The Archipelago: Difference between revisions
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Wikispaces>mbattaglia1 **Imported revision 222628944 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2011-04-25 04:04:51 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>222628944</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | <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;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | ||
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 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. | ||
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[[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. | ||
=Parent Temperaments= | |||
=Island= | |||
Comma: 676/675 | Comma: 676/675 | ||
Map | Map: | ||
<1 0 0 0 0 -1| | <1 0 0 0 0 -1| | ||
<0 2 0 0 0 3| | <0 2 0 0 0 3| | ||
| Line 26: | Line 27: | ||
EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940 | EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940 | ||
[[Optimal patent val]]: [[940edo]] | [[Optimal patent val]]: [[940edo]] | ||
= = | |||
==[[#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 | |||
=Rank four temperaments= | =Rank four temperaments= | ||
==1001/1000== | ==1001/1000== | ||
Commas: 676/675, 1001/1000 | Commas: 676/675, 1001/1000 | ||
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[[Optimal patent val]]: [[940edo]] | [[Optimal patent val]]: [[940edo]] | ||
==49/48== | ==49/48== | ||
Commas: 49/48, 91/90 | Commas: 49/48, 91/90 | ||
| Line 60: | Line 75: | ||
==[[Werckismic temperaments|History]]== | ==[[Werckismic temperaments|History]]== | ||
Commas: 364/363, 441/440, 1001/1000 | Commas: 364/363, 441/440, 1001/1000 | ||
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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 | 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]]== | ==[[Cataharry family|Madagascar]]== | ||
Commas: 351/350, 540/539, 676/675 | Commas: 351/350, 540/539, 676/675 | ||
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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 | 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]]. | 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. | 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== | ==Decitonic== | ||
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224 | Commas: 676/675, 1001/1000, 1716/1715, 4225/4224 | ||
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Badness: 0.0135 | Badness: 0.0135 | ||
==Avicenna== | ==Avicenna== | ||
Commas: 676/675, 1001/1000, 3025/3024, 4096/4095 | Commas: 676/675, 1001/1000, 3025/3024, 4096/4095 | ||
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EDOs: 15, 19, 34, 53, 87, 140, 193, 246 | EDOs: 15, 19, 34, 53, 87, 140, 193, 246 | ||
==Parizekmic== | ==Parizekmic== | ||
Subgroup: 2.3.5.13 | Subgroup: 2.3.5.13 | ||
Commas: 676/675 | Commas: 676/675 | ||
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<0 0 1 1| | <0 0 1 1| | ||
===Music=== | ===Music=== | ||
[[http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3|Petr's Pump]], a comma pump based ditty in Pariekmic temperament. | [[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</pre></div> | EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270</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><!-- ws:start:WikiTextTocRule: | <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><!-- ws:start:WikiTextTocRule:48:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><a href="#Parent Temperaments">Parent Temperaments</a><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --> | <a href="#Island">Island</a><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --> | <a href="#toc2"> </a><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --> | <a href="#Rank four temperaments">Rank four temperaments</a><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --> | <a href="#Rank three temperaments">Rank three temperaments</a><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --> | <a href="#Rank two temperaments">Rank two temperaments</a><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --> | <a href="#Subgroup temperaments">Subgroup temperaments</a><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:73 -->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 /> | ||
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 /> | <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 /> | 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 /> | ||
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<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 /> | ||
<br /> | <br /> | ||
Comma: 676/675<br /> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Parent Temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Parent Temperaments</h1> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Island"></a><!-- ws:end:WikiTextHeadingRule:2 -->Island</h1> | |||
Comma: 676/675<br /> | |||
<br /> | <br /> | ||
Map<br /> | Map:<br /> | ||
&lt;1 0 0 0 0 -1| <br /> | &lt;1 0 0 0 0 -1| <br /> | ||
&lt;0 2 0 0 0 3| <br /> | &lt;0 2 0 0 0 3| <br /> | ||
| Line 184: | Line 200: | ||
EDOs: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940<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 /> | <a class="wiki_link" href="/Optimal%20patent%20val">Optimal patent val</a>: <a class="wiki_link" href="/940edo">940edo</a><br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><!-- ws:end:WikiTextHeadingRule:4 --> </h1> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Island-Barbados"></a><!-- ws:end:WikiTextHeadingRule:6 --><!-- ws:start:WikiTextAnchorRule:74:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Subgroup temperaments-Barbados&quot; title=&quot;Anchor: Subgroup temperaments-Barbados&quot;/&gt; --><a name="Subgroup temperaments-Barbados"></a><!-- ws:end:WikiTextAnchorRule:74 -->Barbados</h2> | |||
Subgroup: 2.3.13/5<br /> | |||
Commas: 676/675<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | 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>: [&lt;1 0 -1|, &lt;0 2 3|]<br /> | |||
EDOs: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362<br /> | |||
Badness: 0.002335<br /> | |||
