28edo: Difference between revisions

Revision as of 16:38, 28 March 2015

IMPORTED REVISION FROM WIKISPACES

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This revision was by author JosephRuhf and made on 2015-03-28 16:38:01 UTC.

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[[toc|flat]]
----

=Basic properties= 
28edo, a multiple of both [[xenharmonic/7edo|7edo]] and [[xenharmonic/14edo|14edo]] (and of course [[xenharmonic/2edo|2edo]] and [[xenharmonic/4edo|4edo]]), has a step size of 42.857 [[xenharmonic/cent|cent]]s. It shares three intervals with [[xenharmonic/12edo|12edo]]: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it [[xenharmonic/tempering out|tempers out]] the [[xenharmonic/greater diesis|greater diesis]] [[xenharmonic/648_625|648:625]]. It does not however temper out the [[xenharmonic/128_125|128:125]] [[xenharmonic/lesser diesis|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 [[xenharmonic/9_7|9/7]] and its inversion [[xenharmonic/14_9|14/9]] are also found in 14edo.

=Subgroups= 
28edo can approximate the [[xenharmonic/7-limit|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 [[xenharmonic/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 [[xenharmonic/augmented triad|augmented triad]] has a very low complexity, so many of them appear in the [[xenharmonic/MOS scales|MOS scales]] for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.

Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39.

=Table of intervals= 
The following table compares it to potentially useful nearby [[xenharmonic/just intervals|just intervals]].

