3776edo
| ← 3775edo | 3776edo | 3777edo → |
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
| 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.
| 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 |
| 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 |