Breedsmic temperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 212554536 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 212556890 - 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>2011-03-21 16: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-21 16:33:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>212556890</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">Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma. | <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]] | ||
Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma. | |||
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. | ||
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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| | Map: [<1 1 -5 -1|, <0 2 25 13|] | ||
EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [239edo|239]], [338edo|338]] | EDOs: [[41edo|41]], [[58edo|58]], [[99edo|99]], [[239edo|239]], [[338edo|338]] | ||
Badness: 0.0222 | Badness: 0.0222 | ||
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Badness: 0.0235 | Badness: 0.0235 | ||
==13-limit== | |||
Commas: 144/143, 196/195, 243/242, 364/363 | Commas: 144/143, 196/195, 243/242, 364/363 | ||
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Badness: 0.0191 | Badness: 0.0191 | ||
=Tertiaseptal= | |||
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&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. [[171edo]] 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. | 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&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. [[171edo]] 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. | ||
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Badness: 0.0130 | Badness: 0.0130 | ||
==11-limit== | |||
Commas: 243/242, 441/440, 65625/65536 | Commas: 243/242, 441/440, 65625/65536 | ||
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Badness: 0.0356 | Badness: 0.0356 | ||
=Harry= | |||
Commas: 2401/2400, 19683/19600 | Commas: 2401/2400, 19683/19600 | ||
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Badness: 0.0341 | Badness: 0.0341 | ||
==11-limit== | |||
Commas: 243/242, 441/440, 4000/3993 | Commas: 243/242, 441/440, 4000/3993 | ||
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Badness: 0.0159 | Badness: 0.0159 | ||
==13-limit== | |||
Commas: 243/242, 351/350, 441/440, 676/675 | Commas: 243/242, 351/350, 441/440, 676/675 | ||
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Badness: 0.0130 | Badness: 0.0130 | ||
=Quasiorwell= | |||
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1>. It has a wedgie <<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&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. | In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1>. It has a wedgie <<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&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. | ||
Adding 3025/3024 extends to the 11-limit and gives <<38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning. | Adding 3025/3024 extends to the 11-limit and gives <<38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning. | ||
=Decoid= | |||
Commas: 2401/2400, 67108864/66976875 | Commas: 2401/2400, 67108864/66976875 | ||
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Badness: 0.0339 | Badness: 0.0339 | ||
==11-limit== | |||
Commas: 2401/2400, 5832/5825, 9801/9800 | Commas: 2401/2400, 5832/5825, 9801/9800 | ||
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Badness: 0.0187 | Badness: 0.0187 | ||
==13-limit== | |||
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</pre></div> | Badness: 0.0135</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>Breedsmic temperaments</title></head><body>Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.<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>Breedsmic temperaments</title></head><body><!-- ws:start:WikiTextTocRule:24:&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:24 --><!-- ws:start:WikiTextTocRule:25: --><a href="#Hemififths">Hemififths</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Tertiaseptal">Tertiaseptal</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Harry">Harry</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Quasiorwell">Quasiorwell</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Decoid">Decoid</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | ||
<!-- ws:end:WikiTextTocRule:37 --><br /> | |||
Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.<br /> | |||
<br /> | <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 /> | 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 /> | ||
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Algebraic generator: (2 + sqrt(2))/2<br /> | Algebraic generator: (2 + sqrt(2))/2<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 1 -5 -1|, &lt;0 2 25 13|<br /> | 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>, | 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>, <a class="wiki_link" href="/239edo">239</a>, <a class="wiki_link" href="/338edo">338</a><br /> | ||
Badness: 0.0222<br /> | Badness: 0.0222<br /> | ||
<br /> | <br /> | ||
| Line 162: | Line 166: | ||
Badness: 0.0235<br /> | Badness: 0.0235<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt; | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Hemififths-13-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->13-limit</h2> | ||
Commas: 144/143, 196/195, 243/242, 364/363<br /> | Commas: 144/143, 196/195, 243/242, 364/363<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0191<br /> | Badness: 0.0191<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt; | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Tertiaseptal"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tertiaseptal</h1> | ||
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 /> | ||
<br /> | <br /> | ||
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Badness: 0.0130<br /> | Badness: 0.0130<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt; | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Tertiaseptal-11-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit</h2> | ||
Commas: 243/242, 441/440, 65625/65536<br /> | Commas: 243/242, 441/440, 65625/65536<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0356<br /> | Badness: 0.0356<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt; | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Harry"></a><!-- ws:end:WikiTextHeadingRule:10 -->Harry</h1> | ||
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:12:&lt; | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Harry-11-limit"></a><!-- ws:end:WikiTextHeadingRule:12 -->11-limit</h2> | ||
Commas: 243/242, 441/440, 4000/3993<br /> | Commas: 243/242, 441/440, 4000/3993<br /> | ||
<br /> | <br /> | ||
| Line 216: | Line 220: | ||
Badness: 0.0159<br /> | Badness: 0.0159<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt; | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Harry-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2> | ||
Commas: 243/242, 351/350, 441/440, 676/675<br /> | Commas: 243/242, 351/350, 441/440, 676/675<br /> | ||
<br /> | <br /> | ||
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Badness: 0.0130<br /> | Badness: 0.0130<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt; | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Quasiorwell"></a><!-- ws:end:WikiTextHeadingRule:16 -->Quasiorwell</h1> | ||
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 /> | |||
<br /> | <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 /> | 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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:18:&lt; | <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Decoid"></a><!-- ws:end:WikiTextHeadingRule:18 -->Decoid</h1> | ||
Commas: 2401/2400, 67108864/66976875<br /> | Commas: 2401/2400, 67108864/66976875<br /> | ||
<br /> | <br /> | ||
| Line 240: | Line 244: | ||
Badness: 0.0339<br /> | Badness: 0.0339<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt; | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Decoid-11-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->11-limit</h2> | ||
Commas: 2401/2400, 5832/5825, 9801/9800<br /> | Commas: 2401/2400, 5832/5825, 9801/9800<br /> | ||
<br /> | <br /> | ||
| Line 249: | Line 253: | ||
Badness: 0.0187<br /> | Badness: 0.0187<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt; | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Decoid-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h2> | ||
Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br /> | Commas: 676/675, 1001/1000, 1716/1715, 4225/4224<br /> | ||
<br /> | <br /> | ||