Ragismic microtemperaments: Difference between revisions

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The ragisma is [[4375/4374]] with a [[monzo]] of |-1 -7 4 1>, the smallest 7-limit [[superparticular]] ratio. Since (10/9)^4=4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

The ragisma is [[4375/4374]] with a [[monzo]] of |-1 -7 4 1>, the smallest 7-limit [[superparticular]] ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

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= Deca =

~~Commas~~: 4375/4374, 165288374272/164794921875

[[Comma list]]: 4375/4374, 165288374272/164794921875

~~POTE generator~~: ~~~460992/390625 = 284.423~~

[[Mapping]]: [<10 4 9 2|, <0 5 6 11|]

~~Map~~: [<~~10 4 2 9|,~~ <~~0 5 6 11~~|]

[[Wedgie]]: <<50 60 110 -21 34 87||

~~EDOs~~: ~~80, 190, 270, 1270, 1540, 1810, 2080~~

POTE generator: ~6/5 = 315.577

~~Badness~~: ~~0.0806~~

[[EDO|Vals]]: {{Val list| 80, 190, 270, 1270, 1540, 1810, 2080 }}

~~==11-limit==~~

[[Badness]]: 0.080637

~~Commas~~: ~~3025/3024, 4375/4374, 422576/421875~~

~~POTE generator~~: ~~~33~~/~~28 = 284.418~~

== 11-limit ==

Comma list: 3025/3024, 4375/4374, 422576/421875

~~Map~~: [<10 4 2 9 18|, <0 5 6 11 7|]

Mapping: [<10 4 9 2 18|, <0 5 6 11 7|]

~~EDOs~~: ~~80, 190, 270, 1000, 1270~~

POTE generator: ~6/5 = 315.582

~~Badness~~: ~~0.0243~~

Vals: {{Val list| 80, 190, 270, 1000, 1270 }}

~~==13-limit==~~

Badness: 0.024329

~~Commas~~: ~~1001/1000, 3025/3024, 4225/4224, 4375/4374~~

~~POTE generator~~: ~~~33~~/~~28 = 284.398~~

== 13-limit ==

Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374

~~Map~~: [<10 4 2 9 18 37|, <0 5 6 11 7 0|]

Mapping: [<10 4 9 2 18 37|, <0 5 6 11 7 0|]

~~EDOs~~: ~~80, 190, 270, 730, 1000~~

POTE generator: ~6/5 = 315.602

Badness: 0.~~0168~~

Vals: {{Val list| 80, 190, 270, 730, 1000 }}

Badness: 0.016810

= Mitonic =

~~Commas~~: 4375/4374, 2100875/2097152

[[Comma list]]: 4375/4374, 2100875/2097152

~~POTE generator~~: ~~~10/9 = 182.458~~

[[Mapping]]: [<1 -1 -3 6|, <0 17 35 -21|]

~~Map:~~ [~~<1 16 32 -15|, <0 -17 -35 21~~|]

[[POTE tuning|POTE generator]]: ~10/9 = 182.458

~~EDOs~~: 46, 125, 171

[[EDO|Vals]]: {{Val list| 46, 125, 171 }}

Badness: 0.~~0252~~

[[Badness]]: 0.025184

=Abigail=

= Abigail =

~~Commas~~: 4375/4374, 2147483648/2144153025

[[Comma list]]: 4375/4374, 2147483648/2144153025

[[~~POTE_tuning|POTE generator~~]]: ~~208.899~~

[[Mapping]]: [<2 7 13 -1|, <0 -11 -24 19|]

~~Map~~: [<~~2 7 13 -1|,~~ <0 -11 -~~24 19~~|]