|| Step # || ET Cents
DMS || Just Interval || Just Cents
DMS || Difference (ET minus Just) ||
|| 1 || 42.86
12°<span style="background-color: #ffffff;">51'26"</span> ||   ||   ||   ||
|| 2 || 85.71
<span style="background-color: #ffffff;">25°42'51"</span> || 21:20 || 84.47
<span style="background-color: #ffffff;">25°20'25"</span> || 1.24
22'26" ||
|| 3 || 128.57
38°<span style="background-color: #ffffff;">34'17"</span> || 14:13 || 128.30
38°29<span style="background-color: #ffffff;">'22"</span> || 0.27
4'55" ||
|| 4 || 171.43
<span style="background-color: #ffffff;">51°25'43"</span> || 11:10 || 165.00
<span style="background-color: #ffffff;">49°30'5"</span> || 6.43
1°55'38° ||
|| 5 || 214.29
64°<span style="background-color: #ffffff;">17'9"</span> || 17:15 || 216.69
65°22" || -2.40
-43'13" ||
|| 6 || 257.14
<span style="background-color: #ffffff;">77°8'34"</span> || 7:6 || 266.87
80°3'41" || -9.73
-2°55'7" ||
|| 7 || 300
90° || 6:5 || 315.64
94°41'33" || -15.64
-4°41'33" ||
|| 8 || 342.86
<span style="background-color: #ffffff;">102°51'26"</span> || 11:9 || 347.41
<span style="background-color: #ffffff;">104°13'21"</span> || -4.55
-1°21'55" ||
|| 9 || 385.71
115°<span style="background-color: #ffffff;">42'51"</span> || 5:4 || 386.31
115°<span style="background-color: #ffffff;">53'39"</span> || -0.60
-10'48" ||
|| 10 || 428.57
<span style="background-color: #ffffff;">128°34'17"</span> || 9:7 || 435.08
130°31'30" || -6.51
-1°57'13" ||
|| 11 || 471.43
141°<span style="background-color: #ffffff;">25'43"</span> || 21:16 || 470.78
141°<span style="background-color: #ffffff;">14'3"</span> || 0.65
11'40" ||
|| 12 || 514.29
<span style="background-color: #ffffff;">154°17'9"</span> || 4:3 || 498.04
149°24<span style="background-color: #ffffff;">'49"</span> || 16.25
4°52'20" ||
|| 13 || 557.14
167°<span style="background-color: #ffffff;">8'34"</span> || 11:8 || 551.32
165°<span style="background-color: #ffffff;">23'43"</span> || 5.82
1°45'9" ||
|| 14 || 600
<span style="background-color: #ffffff;">180°</span> || 7:5 10:7 || 582.51 617.49
174°45'13" <span style="background-color: #ffffff;">185°14'47"</span> || ±17.49
±<span style="background-color: #ffffff;">5°14'47"</span> ||
|| 15 || 642.86
<span style="background-color: #ffffff;">192°51'26"</span> || 16:11 || 648.68
<span style="background-color: #ffffff;">194°36'35"</span> || -5.82
-1°45'9" ||
|| 16 || 685.71
<span style="background-color: #ffffff;">205°42'51"</span> || 3:2 || 701.96
<span style="background-color: #ffffff;">210°35'11"</span> || -16.25
-4°52'20" ||
|| 17 || 728.57
218°<span style="background-color: #ffffff;">34'17"</span> || 32:21 || 729.22
218°45'57" || -0.65
-11'40" ||
|| 18 || 771.43
<span style="background-color: #ffffff;">231°25'43"</span> || 14:9 || 764.92
229°28'30" || 6.51
1°57'13" ||
|| 19 || 814.29
244°<span style="background-color: #ffffff;">17'9"</span> || 5:8 || 813.68
244°<span style="background-color: #ffffff;">6'21"</span> || 0.61
10'48" ||
|| 20 || 857.14
<span style="background-color: #ffffff;">257°8'34"</span> || 18:11 || 852.59
<span style="background-color: #ffffff;">258°30'29"</span> || 4.55
1°21'55" ||
|| 21 || 900
270° || 5:3 || 884.36
265°18'27" || 15.64
4°41'33" ||
|| 22 || 942.86
<span style="background-color: #ffffff;">282°51'26"</span> || 12:7 || 933.13
<span style="background-color: #ffffff;">285°46'33"</span> || 9.73
2°55'7" ||
|| 23 || 985.71
<span style="background-color: #ffffff;">295°42'51"</span> || 30:17 || 983.31
<span style="background-color: #ffffff;">296°26'4"</span> || 2.40
43'13" ||
|| 24 || 1028.57
<span style="background-color: #ffffff;">308°34'17"</span> || 20:11 || 1035.00
310°29'55" || -6.43
-1°55'38° ||
|| 25 || 1071.42
321°<span style="background-color: #ffffff;">25'43"</span> || 13:7 || 1071.70
321°<span style="background-color: #ffffff;">30'38"</span> || -0.27
-4'55" ||
|| 26 || 1114.29
334°17'9" || 40:21 || 1115.53
334°39'35" || -1.24
-22'26" ||
|| 27 || 1157.14
347°<span style="background-color: #ffffff;">8'34"</span> ||   ||   ||   ||
|| 28 || 1200
<span style="background-color: #ffffff;">360°</span> || 2:1 || 1200 || 0 ||
=<span style="background-color: #ffffff;">Rank two temperaments</span>= 

||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\28 ||   ||
|| 1 || 3\28 || [[xenharmonic/Negri|Negri]] ||
|| 1 || 5\28 || [[xenharmonic/Machine|Machine]] ||
|| 1 || 9\28 || [[xenharmonic/Würschmidt family#Worschmidt|Worschmidt]] ||
|| 1 || 11\28 ||   ||
|| 1 || 13\28 || <span style="background-color: #ffffff;">[[xenharmonic/Thuja|Thuja]]</span> ||
|| 2 || 1\28 ||   ||
|| 2 || 3\28 ||   ||
|| 2 || 5\28 || [[antikythera|Antikythera]] ||
|| 4 || 1\28 ||   ||
|| 4 || 2\28 || [[xenharmonic/Diminished#Demolished|Demolished]] ||
|| 4 || 3\28 ||   ||
|| 7 || 1\28 || [[xenharmonic/Apotome family|Whitewood]] ||
|| 14 || 1\28 ||   ||

=Commas= 
28 EDO tempers out the following [[xenharmonic/comma|comma]]s. (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= 
[[xenharmonic/machine5|machine5]]
[[xenharmonic/machine6|machine6]]
[[xenharmonic/machine11|machine11]]

