359edo: Difference between revisions

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== Theory ==

359edo contains a very close approximation of the pure [[3/2]] fifth of 701.955 cents, with the 210\359 step of 701.94986 cents. ~~It provides~~ the [[optimal patent val]] for ~~the~~ 11-limit [[hera]] ~~temperament~~.

359edo contains a very close approximation of the pure [[3/2]] fifth of 701.955 cents, with the 210\359 step of 701.94986 cents. In the 5-limit it tempers out the [[würschmidt comma]] and the [[counterschisma]]; in the 7-limit [[2401/2400]] and [[3136/3125]], supporting [[hemiwürschmidt]]; in the 11-limit, [[8019/8000]], providing the [[optimal patent val]] for 11-limit [[hera]]. Due to the fifth being reached at the extremely divisible number of 210 steps, 359edo turns out to be important as an accurate supporting edo of various temperaments that divide the fifth into multiple parts.

359edo [[support]]s a type of exaggerated Hornbostel mode, with an approximation of the blown fifth that he described of the pan flutes of some regions of South America; the ~~Pythagorean fifth (701~~.~~955¢) minus the~~ Pythagorean ~~comma (23.46¢) = 678.495¢~~; in 359edo this is ~~the step~~ 203~~\359 of~~ 678.~~55153¢~~.

359edo [[support]]s a type of exaggerated Hornbostel mode, with an approximation of the blown fifth that he described of the pan flutes of some regions of South America{{citation needed}}; the 678.495{{c}} [[262144/177147|Pythagorean diminished sixth]]; in 359edo this is reached using 203 steps, or 678.55153{{c}}.

Pythagorean diatonic scale: 61 61 27 61 61 61 27

Exaggerated Hornbostel superdiatonic scale: 47 47 47 15 47 47 47 47 15 (fails in the position of Phi and the square root of Pi [+1\359 step of each one]{{clarify}}).

~~359edo is the 72nd [[prime edo]].~~

=== Prime harmonics ===

=== Subsets and supersets ===

359edo is the 72nd [[prime edo]]. [[718edo]], which doubles it, provides a good correction to the harmonics 5, 13, 17, and 31.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 2.3

| {{monzo| -569 359 }}

| [{{~~val~~| 359 569 }}]

| {{mapping| 359 569 }}

| +0.0016

| 0.0016

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| 2.3.5

| 393216/390625, {{monzo| -69 45 -1 }}

| [{{~~val~~| 359 569 834 }}]

| {{mapping| 359 569 834 }}

| -0.2042

| −0.2042

| 0.2910

| 8.71

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| 2.3.5.7

| 2401/2400, 3136/3125, {{monzo| -18 24 -5 -3 }}

| [{{~~val~~| 359 569 834 1008 }}]

| {{mapping| 359 569 834 1008 }}

| -0.2007

| −0.2007

| 0.2521

| 7.54

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| 2.3.5.7.11

| 2401/2400, 3136/3125, 8019/8000, 42592/42525

| [{{~~val~~| 359 569 834 1008 1242 }}]

| {{mapping| 359 569 834 1008 1242 }}

| -0.1729

| −0.1729

| 0.2322

| 6.95

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| 2.3.5.7.11.13

| 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125

| [{{~~val~~| 359 569 834 1008 1242 1328 }}] (359f)

| {{mapping| 359 569 834 1008 1242 1328 }} (359f)

| -0.2257

| −0.2257

| 0.2426

| 7.26

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=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

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

! Periods per ~~Octave~~

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

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| [[Counterschismic]]

|}

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

== Music ==

; [[Francium]]

* "This Madness Won't Stop!" from ''End Of Sartorius Membranes'' (2024) – [https://open.spotify.com/track/50O9nTxeMafR8AyBtsPSKa Spotify] | [https://francium223.bandcamp.com/track/this-madness-wont-stop Bandcamp] | [https://www.youtube.com/watch?v=UJyIKzgLVQU YouTube]

[[Category:~~Equal divisions of the octave~~|###]]

[[Category:3-limit record edos|###]]

