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{{Infobox ET}}
{{Infobox ET}}
'''360 equal divisions of the octave''' ('''360edo'''), or '''360-tone equal temperament''' ('''360tet'''), '''360 equal temperament''' ('''360et''') when viewed from a [[regular temperament]] perspective, is the tuning system that divides the [[octave]] into 360 [[equal]] parts of {{ExactlyOrAbout| {{#expr: 1200/360 round 16}} }}{{cent}} each, a step size known as '''the Dröbisch angle'''.
{{ED intro}}


360edo is used in the [[wikipedia:Eyeborg|eyeborg]], which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals.
== Theory ==
360edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. It can also be used with 2.5.9.13 subgroup.  


== Theory ==
In the 5-limit, the [[patent val]] [[support]]s the [[misty]] temperament, and in the 7-limit 360edo supports the [[trimisty]] (name proposed by Eliora) 63 & 99 temperament with the comma basis {[[10976/10935]], 2097152/2083725}, which is similar to the misty temperament but has a period of 1/9- rather than 1/3-octave.
{{Primes in edo|360|columns=10}}
360 has many proper divisors: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180. 360 is the 13th [[superabundant EDO]].  


Its 23-limit patent val is <360 571 836 1011 1245 1332 1471 1529 1628|. This val tempers out the kalisma, the triaphonisma, the septendecimal bridge comma, the misty comma, hemimage, dimicomp, 2*(14/15)^10, 289/288, 352/351, 589824/588245 and 2560000000/2542277421.  
360edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for [[degrees]], the {{nowrap|140 & 220}} temperament with period 1\20. Aside from that, it provides the optimal patent val for the {{nowrap|41 & 360}} temperament with comma basis {10976/10935, 16384000000/16209796869}, on which it has lower badness than any other 7-limit temperament for which 360edo gives the optimal patent val. It also supports {{nowrap|12 & 360}} with the comma basis {[[390625/388962]], 67108864/66430125}.


Its 5-limit patent val [[support]]s [[misty]] temperament.  
Aside from the patent val, there is a number of mappings to be considered. The 360d val, {{val|360 571 836 '''1010'''}}, tempers out 3136/3125, 5120/5103, and extends the misty temperament in to the 7-limit. It is also a tuning for the 12th-octave [[magnesium]] temperament.  


In the 7-limit, 360edo supports the [[trimisty]] (name proposed by Eliora) 63&99 temperament with wedgie <<9 -36 9 -78 -11 122|| which tempers out misty but has a period of 1/9 rather than 1/3 octave,. Two other seven limit temperaments it supports and also provides the optimal patent val for are 41&360 = <<11 76 51 95 50 -95|| and 12&360 = <<12 -48 -108 -104 -205 -116||; neither is very good though 41&360 has a TE badness lower than any alternative 7-limit temperament for which 360 gives the optimal patent val. In the 7-limit, 360edo tempers out the [[15/14 equal-step tuning|linus comma]], meaning 15/14 corresponds to 1/10th of the octave, 36 steps.
=== Odd harmonics ===
{{Harmonics in equal|360}}


Much better is [[Hemimage_temperaments#Degrees|degrees temperament]], the 80&140 temperament with period 20, for which 360 supplies the optimal patent val in the 11-limit and which it supports and provides an excellent tuning for in the 13-limit. In the 
=== Subsets and supersets ===
360 is the 13th [[highly composite edo]], with many proper divisors: {{EDOs| 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 }}. One step of 360edo is known as '''the Dröbisch angle''', an [[interval size measure]] first proposed by Moritz Dröbisch in the 19th century at first merely by the name "angle".


In the 360b val, 360edo's fifth is the same as 12edo. Coincidentally, the difference between a just fifth and a 12edo one is known as the grad, being a variant of translation of "degree", and 1/360th of a circle is a degree.
== Table of intervals ==
[[Eliora]] proposes notating 360edo with calendar dates, Jan 1 being the tonic, Jan 2 being the next step, etc, and each month having even 30 days. The notation is convenient because 1 month in this scenario is equal to 1 semitone, and corresponds to [[12edo]].


