3776edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|3776}}
{{ED intro}}
==Theory==
== 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.
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.


It tempers out the [[quartisma]] in the 11-limit, and is a tuning for the rank-3 Van Gogh temperament.
=== Odd harmonics ===
=== Odd harmonics ===
{{harmonics in equal|3776}}
{{harmonics in equal|3776}}


[[Category:Equal divisions of the octave|####]]
{{15-odd-limit|3776|19}}
 
=== Subsets and supersets ===
Since 3776 factors as {{Factorization|3776}}, 3776edo has subset edos {{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 ==
== Regular temperament properties ==
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods<br>per 8ve
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
!Generator<br>(reduced)
!Cents<br>(reduced)
!Associated<br>ratio
!Temperaments
|-
|-
|118
! Periods<br />per 8ve
|1781\3776<br>(21\3776)
! Generator*
|565.995<br>(6.67)
! Cents*
|165/119<br>(?)
! Associated<br />ratio*
|[[Oganesson]]
! Temperaments
|}<!-- 4-digit number -->
|-
| 118
| 1781\3776<br>(21\3776)
| 565.995<br>(6.67)
| 165/119<br>(?)
| [[Oganesson]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:Equal divisions of the octave|####]]
<!-- 4-digit number -->

Latest revision as of 23:15, 20 February 2025

← 3775edo 3776edo 3777edo →
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

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

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