Diminished family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
__FORCETOC__
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>2012-05-09 01:09:29 UTC</tt>.<br>
: The original revision id was <tt>332260622</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>
<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]]
 
The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo.  
The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo.  


Line 16: Line 8:


Map: [&lt;4 0 3|, &lt;0 1 1|]
Map: [&lt;4 0 3|, &lt;0 1 1|]
EDOs: 8, 12
EDOs: 8, 12
Badness: 0.0472
Badness: 0.0472


Line 25: Line 19:


Map: [&lt;4 0 3 8|, &lt;0 2 2 1|]
Map: [&lt;4 0 3 8|, &lt;0 2 2 1|]
Wedgie: &lt;&lt;8 8 4 -6 -16 -13||
Wedgie: &lt;&lt;8 8 4 -6 -16 -13||
EDOs: 24, 52d, 76cd
EDOs: 24, 52d, 76cd
Badness: 0.0864
Badness: 0.0864


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Map: [&lt;4 0 3 8 -2|, &lt;0 2 2 1 5|]
Map: [&lt;4 0 3 8 -2|, &lt;0 2 2 1 5|]
EDOs: 24, 76cde
EDOs: 24, 76cde
Badness: 0.0566
Badness: 0.0566


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Map: [&lt;4 0 3 8 -2 -1|, &lt;0 2 2 1 5 5|]
Map: [&lt;4 0 3 8 -2 -1|, &lt;0 2 2 1 5 5|]
EDOs: 24, 52de, 76cde
EDOs: 24, 52de, 76cde
Badness: 0.039
Badness: 0.039


Line 53: Line 54:


Map: [&lt;8 0 6 -3|, &lt;0 1 1 2|]
Map: [&lt;8 0 6 -3|, &lt;0 1 1 2|]
Wedgie: &lt;&lt;8 8 16 -6 3 15||
Wedgie: &lt;&lt;8 8 16 -6 3 15||
EDOs: 24, 32c, 56c  
EDOs: 24, 32c, 56c  
Badness: 0.1075
Badness: 0.1075


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Map: [&lt;8 0 6 -3 15|, &lt;0 1 1 2 1|]
Map: [&lt;8 0 6 -3 15|, &lt;0 1 1 2 1|]
EDOs: 24, 32c, 56c
EDOs: 24, 32c, 56c
Badness: 0.0476
Badness: 0.0476


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Map: [&lt;8 0 6 -3 15 17|, &lt;0 1 1 2 1 1|]
Map: [&lt;8 0 6 -3 15 17|, &lt;0 1 1 2 1 1|]
EDOs: 24, 32cf, 56cf
EDOs: 24, 32cf, 56cf
Badness: 0.0306
Badness: 0.0306
</pre></div>
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Dimipent family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;a href="#Dimipent"&gt;Dimipent&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Hemidim"&gt;Hemidim&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Semidim"&gt;Semidim&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;
&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;br /&gt;
The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo. &lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Dimipent"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Dimipent&lt;/h1&gt;
Comma: 648/625&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~3/2 =  699.507&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;4 0 3|, &amp;lt;0 1 1|]&lt;br /&gt;
EDOs: 8, 12&lt;br /&gt;
Badness: 0.0472&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Hemidim"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Hemidim&lt;/h1&gt;
Commas: 49/48, 648/625&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~8/7 = 252.555&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;4 0 3 8|, &amp;lt;0 2 2 1|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;8 8 4 -6 -16 -13||&lt;br /&gt;
EDOs: 24, 52d, 76cd&lt;br /&gt;
Badness: 0.0864&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Hemidim-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
Commas: 49/48, 77/75, 243/242&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~8/7 = 251.658&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;4 0 3 8 -2|, &amp;lt;0 2 2 1 5|]&lt;br /&gt;
EDOs: 24, 76cde&lt;br /&gt;
Badness: 0.0566&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Hemidim-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
Commas: 49/48, 66/65, 77/75, 648/625&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~8/7 = 252.225&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;4 0 3 8 -2 -1|, &amp;lt;0 2 2 1 5 5|]&lt;br /&gt;
EDOs: 24, 52de, 76cde&lt;br /&gt;
Badness: 0.039&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Semidim"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Semidim&lt;/h1&gt;
Commas: 245/243, 392/375&lt;br /&gt;
&lt;br /&gt;
POTE generator4: ~3/2 = 707.014&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;8 0 6 -3|, &amp;lt;0 1 1 2|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;8 8 16 -6 3 15||&lt;br /&gt;
EDOs: 24, 32c, 56c &lt;br /&gt;
Badness: 0.1075&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Semidim-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
Commas: 56/55, 77/75, 245/243&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~3/2 = 706.645&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;8 0 6 -3 15|, &amp;lt;0 1 1 2 1|]&lt;br /&gt;
EDOs: 24, 32c, 56c&lt;br /&gt;
Badness: 0.0476&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Semidim-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h2&gt;
Commas: 56/55, 66/65, 77/75, 507/500&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~3/2 = 707.376&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;8 0 6 -3 15 17|, &amp;lt;0 1 1 2 1 1|]&lt;br /&gt;
EDOs: 24, 32cf, 56cf&lt;br /&gt;
Badness: 0.0306&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo.

Dimipent

Comma: 648/625

POTE generator: ~3/2 = 699.507

Map: [<4 0 3|, <0 1 1|]

EDOs: 8, 12

Badness: 0.0472

Hemidim

Commas: 49/48, 648/625

POTE generator: ~8/7 = 252.555

Map: [<4 0 3 8|, <0 2 2 1|]

Wedgie: <<8 8 4 -6 -16 -13||

EDOs: 24, 52d, 76cd

Badness: 0.0864

11-limit

Commas: 49/48, 77/75, 243/242

POTE generator: ~8/7 = 251.658

Map: [<4 0 3 8 -2|, <0 2 2 1 5|]

EDOs: 24, 76cde

Badness: 0.0566

13-limit

Commas: 49/48, 66/65, 77/75, 648/625

POTE generator: ~8/7 = 252.225

Map: [<4 0 3 8 -2 -1|, <0 2 2 1 5 5|]

EDOs: 24, 52de, 76cde

Badness: 0.039

Semidim

Commas: 245/243, 392/375

POTE generator4: ~3/2 = 707.014

Map: [<8 0 6 -3|, <0 1 1 2|]

Wedgie: <<8 8 16 -6 3 15||

EDOs: 24, 32c, 56c

Badness: 0.1075

11-limit

Commas: 56/55, 77/75, 245/243

POTE generator: ~3/2 = 706.645

Map: [<8 0 6 -3 15|, <0 1 1 2 1|]

EDOs: 24, 32c, 56c

Badness: 0.0476

13-limit

Commas: 56/55, 66/65, 77/75, 507/500

POTE generator: ~3/2 = 707.376

Map: [<8 0 6 -3 15 17|, <0 1 1 2 1 1|]

EDOs: 24, 32cf, 56cf

Badness: 0.0306

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