=Compositions= 
[[http://www.youtube.com/watch?v=26UpCbrb3mE|28 tone Prelude]] by Kosmorksy

Original HTML content:

<html><head><title>28edo</title></head><body><!-- ws:start:WikiTextTocRule:14:&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:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#Basic properties">Basic properties</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Subgroups">Subgroups</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Table of intervals">Table of intervals</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Rank two temperaments">Rank two temperaments</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Some scales">Some scales</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: -->
<!-- ws:end:WikiTextTocRule:22 --><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 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7edo">7edo</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/14edo">14edo</a> (and of course <a class="wiki_link" href="http://xenharmonic.wikispaces.com/2edo">2edo</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/4edo">4edo</a>), has a step size of 42.857 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/cent">cent</a>s. It shares three intervals with <a class="wiki_link" href="http://xenharmonic.wikispaces.com/12edo">12edo</a>: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it <a class="wiki_link" href="http://xenharmonic.wikispaces.com/tempering%20out">tempers out</a> the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/greater%20diesis">greater diesis</a> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/648_625">648:625</a>. It does not however temper out the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/128_125">128:125</a> <a class="wiki_link" href="http://xenharmonic.wikispaces.com/lesser%20diesis">lesser diesis</a>, 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 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/9_7">9/7</a> and its inversion <a class="wiki_link" href="http://xenharmonic.wikispaces.com/14_9">14/9</a> 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 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit">7-limit</a> 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="http://xenharmonic.wikispaces.com/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="http://xenharmonic.wikispaces.com/augmented%20triad">augmented triad</a> has a very low complexity, so many of them appear in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/MOS%20scales">MOS scales</a> for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.<br />
<br />
Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39.<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 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/just%20intervals">just intervals</a>.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Step #<br />
</td>
        <td>ET Cents<br />
DMS<br />
</td>
        <td>Just Interval<br />
</td>
        <td>Just Cents<br />
DMS<br />
</td>
        <td>Difference (ET minus Just)<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>42.86<br />
12°<span style="background-color: #ffffff;">51'26&quot;</span><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>85.71<br />
<span style="background-color: #ffffff;">25°42'51&quot;</span><br />
</td>
        <td>21:20<br />
</td>
        <td>84.47<br />
<span style="background-color: #ffffff;">25°20'25&quot;</span><br />
</td>
        <td>1.24<br />
22'26&quot;<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>128.57<br />
38°<span style="background-color: #ffffff;">34'17&quot;</span><br />
</td>
        <td>14:13<br />
</td>
        <td>128.30<br />
38°29<span style="background-color: #ffffff;">'22&quot;</span><br />
</td>
        <td>0.27<br />
4'55&quot;<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>171.43<br />
<span style="background-color: #ffffff;">51°25'43&quot;</span><br />
</td>
        <td>11:10<br />
</td>
        <td>165.00<br />
<span style="background-color: #ffffff;">49°30'5&quot;</span><br />
</td>
        <td>6.43<br />
1°55'38°<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>214.29<br />
64°<span style="background-color: #ffffff;">17'9&quot;</span><br />
</td>
        <td>17:15<br />
</td>
        <td>216.69<br />
65°22&quot;<br />
</td>
        <td>-2.40<br />
-43'13&quot;<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>257.14<br />
<span style="background-color: #ffffff;">77°8'34&quot;</span><br />
</td>
        <td>7:6<br />
</td>
        <td>266.87<br />
80°3'41&quot;<br />
</td>
        <td>-9.73<br />
-2°55'7&quot;<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>300<br />
90°<br />
</td>
        <td>6:5<br />
</td>
        <td>315.64<br />
94°41'33&quot;<br />
</td>
        <td>-15.64<br />
-4°41'33&quot;<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>342.86<br />
<span style="background-color: #ffffff;">102°51'26&quot;</span><br />
</td>
        <td>11:9<br />
</td>
        <td>347.41<br />
<span style="background-color: #ffffff;">104°13'21&quot;</span><br />
</td>
        <td>-4.55<br />
-1°21'55&quot;<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>385.71<br />
115°<span style="background-color: #ffffff;">42'51&quot;</span><br />
</td>
        <td>5:4<br />
</td>
        <td>386.31<br />
115°<span style="background-color: #ffffff;">53'39&quot;</span><br />
</td>
