The Archipelago
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[[toc]] 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 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 [[Optimal patent val]]: [[940edo]] ==Rank four temperaments== ===1001/1000=== Commas: 676/675, 1001/1000 EDOs: 15, 19, 29, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940 [[Optimal patent val]]: [[940edo]] ===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== ===[[Breed family|Greenland]]=== Commas: 676/675, 1001/1000, 1716/1715 Map: [<2 0 1 3 7 -1|, <0 2 1 1 -2 4|, <0 0 2 1 3 2|] Edos: 58, 72, 130, 198, 270, 940 [[Optimal patent val]]: [[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]]=== Commas: 364/363, 441/440, 1001/1000 EDOs: 15, 29, 43, 58, 72, 87, 130, 217, 289 [[Optimal patent val]]: [[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 ===Borneo=== Commas: 676/675, 1001/1000, 3025/3024 Map: [<3 0 0 4 8 -3|, <0 2 0 -4 1 3|, <0 0 1 2 0 1|] EDOs: 15, 72, 87, 111, 159, 183, 198, 270 [[Optimal patent val]]: [[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 ===[[Cataharry family|Madagascar]]=== Commas: 351/350, 540/539, 676/675 EDOs: 19, 53, 58, 72, 111, 130, 183, 313 [[Optimal patent val]]: [[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]] ===Baffin=== Commas: 676/675, 1001/1000, 4225/4224 Map: [<1 0 0 13 -9 1|, <0 2 0 -7 4 3|, <0 0 1 -2 4 1|] EDOs: 34, 43, 53, 87, 130, 183, 217, 270, 940 [[Optimal patent val]]: [[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 ==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 [[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]]: [<1 0 -1|, <0 2 3|] EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 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
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<html><head><title>The Archipelago</title></head><body><!-- ws:start:WikiTextTocRule:38:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 2em;"><a href="#x-Rank four temperaments">Rank four temperaments</a></div> <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 3em;"><a href="#x-Rank four temperaments-1001/1000">1001/1000</a></div> <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 3em;"><a href="#x-Rank four temperaments-49/48">49/48</a></div> <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 3em;"><a href="#x-Rank four temperaments-1716/1715">1716/1715</a></div> <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 3em;"><a href="#x-Rank four temperaments-364/363">364/363</a></div> <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 3em;"><a href="#x-Rank four temperaments-351/350">351/350</a></div> <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#x-Rank three temperaments">Rank three temperaments</a></div> <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 3em;"><a href="#x-Rank three temperaments-Greenland">Greenland</a></div> <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 3em;"><a href="#x-Rank three temperaments-History">History</a></div> <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 3em;"><a href="#x-Rank three temperaments-Borneo">Borneo</a></div> <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 3em;"><a href="#x-Rank three temperaments-Madagascar">Madagascar</a></div> <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 3em;"><a href="#x-Rank three temperaments-Baffin">Baffin</a></div> <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 2em;"><a href="#x-Rank two temperaments">Rank two temperaments</a></div> <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 3em;"><a href="#x-Rank two temperaments-Decitonic">Decitonic</a></div> <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 3em;"><a href="#x-Rank two temperaments-Avicenna">Avicenna</a></div> <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 2em;"><a href="#x-Subgroup temperaments">Subgroup temperaments</a></div> <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 3em;"><a href="#x-Subgroup temperaments-Barbados">Barbados</a></div> <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 3em;"><a href="#x-Subgroup temperaments-Trinidad">Trinidad</a></div> <!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 3em;"><a href="#x-Subgroup temperaments-Parizekmic">Parizekmic</a></div> <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --></div> <!-- ws:end:WikiTextTocRule:58 --><br /> 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 <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 barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.<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 14:18:21 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.<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 /> <a class="wiki_link" href="/Optimal%20patent%20val">Optimal patent val</a>: <a class="wiki_link" href="/940edo">940edo</a><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 /> EDOs: 15, 19, 29, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940<br /> <a class="wiki_link" href="/Optimal%20patent%20val">Optimal patent val</a>: <a class="wiki_link" href="/940edo">940edo</a><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-Greenland"></a><!-- ws:end:WikiTextHeadingRule:14 --><a class="wiki_link" href="/Breed%20family">Greenland</a></h3> Commas: 676/675, 1001/1000, 1716/1715<br /> <br /> Map: [<2 0 1 3 7 -1|, <0 2 1 1 -2 4|, <0 0 2 1 3 2|]<br /> Edos: 58, 72, 130, 198, 270, 940<br /> <a class="wiki_link" href="/Optimal%20patent%20val">Optimal patent val</a>: <a class="wiki_link" href="/940edo">940edo</a><br /> Badness: 0.000433<br /> <br /> <a class="wiki_link" href="/Spectrum%20of%20a%20temperament">Spectrum</a>: 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<br /> <br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="x-Rank three temperaments-History"></a><!-- ws:end:WikiTextHeadingRule:16 --><a class="wiki_link" href="/Werckismic%20temperaments">History</a></h3> Commas: 364/363, 441/440, 1001/1000<br /> <br /> EDOs: 15, 29, 43, 58, 72, 87, 130, 217, 289<br /> <a class="wiki_link" href="/Optimal%20patent%20val">Optimal patent val</a>: <a class="wiki_link" href="/289edo">289edo</a><br /> Badness: 0.000540<br /> <br /> 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<br /> <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 /> Map: [<3 0 0 4 8 -3|, <0 2 0 -4 1 3|, <0 0 1 2 0 1|]<br /> EDOs: 15, 72, 87, 111, 159, 183, 198, 270<br /> <a class="wiki_link" href="/Optimal%20patent%20val">Optimal patent val</a>: <a class="wiki_link" href="/270edo">270edo</a><br /> Badness: 0.000549<br /> <br /> <br /> 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<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h3> --><h3 id="toc10"><a name="x-Rank three temperaments-Madagascar"></a><!-- ws:end:WikiTextHeadingRule:20 --><a class="wiki_link" href="/Cataharry%20family">Madagascar</a></h3> Commas: 351/350, 540/539, 676/675<br /> <br /> EDOs: 19, 53, 58, 72, 111, 130, 183, 313<br /> <a class="wiki_link" href="/Optimal%20patent%20val">Optimal patent val</a>: <a class="wiki_link" href="/313edo">313edo</a><br /> Badness: 0.000560<br /> <br /> 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<br /> <a class="wiki_link" href="/madagascar19">madagascar19</a><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 /> Map: [<1 0 0 13 -9 1|, <0 2 0 -7 4 3|, <0 0 1 -2 4 1|]<br /> EDOs: 34, 43, 53, 87, 130, 183, 217, 270, 940<br /> <a class="wiki_link" href="/Optimal%20patent%20val">Optimal patent val</a>: <a class="wiki_link" href="/940edo">940edo</a><br /> Badness: 0.000604<br /> <br /> 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<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="/111edo">111edo</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, 9, 14, 19, 24, 29, 53, 82, 111, 140, 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>