Breedsmic temperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 202797834 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 203606980 - 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>2011-02- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-21 01:42:56 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>203606980</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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Algebraic generator: (2 + sqrt(2))/2 | Algebraic generator: (2 + sqrt(2))/2 | ||
===11-limit=== | |||
Commas: 243/242, 441/440, 896/891 | |||
POTE generator: ~11/9 = 351.521 | |||
Map: [<1 1 -5 -1 2|, <0 2 25 13 5|] | |||
EDOs: 7, 17, 41, 58, 99 | |||
Badness: 0.0235 | |||
==Tertiaseptal== | ==Tertiaseptal== | ||
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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;h2&gt; --><h2 id=" | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Hemififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h3> | ||
Commas: 243/242, 441/440, 896/891<br /> | |||
<br /> | |||
POTE generator: ~11/9 = 351.521<br /> | |||
<br /> | |||
Map: [&lt;1 1 -5 -1 2|, &lt;0 2 25 13 5|]<br /> | |||
EDOs: 7, 17, 41, 58, 99<br /> | |||
Badness: 0.0235<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Tertiaseptal"></a><!-- ws:end:WikiTextHeadingRule:4 -->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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-Tertiaseptal-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->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: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-Harry"></a><!-- ws:end:WikiTextHeadingRule:8 -->Harry</h2> | ||
Commas: 2401/2400, 19683/19600<br /> | Commas: 2401/2400, 19683/19600<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0341<br /> | Badness: 0.0341<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Harry-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Harry-13-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="x-Quasiorwell"></a><!-- ws:end:WikiTextHeadingRule:14 -->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.</body></html></pre></div> | 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.</body></html></pre></div> | ||