        <td>-0.60<br />
-10'48&quot;<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>428.57<br />
<span style="background-color: #ffffff;">128°34'17&quot;</span><br />
</td>
        <td>9:7<br />
</td>
        <td>435.08<br />
130°31'30&quot;<br />
</td>
        <td>-6.51<br />
-1°57'13&quot;<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>471.43<br />
141°<span style="background-color: #ffffff;">25'43&quot;</span><br />
</td>
        <td>21:16<br />
</td>
        <td>470.78<br />
141°<span style="background-color: #ffffff;">14'3&quot;</span><br />
</td>
        <td>0.65<br />
11'40&quot;<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>514.29<br />
<span style="background-color: #ffffff;">154°17'9&quot;</span><br />
</td>
        <td>4:3<br />
</td>
        <td>498.04<br />
149°24<span style="background-color: #ffffff;">'49&quot;</span><br />
</td>
        <td>16.25<br />
4°52'20&quot;<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>557.14<br />
167°<span style="background-color: #ffffff;">8'34&quot;</span><br />
</td>
        <td>11:8<br />
</td>
        <td>551.32<br />
165°<span style="background-color: #ffffff;">23'43&quot;</span><br />
</td>
        <td>5.82<br />
1°45'9&quot;<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>600<br />
<span style="background-color: #ffffff;">180°</span><br />
</td>
        <td>7:5 10:7<br />
</td>
        <td>582.51 617.49<br />
174°45'13&quot; <span style="background-color: #ffffff;">185°14'47&quot;</span><br />
</td>
        <td>±17.49<br />
±<span style="background-color: #ffffff;">5°14'47&quot;</span><br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>642.86<br />
<span style="background-color: #ffffff;">192°51'26&quot;</span><br />
</td>
        <td>16:11<br />
</td>
        <td>648.68<br />
<span style="background-color: #ffffff;">194°36'35&quot;</span><br />
</td>
        <td>-5.82<br />
-1°45'9&quot;<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>685.71<br />
<span style="background-color: #ffffff;">205°42'51&quot;</span><br />
</td>
        <td>3:2<br />
</td>
        <td>701.96<br />
<span style="background-color: #ffffff;">210°35'11&quot;</span><br />
</td>
        <td>-16.25<br />
-4°52'20&quot;<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>728.57<br />
218°<span style="background-color: #ffffff;">34'17&quot;</span><br />
</td>
        <td>32:21<br />
</td>
        <td>729.22<br />
218°45'57&quot;<br />
</td>
        <td>-0.65<br />
-11'40&quot;<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>771.43<br />
<span style="background-color: #ffffff;">231°25'43&quot;</span><br />
</td>
        <td>14:9<br />
</td>
        <td>764.92<br />
229°28'30&quot;<br />
</td>
        <td>6.51<br />
1°57'13&quot;<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>814.29<br />
244°<span style="background-color: #ffffff;">17'9&quot;</span><br />
</td>
        <td>5:8<br />
</td>
        <td>813.68<br />
244°<span style="background-color: #ffffff;">6'21&quot;</span><br />
</td>
        <td>0.61<br />
10'48&quot;<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>857.14<br />
<span style="background-color: #ffffff;">257°8'34&quot;</span><br />
</td>
        <td>18:11<br />
</td>
        <td>852.59<br />
<span style="background-color: #ffffff;">258°30'29&quot;</span><br />
</td>
        <td>4.55<br />
1°21'55&quot;<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>900<br />
270°<br />
</td>
        <td>5:3<br />
</td>
        <td>884.36<br />
265°18'27&quot;<br />
</td>
        <td>15.64<br />
4°41'33&quot;<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>942.86<br />
<span style="background-color: #ffffff;">282°51'26&quot;</span><br />
</td>
        <td>12:7<br />
</td>
        <td>933.13<br />
<span style="background-color: #ffffff;">285°46'33&quot;</span><br />
</td>
        <td>9.73<br />
2°55'7&quot;<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>985.71<br />
<span style="background-color: #ffffff;">295°42'51&quot;</span><br />
</td>
        <td>30:17<br />
</td>
        <td>983.31<br />
<span style="background-color: #ffffff;">296°26'4&quot;</span><br />
</td>
        <td>2.40<br />
43'13&quot;<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>1028.57<br />
<span style="background-color: #ffffff;">308°34'17&quot;</span><br />
</td>
        <td>20:11<br />
</td>
        <td>1035.00<br />
310°29'55&quot;<br />
</td>
        <td>-6.43<br />
-1°55'38°<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>1071.42<br />
321°<span style="background-color: #ffffff;">25'43&quot;</span><br />
</td>
        <td>13:7<br />
</td>
        <td>1071.70<br />
321°<span style="background-color: #ffffff;">30'38&quot;</span><br />
</td>
        <td>-0.27<br />
-4'55&quot;<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>1114.29<br />
334°17'9&quot;<br />
</td>
        <td>40:21<br />
</td>
        <td>1115.53<br />
334°39'35&quot;<br />
</td>
        <td>-1.24<br />
-22'26&quot;<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>1157.14<br />
347°<span style="background-color: #ffffff;">8'34&quot;</span><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>1200<br />
<span style="background-color: #ffffff;">360°</span><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="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 --><span style="background-color: #ffffff;">Rank two temperaments</span></h1>
 <br />


