The Biosphere
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<span style="display: block; display: none;"> ||~ Details || last edit by <span style="outline: medium none;">[[image:http://www.wikispaces.com/user/pic/1202793136/genewardsmith-sm.jpg width="16" height="16" caption="genewardsmith" link="http://www.wikispaces.com/user/view/genewardsmith"]]</span> <span style="outline: medium none;">[[http://www.wikispaces.com/user/view/genewardsmith|genewardsmith]]</span> [[page/diff/The Archipelago/219253274|Apr 11, 2011 3:26 pm]] - [[page/history/the Archipelago|36 revisions]] || [[image:http://www.wikispaces.com/i/w/W_close.gif caption="hide details" link="the Archipelago#"]] || ||~ Tags || * [[tag/view/listen|listen]] * [[tag/view/theory|theory]] [[the Archipelago#|edit]] listentheory Save[[the Archipelago#|Cancel]] || </span> The biosphere is the name given to the collection of temperaments that are children of or related to **//biome temperament//**, the rank 3 2.3.7.13/10 subgroup temperament eliminating 91/90. The term "biome" loosely means "ecosystem" or "climate." This temperament is so named because temperaments that arise from eliminating 91/90 can evoke synesthetic associations of different "natural" settings, some very familiar and some much less so. This temperament makes the utonal inverse of 6:7:9 out to be 10:13:15, whereas it is normally the much more complex (and arguably more dissonant) 14:18:21. Eliminating 91/90 thus enriches septimal harmony by increasing the concordance of "utonal" septimal triads such as 6:7:9 or tetrads such as 4:6:7:9. Biome temperament, from which is of particular theoretical interest because it generates a rank-3 lattice that is analogous to the 5-limit JI lattice. 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]]Rank four temperaments= ==[[#Rank four temperaments-1001/1000]]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]] ==[[#Rank four temperaments-49/48]]49/48== Commas: 49/48, 91/90 ==[[#Rank four temperaments-1716/1715]]1716/1715== Commas: 676/675, 1716/1715 ==[[#Rank four temperaments-364/363]]364/363== Commas: 364/363, 676/675 ===[[#Rank four temperaments-364/363-351/350]]351/350=== Commas: 351/350, 676/675 =[[#Rank three temperaments]]Rank three temperaments= ==[[#Rank three temperaments-Greenland]][[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 ==[[#Rank three temperaments-History]][[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 ==[[#Rank three temperaments-Borneo]]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 ==[[#Rank three temperaments-Madagascar]][[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]] ==[[#Rank three temperaments-Baffin]]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= 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. ==[[#Rank two temperaments-Decitonic]]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 ==[[#Rank two temperaments-Avicenna]]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]]Subgroup temperaments= ==[[#Subgroup temperaments-Barbados]]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 ==[[#Subgroup temperaments-Trinidad]]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 ==[[#Subgroup temperaments-Parizekmic]]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| ===[[#Subgroup temperaments-Parizekmic-Music]]Music=== [[http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3|Petr's Pump]], a comma pump based ditty in Pariekmic temperament. EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270
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The biosphere is the name given to the collection of temperaments that are children of or related to <strong><em>biome temperament</em></strong>, the rank 3 2.3.7.13/10 subgroup temperament eliminating 91/90. The term "biome" loosely means "ecosystem" or "climate." This temperament is so named because temperaments that arise from eliminating 91/90 can evoke synesthetic associations of different "natural" settings, some very familiar and some much less so.<br />
<br />
This temperament makes the utonal inverse of 6:7:9 out to be 10:13:15, whereas it is normally the much more complex (and arguably more dissonant) 14:18:21. Eliminating 91/90 thus enriches septimal harmony by increasing the concordance of "utonal" septimal triads such as 6:7:9 or tetrads such as 4:6:7:9.<br />
<br />
Biome temperament, from which is of particular theoretical interest because it generates a rank-3 lattice that is analogous to the 5-limit JI lattice.<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 />
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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 />
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<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Rank four temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 --><!-- ws:start:WikiTextAnchorRule:40:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank four temperaments" title="Anchor: Rank four temperaments"/> --><a name="Rank four temperaments"></a><!-- ws:end:WikiTextAnchorRule:40 -->Rank four temperaments</h1>
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<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Rank four temperaments-1001/1000"></a><!-- ws:end:WikiTextHeadingRule:2 --><!-- ws:start:WikiTextAnchorRule:41:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank four temperaments-1001/1000" title="Anchor: Rank four temperaments-1001/1000"/> --><a name="Rank four temperaments-1001/1000"></a><!-- ws:end:WikiTextAnchorRule:41 -->1001/1000</h2>
