The Archipelago: Difference between revisions

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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]] 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

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

===1001/1000===

Commas: 676/675, 1001/1000

Commas: 49/48, 91/90

===1716/1715===  

Commas: 676/675, 1716/1715

Commas: 364/363, 676/675

Commas: 351/350, 676/675

==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

===[[Werckismic temperaments|History]]===

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

===Borneo===

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

===[[Cataharry family|Madagascar]]===

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

[[madagascar19]]

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.

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

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

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

EDOs: 130, 270, 940, 1480

===Avicenna===  

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

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

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

EDOs: 87, 183, 270

Badness: 0.0156

==Subgroup temperaments==

===Barbados===  

Subgroup: 2.3.13/5

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 [[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

===Trinidad===  

Subgroup: 2.3.5.13

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

===Parizekmic===  

Subgroup: 2.3.5.13

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</pre></div>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;The Archipelago&lt;/title&gt;&lt;/head&gt;&lt;body&gt;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 &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;just intonation subgroup&lt;/a&gt; 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.&lt;br /&gt;

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 &lt;em&gt;ultramajor&lt;/em&gt; triad, with a third sharper even than the 9/7 supermajor third.&lt;br /&gt;

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.&lt;br /&gt;

&lt;a class="wiki_link" href="/24edo"&gt;24edo&lt;/a&gt; approximates this triad to within an error of four cents, and &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt; does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below. &lt;br /&gt;

Comma: 676/675&lt;br /&gt;

&amp;lt;1 0 0 0 0 -1| &lt;br /&gt;

&amp;lt;0 2 0 0 0 3| &lt;br /&gt;

&amp;lt;0 0 0 1 0 0| &lt;br /&gt;

&amp;lt;0 0 0 0 1 0|&lt;br /&gt;

EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Rank four temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Rank four temperaments&lt;/h2&gt;

&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Rank four temperaments-1001/1000"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;1001/1000&lt;/h3&gt;

Commas: 676/675, 1001/1000&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Rank four temperaments-49/48"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;49/48&lt;/h3&gt;

Commas: 49/48, 91/90&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Rank four temperaments-1716/1715"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;1716/1715&lt;/h3&gt;

Commas: 676/675, 1716/1715&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Rank four temperaments-364/363"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;364/363&lt;/h3&gt;

Commas: 364/363, 676/675&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Rank four temperaments-351/350"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;351/350&lt;/h3&gt;

Commas: 351/350, 676/675&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-Rank three temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Rank three temperaments&lt;/h2&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Rank three temperaments-Greenland"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;a class="wiki_link" href="/Breed%20family"&gt;Greenland&lt;/a&gt;&lt;/h3&gt;

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

Edos: 58, 72, 130, 198, 270, 940&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Rank three temperaments-History"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;a class="wiki_link" href="/Werckismic%20temperaments"&gt;History&lt;/a&gt;&lt;/h3&gt;

Commas: 364/363, 441/440, 1001/1000&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-Rank three temperaments-Borneo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Borneo&lt;/h3&gt;

Commas: 676/675, 1001/1000, 3025/3024&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="x-Rank three temperaments-Madagascar"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;a class="wiki_link" href="/Cataharry%20family"&gt;Madagascar&lt;/a&gt;&lt;/h3&gt;

Commas: 351/350, 540/539, 676/675&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="x-Rank three temperaments-Baffin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Baffin&lt;/h3&gt;

Commas: 676/675, 1001/1000, 4225/4224&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="x-Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Rank two temperaments&lt;/h2&gt;

Rank two temperaments tempering out 676/675 include the 13-limit versions of &lt;a class="wiki_link" href="/Ragismic%20microtemperaments"&gt;hemiennealimmal&lt;/a&gt;, &lt;a class="wiki_link" href="/Breedsmic%20temperaments"&gt;harry&lt;/a&gt;, &lt;a class="wiki_link" href="/Kleismic%20family"&gt;tritikleismic&lt;/a&gt;, &lt;a class="wiki_link" href="/Kleismic%20family"&gt;catakleimsic&lt;/a&gt;, &lt;a class="wiki_link" href="/Marvel%20temperaments"&gt;negri&lt;/a&gt;, &lt;a class="wiki_link" href="/Hemifamity%20temperaments"&gt;mystery&lt;/a&gt;, &lt;a class="wiki_link" href="/Hemifamity%20temperaments"&gt;buzzard&lt;/a&gt;, &lt;a class="wiki_link" href="/Kleismic%20family"&gt;quadritikleismic&lt;/a&gt;. &lt;br /&gt;

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 &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt; alone.&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="x-Rank two temperaments-Decitonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Decitonic&lt;/h3&gt;