<table class="wiki_table">
    <tr>
        <th>Periods<br />
per octave<br />
</th>
        <th>Generator<br />
</th>
        <th>Temperaments<br />
</th>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1\28<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>3\28<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Negri">Negri</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>5\28<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Machine">Machine</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>9\28<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/W%C3%BCrschmidt%20family#Worschmidt">Worschmidt</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>11\28<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>13\28<br />
</td>
        <td><span style="background-color: #ffffff;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Thuja">Thuja</a></span><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>1\28<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>3\28<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>5\28<br />
</td>
        <td><a class="wiki_link" href="/antikythera">Antikythera</a><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>1\28<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>2\28<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Diminished#Demolished">Demolished</a><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>3\28<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>1\28<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Apotome%20family">Whitewood</a><br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>1\28<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:8 -->Commas</h1>
 28 EDO tempers out the following <a class="wiki_link" href="http://xenharmonic.wikispaces.com/comma">comma</a>s. (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:10:&lt;h1&gt; --><h1 id="toc5"><a name="Some scales"></a><!-- ws:end:WikiTextHeadingRule:10 -->Some scales</h1>
 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/machine5">machine5</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/machine6">machine6</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/machine11">machine11</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:12 -->Compositions</h1>
 <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=26UpCbrb3mE" rel="nofollow">28 tone Prelude</a> by Kosmorksy</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:keenanpepper|keenanpepper]] and made on <tt>2012-05-04 18:41:52 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-03-28 16:38:01 UTC</tt>.<br>
-: The original revision id was <tt>330243264</tt>.<br>
+: The original revision id was <tt>545657998</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 20: / Line 20: @@
 The following table compares it to potentially useful nearby [[xenharmonic/just intervals|just intervals]].
-|| Step # || ET Cents || Just Interval || Just Cents || Difference (ET minus Just) ||
+|| Step # || ET Cents
-||   ||   ||   ||   ||   ||
