28edo

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This revision was by author hstraub and made on 2011-06-22 05:22:12 UTC.

The original revision id was 238134791.

The revision comment was:

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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 ||
=Commas= 
28 EDO tempers out the following commas. (Note: This assumes the val < 28 44 65 79 97 104 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
||= 2187/2048 || | -11 7 > ||> 113.69 ||= Apotome ||=   ||
||= 648/625 || | 3 4 -4 > ||> 62.57 ||= Major Diesis ||= Diminished Comma ||
||= 16875/16384 || | -14 3 4 > ||> 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||
||= 393216/390625 || | 17 1 -8 > ||> 11.45 ||= Wuerschmidt Comma ||=   ||
||= 36/35 || | 2 2 -1 -1 > ||> 48.77 ||= Septimal Quarter Tone ||=   ||
||= 50/49 || | 1 0 2 -2 > ||> 34.98 ||= Tritonic Diesis ||= Jubilisma ||
||= 3125/3087 || | 0 -2 5 -3 > ||> 21.18 ||= Gariboh ||=   ||
||= 126/125 || | 1 2 -3 1 > ||> 13.79 ||= Septimal Semicomma ||= Starling Comma ||
||= 65625/65536 || | -16 1 5 1 > ||> 2.35 ||= Horwell ||=   ||
||= 394839/394762 || | 47 -7 -7 -7 > ||> 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||
||= 176/175 || | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||=   ||
||= 441/440 || | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||=   ||
||= 4000/3993 || | 5 -1 3 0 -3 > ||> 3.03 ||= Wizardharry ||=   ||

=Some scales= 
[[machine5]]
[[machine6]]
[[machine11]]

Original HTML content:

<html><head><title>28edo</title></head><body><!-- ws:start:WikiTextTocRule:10:&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:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Basic properties">Basic properties</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Subgroups">Subgroups</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#Table of intervals">Table of intervals</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Some scales">Some scales</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: -->
<!-- ws:end:WikiTextTocRule:16 --><hr />
<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>

<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h1>
 28 EDO tempers out the following commas. (Note: This assumes the val &lt; 28 44 65 79 97 104 |.)<br />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">2187/2048<br />
</td>
        <td>| -11 7 &gt;<br />
</td>
        <td style="text-align: right;">113.69<br />
</td>
        <td style="text-align: center;">Apotome<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">648/625<br />
</td>
        <td>| 3 4 -4 &gt;<br />
</td>
        <td style="text-align: right;">62.57<br />
</td>
        <td style="text-align: center;">Major Diesis<br />
</td>
        <td style="text-align: center;">Diminished Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16875/16384<br />
</td>
        <td>| -14 3 4 &gt;<br />
</td>
        <td style="text-align: right;">51.12<br />
</td>
        <td style="text-align: center;">Negri Comma<br />
</td>
        <td style="text-align: center;">Double Augmentation Diesis<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">393216/390625<br />
</td>
        <td>| 17 1 -8 &gt;<br />
</td>
        <td style="text-align: right;">11.45<br />
</td>
        <td style="text-align: center;">Wuerschmidt Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36/35<br />
</td>
        <td>| 2 2 -1 -1 &gt;<br />
</td>
        <td style="text-align: right;">48.77<br />
</td>
        <td style="text-align: center;">Septimal Quarter Tone<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">50/49<br />
</td>
        <td>| 1 0 2 -2 &gt;<br />
</td>
        <td style="text-align: right;">34.98<br />
</td>
        <td style="text-align: center;">Tritonic Diesis<br />
</td>
        <td style="text-align: center;">Jubilisma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3125/3087<br />
</td>
        <td>| 0 -2 5 -3 &gt;<br />
</td>
        <td style="text-align: right;">21.18<br />
</td>
        <td style="text-align: center;">Gariboh<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">126/125<br />
</td>
        <td>| 1 2 -3 1 &gt;<br />
</td>
        <td style="text-align: right;">13.79<br />
</td>
        <td style="text-align: center;">Septimal Semicomma<br />
</td>
        <td style="text-align: center;">Starling Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">65625/65536<br />
</td>
        <td>| -16 1 5 1 &gt;<br />
</td>
        <td style="text-align: right;">2.35<br />
</td>
        <td style="text-align: center;">Horwell<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">394839/394762<br />
</td>
        <td>| 47 -7 -7 -7 &gt;<br />
</td>
        <td style="text-align: right;">0.34<br />
</td>
        <td style="text-align: center;">Akjaysma<br />
</td>
        <td style="text-align: center;">5\7 Octave Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">176/175<br />
</td>
        <td>| 4 0 -2 -1 1 &gt;<br />
</td>
        <td style="text-align: right;">9.86<br />
</td>
        <td style="text-align: center;">Valinorsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">441/440<br />
</td>
        <td>| -3 2 -1 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">3.93<br />
</td>
        <td style="text-align: center;">Werckisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4000/3993<br />
</td>
        <td>| 5 -1 3 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">3.03<br />
</td>
        <td style="text-align: center;">Wizardharry<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Some scales"></a><!-- ws:end:WikiTextHeadingRule:8 -->Some scales</h1>
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