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=== Subgroup: 2.3.13/5 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]], [[29edo]], [[53edo]] and [[135edo]], 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]]: [<1 0 -1|, <0 2 3|] EDOs: 5, 14, 19, 24, 29, 82, 111, 251, 362 Badness: 0.002335 ===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]]: [<1 0 1 0 |, <0 6 5 14|] EDOs: 15, 19, 34, 53, 87, 140, 193, 246 ===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]] <1 0 0 -1| <0 2 0 3| <0 0 1 1| EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270
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 /> <1 0 0 0 0 -1| <br /> <0 2 0 0 0 3| <br /> <0 0 1 0 0 1| <br /> <0 0 0 1 0 0| <br /> <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:<h2> --><h2 id="toc0"><a name="x-Rank four temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Rank four temperaments</h2> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><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:<h3> --><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:<h3> --><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:<h3> --><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:<h3> --><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:<h2> --><h2 id="toc6"><a name="x-Rank three temperaments"></a><!-- ws:end:WikiTextHeadingRule:12 -->Rank three temperaments</h2> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h3> --><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:<h3> --><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:<h3> --><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:<h3> --><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:<h3> --><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:<h2> --><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:<h3> --><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: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]<br /> EDOs: 130, 270, 940, 1480<br /> Badness: 0.0135<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h3> --><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: [<3 2 8 16 9 8|, <0 8 -3 -22 4 9|]<br /> EDOs: 87, 183, 270<br /> Badness: 0.0156<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc15"><a name="x-Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:30 -->Subgroup temperaments</h2> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h3> --><h3 id="toc16"><a name="x-Subgroup temperaments-Barbados"></a><!-- ws:end:WikiTextHeadingRule:32 -->Barbados</h3> Subgroup: 2.3.13/5<br /> Commas: 676/675<br /> <br /> 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 /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~15/13 = 248.621<br /> <br /> <a class="wiki_link" href="/Smonzos%20and%20Svals">Sval map</a>: [<1 0 -1|, <0 2 3|]<br /> EDOs: 5, 14, 19, 24, 29, 82, 111, 251, 362<br /> Badness: 0.002335<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h3> --><h3 id="toc17"><a name="x-Subgroup temperaments-Trinidad"></a><!-- ws:end:WikiTextHeadingRule:34 -->Trinidad</h3> Subgroup: 2.3.5.13<br /> Commas: 325/324, 625/624<br /> <br /> 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 /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 317.076<br /> <br /> <a class="wiki_link" href="/Smonzos%20and%20Svals">Sval map</a>: [<1 0 1 0 |, <0 6 5 14|]<br /> EDOs: 15, 19, 34, 53, 87, 140, 193, 246<br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h3> --><h3 id="toc18"><a name="x-Subgroup temperaments-Parizekmic"></a><!-- ws:end:WikiTextHeadingRule:36 -->Parizekmic</h3> Subgroup: 2.3.5.13<br /> Commas: 676/675<br /> <br /> 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.<br /> <br /> <a class="wiki_link" href="/Smonzos%20and%20Svals">Sval map</a><br /> <1 0 0 -1|<br /> <0 2 0 3|<br /> <0 0 1 1|<br /> EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270</body></html>