360edo: Difference between revisions

Line 3:

== Theory ==

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 is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. Its 5-limit [[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. 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 [[linus comma]], meaning 15/14 corresponds to 1/10 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 140 & 220 temperament with period 1\20.

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.

~~=== As an interval size measure ===~~

~~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.~~

~~One step of 360edo is known as '''the Dröbisch angle''', being first proposed by Moritz Dröbisch in the 19th century at first merely by the name "angle".~~

=== 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 ==

Line 380:

Line 381:

}}

==Regular temperament properties==

== Regular temperament properties ==

===Rank-2 temperaments ===

=== Rank-2 temperaments ===

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

!Periods per 8ve

! Periods per 8ve

!Generator~~ (reduced)~~

! Generator*

!Cents~~ (reduced)~~

! Cents*

!Associated ~~ratio~~

! Associated Ratio*

!Temperaments

! Temperaments

|-

|1

| 1

|119\360

| 119\360

|396.67

| 396.67

|44/35

| 44/35

|[[Squarschmidt]]

| [[Squarschmidt]]

|-

|2

| 2

|53\360

| 53\360

|176.67

| 176.67

|448/405

| 448/405

|~~[[Ragismic microtemperaments#~~Quatracot~~|Quatracot]]~~

| Quatracot

|-

|3

| 3

|211\360 (91\360)

| 211\360 (91\360)

|703.33 (303.33)

| 703.33 (303.33)

|3/2

| 3/2

|[[Misty]]

| [[Misty]]

|-

|4

| 4

|23\360

| 23\360

|76.67

| 76.67

|4302592/4100625

| 4302592/4100625

|[[Reenactment]]

| [[Reenactment]]

|-

|9

| 9

|211\360 (11\360)

| 211\360 (11\360)

|703.33 (36.67)

| 703.33 (36.67)

|3/2

| 3/2

|[[Trimisty]]

| [[Trimisty]]

|-

|20

| 20

|211\360 (13\360)

| 211\360 (13\360)

|703.33 (43.33)

| 703.33 (43.33)

|3/2 (45/44)

| 3/2 (45/44)

|[[Degrees]]

| [[Degrees]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==

; [[User:Eliora|Eliora]]

* [https://www.youtube.com/watch?v=VSKqwJkWu_U ''Idyllic Tribe''] (2022)

~~==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.

~~==References==~~

* [https://www.huygens-fokker.org/docs/measures.html Logarithmic size measures]

==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:Sonifications]]

[[Category:Listen]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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.
+edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. Its 5-limit [[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. 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 [[linus comma]], meaning 15/14 corresponds to 1/10 of the octave, 36 steps.
+edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for [[degrees]], the 140 &amp; 220 temperament with period 1\20.
-edo 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&amp;140 temperament with period 20.
-=== As an interval size measure ===
-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.
-One step of 360edo is known as '''the Dröbisch angle''', being first proposed by Moritz Dröbisch in the 19th century at first merely by the name "angle".
 === Odd harmonics ===
-{{harmonics in equal|360}}
+{{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 ==
@@ Line 380: / Line 381: @@
 }}
-==Regular temperament properties==
+== Regular temperament properties ==
-===Rank-2 temperaments ===
+=== Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-!Periods<br>per 8ve
+! Periods<br>per 8ve
-!Generator<br>(reduced)
+! Generator*
-!Cents<br>(reduced)
+! Cents*
-!Associated<br>ratio
+! Associated<br>Ratio*
-!Temperaments
+! Temperaments
 |-
-|1
+| 1
-|119\360
+| 119\360
-|396.67
+| 396.67
-|44/35
+| 44/35
-|[[Squarschmidt]]
+| [[Squarschmidt]]
 |-
-|2
+| 2
-|53\360
+| 53\360
-|176.67
+| 176.67
-|448/405
+| 448/405
-|[[Ragismic microtemperaments#Quatracot|Quatracot]]
+| Quatracot
 |-
-|3
+| 3
-|211\360<br>(91\360)
+| 211\360<br>(91\360)
-|703.33<br>(303.33)
+| 703.33<br>(303.33)
-|3/2
+| 3/2
-|[[Misty]]
+| [[Misty]]
 |-
-|4
+| 4
-|23\360
+| 23\360
-|76.67
+| 76.67
-|4302592/4100625
+| 4302592/4100625
-|[[Reenactment]]
+| [[Reenactment]]
 |-
-|9
+| 9
-|211\360<br>(11\360)
+| 211\360<br>(11\360)
-|703.33<br>(36.67)
+| 703.33<br>(36.67)
-|3/2
+| 3/2
-|[[Trimisty]]
+| [[Trimisty]]
 |-
-|20
+| 20
-|211\360<br>(13\360)
+| 211\360<br>(13\360)
-|703.33<br>(43.33)
+| 703.33<br>(43.33)
-|3/2<br>(45/44)
+| 3/2<br>(45/44)
-|[[Degrees]]
+| [[Degrees]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+== Music ==
+; [[User:Eliora|Eliora]]
+* [https://www.youtube.com/watch?v=VSKqwJkWu_U ''Idyllic Tribe''] (2022)
-==Music==
+== Application as a logarithmic scale outside of music ==
-* [https://www.youtube.com/watch?v=VSKqwJkWu_U Idyllic Tribe] by [[User:Eliora|Eliora]]
+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.
-==References==
-* [https://www.huygens-fokker.org/docs/measures.html Logarithmic size measures]
-==Application as a logarithmic scale outside of music==
-edo 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:Sonifications]]
 [[Category:Listen]]
+{{Todo| cleanup |comment=move trimisty away}}