[[Wedgie]]: <<22 48 -38 25 -122 -223||

~~Wedgie~~: ~~<<22 48 -38 25 -122 -223||~~

[[POTE tuning|POTE generator]]: ~6912/6125 = 208.899

~~EDOs~~: 46, 132, 178, 224, 270, 494, 764, 1034, 1798

[[EDO|Vals]]: {{Val list| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}

Badness: 0.~~0370~~

[[Badness]]: 0.037000

==11-limit==

== 11-limit ==

Comma: 3025/3024, 4375/4374, 20614528/20588575

Comma list: 3025/3024, 4375/4374, 20614528/20588575

[~~[POTE_tuning~~|~~POTE generator~~]~~]: 208.901~~

Mapping: [<2 7 13 -1 1|, <0 -11 -24 19 17|]

~~Map~~: ~~[<2 7 13 -1 1|, <0 -11 -24 19 17|]~~

POTE generator: ~1155/1024 = 208.901

~~EDOs~~: 46, 132, 178, 224, 270, 494, 764

Vals: {{Val list| 46, 132, 178, 224, 270, 494, 764 }}

Badness: 0.~~0129~~

Badness: 0.012860

==13-limit==

== 13-limit ==

~~Commas~~: 1716/1715, 2080/2079, 3025/3024, 4096/4095

Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095

[~~[POTE_tuning~~|~~POTE generator~~]~~]: 208.903~~

Map: [<2 7 13 -1 1 -2|, <0 -11 -24 19 17 27|]

~~Map~~: ~~[<2 7 13 -1 1 -2|, <0 -11 -24 19 17 27|]~~

POTE generator: ~44/39 = 208.903

~~EDOs~~: 46, 178, 224, 270, 494, 764, 1258

Vals: {{Val list| 46, 178, 224, 270, 494, 764, 1258 }}

Badness: 0.~~00886~~

Badness: 0.008856

=Semidimi=

= Semidimi =

The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit |-12 -73 55> and 7-limit 3955078125/3954653486, as well as 4375/4374.

