1080edo: Difference between revisions

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== Theory ==

~~1080~~ is ~~a largely composite EDO, meaning it's notable for its divisors. Its~~ [[~~number of the divisors~~|~~32 divisors~~]] ~~are 1~~, 2, 3, 4, ~~5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540. 1080's abundancy index is 2.33..., or exactly 7~~/3.

1080edo is [[enfactoring|enfactored]] in the 13-limit, with the same tuning as [[270edo]]. In the 1080e val, which puts the 11th harmonic on 3737 steps, it [[Tempering out|tempers out]] 114345/114244, and in the 1080ef val it tempers out [[2080/2079]].

~~Notable subsets of 1080edo are [[270edo]] and [[72edo]], as they both belong to [[The Riemann Zeta Function and Tuning#Zeta EDO lists~~|the ''zeta peak edos'', ''zeta integral edos'' and ''zeta gap edos'' sequences]], however the patent val of 1080edo does not consist of their approximation alone, as the 17th harmonic comes from [[540edo]]. In addition, [[12edo]] is the dominant tuning system in the world, and [[360edo]] is a highly composite EDO.

=== Odd harmonics ===

~~As every 4th step of [[4320edo]]~~, it is ~~a good tuning~~ for ~~the 2.5/3~~.~~7 subgroup, and has strong representation for~~ [[~~19/12~~]], ~~[[19/~~10]], ~~[[17/13]]~~, ~~[[23/13]]~~, and ~~[[23~~/~~17]]~~.

=== Subsets and supersets ===

1080 is a largely composite edo, meaning it is notable for its divisors. Its 32 [[number of the divisors|divisors]] are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, and 1080. 1080's abundancy index is 2.33…, or exactly 7/3.

~~=== Regular temperament theory ===~~

Notable subsets of 1080edo are [[270edo]] and [[72edo]], as they both belong to the [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edos, zeta integral edos and zeta gap edos]]. However, the [[patent val]] of 1080edo does not consist of their approximation alone, as the 17th harmonic comes from [[540edo]]. In addition, [[12edo]] is the dominant tuning system in the world, and [[360edo]] is a highly composite edo.

~~In the 13-limit,~~ 1080edo ~~is contorted order-4~~, ~~with~~ the ~~same tuning as~~ [[~~270edo~~]]. In the ~~1080e~~ val, ~~which puts~~ the ~~11th~~ harmonic ~~on 3737~~, ~~it tempers out 114345/114244~~, and ~~in the 1080ef val it tempers out~~ [[~~2080/2079~~]].

~~=== Harmonics ===~~

As every 4th step of [[4320edo]], it is a good tuning for the 2.5/3.7 subgroup, and has strong representation for [[19/12]], [[19/10]], [[17/13]], [[23/13]], and [[23/17]].

~~{{Harmonics in equal|1080}}~~

== ~~Table of~~ intervals ==

== Selected intervals ==

{| class="wikitable"

{| class="wikitable mw-collapsible mw-collapsed"

|+

!Step

! Step

!~~Name~~

! Eliora's Naming System

!~~Associated ratio~~

! Approximate Ratio

!Comments

! Comments

|-

|0

@@ Line 3: / Line 3: @@
 == Theory ==
-is a largely composite EDO, meaning it's notable for its divisors. Its [[number of the divisors|32 divisors]] are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540. 1080's abundancy index is 2.33..., or exactly 7/3.
+edo is [[enfactoring|enfactored]] in the 13-limit, with the same tuning as [[270edo]]. In the 1080e val, which puts the 11th harmonic on 3737 steps, it [[Tempering out|tempers out]] 114345/114244, and in the 1080ef val it tempers out [[2080/2079]].
-Notable subsets of 1080edo are [[270edo]] and [[72edo]], as they both belong to [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the ''zeta peak edos'', ''zeta integral edos'' and ''zeta gap edos'' sequences]], however the patent val of 1080edo does not consist of their approximation alone, as the 17th harmonic comes from [[540edo]]. In addition, [[12edo]] is the dominant tuning system in the world, and [[360edo]] is a highly composite EDO.
+=== Odd harmonics ===
+{{Harmonics in equal|1080}}
-As every 4th step of [[4320edo]], it is a good tuning for the 2.5/3.7 subgroup, and has strong representation for [[19/12]], [[19/10]], [[17/13]], [[23/13]], and [[23/17]].
+=== Subsets and supersets ===
+is a largely composite edo, meaning it is notable for its divisors. Its 32 [[number of the divisors|divisors]] are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, and 1080. 1080's abundancy index is 2.33…, or exactly 7/3.
-=== Regular temperament theory ===
+Notable subsets of 1080edo are [[270edo]] and [[72edo]], as they both belong to the [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edos, zeta integral edos and zeta gap edos]]. However, the [[patent val]] of 1080edo does not consist of their approximation alone, as the 17th harmonic comes from [[540edo]]. In addition, [[12edo]] is the dominant tuning system in the world, and [[360edo]] is a highly composite edo.
-In the 13-limit, 1080edo is contorted order-4, with the same tuning as [[270edo]]. In the 1080e val, which puts the 11th harmonic on 3737, it tempers out 114345/114244, and in the 1080ef val it tempers out [[2080/2079]].
-=== Harmonics ===
+As every 4th step of [[4320edo]], it is a good tuning for the 2.5/3.7 subgroup, and has strong representation for [[19/12]], [[19/10]], [[17/13]], [[23/13]], and [[23/17]].
-{{Harmonics in equal|1080}}
-== Table of intervals ==
+== Selected intervals ==
-{| class="wikitable"
+{| class="wikitable mw-collapsible mw-collapsed"
 |+
-!Step
+! Step
-!Name
+! Eliora's Naming System
-!Associated ratio
+! Approximate Ratio
-!Comments
+! Comments
 |-
 |0