28edo: Difference between revisions

Revision as of 22:12, 16 May 2011

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author guest and made on 2011-05-16 22:12:37 UTC.

The original revision id was 229063642.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

[[toc|flat]]

=Basic properties= 
28edo, a multiple of both 7edo and 14edo, has a step size of 42.86 cents. It shares three intervals with 12edo: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it tempers out the greater diesis 648:625. It does not however temper out the 128:125 lesser diesis, as its major third is less than 1 cent flat (and its inversion the minor sixth less than 1 cent sharp). It has the same perfect fourth and fifth as 14edo. It also has decent approximations of several septimal intervals, of which 9/7 and its inversion 14/9 are also found in 14edo.

=Subgroups= 
28edo can approximate the 7-limit subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to [[Semicomma family|orwell temperament]] now has a major third as generator, though as before 225/224, 1728/1715 and 6144/6125 are tempered out. The 225/224-tempered version of the [[augmented triad]] has a very low complexity, so many of them appear in the MOS scales for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.

=Table of intervals= 
The following table compares it to potentially useful nearby just intervals.

|| Step # || ET Cents || Just Interval || Just Cents || Difference (ET minus Just) ||
||   ||   ||   ||   ||   ||
|| 1 || 42.86 ||   ||   ||   ||
|| 2 || 85.71 || 21:20 || 84.47 || 1.24 ||
|| 3 || 128.57 || 14:13 || 128.30 || 0.27 ||
|| 4 || 171.43 || 11:10 || 165.00 || 6.43 ||
|| 5 || 214.29 || 17:15 || 216.69 || -2.40 ||
|| 6 || 257.14 || 7:6 || 266.87 || -9.73 ||
|| 7 || 300 || 6:5 || 315.64 || -15.64 ||
|| 8 || 342.86 || 11:9 || 347.41 || -4.55 ||
|| 9 || 385.71 || 5:4 || 386.31 || -0.60 ||
|| 10 || 428.57 || 9:7 || 435.08 || -6.51 ||
|| 11 || 471.43 || 21:16 || 470.78 || 0.65 ||
|| 12 || 514.29 || 4:3 || 498.04 || 16.25 ||
|| 13 || 557.14 || 11:8 || 551.32 || 5.82 ||
|| 14 || 600 || 7:5 || 582.51 || 17.49 ||
|| 15 || 642.86 || 16:11 || 648.68 || -5.82 ||
|| 16 || 685.71 || 3:2 || 701.96 || -16.25 ||
|| 17 || 728.57 || 32:21 || 729.22 || -0.65 ||
|| 18 || 771.43 || 14:9 || 764.92 || 6.51 ||
|| 19 || 814.29 || 5:8 || 813.68 || 0.61 ||
|| 20 || 857.14 || 18:11 || 852.59 || 4.55 ||
|| 21 || 900 || 5:3 || 884.36 || 15.64 ||
|| 22 || 942.86 || 12:7 || 933.13 || 9.73 ||
|| 23 || 985.71 || 30:17 || 983.31 || 2.40 ||
|| 24 || 1028.57 || 20:11 || 1035.00 || -6.43 ||
|| 25 || 1071.42 || 13:7 || 1071.70 || -0.27 ||
|| 26 || 1114.29 || 40:21 || 1115.53 || -1.24 ||
|| 27 || 1157.14 ||   ||   ||   ||
|| 28 || 1200 || 2:1 || 1200 || 0 ||

Original HTML content:

<html><head><title>28edo</title></head><body><!-- ws:start:WikiTextTocRule:6:&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:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Basic properties">Basic properties</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Subgroups">Subgroups</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Table of intervals">Table of intervals</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: -->
<!-- ws:end:WikiTextTocRule:10 --><br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Basic properties"></a><!-- ws:end:WikiTextHeadingRule:0 -->Basic properties</h1>
 28edo, a multiple of both 7edo and 14edo, has a step size of 42.86 cents. It shares three intervals with 12edo: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it tempers out the greater diesis 648:625. It does not however temper out the 128:125 lesser diesis, as its major third is less than 1 cent flat (and its inversion the minor sixth less than 1 cent sharp). It has the same perfect fourth and fifth as 14edo. It also has decent approximations of several septimal intervals, of which 9/7 and its inversion 14/9 are also found in 14edo.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Subgroups"></a><!-- ws:end:WikiTextHeadingRule:2 -->Subgroups</h1>
 28edo can approximate the 7-limit subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to <a class="wiki_link" href="/Semicomma%20family">orwell temperament</a> now has a major third as generator, though as before 225/224, 1728/1715 and 6144/6125 are tempered out. The 225/224-tempered version of the <a class="wiki_link" href="/augmented%20triad">augmented triad</a> has a very low complexity, so many of them appear in the MOS scales for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Table of intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Table of intervals</h1>
 The following table compares it to potentially useful nearby just intervals.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Step #<br />
</td>
        <td>ET Cents<br />
</td>
        <td>Just Interval<br />
</td>
        <td>Just Cents<br />
</td>
        <td>Difference (ET minus Just)<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>42.86<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>85.71<br />
</td>
        <td>21:20<br />
</td>
        <td>84.47<br />
</td>
        <td>1.24<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>128.57<br />
</td>
        <td>14:13<br />
</td>
        <td>128.30<br />
</td>
        <td>0.27<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>171.43<br />
</td>
        <td>11:10<br />
</td>
        <td>165.00<br />
</td>
        <td>6.43<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>214.29<br />
</td>
        <td>17:15<br />
</td>
        <td>216.69<br />
</td>
        <td>-2.40<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>257.14<br />
</td>
        <td>7:6<br />
</td>
        <td>266.87<br />
</td>
        <td>-9.73<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>300<br />
</td>
        <td>6:5<br />
</td>
        <td>315.64<br />
</td>
        <td>-15.64<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>342.86<br />
</td>
        <td>11:9<br />
</td>
        <td>347.41<br />
</td>
        <td>-4.55<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>385.71<br />
</td>
        <td>5:4<br />
</td>
        <td>386.31<br />
</td>
        <td>-0.60<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>428.57<br />
</td>
        <td>9:7<br />
</td>
        <td>435.08<br />
</td>
        <td>-6.51<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>471.43<br />
</td>
        <td>21:16<br />
</td>
        <td>470.78<br />
</td>
        <td>0.65<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>514.29<br />
</td>
        <td>4:3<br />
</td>
        <td>498.04<br />
</td>
        <td>16.25<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>557.14<br />
</td>
        <td>11:8<br />
</td>
        <td>551.32<br />
</td>
        <td>5.82<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>600<br />
</td>
        <td>7:5<br />
</td>
        <td>582.51<br />
</td>
        <td>17.49<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>642.86<br />
</td>
        <td>16:11<br />
</td>
        <td>648.68<br />
</td>
        <td>-5.82<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>685.71<br />
</td>
        <td>3:2<br />
</td>
        <td>701.96<br />
</td>
        <td>-16.25<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>728.57<br />
</td>
        <td>32:21<br />
</td>
        <td>729.22<br />
</td>
        <td>-0.65<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>771.43<br />
</td>
        <td>14:9<br />
</td>
        <td>764.92<br />
</td>
        <td>6.51<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>814.29<br />
</td>
        <td>5:8<br />
</td>
        <td>813.68<br />
</td>
        <td>0.61<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>857.14<br />
</td>
        <td>18:11<br />
</td>
        <td>852.59<br />
</td>
        <td>4.55<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>900<br />
</td>
        <td>5:3<br />
</td>
        <td>884.36<br />
</td>
        <td>15.64<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>942.86<br />
</td>
        <td>12:7<br />
</td>
        <td>933.13<br />
</td>
        <td>9.73<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>985.71<br />
</td>
        <td>30:17<br />
</td>
        <td>983.31<br />
</td>
        <td>2.40<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>1028.57<br />
</td>
        <td>20:11<br />
</td>
        <td>1035.00<br />
</td>
        <td>-6.43<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>1071.42<br />
</td>
        <td>13:7<br />
</td>
        <td>1071.70<br />
</td>
        <td>-0.27<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>1114.29<br />
</td>
        <td>40:21<br />
</td>
        <td>1115.53<br />
</td>
        <td>-1.24<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>1157.14<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>1200<br />
</td>
        <td>2:1<br />
</td>
        <td>1200<br />
</td>
        <td>0<br />
</td>
    </tr>
</table>

