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. 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]].

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{{citation needed}}; 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

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=== 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 ==

{~~{comma basis begin}}~~

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

|-

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

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

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

! rowspan="2" | Optimal<br />8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

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

| {{mapping| 359 569 834 }}

| ~~−0~~.2042

| −0.2042

| 0.2910

| 8.71

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

| {{mapping| 359 569 834 1008 }}

| ~~−0~~.2007

| −0.2007

| 0.2521

| 7.54

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

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

| ~~−0~~.1729

| −0.1729

| 0.2322

| 6.95

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

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

| ~~−0~~.2257

| −0.2257

| 0.2426

| 7.26

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| 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

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| 4/3

| [[Counterschismic]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<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]

* "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:3-limit record edos|###]]

[[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. 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]].
+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{{citation needed}}; 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
@@ Line 15: / Line 15: @@
 === 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 ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
@@ Line 30: / Line 39: @@
 | 393216/390625, {{monzo| -69 45 -1 }}
 | {{mapping| 359 569 834 }}
-| &minus;0.2042
+| −0.2042
 | 0.2910
 | 8.71
@@ Line 37: / Line 46: @@
 | 2401/2400, 3136/3125, {{monzo| -18 24 -5 -3 }}
 | {{mapping| 359 569 834 1008 }}
-| &minus;0.2007
+| −0.2007
 | 0.2521
 | 7.54
@@ Line 44: / Line 53: @@
 | 2401/2400, 3136/3125, 8019/8000, 42592/42525
 | {{mapping| 359 569 834 1008 1242 }}
-| &minus;0.1729
+| −0.1729
 | 0.2322
 | 6.95
@@ Line 51: / Line 60: @@
 | 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125
 | {{mapping| 359 569 834 1008 1242 1328 }} (359f)
-| &minus;0.2257
+| −0.2257
 | 0.2426
 | 7.26
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| 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
@@ Line 76: / Line 92: @@
 | 4/3
 | [[Counterschismic]]
-{{rank-2 end}}
+|}
-{{orf}}
+<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) &ndash; [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]
+* "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:3-limit record edos|###]] <!-- 3-digit number -->
 [[Category:Hera]]
 [[Category:Listen]]