<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Rank four temperaments"></a><!-- ws:end:WikiTextHeadingRule:8 -->Rank four temperaments</h1> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Rank four temperaments-1001/1000"></a><!-- ws:end:WikiTextHeadingRule:10 -->1001/1000</h2> | ||
Commas: 676/675, 1001/1000<br /> | Commas: 676/675, 1001/1000<br /> | ||
<br /> | <br /> | ||
EDOs: 15, 19, 29, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940<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 /> | <a class="wiki_link" href="/Optimal%20patent%20val">Optimal patent val</a>: <a class="wiki_link" href="/940edo">940edo</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Rank four temperaments-49/48"></a><!-- ws:end:WikiTextHeadingRule:12 -->49/48</h2> | ||
Commas: 49/48, 91/90<br /> | Commas: 49/48, 91/90<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Rank four temperaments-1716/1715"></a><!-- ws:end:WikiTextHeadingRule:14 -->1716/1715</h2> | ||
Commas: 676/675, 1716/1715<br /> | Commas: 676/675, 1716/1715<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Rank four temperaments-364/363"></a><!-- ws:end:WikiTextHeadingRule:16 -->364/363</h2> | ||
Commas: 364/363, 676/675<br /> | Commas: 364/363, 676/675<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Rank four temperaments-364/363-351/350"></a><!-- ws:end:WikiTextHeadingRule:18 -->351/350</h3> | ||
Commas: 351/350, 676/675<br /> | Commas: 351/350, 676/675<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Rank three temperaments"></a><!-- ws:end:WikiTextHeadingRule:20 -->Rank three temperaments</h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Rank three temperaments-Greenland"></a><!-- ws:end:WikiTextHeadingRule:22 --><a class="wiki_link" href="/Breed%20family">Greenland</a></h2> | ||
Commas: 676/675, 1001/1000, 1716/1715<br /> | Commas: 676/675, 1001/1000, 1716/1715<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Rank three temperaments-History"></a><!-- ws:end:WikiTextHeadingRule:24 --><a class="wiki_link" href="/Werckismic%20temperaments">History</a></h2> | ||
Commas: 364/363, 441/440, 1001/1000<br /> | Commas: 364/363, 441/440, 1001/1000<br /> | ||
<br /> | <br /> | ||
EDOs: 15, 29, 43, 58, 72, 87, 130, 217, 289<br /> | EDOs: 15, 29, 43, 58, 72, 87, 130, 217, 289<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Rank three temperaments-Borneo"></a><!-- ws:end:WikiTextHeadingRule:26 -->Borneo</h2> | ||
Commas: 676/675, 1001/1000, 3025/3024<br /> | Commas: 676/675, 1001/1000, 3025/3024<br /> | ||
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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 /> | 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: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="Rank three temperaments-Madagascar"></a><!-- ws:end:WikiTextHeadingRule:28 --><a class="wiki_link" href="/Cataharry%20family">Madagascar</a></h2> | ||
Commas: 351/350, 540/539, 676/675<br /> | Commas: 351/350, 540/539, 676/675<br /> | ||
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EDOs: 19, 53, 58, 72, 111, 130, 183, 313<br /> | EDOs: 19, 53, 58, 72, 111, 130, 183, 313<br /> | ||
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<a class="wiki_link" href="/madagascar19">madagascar19</a><br /> | <a class="wiki_link" href="/madagascar19">madagascar19</a><br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Rank three temperaments-Baffin"></a><!-- ws:end:WikiTextHeadingRule:30 -->Baffin</h2> | ||
Commas: 676/675, 1001/1000, 4225/4224<br /> | Commas: 676/675, 1001/1000, 4225/4224<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 /> | 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: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:32 -->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 /> | 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 /> | 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: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Rank two temperaments-Decitonic"></a><!-- ws:end:WikiTextHeadingRule:34 -->Decitonic</h2> | ||
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br /> | 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 /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~15/13 = 248.917<br /> | ||
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Badness: 0.0135<br /> | Badness: 0.0135<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Rank two temperaments-Avicenna"></a><!-- ws:end:WikiTextHeadingRule:36 -->Avicenna</h2> | ||
Commas: 676/675, 1001/1000, 3025/3024, 4096/4095<br /> | 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 /> | <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~13/12 = 137.777<br /> | ||
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Badness: 0.0156<br /> | Badness: 0.0156<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:38 -->Subgroup temperaments</h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Subgroup temperaments-Barbados"></a><!-- ws:end:WikiTextHeadingRule:40 -->Barbados</h2> | ||
Subgroup: 2.3.13/5<br /> | Subgroup: 2.3.13/5<br /> | ||
Commas: 676/675<br /> | Commas: 676/675<br /> | ||
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Badness: 0.002335<br /> | Badness: 0.002335<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><a name="Subgroup temperaments-Trinidad"></a><!-- ws:end:WikiTextHeadingRule:42 -->Trinidad</h2> | ||
Subgroup: 2.3.5.13<br /> | Subgroup: 2.3.5.13<br /> | ||
Commas: 325/324, 625/624<br /> | Commas: 325/324, 625/624<br /> | ||
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EDOs: 15, 19, 34, 53, 87, 140, 193, 246<br /> | EDOs: 15, 19, 34, 53, 87, 140, 193, 246<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc22"><a name="Subgroup temperaments-Parizekmic"></a><!-- ws:end:WikiTextHeadingRule:44 -->Parizekmic</h2> | ||
Subgroup: 2.3.5.13<br /> | Subgroup: 2.3.5.13<br /> | ||
Commas: 676/675<br /> | Commas: 676/675<br /> | ||
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&lt;0 0 1 1|<br /> | &lt;0 0 1 1|<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:46:&lt;h3&gt; --><h3 id="toc23"><a name="Subgroup temperaments-Parizekmic-Music"></a><!-- ws:end:WikiTextHeadingRule:46 -->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 /> | <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></pre></div> | EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270</body></html></pre></div> | ||