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.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-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]].
+edo is [[enfactoring|enfactored]] in the 13-limit, with the same tuning as [[270edo]]. Aside from the patent val, there are a number of mappings to be cosidered. The 1080e val, {{mapping|1080 1712 2508 3032 '''3737'''}}, [[Tempering out|tempers out]] 114345/114244, and the 1080ef val, {{mapping|1080 1712 2508 3032 '''3737 3997'''}} it tempers out [[2080/2079]].
-== Divisors ==
+=== Odd harmonics ===
-The prime factorization of 1080 is
+{{Harmonics in equal|1080}}
-<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.
+=== 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.
-[[Category:edo]]
+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.
+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]].
+== Selected intervals ==
+{| class="wikitable mw-collapsible mw-collapsed"
+|-
+! Step
+! Eliora's Naming System
+! Approximate Ratio
+! Comments
+|-
+| 0
+| Prime
+|
+|
+|-
+| 3
+| Degree
+|
+| Derives from 360edo.
+|-
+| 4
+| Ducentiseptuagesima
+|
+| Derives from 270edo
+|-
+| 7
+| Septimal kelisma
+|
+|
+|-
+| 15
+| Moria
+|
+| Derives form 72edo.
+|-
+| 90
+| Dodecaphonic semitone
+|
+|
+|-
+| 94
+| Septendecimal semitone
+| 17/16
+|
+|-
+| 240
+| Septimal submajor second
+| 7/6
+| Derives form 9edo.
+|-
+| 360
+| Landscape major third
+| 63/50
+|
+|-
+| 495
+| 24-phonic superfourth
+|
+| Derives from 24edo.
+|-
+| 496
+| Undecimal superfourth
+| 11/8
+|
+|-
+| 630
+| Dodecaphonic fifth
+|
+|
+|-
+| 632
+| 135-phonic Fifth
+| 3/2
+|
+|-
+| 756
+| Tridecimal neutral sixth, 13th harmonic
+| 13/8
+| Derives from 10edo.
+|-
+| 1080
+| Octave
+|
+|
+|}
+== Music ==
+; [[No Clue Music]]
+* [https://www.youtube.com/watch?v=hQOvnQhAcKU ''Not Torture Music''] (2024)
+[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->