The Archipelago: Difference between revisions
Wikispaces>genewardsmith **Imported revision 199627174 - Original comment: ** |
Wikispaces>guest **Imported revision 199663974 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:guest|guest]] and made on <tt>2011-02-08 08:08:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>199663974</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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. | 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. | 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. | [[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. | ||
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Badness: 0.0156 | Badness: 0.0156 | ||
==Subgroup temperaments== | ==Subgroup temperaments== | ||
===Barbados=== | ===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. | 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=== | ||
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. | 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. | ||
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EDOs: 15, 19, 34, 53, 87, 193, 246 | EDOs: 15, 19, 34, 53, 87, 193, 246 | ||
===Parizekmic=== | ===Parizekmic=== | ||
Closely related to barbados temperament is parizekmic, the rank three 2.3.5.13 subgroup temperament tempering out 676/675. | Closely related to barbados temperament is parizekmic, the rank three 2.3.5.13 subgroup temperament tempering out 676/675. | ||
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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 /> | 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 /> | <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.<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 /> | <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 /> | <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 /> | ||
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<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> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="x-Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:30 -->Subgroup temperaments</h2> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="x-Subgroup temperaments-Barbados"></a><!-- ws:end:WikiTextHeadingRule:32 -->Barbados</h3> | <!-- 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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="x-Subgroup temperaments-Trinidad"></a><!-- ws:end:WikiTextHeadingRule:34 -->Trinidad</h3> | <!-- 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 /> | 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 /> | <br /> | ||
Commas: 325/324, 625/624 in 2.3.5.13<br /> | Commas: 325/324, 625/624 in 2.3.5.13<br /> | ||
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<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> | <!-- 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 /> | Closely related to barbados temperament is parizekmic, the rank three 2.3.5.13 subgroup temperament tempering out 676/675. <br /> | ||
<br /> | <br /> | ||
Map<br /> | Map<br /> | ||