The Archipelago: Difference between revisions

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_subgroups|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 dyad, 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.

[[24edo|24edo]] approximates this triad to within an error of four cents, and [[29edo|29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below.

=Parent Temperaments=

 

=Island=

[[Optimal_patent_val|Optimal patent val]]: [[940edo|940edo]]

 

==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_subgroups|just intontation subgroup]]. The minimax tuning for this makes the generator 2/sqrt(3), or 249.0225 cents. EDOs which may be used for it are [[24edo|24edo]], [[29edo|29edo]], [[53edo|53edo]] and [[111edo|111edo]], with MOS of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

 

[[Smonzos_and_Svals|Sval map]]: [<1 0 -1|, <0 2 3|]

==1001/1000==

 

==49/48==

==1716/1715==

==364/363==

===351/350===

=Rank three temperaments=

==[[Breed_family|Greenland]]==

 

[[Optimal_patent_val|Optimal patent val]]: [[940edo|940edo]]

 

[[Spectrum_of_a_temperament|Spectrum]]: 15/13, 7/5, 8/7, 7/6, 4/3, 15/14, 5/4, 18/13, 13/12, 14/13, 13/10, 6/5, 16/15, 11/10, 9/7, 9/8, 16/13, 10/9, 14/11, 11/8, 15/11, 12/11, 13/11, 11/9

==[[Werckismic_temperaments|History]]==

 

[[Optimal_patent_val|Optimal patent val]]: [[289edo|289edo]]

 

==Borneo==

 

[[Optimal_patent_val|Optimal patent val]]: [[270edo|270edo]]

 

==Sumatra==

 

Optimal patent val: [[299edo|299edo]]

 

==[[Cataharry_family|Madagascar]]==

 

[[Optimal_patent_val|Optimal patent val]]: [[313edo|313edo]]

 

 

==Baffin==

 

[[Optimal_patent_val|Optimal patent val]]: [[940edo|940edo]]

 

==Kujuku==

 

[[Optimal_patent_val|Optimal patent val]]: [[208edo|208edo]]

 

=Rank two temperaments=

Rank two temperaments tempering out 676/675 include the 13-limit versions of [[Ragismic_microtemperaments|hemiennealimmal]], [[Breedsmic_temperaments|harry]], [[Kleismic_family|tritikleismic]], [[Kleismic_family|catakleimsic]], [[Marvel_temperaments|negri]], [[Hemifamity_temperaments|mystery]], [[Hemifamity_temperaments|buzzard]], [[Kleismic_family|quadritikleismic]].

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|29edo]] alone.

==Decitonic==

[[POTE_tuning|POTE generator]]: ~15/13 = 248.917

==Avicenna==

[[POTE_tuning|POTE generator]]: ~13/12 = 137.777

=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_subgroups|just intontation subgroup]]. The minimax tuning for this makes the generator 2/sqrt(3), or 249.0225 cents. EDOs which may be used for it are [[24edo|24edo]], [[29edo|29edo]], [[53edo|53edo]] and [[111edo|111edo]], with MOS of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

 

[[Smonzos_and_Svals|Sval map]]: [<1 0 -1|, <0 2 3|]

===Music===

[http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 Desert Island Rain] in 313et tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

==Trinidad==

Trinidad may be viewed as the reduction of [[Kleismic_family|catakleismic temperament]] to the 2.3.5.13 subgroup. Another way to put it is that it is the rank two 2.3.5.13 subgroup temperament tempering out 325/324, 625/624 and hence also 676/675.

 

[[Smonzos_and_Svals|Sval map]]: [<1 0 1 0 |, <0 6 5 14|]

==[[Chromatic_pairs#Tobago|Tobago]]==

==Parizekmic==

[[Smonzos_and_Svals|Sval map]]

 

===Music===

[http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3 Petr's Pump], a comma pump based ditty in Parizekmic temperament.

 

EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270

[[Category:archipelago]]

[[Category:listen]]

[[Category:overview]]

[[Category:temperaments]]

[[Category:theory]]