Eliora proposes a 7-limit reenactment temperament for 360edo, defined as 188 & 360 and named after the YouTubers cs188 and radicalfaith360. It tempers out 2097152/2083725 and {{Monzo|0, -19, -10, 19}}.
Any other notation system involving the number 360 can also be used.


==== Proposed notation ====
See: [[Table of 360edo intervals]]
Eliora proposes notating 360edo with calendar dates, Jan 1 being the tonic, Jan 2 being the next step, etc, and each month having even 30 days. The notation is convenient because 1 month in this scenario is equal to 1 semitone, and corresponds to [[12edo]].


== Rank two temperaments by generator ==
== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
per octave
!Generator
(reduced)
!Cents
(reduced)
!Associated
ratio
!Temperaments
|-
|1
|119\360
|396.67
|44/35
|[[Squarschmidt]]
|-
|2
|53\360
|176.67
|448/405
|[[Ragismic microtemperaments#Quatracot|Quatracot]]
|-
|3
|211\360<br>(91\360)
|703.33<br>(303.33)
|3/2
|[[Misty]]
|-
|4
|23\360
|76.67
|4302592/4100625
|[[Reenactment]]
|-
|9
|211\360<br>(11\360)
|703.33<br>(36.67)
|3/2
|[[Trimisty]]
|-
|20
|211\360<br>(13\360)
|703.33<br>(43.33)
|3/2<br>(45/44)
|[[Degrees]]
|}
 
== Table of intervals ==
{| class="wikitable"
|+Table of selected intervals
!Step
!Name
!Calendar notation (if unison is Jan 1)
!Ratio
|-
|0
|Prime, unison
|January 1
|1/1
|-
|1
|Degree, grad, schisma
|January 2
|32805/32768
|-
|30
|Dodecaphonic semitone
|February 1
|89/84
|-
|36
|Septimal diatonic semitone, decioctave
|February 6
|[[15/14]]
|-
|60
|Dodecaphonic major second
|March 1
|
|-
|90
|Dodecaphonic minor third
|April 1
|
|-
|116
|Classical major third
|April 26
|
|-
|120
|
|May 1
|
|-
|150
|
|June 1
|
|-
|180
|Symmetric tritone
|July 1
|
|-
|-
|210
! Periods<br />per 8ve
|Dodecaphonic perfect fifth
! Generator*
|August 1
! Cents*
|442/295
! Associated<br />ratio*
! Temperaments
|-
|-
|211
| 1
|Just perfect fifth
| 119\360
|August 2
| 396.67
|3/2
| 44/35
| [[Squarschmidt]]
|-
|-
|240
| 2
|
| 53\360
|September 1
| 176.67
|
| 448/405
| Quatracot
|-
|-
|270
| 3
|
| 149\360<br />(29\360)
|October 1
| 703.33<br />(303.33)
|
| 4/3<br />(135/128)
| [[Misty]]
|-
|-
|291
| 4
|Harmonic seventh
| 23\360
|October 21
| 76.67
|
| 4302592/4100625
| [[Reenactment]]
|-
|-
|300
| 9
|
| 149\360<br />(29\360)
|November 1
| 703.33<br />(36.67)
|
| 4/3<br />(135/128)
| [[Trimisty]]
|-
|-
|330
| 12
|
| 73\360<br />(13\360)
|December 1
| 243.333<br />(43.333)
|
| 3145728/2734375<br />(?)
| [[Magnesium]] (360d)
|-
|-
|360
| 20
|Octave
| 149\360<br />(5\360)
|January 1
| 703.33<br />(43.33)
|
| 4/3<br />(126/125)
| [[Degrees]]
|}
|}
==Music==
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
* [https://www.youtube.com/watch?v=VSKqwJkWu_U Idyllic Tribe] by [[User:Eliora|Eliora]]
 
== Music ==
; [[User:Eliora|Eliora]]
* [https://www.youtube.com/watch?v=VSKqwJkWu_U ''Idyllic Tribe''] (2022)
 
== Application as a logarithmic scale outside of music ==
360edo is used in the {{w|eyeborg}}, which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals.


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Sonifications]]
[[Category:Highly composite]]
[[Category:Real-life sonifications]]
[[Category:Listen]]
[[Category:Listen]]
{{Todo| cleanup |comment=move trimisty away}}
Retrieved from "https://en.xen.wiki/w/360edo"