Breedsmic temperaments: Difference between revisions
Wikispaces>Natebedell **Imported revision 251319616 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 268879194 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:genewardsmith|genewardsmith]] and made on <tt>2011-10-26 14:54:44 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>268879194</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]] | ||
Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4> = 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. | Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4> = 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. | ||
| Line 157: | Line 159: | ||
Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|] | Map: [<10 0 47 36 98 37|, <0 2 -3 -1 -8 0|] | ||
EDOs: 130, 270, 940, 1480 | EDOs: 130, 270, 940, 1480 | ||
Badness: 0.0135</pre></div> | Badness: 0.0135 | ||
=Neominor= | |||
Commas: 2401/2400, 177147/175616 | |||
POTE generator: ~189/160 = 283.280 | |||
Map: [<1 3 12 8|, <0 -6 -41 -22|] | |||
Weggie: <<6 41 22 51 18 -64|| | |||
EDOs: 72, 161, 233, 305 | |||
Badness: 0.0882 | |||
==11-limit== | |||
Commas: 243/242, 441/440, 35937/35840 | |||
POTE: ~33/28 = 283.276 | |||
Map: [<1 3 12 8 7|, <0 -6 -41 -22 -15|] | |||
EDOs: 72, 161, 233, 305 | |||
Badness: 0.0280 | |||
==13-limit== | |||
Commas: 169/168, 243/242, 364/363, 441/440 | |||
POTE generator: ~13/11 = 283.294 | |||
Map: [<1 3 12 8 7 7|, <0 -6 -41 -22 -15 -14|] | |||
EDOs: 72, 161f, 233f | |||
Badness: 0.0269 | |||
</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><!-- 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>Breedsmic temperaments</title></head><body><!-- ws:start:WikiTextTocRule:34:&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:34 --><!-- ws:start:WikiTextTocRule:35: --><a href="#Hemififths">Hemififths</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --> | <a href="#Tertiaseptal">Tertiaseptal</a><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <a href="#Harry">Harry</a><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --> | <a href="#Quasiorwell">Quasiorwell</a><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --> | <a href="#Decoid">Decoid</a><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --> | <a href="#Neominor">Neominor</a><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><br /> | ||
<br /> | |||
Breedsmic temperaments are rank two temperaments tempering out the breedsma, |-5 -1 -2 4&gt; = 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 /> | ||
| Line 310: | Line 344: | ||
Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]<br /> | Map: [&lt;10 0 47 36 98 37|, &lt;0 2 -3 -1 -8 0|]<br /> | ||
EDOs: 130, 270, 940, 1480<br /> | EDOs: 130, 270, 940, 1480<br /> | ||
Badness: 0.0135</body></html></pre></div> | Badness: 0.0135<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Neominor"></a><!-- ws:end:WikiTextHeadingRule:28 -->Neominor</h1> | |||
Commas: 2401/2400, 177147/175616<br /> | |||
<br /> | |||
POTE generator: ~189/160 = 283.280<br /> | |||
<br /> | |||
Map: [&lt;1 3 12 8|, &lt;0 -6 -41 -22|]<br /> | |||
Weggie: &lt;&lt;6 41 22 51 18 -64||<br /> | |||
EDOs: 72, 161, 233, 305<br /> | |||
Badness: 0.0882<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Neominor-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2> | |||
Commas: 243/242, 441/440, 35937/35840<br /> | |||
<br /> | |||
POTE: ~33/28 = 283.276<br /> | |||
<br /> | |||
Map: [&lt;1 3 12 8 7|, &lt;0 -6 -41 -22 -15|]<br /> | |||
EDOs: 72, 161, 233, 305<br /> | |||
Badness: 0.0280<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Neominor-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2> | |||
Commas: 169/168, 243/242, 364/363, 441/440<br /> | |||
<br /> | |||
POTE generator: ~13/11 = 283.294<br /> | |||
<br /> | |||
Map: [&lt;1 3 12 8 7 7|, &lt;0 -6 -41 -22 -15 -14|]<br /> | |||
EDOs: 72, 161f, 233f<br /> | |||
Badness: 0.0269</body></html></pre></div> | |||