Commas: 676/675, 1001/1000<br />
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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 />
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<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Rank four temperaments-49/48"></a><!-- ws:end:WikiTextHeadingRule:4 --><!-- ws:start:WikiTextAnchorRule:42:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank four temperaments-49/48" title="Anchor: Rank four temperaments-49/48"/> --><a name="Rank four temperaments-49/48"></a><!-- ws:end:WikiTextAnchorRule:42 -->49/48</h2>
Commas: 49/48, 91/90<br />
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<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Rank four temperaments-1716/1715"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:43:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank four temperaments-1716/1715" title="Anchor: Rank four temperaments-1716/1715"/> --><a name="Rank four temperaments-1716/1715"></a><!-- ws:end:WikiTextAnchorRule:43 -->1716/1715</h2>
Commas: 676/675, 1716/1715<br />
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<!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Rank four temperaments-364/363"></a><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextAnchorRule:44:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank four temperaments-364/363" title="Anchor: Rank four temperaments-364/363"/> --><a name="Rank four temperaments-364/363"></a><!-- ws:end:WikiTextAnchorRule:44 -->364/363</h2>
Commas: 364/363, 676/675<br />
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<!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="Rank four temperaments-364/363-351/350"></a><!-- ws:end:WikiTextHeadingRule:10 --><!-- ws:start:WikiTextAnchorRule:45:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank four temperaments-364/363-351/350" title="Anchor: Rank four temperaments-364/363-351/350"/> --><a name="Rank four temperaments-364/363-351/350"></a><!-- ws:end:WikiTextAnchorRule:45 -->351/350</h3>
Commas: 351/350, 676/675<br />
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<!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Rank three temperaments"></a><!-- ws:end:WikiTextHeadingRule:12 --><!-- ws:start:WikiTextAnchorRule:46:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank three temperaments" title="Anchor: Rank three temperaments"/> --><a name="Rank three temperaments"></a><!-- ws:end:WikiTextAnchorRule:46 -->Rank three temperaments</h1>
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<!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Rank three temperaments-Greenland"></a><!-- ws:end:WikiTextHeadingRule:14 --><!-- ws:start:WikiTextAnchorRule:47:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank three temperaments-Greenland" title="Anchor: Rank three temperaments-Greenland"/> --><a name="Rank three temperaments-Greenland"></a><!-- ws:end:WikiTextAnchorRule:47 --><a class="wiki_link" href="/Breed%20family">Greenland</a></h2>
Commas: 676/675, 1001/1000, 1716/1715<br />
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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 />
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<!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="Rank three temperaments-History"></a><!-- ws:end:WikiTextHeadingRule:16 --><!-- ws:start:WikiTextAnchorRule:48:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank three temperaments-History" title="Anchor: Rank three temperaments-History"/> --><a name="Rank three temperaments-History"></a><!-- ws:end:WikiTextAnchorRule:48 --><a class="wiki_link" href="/Werckismic%20temperaments">History</a></h2>
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 />
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<!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Rank three temperaments-Borneo"></a><!-- ws:end:WikiTextHeadingRule:18 --><!-- ws:start:WikiTextAnchorRule:49:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank three temperaments-Borneo" title="Anchor: Rank three temperaments-Borneo"/> --><a name="Rank three temperaments-Borneo"></a><!-- ws:end:WikiTextAnchorRule:49 -->Borneo</h2>
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 />
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<!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="Rank three temperaments-Madagascar"></a><!-- ws:end:WikiTextHeadingRule:20 --><!-- ws:start:WikiTextAnchorRule:50:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank three temperaments-Madagascar" title="Anchor: Rank three temperaments-Madagascar"/> --><a name="Rank three temperaments-Madagascar"></a><!-- ws:end:WikiTextAnchorRule:50 --><a class="wiki_link" href="/Cataharry%20family">Madagascar</a></h2>
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:<h2> --><h2 id="toc11"><a name="Rank three temperaments-Baffin"></a><!-- ws:end:WikiTextHeadingRule:22 --><!-- ws:start:WikiTextAnchorRule:51:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank three temperaments-Baffin" title="Anchor: Rank three temperaments-Baffin"/> --><a name="Rank three temperaments-Baffin"></a><!-- ws:end:WikiTextAnchorRule:51 -->Baffin</h2>
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 />
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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<br />