</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-03-28 23:14:59 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-05-16 22:12:37 UTC</tt>.<br>
-: The original revision id was <tt>214918260</tt>.<br>
+: The original revision id was <tt>229063642</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>
@@ Line 8: / Line 8: @@
 <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]]
+=Basic properties=
-=Basic properties=
 edo, a multiple of both 7edo and 14edo, has a step size of 42.86 cents. It shares three intervals with 12edo: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it tempers out the greater diesis 648:625. It does not however temper out the 128:125 lesser diesis, as its major third is less than 1 cent flat (and its inversion the minor sixth less than 1 cent sharp). It has the same perfect fourth and fifth as 14edo. It also has decent approximations of several septimal intervals, of which 9/7 and its inversion 14/9 are also found in 14edo.
 =Subgroups=
 edo can approximate the 7-limit subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to [[Semicomma family|orwell temperament]] now has a major third as generator, though as before 225/224, 1728/1715 and 6144/6125 are tempered out. The 225/224-tempered version of the [[augmented triad]] has a very low complexity, so many of them appear in the MOS scales for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.
 =Table of intervals=
 The following table compares it to potentially useful nearby just intervals.
@@ Line 30: / Line 29: @@
 || 9 || 385.71 || 5:4 || 386.31 || -0.60 ||
 || 10 || 428.57 || 9:7 || 435.08 || -6.51 ||
-|| 11 || 471.43 ||   ||   ||   ||
+|| 11 || 471.43 || 21:16 || 470.78 || 0.65 ||
 || 12 || 514.29 || 4:3 || 498.04 || 16.25 ||
-|| 13 || 557.14 ||   ||   ||   ||
+|| 13 || 557.14 || 11:8 || 551.32 || 5.82 ||
 || 14 || 600 || 7:5 || 582.51 || 17.49 ||
-|| 15 || 642.86 ||   ||   ||   ||
+|| 15 || 642.86 || 16:11 || 648.68 || -5.82 ||
 || 16 || 685.71 || 3:2 || 701.96 || -16.25 ||
-|| 17 || 728.57 ||   ||   ||   ||
+|| 17 || 728.57 || 32:21 || 729.22 || -0.65 ||
 || 18 || 771.43 || 14:9 || 764.92 || 6.51 ||
 || 19 || 814.29 || 5:8 || 813.68 || 0.61 ||
@@ Line 51: / Line 50: @@
 <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;28edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&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:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Basic properties"&gt;Basic properties&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#Subgroups"&gt;Subgroups&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#Table of intervals"&gt;Table of intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:10 --&gt;&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="Basic properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Basic properties&lt;/h1&gt;
 edo, a multiple of both 7edo and 14edo, has a step size of 42.86 cents. It shares three intervals with 12edo: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it tempers out the greater diesis 648:625. It does not however temper out the 128:125 lesser diesis, as its major third is less than 1 cent flat (and its inversion the minor sixth less than 1 cent sharp). It has the same perfect fourth and fifth as 14edo. It also has decent approximations of several septimal intervals, of which 9/7 and its inversion 14/9 are also found in 14edo.&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="Subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Subgroups&lt;/h1&gt;
 edo can approximate the 7-limit subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to &lt;a class="wiki_link" href="/Semicomma%20family"&gt;orwell temperament&lt;/a&gt; now has a major third as generator, though as before 225/224, 1728/1715 and 6144/6125 are tempered out. The 225/224-tempered version of the &lt;a class="wiki_link" href="/augmented%20triad"&gt;augmented triad&lt;/a&gt; has a very low complexity, so many of them appear in the MOS scales for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Table of intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Table of intervals&lt;/h1&gt;
-The following table compares it to potentially useful nearby just intervals.&lt;br /&gt;
+ The following table compares it to potentially useful nearby just intervals.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 213: / Line 211: @@
          &lt;td&gt;471.43&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;21:16&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;470.78&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;0.65&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 237: / Line 235: @@
          &lt;td&gt;557.14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;11:8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;551.32&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;5.82&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 261: / Line 259: @@
          &lt;td&gt;642.86&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;16:11&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;648.68&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;-5.82&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 285: / Line 283: @@
          &lt;td&gt;728.57&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;32:21&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;729.22&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;-0.65&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;

28edo: Difference between revisions

Revision as of 22:12, 16 May 2011

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

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