The Archipelago

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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, and the barbados tetrad, 1-13/10-3/2-26/15. This is because the [[Just intonation subgroups|just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the triad is found in that.

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 14:19:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:19:21), 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]] 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. 

Comma: 676/675

Map
<1 0 0 0 0 -1| 
<0 2 0 0 0 3| 
<0 0 1 0 0 1| 
<0 0 0 1 0 0| 
<0 0 0 0 1 0|
EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940

==Rank four temperaments== 

===1001/1000=== 
Commas: 676/675, 1001/1000

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

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

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

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

==Rank three temperaments== 

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

===Papua=== 
Commas: 364/363, 441/440, 1001/1000

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

===Madagascar=== 
Commas: 351/350, 540/539, 676/675

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

==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]] alone.

===Decitonic=== 
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224

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

Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]
EDOs: 130, 270, 940, 1480
Badness: 0.0135

===Avicenna=== 
Commas: 676/675, 1001/1000, 3025/3024, 4096/4095

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

Map: [<3 2 8 16 9 8|, <0 8 -3 -22 4 9|]
EDOs: 87, 183, 270
Badness: 0.0156

==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]], [[29edo]], [[53edo]] and [[135edo]], with MOS of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

===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.

Commas: 325/324, 625/624 in 2.3.5.13

[[POTE tuning|POTE generator]]: 317.076

Map: [<1 0 1 0 0 0|, <0 6 5 0 0 14|]
EDOs: 15, 19, 34, 53, 87, 193, 246

===Parizekmic=== 
Closely related to barbados temperament is parizekmic, the rank three 2.3.5.13 subgroup temperament tempering out 676/675. 

Map
<1 0 0 0 0 -1|
<0 2 0 0 0 3|
<0 0 1 0 0 1|

Original HTML content:

<html><head><title>The Archipelago</title></head><body>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, and the barbados tetrad, 1-13/10-3/2-26/15. This is because the <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> generated by 2, 4/3 and 15/13 is 2.3.13/5, and the triad is found in that.<br />
<br />
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 <em>ultramajor</em> triad, with a third sharper even than the 9/7 supermajor third.<br />
<br />
Compared to the 7-limit 14:19:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:19:21), 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.<br />
<br />
<a class="wiki_link" href="/24edo">24edo</a> approximates this triad to within an error of four cents, and <a class="wiki_link" href="/29edo">29edo</a> does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below. <br />
<br />
Comma: 676/675<br />
<br />
Map<br />
&lt;1 0 0 0 0 -1| <br />
&lt;0 2 0 0 0 3| <br />
&lt;0 0 1 0 0 1| <br />
&lt;0 0 0 1 0 0| <br />
&lt;0 0 0 0 1 0|<br />
EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Rank four temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Rank four temperaments</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Rank four temperaments-1001/1000"></a><!-- ws:end:WikiTextHeadingRule:2 -->1001/1000</h3>
 Commas: 676/675, 1001/1000<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Rank four temperaments-49/48"></a><!-- ws:end:WikiTextHeadingRule:4 -->49/48</h3>
 Commas: 49/48, 91/90<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-Rank four temperaments-1716/1715"></a><!-- ws:end:WikiTextHeadingRule:6 -->1716/1715</h3>
 Commas: 676/675, 1716/1715<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Rank four temperaments-364/363"></a><!-- ws:end:WikiTextHeadingRule:8 -->364/363</h3>
 Commas: 364/363, 676/675<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Rank four temperaments-351/350"></a><!-- ws:end:WikiTextHeadingRule:10 -->351/350</h3>
 Commas: 351/350, 676/675<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x-Rank three temperaments"></a><!-- ws:end:WikiTextHeadingRule:12 -->Rank three temperaments</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Rank three temperaments-Kalaallisut"></a><!-- ws:end:WikiTextHeadingRule:14 -->Kalaallisut</h3>
 Commas: 676/675, 1001/1000, 1716/1715<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Rank three temperaments-Papua"></a><!-- ws:end:WikiTextHeadingRule:16 -->Papua</h3>
 Commas: 364/363, 441/440, 1001/1000<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Rank three temperaments-Borneo"></a><!-- ws:end:WikiTextHeadingRule:18 -->Borneo</h3>
 Commas: 676/675, 1001/1000, 3025/3024<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="x-Rank three temperaments-Madagascar"></a><!-- ws:end:WikiTextHeadingRule:20 -->Madagascar</h3>
 Commas: 351/350, 540/539, 676/675<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="x-Rank three temperaments-Baffin"></a><!-- ws:end:WikiTextHeadingRule:22 -->Baffin</h3>
 Commas: 676/675, 1001/1000, 4225/4224<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="x-Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:24 -->Rank two temperaments</h2>
 Rank two temperaments tempering out 676/675 include the 13-limit versions of <a class="wiki_link" href="/Ragismic%20microtemperaments">hemiennealimmal</a>, <a class="wiki_link" href="/Breedsmic%20temperaments">harry</a>, <a class="wiki_link" href="/Kleismic%20family">tritikleismic</a>, <a class="wiki_link" href="/Kleismic%20family">catakleimsic</a>, <a class="wiki_link" href="/Marvel%20temperaments">negri</a>, <a class="wiki_link" href="/Hemifamity%20temperaments">mystery</a>, <a class="wiki_link" href="/Hemifamity%20temperaments">buzzard</a>, <a class="wiki_link" href="/Kleismic%20family">quadritikleismic</a>. <br />
<br />
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 <a class="wiki_link" href="/29edo">29edo</a> alone.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="x-Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextHeadingRule:26 -->Decitonic</h3>
 Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~15/13 = 248.917<br />
<br />
Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]<br />
EDOs: 130, 270, 940, 1480<br />
Badness: 0.0135<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="x-Rank two temperaments-Avicenna"></a><!-- ws:end:WikiTextHeadingRule:28 -->Avicenna</h3>
 Commas: 676/675, 1001/1000, 3025/3024, 4096/4095<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~13/12 = 137.777<br />
<br />
Map: [&lt;3 2 8 16 9 8|, &lt;0 8 -3 -22 4 9|]<br />
EDOs: 87, 183, 270<br />
Badness: 0.0156<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="x-Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:30 -->Subgroup temperaments</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="x-Subgroup temperaments-Barbados"></a><!-- ws:end:WikiTextHeadingRule:32 -->Barbados</h3>
 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 <a class="wiki_link" href="/Just%20intonation%20subgroups">just intontation subgroup</a>. The minimax tuning for this makes the generator 2/sqrt(3), or 249.0225 cents. EDOs which may be used for it are <a class="wiki_link" href="/24edo">24edo</a>, <a class="wiki_link" href="/29edo">29edo</a>, <a class="wiki_link" href="/53edo">53edo</a> and <a class="wiki_link" href="/135edo">135edo</a>, with MOS of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="x-Subgroup temperaments-Trinidad"></a><!-- ws:end:WikiTextHeadingRule:34 -->Trinidad</h3>
 Trinidad may be viewed as the reduction of <a class="wiki_link" href="/Kleismic%20family">catakleismic temperament</a> 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.<br />
<br />
Commas: 325/324, 625/624 in 2.3.5.13<br />
<br />
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 317.076<br />
<br />
Map: [&lt;1 0 1 0 0 0|, &lt;0 6 5 0 0 14|]<br />
EDOs: 15, 19, 34, 53, 87, 193, 246<br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="x-Subgroup temperaments-Parizekmic"></a><!-- ws:end:WikiTextHeadingRule:36 -->Parizekmic</h3>
 Closely related to barbados temperament is parizekmic, the rank three 2.3.5.13 subgroup temperament tempering out 676/675. <br />
<br />
Map<br />
&lt;1 0 0 0 0 -1|<br />
&lt;0 2 0 0 0 3|<br />
&lt;0 0 1 0 0 1|</body></html>