The Archipelago: Difference between revisions
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The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup. | The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup. | ||
The barbados triad is of particular theoretical interest because, when reduced to lowest terms, it is the 10:13:15 triad. Thus, this triad is only slightly higher in complexity than the 5-limit 10:12:15 minor triad, which means it may be of distinct value as a relatively unexplored musical consonance. It is one of only a few low-complexity triads with a 3/2 on the outer | The barbados triad is of particular theoretical interest because, when reduced to lowest terms, it is the 10:13:15 triad. Thus, this triad is only slightly higher in complexity than the 5-limit 10:12:15 minor triad, which means it may be of distinct value as a relatively unexplored musical consonance. It is one of only a few low-complexity triads with a 3/2 on the outer interval, some others being 4:5:6, 6:7:9, and 10:12:15. It works out to 0-454-702 cents, which means that it is an ''ultramajor'' triad, with a third sharper even than the 9/7 supermajor third. | ||
Compared to the 7-limit 14:18:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:18:21), but contains | Compared to the 7-limit 14:18:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:18:21), but contains intervals that are on average higher in complexity (9/7 vs 13/10 and 7/6 vs 15/13). Its inverse, however, is the ultraminor 26:30:39, which is far more complex than the 7-limit subminor 6:7:9. [[The Biosphere|Temperaments in which 91/90 vanishes]] equate the two types of triads. | ||
[[24edo]] approximates this triad to within an error of four cents, and [[29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below. | [[24edo]] approximates this triad to within an error of four cents, and [[29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below. | ||
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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]] → [[ | * [[Borneo]] → [[Landscape family #Borneo|Landscape family]] | ||
: +1001/1000, 3025/3024 | : +1001/1000, 3025/3024 | ||
* [[Enlil|Enlil | * [[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]] | ||
: +351/350, 540/539 | : +351/350, 540/539 | ||
* [[Baffin]] → [[ | * [[Hagrid]] → [[Cataharry family #Hagrid|Cataharry family]] | ||
: +243/242, 351/350 | |||
* [[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]] → [[Aberschismic family #Namaka|Aberschismic family]] | |||
: +352/351, 847/845 | |||
== Rank-2 temperaments == | == Rank-2 temperaments == | ||
Rank | 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 | 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. | ||
== 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]]): ~ | [[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 }} | ||
[[Tp tuning#T2 tuning|RMS error]]: 0.3533 cents | [[Tp tuning#T2 tuning|RMS error]]: 0.3533 cents | ||
==== Pakkanian hemipyth ==== | |||
[[Subgroup]]: 2.3.11.13/5.17 | |||
[[Comma list]]: 221/220, 243/242, 289/288 | |||
{{Mapping|legend=3| 2 0 -1 -2 5 | 0 2 5 3 2 }} | |||
[[Optimal tuning]]s: | |||
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344) | |||
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989) | |||
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }} | |||
<nowiki>*</nowiki> wart for 13/5 | |||
=== Cata === | === Cata === | ||
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{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }} | {{Optimal ET sequence|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }} | ||
[[Badness]]: 0. | [[Badness]]: 0.00394 | ||
=== Taylor === | === Taylor === | ||