/10, 20/11 0.068 21.3
5/3, 6/5 0.069 21.8
13/9, 18/13 0.071 22.4
11/7, 14/11 0.077 24.2
19/15, 30/19 0.078 24.5
7/6, 12/7 0.078 24.6
17/16, 32/17 0.083 26.0
13/10, 20/13 0.083 26.0
13/7, 14/13 0.092 28.8
17/14, 28/17 0.099 31.2
19/13, 26/19 0.100 31.4
17/10, 20/17 0.108 34.1
19/12, 24/19 0.113 35.6
19/11, 22/19 0.115 36.0
9/8, 16/9 0.115 36.3
17/9, 18/17 0.120 37.7
15/11, 22/15 0.125 39.5
19/14, 28/19 0.126 39.8
17/13, 26/17 0.127 39.9
5/4, 8/5 0.127 40.0
15/8, 16/15 0.133 41.9
19/10, 20/19 0.135 42.6
7/4, 8/7 0.136 42.8
17/12, 24/17 0.140 44.1
15/13, 26/15 0.140 44.1
17/11, 22/17 0.142 44.6
19/18, 36/19 0.147 46.3
19-odd-limit intervals in 3776edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/6, 12/11 0.001 0.5
7/5, 10/7 0.009 2.8
9/5, 10/9 0.012 3.6
13/12, 24/13 0.013 4.2
13/11, 22/13 0.015 4.7
9/7, 14/9 0.021 6.5
19/17, 34/19 0.027 8.5
13/8, 16/13 0.044 14.0
15/14, 28/15 0.049 15.3
19/16, 32/19 0.055 17.4
11/9, 18/11 0.056 17.7
3/2, 4/3 0.058 18.2
11/8, 16/11 0.059 18.6
11/10, 20/11 0.068 21.3
5/3, 6/5 0.069 21.8
13/9, 18/13 0.071 22.4
11/7, 14/11 0.077 24.2
7/6, 12/7 0.078 24.6
17/16, 32/17 0.083 26.0
13/10, 20/13 0.083 26.0
13/7, 14/13 0.092 28.8
19/13, 26/19 0.100 31.4
19/12, 24/19 0.113 35.6
19/11, 22/19 0.115 36.0
9/8, 16/9 0.115 36.3
15/11, 22/15 0.125 39.5
17/13, 26/17 0.127 39.9
5/4, 8/5 0.127 40.0
7/4, 8/7 0.136 42.8
17/12, 24/17 0.140 44.1
15/13, 26/15 0.140 44.1
17/11, 22/17 0.142 44.6
19/18, 36/19 0.171 53.7
19/10, 20/19 0.182 57.4
15/8, 16/15 0.185 58.1
19/14, 28/19 0.191 60.2
17/9, 18/17 0.198 62.3
17/10, 20/17 0.209 65.9
17/14, 28/17 0.218 68.8
19/15, 30/19 0.240 75.5
17/15, 30/17 0.267 84.1

Subsets and supersets

Since 3776 factors as 26 × 59, 3776edo has subset edos 2, 4, 8, 16, 32, 59, 64, 118, 236, 472, 944, 1888, of which 16edo, 118edo and 472edo are particularly notable.

Regular temperament properties

Rank-2 temperaments

Periods
per 8ve
Generator
(reduced)
Cents
(reduced)
Associated
ratio
Temperaments
118 1781\3776
(21\3776)
565.995
(6.67)
165/119
(?)
Oganesson