1080edo: Difference between revisions

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~~'''1080 tone equal temperament''', also called '''1080-~~EDO~~''' divides the octave in~~ 1080 ~~equal steps of approximately 1.11 [[cent]]s.~~

== Theory ==

~~{{primes in edo|~~1080~~|columns=15}}~~

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.

~~Since 1080 = 4 * 270 and 1080 = 15 * 72~~, it ~~contains~~ [[~~270edo~~]] ~~and [[72edo]] as subsets~~, ~~both belonging to [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the ''zeta peak edos'~~', ~~''zeta integral edos'' and ''zeta gap edos'' sequences]]~~.

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 1080 itself cannot be consistently read through their approximation alone. In addition, [[12edo]] is the dominant tuning system in the world, and [[360edo]] is a highly composite EDO.

=== Regular temperament theory ===

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]].

~~=== Divisors ===~~

~~The prime factorization of 1080 is~~

~~<math>1080 = 2^{3} \cdot 3^{3} \cdot 5</math>~~

~~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.~~

~~1080's abundancy index is 2.33333... = exactly 7/3.~~

== Table of intervals ==

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|

|Derives form 72edo.

|-

~~|21~~

~~|Pythagorean comma~~

|

|-

|90

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|-

|632

|~~270~~-phonic Fifth

|135-phonic Fifth

|3/2

|

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''1080 tone equal temperament''', also called '''1080-EDO''' divides the octave in 1080 equal steps of approximately 1.11 [[cent]]s.
+{{EDO intro|1080}}
 == Theory ==
-{{primes in edo|1080|columns=15}}
+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.
-Since 1080 = 4 * 270 and 1080 = 15 * 72, it contains [[270edo]] and [[72edo]] as subsets, both belonging to [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the ''zeta peak edos'', ''zeta integral edos'' and ''zeta gap edos'' sequences]].
+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 1080 itself cannot be consistently read through their approximation alone. In addition, [[12edo]] is the dominant tuning system in the world, and [[360edo]] is a highly composite EDO.
+=== Regular temperament theory ===
 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]].
-=== Divisors ===
-The prime factorization of 1080 is
-<math>1080 = 2^{3} \cdot 3^{3} \cdot 5</math>
-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 2.33333... = exactly 7/3.
 == Table of intervals ==
@@ Line 48: / Line 42: @@
 |
 |Derives form 72edo.
-|-
-|21
-|Pythagorean comma
-|
-|
 |-
 |90
@@ Line 90: / Line 79: @@
 |-
 |632
-|270-phonic Fifth
+|135-phonic Fifth
 |3/2
 |