Xenharmonic Wiki

360edo: Difference between revisions

← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,387 edits
→Table of intervals: more complete
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
(17 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|360}}
{{ED intro}}


One step of 360edo is known as '''the Dröbisch angle'''.
== 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.
360edo is consistent in the 7-limit. Its 5-limit patent val [[support]]s [[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. 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.
 
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 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. 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}}.
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.  


=== Odd harmonics ===
=== Odd harmonics ===
{{harmonics in equal|360}}
{{Harmonics in equal|360}}


=== Subsets and supersets ===
=== Subsets and supersets ===
360 is the 13th [[highly composite EDO]], with 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]], 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".


== Table of intervals ==
== 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]].
 
Any other notation system involving the number 360 can also be used.


Any other notation system involving the number 360 can also be used, such as calling steps degrees, deriving them from Moritz Dröbisch's proposal of calling the step an angle.
See: [[Table of 360edo intervals]]


{| class="wikitable mw-collapsible mw-collapsed"
== Regular temperament properties ==
|+Table of selected intervals
=== Rank-2 temperaments ===
!Degree
{| class="wikitable center-all left-5"
!Name
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
!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
|
|}
|}
<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


==Regular temperament properties==
== Music ==
===Rank-2 temperaments ===
; [[User:Eliora|Eliora]]
{| class="wikitable center-all left-5"
* [https://www.youtube.com/watch?v=VSKqwJkWu_U ''Idyllic Tribe''] (2022)
!Periods<br>per 8ve
!Generator<br>(reduced)
!Cents<br>(reduced)
!Associated<br>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]]
|}


==Music==
== Application as a logarithmic scale outside of music ==
* [https://www.youtube.com/watch?v=VSKqwJkWu_U Idyllic Tribe] by [[User:Eliora|Eliora]]
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.
==Application as a logarithmic scale outside of music==
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.


[[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"