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.

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 dyads 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. Temperaments in which 91/90 vanishes equate the two types of triads.

[[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.

=Rank five temperament=

Comma: 676/675

&lt;0 0 0 1 0 0|<br>

&lt;0 0 0 0 1 0|

EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940

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

==1001/1000==

Commas: 676/675, 1001/1000

Map: [&lt;1 0 0 0 4 -1|, &lt;0 2 0 0 -3 3|, &lt;0 0 1 0 2 1|, &lt;0 0 0 1 -1 0|]

EDOs: 15, 19, 29, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940

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

==49/48==

Commas: 49/48, 91/90

Map: [&lt;1 0 0 2 0 -1|, &lt;0 2 0 1 0 3|, &lt;0 0 1 0 0 1|, &lt;0 0 0 0 1 0|]

EDOs: 5, 9, 10, 15, 19, 24

Commas: 676/675, 1716/1715

Map: [&lt;1 0 0 0 -1 -1|, &lt;0 2 0 0 -5 3|, &lt;0 0 1 0 0 1|, &lt;0 0 0 1 3 0|]

EDOs: 58, 72, 77, 121, 130, 140, 149, 198, 212, 270

Commas: 364/363, 676/675

Map: [&lt;1 0 0 -1 0 -1|, &lt;0 2 0 1 1 3|, &lt;0 0 1 1 1 1|, &lt;0 0 0 2 1 0|]

EDOs: 9, 15, 29, 43, 58, 72, 87, 121, 130

==351/350==

Commas: 351/350, 676/675

Map: [&lt;1 0 0 -2 0 -1|, &lt;0 2 0 9 0 3|, &lt;0 0 1 -1 0 1|, &lt;0 0 0 0 1 0|]

EDOs: 19, 53, 58, 72, 77, 111, 130

==352/351==

Commas: 352/351, 676/675

Map: [&lt;1 0 0 0 -6 -1|, &lt;0 2 0 0 9 3|, &lt;0 0 1 0 1 1|, &lt;0 0 0 1 0 0|]

EDOs: 29, 34, 53, 58, 63, 77, 87, 111, 121

Commas: 540/539, 676/675

Map: [&lt;1 0 0 0 2 -1|, &lt;0 2 0 0 6 3|, &lt;0 0 1 0 1 1|, &lt;0 0 0 1 -2 0|]

EDOs: 9, 19, 53, 58, 63, 72, 111, 121, 183

==847/845==

Commas: 676/675, 847/845

Map: [&lt;1 0 0 0 -1 -1|, &lt;0 2 0 0 3 3|, &lt;0 0 1 0 1 1|, &lt;0 0 0 2 -1 0|]

EDOs: 9, 29, 53, 58, 87, 111, 140, 149, 198

=Rank three temperaments=

==[[Breed_family|Greenland]]==

Commas: 676/675, 1001/1000, 1716/1715

Map: [&lt;2 0 1 3 7 -1|, &lt;0 2 1 1 -2 4|, &lt;0 0 2 1 3 2|]

Edos: 58, 72, 130, 198, 270, 940

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

Badness: 0.000433

[[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]]

Badness: 0.000540

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

Commas: 676/675, 1001/1000, 3025/3024

Map: [&lt;3 0 0 4 8 -3|, &lt;0 2 0 -4 1 3|, &lt;0 0 1 2 0 1|]

EDOs: 15, 72, 87, 111, 159, 183, 198, 270

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

Badness: 0.000549

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

Commas: 325/324, 385/384, 625/624

EDOs: 15, 19, 34, 53, 72, 87, 140, 159, 212, 299

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

Badness: 0.000680

==[[Cataharry_family|Madagascar]]==

EDOs: 19, 53, 58, 72, 111, 130, 183, 313

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

Badness: 0.000560

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

[[madagascar19|madagascar19]]

Commas: 676/675, 1001/1000, 4225/4224

Map: [&lt;1 0 0 13 -9 1|, &lt;0 2 0 -7 4 3|, &lt;0 0 1 -2 4 1|]

EDOs: 34, 43, 53, 87, 130, 183, 217, 270, 940

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

Badness: 0.000604

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

==Kujuku==

Commas: 352/351, 364/363, 676/675

Map: [&lt;1 0 0 -13 -6 -1|, &lt;0 2 0 17 9 3|, &lt;0 0 1 1 1 1|]

EDOs: 24, 29, 58, 87, 121, 145, 208, 266ef, 474bef

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

Badness: 0.001060

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

=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 (aka Decoid)==

{{see also|Breedsmic temperaments#Decoid}}

[[Comma list]]: 676/675, 1001/1000, 1716/1715, 4225/4224

[[Mapping]]: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]

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

{{Val list|legend=1| 130, 270, 940, 1210f, 1480cf }}

[[Badness]]: 0.013475

[[Comma list]]: 676/675, 1001/1000, 3025/3024, 4096/4095

[[Mapping]]: [&lt;3 2 8 16 9 8|, &lt;0 8 -3 -22 4 9|]

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

{{Val list|legend=1| 87, 183, 270 }}

Comma list: 676/675, 1716/1715, 3025/3024, 4225/4224

Mapping: [&lt;1 -4 2 -6 -9 -5|, &lt;0 52 3 82 116 81|]

POTE generator: ~14/13 = 128.8902

Vals: {{Val list| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}

===17-limit===

Mapping: [&lt;1 -4 2 -6 -9 -5 -3|, &lt;0 52 3 82 116 81 66|]

POTE generator: ~14/13 = 128.8912

Vals: {{Val list| 121, 149, 270 }}

Badness: 0.019107

=Subgroup temperaments=

==Barbados==

Commas: 676/675

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.

[[POTE_tuning|POTE generator]]: ~15/13 = 248.621

[[Smonzos_and_Svals|Sval map]]: [&lt;1 0 -1|, &lt;0 2 3|]

EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362

Badness: 0.002335

===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==

Subgroup: 2.3.5.13

Commas: 325/324, 625/624

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.

[[POTE_tuning|POTE generator]]: 317.076

[[Smonzos_and_Svals|Sval map]]: [&lt;1 0 1 0 |, &lt;0 6 5 14|]

EDOs: 15, 19, 34, 53, 87, 140, 193, 246

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

Subgroup: 2.3.11.13/5

Commas: 243/242, 676/675

POT2 generator: ~15/13 = 249.312

Map: [<2 0 -1 -2], <0 2 5 3]]

EDOs: 10, 14, 24, 58, 82, 130

==Parizekmic==

Subgroup: 2.3.5.13

Commas: 676/675

Closely related to barbados temperament is parizekmic, the rank three 2.3.5.13 subgroup temperament tempering out 676/675. This is generated by 2, 5, and 15/13, where the minimax tuning makes 2 and 5 pure, and 15/13 sharp by sqrt(676/675), or 1.28145 cents. This is, in other words, the same sqrt(4/3) generator as the minimax tuning for barbados, and it gives parizekmic a just 5-limit, with barbados triads where the 13/10 is a cent flat.

[[Smonzos_and_Svals|Sval map]]: [&lt;1 0 0 -1|, &lt;0 2 0 3|, &lt;0 0 1 1|]

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

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

[[Category:archipelago]]

[[Category:temperament]]

[[Category:theory]]