The Archipelago: Difference between revisions

Revision as of 17:47, 7 February 2011

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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.

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

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 />
Comma: 676/675<br />
<br />
Map<br />
&lt;1 0 0 0 0 -1| <br />
&lt;0 2 0 0 0 3| <br />
&lt;0 0 1 0 0 1| <br />
&lt;0 0 0 1 0 0| <br />
&lt;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:&lt;h2&gt; --><h2 id="toc0"><a name="x-Rank four temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Rank four temperaments</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><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:&lt;h3&gt; --><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:&lt;h3&gt; --><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:&lt;h3&gt; --><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:&lt;h3&gt; --><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:&lt;h2&gt; --><h2 id="toc6"><a name="x-Rank three temperaments"></a><!-- ws:end:WikiTextHeadingRule:12 -->Rank three temperaments</h2>
 <br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><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:&lt;h3&gt; --><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:&lt;h3&gt; --><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:&lt;h3&gt; --><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:&lt;h3&gt; --><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:&lt;h2&gt; --><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:&lt;h3&gt; --><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: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]<br />
EDOs: 130, 270, 940, 1480<br />
Badness: 0.0135<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><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: [&lt;3 2 8 16 9 8|, &lt;0 8 -3 -22 4 9|]<br />
EDOs: 87, 183, 270<br />
Badness: 0.0156</body></html>

The Archipelago: Difference between revisions

Revision as of 17:47, 7 February 2011

IMPORTED REVISION FROM WIKISPACES

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