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[[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.
[[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=
= Rank five temperament =
== Island ==


==Island==
Subgroup: 2.3.5.7.11.13


Comma: 676/675
[[Comma list]]: [[676/675]]


Map:<br>
Map:<br>
&lt;1 0 0 0 0 -1|<br>
{{val| 1 0 0 0 0 -1 }}<br>
&lt;0 2 0 0 0 3|<br>
{{val| 0 2 0 0 0 3 }}<br>
&lt;0 0 1 0 0 1|<br>
{{val| 0 0 1 0 0 1 }}<br>
&lt;0 0 0 1 0 0|<br>
{{val| 0 0 0 1 0 0 }}<br>
&lt;0 0 0 0 1 0|
{{val| 0 0 0 0 1 0 }}


EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940
Vals: 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]]
[[Optimal patent val]]: [[940edo]]
 
 
=Rank four temperaments=


==1001/1000==
= Rank four temperaments=
== 1001/1000 ==
Commas: 676/675, 1001/1000
Commas: 676/675, 1001/1000


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


==49/48==
== 49/48 ==
Commas: 49/48, 91/90
Commas: 49/48, 91/90


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


==1716/1715==
== 1716/1715 ==
Commas: 676/675, 1716/1715
Commas: 676/675, 1716/1715


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


==364/363==
== 364/363 ==
Commas: 364/363, 676/675
Commas: 364/363, 676/675


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


==351/350==
== 351/350 ==
Commas: 351/350, 676/675
Commas: 351/350, 676/675


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


==352/351==
== 352/351 ==
Commas: 352/351, 676/675
Commas: 352/351, 676/675


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


==540/539==
== 540/539 ==
Commas: 540/539, 676/675
Commas: 540/539, 676/675


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


==847/845==
== 847/845 ==
Commas: 676/675, 847/845
Commas: 676/675, 847/845


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


=Rank three temperaments=
= Rank three temperaments =
 
== [[Breed_family|Greenland]] ==
==[[Breed_family|Greenland]]==
Commas: 676/675, 1001/1000, 1716/1715
Commas: 676/675, 1001/1000, 1716/1715


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[[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
[[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]]==
== [[Werckismic_temperaments|History]] ==
Commas: 364/363, 441/440, 1001/1000
Commas: 364/363, 441/440, 1001/1000


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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
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


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


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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
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


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


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Badness: 0.000680
Badness: 0.000680


==[[Cataharry_family|Madagascar]]==
== [[Cataharry_family|Madagascar]] ==
Commas: 351/350, 540/539, 676/675
Commas: 351/350, 540/539, 676/675


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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
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]]
[[madagascar19]]


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


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


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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
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 =
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]].
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.
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)==
== Decitonic (aka Decoid) ==
{{see also|Breedsmic temperaments#Decoid}}
{{see also|Breedsmic temperaments#Decoid}}


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[[Badness]]: 0.013475
[[Badness]]: 0.013475


==Avicenna==
== Avicenna ==
{{see also|Landscape microtemperaments#Avicenna}}
{{see also|Landscape microtemperaments#Avicenna}}


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[[Badness]]: 0.015557
[[Badness]]: 0.015557


==Tertiathirds==
== Tertiathirds ==
{{see also|Wizmic microtemperaments#Tertiathirds}}
{{see also|Wizmic microtemperaments#Tertiathirds}}


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[[Badness]]: 0.019494
[[Badness]]: 0.019494


===17-limit===
=== 17-limit ===
Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224
Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224


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Badness: 0.019107
Badness: 0.019107


=Subgroup temperaments=
= Subgroup temperaments =
 
== Barbados ==
==Barbados==
Subgroup: 2.3.13/5
Subgroup: 2.3.13/5


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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 the cube root of 20/13, or 248.5953 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.
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 the cube root of 20/13, or 248.5953 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
[[POTE generator]]: ~15/13 = 248.621


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


EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362
EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362
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Badness: 0.002335
Badness: 0.002335


===Music===
=== 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]
* [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==
Subgroup: 2.3.5.13
Subgroup: 2.3.5.13


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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.
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
[[POTE generator]]: 317.076


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


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


==[[Chromatic_pairs#Tobago|Tobago]]==
== [[Chromatic pairs #Tobago|Tobago]] ==
Subgroup: 2.3.11.13/5
Subgroup: 2.3.11.13/5


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


 
== Parizekmic ==
==Parizekmic==
Subgroup: 2.3.5.13
Subgroup: 2.3.5.13


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


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


 
[[Category:Regular temperament theory]]
[[Category:archipelago]]
[[Category:Commatic realm]]
[[Category:listen]]
[[Category:overview]]
[[Category:temperament]]
[[Category:Island]]
[[Category:Island]]
[[Category:theory]]
[[Category:Listen]]
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