The Archipelago: Difference between revisions
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== Rank-2 temperaments == | == Rank-2 temperaments == | ||
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]], [[ | 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]], [[septiennealimmic 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 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. | 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. | ||
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=== Barbados === | === Barbados === | ||
{{See also|Extraclassical tonality}} | {{See also|Extraclassical tonality}} | ||
[[Subgroup]] | See [[Subgroup temperaments#Barbados]]. | ||
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]]. | |||
[[ | |||
==== Pinkan ==== | ==== Pinkan ==== | ||
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[[Sval]] [[mapping]]: [{{val| 1 0 -1 -7 }}, {{val| 0 2 3 10 }}] | [[Sval]] [[mapping]]: [{{val| 1 0 -1 -7 }}, {{val| 0 2 3 10 }}] | ||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/ | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~26/15 = 951.132 | ||
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~26/15 = 951.097 | |||
{{Optimal ET sequence|legend=1| 5, 24, 29, 53, 82, 111, 135 }} | {{Optimal ET sequence|legend=1| 5, 24, 29, 53, 82, 111, 135 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 0.039 | ||
==== Tobago ==== | ==== Tobago ==== | ||
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==== Crete ==== | ==== Crete ==== | ||
Crete is a strong extension of Barbados tempering out [[64/63]], thus mapping [[7/4]] to a stack of 4 semifourths, and favoring a flat semifourth near 1 step of [[5edo]]. It is a weak extension of [[archy]]. This is a somewhat peculiar temperament as most archy systems do not have interseptimals or equate them with septimal intervals as in [[oceanfront]], but crete can be used in [[34d-edo]], [[39edo|39d-edo]], [[44edo]] and [[49edo]] (with 34d and 39d | Crete is a strong extension of Barbados tempering out [[64/63]], thus mapping [[7/4]] to a stack of 4 semifourths, and favoring a flat semifourth near 1 step of [[5edo]]. It is a weak extension of [[archy]]. This is a somewhat peculiar temperament as most archy systems do not have interseptimals or equate them with septimal intervals as in [[oceanfront]], but crete can be used in [[34edo|34d-edo]], [[39edo|39d-edo]], [[44edo]], and [[49edo]] (with 34d and 39d often considered better than 34edo and 39edo's [[semaphore]] patent vals). Some other notable non-patent vals are [[54edo]] using its second-best 13/5 and [[93edo|93b-edo]]. Crete can be seen as a version of [[pseudo-semaphore]] with a singular mapping for 3, making pseudo-semaphore's flat fifth [[52/35]] and pseudo-semaphore's sharp fifth [[3/2]]. It was discovered and named by {{u|CompactStar}} in 2026. Since it is the combination of [[Archy]] (which is named after an Ancient Greek music theorist) and island, Crete is named after a Greek island. | ||
[[Subgroup]]: 2.3.7.13/5 | [[Subgroup]]: 2.3.7.13/5 | ||
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[[Sval]] [[mapping]]: [{{val| 1 0 6 -1 }}, {{val| 0 2 -4 3 }}] | [[Sval]] [[mapping]]: [{{val| 1 0 6 -1 }}, {{val| 0 2 -4 3 }}] | ||
: Mapping generators: ~2, ~26/15 | |||
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~26/15 = 954.438 | [[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~26/15 = 954.438 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1|5, 24d, 29d, 34d, 39d, 44}} | ||
[[Badness]] (Sintel): 0.457 | [[Badness]] (Sintel): 0.457 | ||