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360edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>2</sup> × 3<sup>2</sup> × 5
{{ED intro}}
| Step size = 3.333333¢
| Fifth = 211\360 (703.333¢)
| Semitones = 37:25 (123.333¢ : 83.333¢)
| Consistency = 7
}}
'''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'''.


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]].  
360edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for [[degrees]], the {{nowrap|140 &amp; 220}} temperament with period 1\20. Aside from that, it provides the optimal patent val for the {{nowrap|41 &amp; 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 &amp; 360}} with the comma basis {[[390625/388962]], 67108864/66430125}.


Its 23-limit patent val is &lt;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.  
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.  


Its 5-limit patent val [[support]]s [[misty]] temperament.
=== Odd harmonics ===
{{Harmonics in equal|360}}


In the 7-limit, 360edo supports the [[trimisty]] (name proposed by Eliora) 63&amp;99 temperament with wedgie &lt;&lt;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&amp;360 = &lt;&lt;11 76 51 95 50 -95|| and 12&amp;360 = &lt;&lt;12 -48 -108 -104 -205 -116||; neither is very good though 41&amp;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.  
=== 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".


Much better is [[Hemimage_temperaments#Degrees|degrees temperament]], the 80&amp;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 
== 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]].


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.
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
|-
|-
|3
! Periods<br />per 8ve
|211\360
! Generator*
(91\360)
! Cents*
|703.33
! Associated<br />ratio*
(303.33)
! Temperaments
|3/2
|[[Misty]]
|-
|-
|9
| 1
|211\360
| 119\360
(11\360)
| 396.67
|703.33
| 44/35
(36.67)
| [[Squarschmidt]]
|3/2
|[[Trimisty]]
|-
|-
|20
| 2
|211\360
| 53\360
(13\360)
| 176.67
|703.33
| 448/405
(43.33)
| Quatracot
|3/2
(45/44)
|[[Degrees]]
|}
== Table of intervals ==
{| class="wikitable"
|+Table of selected intervals
!Step
!Name
!Calendar notation (if unison is Jan 1)
!Ratio
|-
|-
|0
| 3
|Prime, unison
| 149\360<br />(29\360)
|January 1
| 703.33<br />(303.33)
|1/1
| 4/3<br />(135/128)
| [[Misty]]
|-
|-
|1
| 4
|Degree, grad, schisma
| 23\360
|January 2
| 76.67
|32805/32768
| 4302592/4100625
| [[Reenactment]]
|-
|-
|30
| 9
|Dodecaphonic semitone
| 149\360<br />(29\360)
|February 1
| 703.33<br />(36.67)
|89/84
| 4/3<br />(135/128)
| [[Trimisty]]
|-
|-
|36
| 12
|Septimal diatonic semitone, decioctave
| 73\360<br />(13\360)
|February 6
| 243.333<br />(43.333)
|[[15/14]]
| 3145728/2734375<br />(?)
| [[Magnesium]] (360d)
|-
|-
|60
| 20
|Dodecaphonic major second
| 149\360<br />(5\360)
|March 1
| 703.33<br />(43.33)
|
| 4/3<br />(126/125)
|-
| [[Degrees]]
|90
|Dodecaphonic minor third
|April 1
|
|-
|116
|Classical major third
|April 26
|
|-
|120
|
|May 1
|
|-
|150
|
|June 1
|
|-
|180
|Symmetric tritone
|July 1
|
|-
|210
|Dodecaphonic perfect fifth
|August 1
|442/295
|-
|211
|Just perfect fifth
|August 2
|3/2
|-
|240
|
|September 1
|
|-
|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 melodic]]
[[Category:Real-life sonifications]]
[[Category:Listen]]
[[Category:Listen]]
{{Todo| cleanup |comment=move trimisty away}}
Retrieved from "https://en.xen.wiki/w/360edo"