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 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.

* [[Borneo]] → [[Lehmerismic temperaments #Borneo|Lehmerismic temperaments]]

: +1001/1000, 3025/3024

: +325/324, 385/384

* [[Madagascar]] → [[Cataharry family #Madagascar|Cataharry family]]

: +351/350, 540/539

* [[Baffin]] → [[Olympic clan #Baffin|Olympic clan]]

: +1001/1000, 4096/4095

* [[Kujuku]] → [[Pentacircle clan #Kujuku|Pentacircle clan]]

: +352/351, 364/363

== Rank-2 temperaments ==

== 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]]. 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.

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

{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}

[[Tp tuning#T2 tuning|RMS error]]: 0.3533 cents

=== Cata ===