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<!-- ws:start:WikiTextHeadingRule:24:<h1> --><h1 id="toc12"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:24 --><!-- ws:start:WikiTextAnchorRule:52:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank two temperaments" title="Anchor: Rank two temperaments"/> --><a name="Rank two temperaments"></a><!-- ws:end:WikiTextAnchorRule:52 -->Rank two temperaments</h1>
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 />
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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 />
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<!-- ws:start:WikiTextHeadingRule:26:<h2> --><h2 id="toc13"><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextHeadingRule:26 --><!-- ws:start:WikiTextAnchorRule:53:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank two temperaments-Decitonic" title="Anchor: Rank two temperaments-Decitonic"/> --><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextAnchorRule:53 -->Decitonic</h2>
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br />
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~15/13 = 248.917<br />
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Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|]<br />
EDOs: 130, 270, 940, 1480<br />
Badness: 0.0135<br />
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<!-- ws:start:WikiTextHeadingRule:28:<h2> --><h2 id="toc14"><a name="Rank two temperaments-Avicenna"></a><!-- ws:end:WikiTextHeadingRule:28 --><!-- ws:start:WikiTextAnchorRule:54:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Rank two temperaments-Avicenna" title="Anchor: Rank two temperaments-Avicenna"/> --><a name="Rank two temperaments-Avicenna"></a><!-- ws:end:WikiTextAnchorRule:54 -->Avicenna</h2>
Commas: 676/675, 1001/1000, 3025/3024, 4096/4095<br />
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~13/12 = 137.777<br />
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Map: [<3 2 8 16 9 8|, <0 8 -3 -22 4 9|]<br />
EDOs: 87, 183, 270<br />
Badness: 0.0156<br />
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<!-- ws:start:WikiTextHeadingRule:30:<h1> --><h1 id="toc15"><a name="Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:30 --><!-- ws:start:WikiTextAnchorRule:55:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Subgroup temperaments" title="Anchor: Subgroup temperaments"/> --><a name="Subgroup temperaments"></a><!-- ws:end:WikiTextAnchorRule:55 -->Subgroup temperaments</h1>
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<!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc16"><a name="Subgroup temperaments-Barbados"></a><!-- ws:end:WikiTextHeadingRule:32 --><!-- ws:start:WikiTextAnchorRule:56:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Subgroup temperaments-Barbados" title="Anchor: Subgroup temperaments-Barbados"/> --><a name="Subgroup temperaments-Barbados"></a><!-- ws:end:WikiTextAnchorRule:56 -->Barbados</h2>
Subgroup: 2.3.13/5<br />
Commas: 676/675<br />
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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 <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 />
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~15/13 = 248.621<br />
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<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 />
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<!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc17"><a name="Subgroup temperaments-Trinidad"></a><!-- ws:end:WikiTextHeadingRule:34 --><!-- ws:start:WikiTextAnchorRule:57:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Subgroup temperaments-Trinidad" title="Anchor: Subgroup temperaments-Trinidad"/> --><a name="Subgroup temperaments-Trinidad"></a><!-- ws:end:WikiTextAnchorRule:57 -->Trinidad</h2>
Subgroup: 2.3.5.13<br />
Commas: 325/324, 625/624<br />
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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 />
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 317.076<br />
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<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 />
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<!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc18"><a name="Subgroup temperaments-Parizekmic"></a><!-- ws:end:WikiTextHeadingRule:36 --><!-- ws:start:WikiTextAnchorRule:58:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Subgroup temperaments-Parizekmic" title="Anchor: Subgroup temperaments-Parizekmic"/> --><a name="Subgroup temperaments-Parizekmic"></a><!-- ws:end:WikiTextAnchorRule:58 -->Parizekmic</h2>
Subgroup: 2.3.5.13<br />
Commas: 676/675<br />
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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.<br />
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<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 />
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<!-- ws:start:WikiTextHeadingRule:38:<h3> --><h3 id="toc19"><a name="Subgroup temperaments-Parizekmic-Music"></a><!-- ws:end:WikiTextHeadingRule:38 --><!-- ws:start:WikiTextAnchorRule:59:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Subgroup temperaments-Parizekmic-Music" title="Anchor: Subgroup temperaments-Parizekmic-Music"/> --><a name="Subgroup temperaments-Parizekmic-Music"></a><!-- ws:end:WikiTextAnchorRule:59 -->Music</h3>
<a class="wiki_link_ext" href="http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3" rel="nofollow">Petr's Pump</a>, a comma pump based ditty in Pariekmic temperament.<br />
EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270</body></html>