The Archipelago: Difference between revisions
Wikispaces>guest **Imported revision 199554484 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 199619078 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:genewardsmith|genewardsmith]] and made on <tt>2011-02-08 01:46:59 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>199619078</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> | ||
| Line 12: | Line 12: | ||
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. | ||
[[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 | Comma: 676/675 | ||
| Line 79: | Line 79: | ||
Map: [<3 2 8 16 9 8|, <0 8 -3 -22 4 9|] | Map: [<3 2 8 16 9 8|, <0 8 -3 -22 4 9|] | ||
EDOs: 87, 183, 270 | EDOs: 87, 183, 270 | ||
Badness: 0.0156</pre></div> | Badness: 0.0156 | ||
==Subgroup temperaments== | |||
===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]] and [[53edo]], with MOS of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales. | |||
</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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"><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 /> | <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"><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 /> | ||
| Line 87: | Line 93: | ||
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.<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 /> | |||
<br /> | <br /> | ||
Comma: 676/675<br /> | Comma: 676/675<br /> | ||
| Line 154: | Line 160: | ||
Map: [&lt;3 2 8 16 9 8|, &lt;0 8 -3 -22 4 9|]<br /> | Map: [&lt;3 2 8 16 9 8|, &lt;0 8 -3 -22 4 9|]<br /> | ||
EDOs: 87, 183, 270<br /> | EDOs: 87, 183, 270<br /> | ||
Badness: 0.0156</body></html></pre></div> | Badness: 0.0156<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> | |||
<br /> | |||
<!-- 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> and <a class="wiki_link" href="/53edo">53edo</a>, with MOS of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.</body></html></pre></div> | |||