Breedsmic temperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 205205922 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 212554536 - 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:genewardsmith|genewardsmith]] and made on <tt>2011- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-21 16:26:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>212554536</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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It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system. | It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system. | ||
=Hemififths= | |||
Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with [[99edo]] and [[140edo]] providing good tunings, and [[239edo]] an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&58 temperament and has wedgie <<2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS. | Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with [[99edo]] and [[140edo]] providing good tunings, and [[239edo]] an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&58 temperament and has wedgie <<2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS. | ||
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Algebraic generator: (2 + sqrt(2))/2 | Algebraic generator: (2 + sqrt(2))/2 | ||
Map: [<1 1 -5 -1|, <0 2 25 13| | |||
EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [239edo|239]], [338edo|338]] | |||
Badness: 0.0222 | |||
==11-limit== | |||
Commas: 243/242, 441/440, 896/891 | Commas: 243/242, 441/440, 896/891 | ||
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It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.<br /> | It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt; | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Hemififths"></a><!-- ws:end:WikiTextHeadingRule:0 -->Hemififths</h1> | ||
Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with <a class="wiki_link" href="/99edo">99edo</a> and <a class="wiki_link" href="/140edo">140edo</a> providing good tunings, and <a class="wiki_link" href="/239edo">239edo</a> an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&amp;58 temperament and has wedgie &lt;&lt;2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS.<br /> | Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with <a class="wiki_link" href="/99edo">99edo</a> and <a class="wiki_link" href="/140edo">140edo</a> providing good tunings, and <a class="wiki_link" href="/239edo">239edo</a> an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&amp;58 temperament and has wedgie &lt;&lt;2 25 13 35 15 -40||, which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS.<br /> | ||
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Algebraic generator: (2 + sqrt(2))/2<br /> | Algebraic generator: (2 + sqrt(2))/2<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt; | Map: [&lt;1 1 -5 -1|, &lt;0 2 25 13|<br /> | ||
EDOs: <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/58edo">58</a>, <a class="wiki_link" href="/99edo">99</a>, [239edo|239]], [338edo|338]]<br /> | |||
Badness: 0.0222<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Hemififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h2> | |||
Commas: 243/242, 441/440, 896/891<br /> | Commas: 243/242, 441/440, 896/891<br /> | ||
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Badness: 0.0235<br /> | Badness: 0.0235<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name=" | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Hemififths-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h3> | ||
Commas: 144/143, 196/195, 243/242, 364/363<br /> | Commas: 144/143, 196/195, 243/242, 364/363<br /> | ||
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Badness: 0.0191<br /> | Badness: 0.0191<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name=" | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Hemififths-Tertiaseptal"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tertiaseptal</h2> | ||
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. <a class="wiki_link" href="/171edo">171edo</a> makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.<br /> | Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. <a class="wiki_link" href="/171edo">171edo</a> makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.<br /> | ||
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Badness: 0.0130<br /> | Badness: 0.0130<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name=" | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Hemififths-Tertiaseptal-11-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit</h3> | ||
Commas: 243/242, 441/440, 65625/65536<br /> | Commas: 243/242, 441/440, 65625/65536<br /> | ||
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Badness: 0.0356<br /> | Badness: 0.0356<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name=" | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Hemififths-Harry"></a><!-- ws:end:WikiTextHeadingRule:10 -->Harry</h2> | ||
Commas: 2401/2400, 19683/19600<br /> | Commas: 2401/2400, 19683/19600<br /> | ||
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Badness: 0.0341<br /> | Badness: 0.0341<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name=" | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Hemififths-Harry-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h3> | ||
Commas: 243/242, 441/440, 4000/3993<br /> | Commas: 243/242, 441/440, 4000/3993<br /> | ||
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Badness: 0.0159<br /> | Badness: 0.0159<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name=" | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Hemififths-Harry-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h3> | ||
Commas: 243/242, 351/350, 441/440, 676/675<br /> | Commas: 243/242, 351/350, 441/440, 676/675<br /> | ||
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Badness: 0.0130<br /> | Badness: 0.0130<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name=" | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Hemififths-Quasiorwell"></a><!-- ws:end:WikiTextHeadingRule:16 -->Quasiorwell</h2> | ||
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.<br /> | In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.<br /> | ||
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Adding 3025/3024 extends to the 11-limit and gives &lt;&lt;38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.<br /> | Adding 3025/3024 extends to the 11-limit and gives &lt;&lt;38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name=" | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Hemififths-Decoid"></a><!-- ws:end:WikiTextHeadingRule:18 -->Decoid</h2> | ||
Commas: 2401/2400, 67108864/66976875<br /> | Commas: 2401/2400, 67108864/66976875<br /> | ||
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Badness: 0.0339<br /> | Badness: 0.0339<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name=" | <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Hemififths-Decoid-11-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->11-limit</h3> | ||
Commas: 2401/2400, 5832/5825, 9801/9800<br /> | Commas: 2401/2400, 5832/5825, 9801/9800<br /> | ||
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Badness: 0.0187<br /> | Badness: 0.0187<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name=" | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Hemififths-Decoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h3> | ||
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br /> | Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br /> | ||
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