3776edo

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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 2^1/3776, 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

Table of rank-2 temperaments by generator

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

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