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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}}


== Theory ==
== Theory ==
{{Primes in edo|360|columns=10}}
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.
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 [[highly composite EDO]].  
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.


Its 5-limit patent val [[support]]s [[misty]] temperament.  
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}.


360edo is consistent in the 7-limit. In it, it 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. In addition, 360edo provides the optimal patent val for the 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 12&360 with the comma basis 390625/388962, 67108864/66430125. 360edo tempers out the [[15/14 equal-step tuning|linus comma]], meaning 15/14 corresponds to 1/10th of the octave, 36 steps.  
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.  


360edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for the [[Hemimage_temperaments#Degrees|degrees temperament]], the 80&140 temperament with period 20. 
=== Odd harmonics ===
{{Harmonics in equal|360}}


Eliora proposes a 7-limit reenactment temperament for 360edo, defined as 188 & 360 and named after the YouTubers cs188 and radicalfaith360. It has a comma basis 2097152/2083725 and {{Monzo|0, -19, -10, 19}}.
=== 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".


=== Proposed notation ===
== 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 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]].


=== Miscellaneous properties ===
Any other notation system involving the number 360 can also be used.
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.


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.
See: [[Table of 360edo intervals]]


== 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
! Periods<br />per 8ve
|Dodecaphonic minor third
! Generator*
|April 1
! Cents*
|
! Associated<br />ratio*
! Temperaments
|-
|-
|116
| 1
|Classical major third
| 119\360
|April 26
| 396.67
|
| 44/35
| [[Squarschmidt]]
|-
|-
|120
| 2
|
| 53\360
|May 1
| 176.67
|
| 448/405
| Quatracot
|-
|-
|150
| 3
|
| 149\360<br />(29\360)
|June 1
| 703.33<br />(303.33)
|
| 4/3<br />(135/128)
| [[Misty]]
|-
|-
|180
| 4
|Symmetric tritone
| 23\360
|July 1
| 76.67
|
| 4302592/4100625
| [[Reenactment]]
|-
|-
|210
| 9
|Dodecaphonic perfect fifth
| 149\360<br />(29\360)
|August 1
| 703.33<br />(36.67)
|442/295
| 4/3<br />(135/128)
| [[Trimisty]]
|-
|-
|211
| 12
|Just perfect fifth
| 73\360<br />(13\360)
|August 2
| 243.333<br />(43.333)
|3/2
| 3145728/2734375<br />(?)
| [[Magnesium]] (360d)
|-
|-
|240
| 20
|
| 149\360<br />(5\360)
|September 1
| 703.33<br />(43.33)
|
| 4/3<br />(126/125)
|-
| [[Degrees]]
|270
|
|October 1
|
|-
|291
|Harmonic seventh
|October 21
|
|-
|300
|
|November 1
|
|-
|330
|
|December 1
|
|-
|360
|Octave
|January 1
|
|}
|}
==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"