360edo: Difference between revisions

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~~360edo divides the octave into steps of 3 1/3 cents. 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. Its 5-limit patent val supports misty temperament, and the 7-limit patent val supports the 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. Much better is~~ [[~~Hemimage_temperaments#Degrees~~|~~degrees temperament~~]]~~, the 80&140 temperament, 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~~. ~~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~~.

== 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.

360edo is used in ~~the~~ [https://en.~~wikipedia~~.~~org~~/~~wiki/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.

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 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}.

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 ===

=== 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".

== 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]].

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

See: [[Table of 360edo intervals]]

== Regular temperament properties ==

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 119\360

| 396.67

| 44/35

| [[Squarschmidt]]

|-

| 2

| 53\360

| 176.67

| 448/405

| Quatracot

|-

| 3

| 149\360 (29\360)

| 703.33 (303.33)

| 4/3 (135/128)

| [[Misty]]

|-

| 4

| 23\360

| 76.67

| 4302592/4100625

| [[Reenactment]]

|-

| 9

| 149\360 (29\360)

| 703.33 (36.67)

| 4/3 (135/128)

| [[Trimisty]]

|-

| 12

| 73\360 (13\360)

| 243.333 (43.333)

| 3145728/2734375 (?)

| [[Magnesium]] (360d)

|-

| 20

| 149\360 (5\360)

| 703.33 (43.33)

| 4/3 (126/125)

| [[Degrees]]

|}

<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

== 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:Sonifications]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
-edo divides the octave into steps of 3 1/3 cents. Its 23-limit patent val is &lt;360 571 836 1011 1245 1332 1471 1529 1628|.
+{{Infobox ET}}
+{{ED intro}}
-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. Its 5-limit patent val supports misty temperament, and the 7-limit patent val supports the 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. Much better is [[Hemimage_temperaments#Degrees|degrees temperament]], the 80&amp;140 temperament, 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. 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.
+== Theory ==
+edo 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.
-edo is used in the [https://en.wikipedia.org/wiki/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.
+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.
+edo 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}.
+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 ===
+{{Harmonics in equal|360}}
+=== Subsets and supersets ===
+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 ==
+[[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.
+See: [[Table of 360edo intervals]]
+== Regular temperament properties ==
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 119\360
+| 396.67
+| 44/35
+| [[Squarschmidt]]
+|-
+| 2
+| 53\360
+| 176.67
+| 448/405
+| Quatracot
+|-
+| 3
+| 149\360<br />(29\360)
+| 703.33<br />(303.33)
+| 4/3<br />(135/128)
+| [[Misty]]
+|-
+| 4
+| 23\360
+| 76.67
+| 4302592/4100625
+| [[Reenactment]]
+|-
+| 9
+| 149\360<br />(29\360)
+| 703.33<br />(36.67)
+| 4/3<br />(135/128)
+| [[Trimisty]]
+|-
+| 12
+| 73\360<br />(13\360)
+| 243.333<br />(43.333)
+| 3145728/2734375<br />(?)
+| [[Magnesium]] (360d)
+|-
+| 20
+| 149\360<br />(5\360)
+| 703.33<br />(43.33)
+| 4/3<br />(126/125)
+| [[Degrees]]
+|}
+<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
+== Music ==
+; [[User:Eliora|Eliora]]
+* [https://www.youtube.com/watch?v=VSKqwJkWu_U ''Idyllic Tribe''] (2022)
+== Application as a logarithmic scale outside of music ==
+edo 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:Sonifications]]
+[[Category:Listen]]
+{{Todo| cleanup |comment=move trimisty away}}