3776edo: Difference between revisions

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==Theory==

== Theory ==

3776edo is a ~~tuning~~ for the [[oganesson]] temperament in the 17-limit, ~~which sets 1/118th of~~ the ~~octave to an interval that represents 169/168~170/169 tempered together~~.

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.

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.

~~While it does tune both 13th and 17th prime harmonic resonably, it is no longer consistent in the [[15-odd-limit]].~~

=== Odd harmonics ===

[[~~Category:Equal divisions of the octave|####~~]]

=== 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 ==

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

!Periods per 8ve

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

!Generator~~ (reduced)~~

|-

!Cents~~ (reduced)~~

! Periods per 8ve

!Associated ratio

! Generator*

!Temperaments

! Cents*

! Associated ratio*

! Temperaments

|-

|118

| 118

|1781\3776 (21\3776)

| 1781\3776 (21\3776)

|565.995 (6.67)

| 565.995 (6.67)

|165/119 (?)

| 165/119 (?)

|[[Oganesson]]

| [[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|####]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|3776}}
+{{ED intro}}
-==Theory==
+== Theory ==
-edo is a tuning for the [[oganesson]] temperament in the 17-limit, which sets 1/118th of the octave to an interval that represents 169/168~170/169 tempered together.
+edo 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.
-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.
-While it does tune both 13th and 17th prime harmonic resonably, it is no longer consistent in the [[15-odd-limit]].
 === Odd harmonics ===
 {{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 ==
 === Rank-2 temperaments ===
 {| 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)
+! Periods<br />per 8ve
-!Associated<br>ratio
+! Generator*
-!Temperaments
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
-|118
+| 118
-|1781\3776<br>(21\3776)
+| 1781\3776<br>(21\3776)
-|565.995<br>(6.67)
+| 565.995<br>(6.67)
-|165/119<br>(?)
+| 165/119<br>(?)
-|[[Oganesson]]
+| [[Oganesson]]
-|}<!-- 4-digit number -->
+|}
+<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 -->