~~[[Category:Prime EDO]]~~

[[Category:Hera]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|359}}
+{{ED intro}}
 == Theory ==
-edo contains a very close approximation of the pure [[3/2]] fifth of 701.955 cents, with the 210\359 step of 701.94986 cents. It provides the [[optimal patent val]] for the 11-limit [[hera]] temperament.
+edo contains a very close approximation of the pure [[3/2]] fifth of 701.955 cents, with the 210\359 step of 701.94986 cents. In the 5-limit it tempers out the [[würschmidt comma]] and the [[counterschisma]]; in the 7-limit [[2401/2400]] and [[3136/3125]], supporting [[hemiwürschmidt]]; in the 11-limit, [[8019/8000]], providing the [[optimal patent val]] for 11-limit [[hera]]. Due to the fifth being reached at the extremely divisible number of 210 steps, 359edo turns out to be important as an accurate supporting edo of various temperaments that divide the fifth into multiple parts.
-edo [[support]]s a type of exaggerated Hornbostel mode, with an approximation of the blown fifth that he described of the pan flutes of some regions of South America; the Pythagorean fifth (701.955¢) minus the Pythagorean comma (23.46¢) = 678.495¢; in 359edo this is the step 203\359 of 678.55153¢.
+edo [[support]]s a type of exaggerated Hornbostel mode, with an approximation of the blown fifth that he described of the pan flutes of some regions of South America{{citation needed}}; the 678.495{{c}} [[262144/177147|Pythagorean diminished sixth]]; in 359edo this is reached using 203 steps, or 678.55153{{c}}.
 Pythagorean diatonic scale: 61 61 27 61 61 61 27
 Exaggerated Hornbostel superdiatonic scale: 47 47 47 15 47 47 47 47 15 (fails in the position of Phi and the square root of Pi [+1\359 step of each one]{{clarify}}).
-edo is the 72nd [[prime edo]].
 === Prime harmonics ===
 {{Harmonics in equal|359|columns=11}}
+=== Subsets and supersets ===
+edo is the 72nd [[prime edo]]. [[718edo]], which doubles it, provides a good correction to the harmonics 5, 13, 17, and 31.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 29: / Line 31: @@
 | 2.3
 | {{monzo| -569 359 }}
-| [{{val| 359 569 }}]
+| {{mapping| 359 569 }}
 | +0.0016
 | 0.0016
@@ Line 36: / Line 38: @@
 | 2.3.5
 | 393216/390625, {{monzo| -69 45 -1 }}
-| [{{val| 359 569 834 }}]
+| {{mapping| 359 569 834 }}
-| -0.2042
+| −0.2042
 | 0.2910
 | 8.71
@@ Line 43: / Line 45: @@
 | 2.3.5.7
 | 2401/2400, 3136/3125, {{monzo| -18 24 -5 -3 }}
-| [{{val| 359 569 834 1008 }}]
+| {{mapping| 359 569 834 1008 }}
-| -0.2007
+| −0.2007
 | 0.2521
 | 7.54
@@ Line 50: / Line 52: @@
 | 2.3.5.7.11
 | 2401/2400, 3136/3125, 8019/8000, 42592/42525
-| [{{val| 359 569 834 1008 1242 }}]
+| {{mapping| 359 569 834 1008 1242 }}
-| -0.1729
+| −0.1729
 | 0.2322
 | 6.95
@@ Line 57: / Line 59: @@
 | 2.3.5.7.11.13
 | 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125
-| [{{val| 359 569 834 1008 1242 1328 }}] (359f)
+| {{mapping| 359 569 834 1008 1242 1328 }} (359f)
-| -0.2257
+| −0.2257
 | 0.2426
 | 7.26
@@ Line 65: / Line 67: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per Octave
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 90: / Line 93: @@
 | [[Counterschismic]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Music ==
+; [[Francium]]
+* "This Madness Won't Stop!" from ''End Of Sartorius Membranes'' (2024) – [https://open.spotify.com/track/50O9nTxeMafR8AyBtsPSKa Spotify] | [https://francium223.bandcamp.com/track/this-madness-wont-stop Bandcamp] | [https://www.youtube.com/watch?v=UJyIKzgLVQU YouTube]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+[[Category:3-limit record edos|###]] <!-- 3-digit number -->
-[[Category:Prime EDO]]
 [[Category:Hera]]
+[[Category:Listen]]