@@ Line 1: / Line 1: @@
-The ragisma is [[4375/4374]] with a [[monzo]] of |-1 -7 4 1&gt;, the smallest 7-limit [[superparticular]] ratio. Since (10/9)^4=4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
+The ragisma is [[4375/4374]] with a [[monzo]] of |-1 -7 4 1&gt;, the smallest 7-limit [[superparticular]] ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
 Temperaments not discussed here include [[Jubilismic clan #Crepuscular|crepuscular]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Hystrix|hystrix]], [[Starling temperaments #Sensi|sensi]], [[Gamelismic clan #Unidec|unidec]], [[Orwellismic temperaments #Quartonic|quartonic]], [[Kleismic family #Catakleismic|catakleismic]], [[Tetracot family #Modus|modus]], [[Schismatic family #Pontiac|pontiac]], [[Würschmidt family #Whirrschmidt|whirrschmidt]],  [[Gravity family #Zarvo|zarvo]], [[Vishnuzmic family #Vishnu|vishnu]], and [[Vulture family #Vulture|vulture]].
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 = Deca =
-Commas: 4375/4374, 165288374272/164794921875
+[[Comma list]]: 4375/4374, 165288374272/164794921875
-POTE generator: ~460992/390625 = 284.423
+[[Mapping]]: [&lt;10 4 9 2|, &lt;0 5 6 11|]
-Map: [&lt;10 4 2 9|, &lt;0 5 6 11|]
+[[Wedgie]]: &lt;&lt;50 60 110 -21 34 87||
-EDOs: 80, 190, 270, 1270, 1540, 1810, 2080
+POTE generator: ~6/5 = 315.577
-Badness: 0.0806
+[[EDO|Vals]]: {{Val list| 80, 190, 270, 1270, 1540, 1810, 2080 }}
-==11-limit==
+[[Badness]]: 0.080637
-Commas: 3025/3024, 4375/4374, 422576/421875
-POTE generator: ~33/28 = 284.418
+== 11-limit ==
+Comma list: 3025/3024, 4375/4374, 422576/421875
-Map: [&lt;10 4 2 9 18|, &lt;0 5 6 11 7|]
+Mapping: [&lt;10 4 9 2 18|, &lt;0 5 6 11 7|]
-EDOs: 80, 190, 270, 1000, 1270
+POTE generator: ~6/5 = 315.582
-Badness: 0.0243
+Vals: {{Val list| 80, 190, 270, 1000, 1270 }}
-==13-limit==
+Badness: 0.024329
-Commas: 1001/1000, 3025/3024, 4225/4224, 4375/4374
-POTE generator: ~33/28 = 284.398
+== 13-limit ==
+Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
-Map: [&lt;10 4 2 9 18 37|, &lt;0 5 6 11 7 0|]
+Mapping: [&lt;10 4 9 2 18 37|, &lt;0 5 6 11 7 0|]
-EDOs: 80, 190, 270, 730, 1000
+POTE generator: ~6/5 = 315.602
-Badness: 0.0168
+Vals: {{Val list| 80, 190, 270, 730, 1000 }}
+Badness: 0.016810
 = Mitonic =
 {{see also|Minortonic family #Mitonic}}
-Commas: 4375/4374, 2100875/2097152
+[[Comma list]]: 4375/4374, 2100875/2097152
-POTE generator: ~10/9 = 182.458
+[[Mapping]]: [&lt;1 -1 -3 6|, &lt;0 17 35 -21|]
-Map: [&lt;1 16 32 -15|, &lt;0 -17 -35 21|]
+[[POTE tuning|POTE generator]]: ~10/9 = 182.458
-EDOs: 46, 125, 171
+[[EDO|Vals]]: {{Val list| 46, 125, 171 }}
-Badness: 0.0252
+[[Badness]]: 0.025184
-=Abigail=
+= Abigail =
-Commas: 4375/4374, 2147483648/2144153025
+[[Comma list]]: 4375/4374, 2147483648/2144153025
-[[POTE_tuning|POTE generator]]: 208.899
+[[Mapping]]: [&lt;2 7 13 -1|, &lt;0 -11 -24 19|]
-Map: [&lt;2 7 13 -1|, &lt;0 -11 -24 19|]
+[[Wedgie]]: &lt;&lt;22 48 -38 25 -122 -223||
-Wedgie: &lt;&lt;22 48 -38 25 -122 -223||
+[[POTE tuning|POTE generator]]: ~6912/6125 = 208.899
-EDOs: 46, 132, 178, 224, 270, 494, 764, 1034, 1798
+[[EDO|Vals]]: {{Val list| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
-Badness: 0.0370
+[[Badness]]: 0.037000
-==11-limit==
+== 11-limit ==
-Comma: 3025/3024, 4375/4374, 20614528/20588575
+Comma list: 3025/3024, 4375/4374, 20614528/20588575
-[[POTE_tuning|POTE generator]]: 208.901
+Mapping: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]
-Map: [&lt;2 7 13 -1 1|, &lt;0 -11 -24 19 17|]
+POTE generator: ~1155/1024 = 208.901
-EDOs: 46, 132, 178, 224, 270, 494, 764
+Vals: {{Val list| 46, 132, 178, 224, 270, 494, 764 }}
-Badness: 0.0129
+Badness: 0.012860
-==13-limit==
+== 13-limit ==
-Commas: 1716/1715, 2080/2079, 3025/3024, 4096/4095
+Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
-[[POTE_tuning|POTE generator]]: 208.903
+Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]
-Map: [&lt;2 7 13 -1 1 -2|, &lt;0 -11 -24 19 17 27|]
+POTE generator: ~44/39 = 208.903
-EDOs: 46, 178, 224, 270, 494, 764, 1258
+Vals: {{Val list| 46, 178, 224, 270, 494, 764, 1258 }}
-Badness: 0.00886
+Badness: 0.008856
-=Semidimi=
+= Semidimi =
 The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit |-12 -73 55&gt; and 7-limit 3955078125/3954653486, as well as 4375/4374.