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The Archipelago: Difference between revisions

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* [[History (temperament)|History]] → [[Werckismic temperaments #History|Werckismic temperaments]]  
* [[History (temperament)|History]] → [[Werckismic temperaments #History|Werckismic temperaments]]  
: +364/363, 441/440
: +364/363, 441/440
* [[Borneo]] → [[Lehmerismic temperaments #Borneo|Lehmerismic temperaments]]
* [[Borneo]] → [[Landscape family #Borneo|Landscape family]]
: +1001/1000, 3025/3024
: +1001/1000, 3025/3024
* [[Enlil|Enlil aka sumatra]] → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]]
* [[Enlil|Enlil a.k.a. sumatra]] → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]]
: +325/324, 385/384
: +325/324, 385/384
* [[Madagascar]] → [[Cataharry family #Madagascar|Cataharry family]]
* [[Madagascar]] → [[Cataharry family #Madagascar|Cataharry family]]
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* [[Hagrid]] → [[Cataharry family #Hagrid|Cataharry family]]
* [[Hagrid]] → [[Cataharry family #Hagrid|Cataharry family]]
: +243/242, 351/350
: +243/242, 351/350
* [[Baffin]] → [[Olympic clan #Baffin|Olympic clan]]
* [[Baffin]] → [[Vishdelismic clan #Baffin|Vishdelismic clan]]
: +1001/1000, 4096/4095
: +1001/1000, 4096/4095
* [[Kujuku]] → [[Pentacircle clan #Kujuku|Pentacircle clan]]
* [[Kujuku]] → [[Pentacircle clan #Kujuku|Pentacircle clan]]
: +352/351, 364/363
: +352/351, 364/363
* [[Namaka]] → [[Hemifamity family#Namaka|Hemifamity family]]
* [[Namaka]] → [[Aberschismic family #Namaka|Aberschismic family]]
: +352/351, 1001/1000
: +352/351, 847/845


== Rank-2 temperaments ==
== Rank-2 temperaments ==
Rank two temperaments tempering out 676/675 include the 13-limit versions of [[Ragismic microtemperaments #Hemiennealimmal|hemiennealimmal]], [[Breedsmic temperaments #Harry|harry]], [[Kleismic family #Tritikleismic|tritikleismic]], [[Kleismic family #Catakleismic|catakleimsic]], [[Marvel temperaments #Negri|negri]], [[Hemifamity temperaments #Mystery|mystery]], [[Hemifamity temperaments #Buzzard|buzzard]], [[Kleismic family #Quadritikleismic|quadritikleismic]].
Rank-2 temperaments tempering out 676/675 include the 13-limit versions of [[semaphoresmic clan #Negri|negri]], [[buzzardsmic clan #Buzzard|buzzard]], [[kleismic family #Catakleismic|catakleismic]], [[gravity family #Harry|harry]], [[kleismic family #Tritikleismic|tritikleismic]], [[quintosec family #Decoid|decoid]], [[kleismic family #Quadritikleismic|quadritikleismic]], [[landscape microtemperaments #Avicenna|avicenna]], [[septiennealimmal clan #Hemiennealimmal|hemiennealimmal]], [[wizmic microtemperaments #Tertiathirds|tertiathirds]], and [[aberschismic temperaments #Mystery|mystery]].


It is interesting to note the Graham complexity of 15/13 in these temperaments. This is 18 in hemiennealimmal, 6 in harry, 9 in tritikleismic, 3 in catakleismic, 2 in negri, 2 in buzzard, 12 in quadritikleismic. Catakleismic and buzzard are particularly interesting from an archipelago point of view. Mystery is special case, since the 15/13 part of it belongs to [[29edo]] alone.
It is interesting to note the Graham complexity of 15/13 in these temperaments. This is 2 in negri, 2 in buzzard, 3 in catakleismic, 6 in harry, 9 in tritikleismic, 10 in decoid, 12 in quadritikleismic, 12 in avicenna, 18 in hemiennealimmal, 26 in tertiathirds. Catakleismic and buzzard are particularly interesting from an archipelago point of view. Mystery is special case, since the 15/13 part of it belongs to [[29edo]] alone.
 
=== Decitonic aka decoid ===
{{see also| Breedsmic temperaments #Decoid}}
 
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 676/675, 1001/1000, 1716/1715, 4096/4095
 
[[Mapping]]: [{{val| 10 0 47 36 98 37 }}, {{val| 0 2 -3 -1 -8 0 }}]
 
[[POTE generator]]: ~15/13 = 248.917
 
{{Optimal ET sequence|legend=1| 130, 270, 940, 1210f }}
 
[[Badness]]: 0.013475
 
=== Avicenna ===
{{see also| Landscape microtemperaments #Avicenna }}
 
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 676/675, 1001/1000, 3025/3024, 4096/4095
 
[[Mapping]]: [{{val| 3 2 8 16 9 8 }}, {{val| 0 8 -3 -22 4 9 }}]
 
[[CTE|CTE generator]]: ~13/12 = 137.777
 
[[POTE generator]]: ~13/12 = 137.777
 
{{Optimal ET sequence|legend=1| 87, 183, 270 }}
 
[[Badness]]: 0.015557
 
=== Tertiathirds ===
{{see also| Wizmic microtemperaments #Tertiathirds }}
 
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 676/675, 1716/1715, 3025/3024, 4225/4224
 
[[Mapping]]: [{{val| 1 -4 2 -6 -9 -5 }}, {{val| 0 52 3 82 116 81 }}]
 
[[POTE generator]]: ~14/13 = 128.8902
 
{{Optimal ET sequence|legend=1| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}
 
[[Badness]]: 0.019494
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224
 
Mapping: [{{val| 1 -4 2 -6 -9 -5 -3 }}, {{val| 0 52 3 82 116 81 66 }}]
 
POTE generator: ~14/13 = 128.8912
 
{{Optimal ET sequence|legend=1| 121, 149, 270 }}
 
Badness: 0.019107


== Subgroup temperaments ==
== Subgroup temperaments ==
=== Barbados ===
=== Barbados ===
{{See also|Extraclassical tonality}}
Perhaps the simplest method of making use of the barbados triad and other characteristic island harmonies is to strip things down to essentials by tempering the 2.3.13/5 [[just intonation subgroup]]. The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
Perhaps the simplest method of making use of the barbados triad and other characteristic island harmonies is to strip things down to essentials by tempering the 2.3.13/5 [[just intonation subgroup]]. The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.


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: [[gencom]]: [55/39 15/13; 243/242 676/675]
: [[gencom]]: [55/39 15/13; 243/242 676/675]


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 249.312
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~55/39 = 1\2, ~15/13 = 249.312


{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
Line 248: Line 189:
[[Comma list]]: 221/220, 243/242, 289/288
[[Comma list]]: 221/220, 243/242, 289/288


{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}
{{Mapping|legend=3| 2 0 -1 -2 5 | 0 2 5 3 2 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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