+DMS || Just Interval || Just Cents
-|| 1 || 42.86 ||   ||   ||   ||
+DMS || Difference (ET minus Just) ||
-|| 2 || 85.71 || 21:20 || 84.47 || 1.24 ||
+|| 1 || 42.86
-|| 3 || 128.57 || 14:13 || 128.30 || 0.27 ||
+°&lt;span style="background-color: #ffffff;"&gt;51'26"&lt;/span&gt; ||   ||   ||   ||
-|| 4 || 171.43 || 11:10 || 165.00 || 6.43 ||
+|| 2 || 85.71
-|| 5 || 214.29 || 17:15 || 216.69 || -2.40 ||
+&lt;span style="background-color: #ffffff;"&gt;25°42'51"&lt;/span&gt; || 21:20 || 84.47
-|| 6 || 257.14 || 7:6 || 266.87 || -9.73 ||
+&lt;span style="background-color: #ffffff;"&gt;25°20'25"&lt;/span&gt; || 1.24
-|| 7 || 300 || 6:5 || 315.64 || -15.64 ||
+'26" ||
-|| 8 || 342.86 || 11:9 || 347.41 || -4.55 ||
+|| 3 || 128.57
-|| 9 || 385.71 || 5:4 || 386.31 || -0.60 ||
+°&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; || 14:13 || 128.30
-|| 10 || 428.57 || 9:7 || 435.08 || -6.51 ||
+°29&lt;span style="background-color: #ffffff;"&gt;'22"&lt;/span&gt; || 0.27
-|| 11 || 471.43 || 21:16 || 470.78 || 0.65 ||
+'55" ||
-|| 12 || 514.29 || 4:3 || 498.04 || 16.25 ||
+|| 4 || 171.43
-|| 13 || 557.14 || 11:8 || 551.32 || 5.82 ||
+&lt;span style="background-color: #ffffff;"&gt;51°25'43"&lt;/span&gt; || 11:10 || 165.00
-|| 14 || 600 || 7:5 || 582.51 || 17.49 ||
+&lt;span style="background-color: #ffffff;"&gt;49°30'5"&lt;/span&gt; || 6.43
-|| 15 || 642.86 || 16:11 || 648.68 || -5.82 ||
+°55'38° ||
-|| 16 || 685.71 || 3:2 || 701.96 || -16.25 ||
+|| 5 || 214.29
-|| 17 || 728.57 || 32:21 || 729.22 || -0.65 ||
+°&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; || 17:15 || 216.69
-|| 18 || 771.43 || 14:9 || 764.92 || 6.51 ||
+°22" || -2.40
-|| 19 || 814.29 || 5:8 || 813.68 || 0.61 ||
+-43'13" ||
-|| 20 || 857.14 || 18:11 || 852.59 || 4.55 ||
+|| 6 || 257.14
-|| 21 || 900 || 5:3 || 884.36 || 15.64 ||
+&lt;span style="background-color: #ffffff;"&gt;77°8'34"&lt;/span&gt; || 7:6 || 266.87
-|| 22 || 942.86 || 12:7 || 933.13 || 9.73 ||
+°3'41" || -9.73
-|| 23 || 985.71 || 30:17 || 983.31 || 2.40 ||
+-2°55'7" ||
-|| 24 || 1028.57 || 20:11 || 1035.00 || -6.43 ||
+|| 7 || 300
-|| 25 || 1071.42 || 13:7 || 1071.70 || -0.27 ||
+° || 6:5 || 315.64
-|| 26 || 1114.29 || 40:21 || 1115.53 || -1.24 ||
+°41'33" || -15.64
-|| 27 || 1157.14 ||   ||   ||   ||
+-4°41'33" ||
-|| 28 || 1200 || 2:1 || 1200 || 0 ||
+|| 8 || 342.86
+&lt;span style="background-color: #ffffff;"&gt;102°51'26"&lt;/span&gt; || 11:9 || 347.41
+&lt;span style="background-color: #ffffff;"&gt;104°13'21"&lt;/span&gt; || -4.55
+-1°21'55" ||
+|| 9 || 385.71
+°&lt;span style="background-color: #ffffff;"&gt;42'51"&lt;/span&gt; || 5:4 || 386.31
+°&lt;span style="background-color: #ffffff;"&gt;53'39"&lt;/span&gt; || -0.60
+-10'48" ||
+|| 10 || 428.57
+&lt;span style="background-color: #ffffff;"&gt;128°34'17"&lt;/span&gt; || 9:7 || 435.08
+°31'30" || -6.51
+-1°57'13" ||
+|| 11 || 471.43
+°&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; || 21:16 || 470.78
+°&lt;span style="background-color: #ffffff;"&gt;14'3"&lt;/span&gt; || 0.65
+'40" ||
+|| 12 || 514.29
+&lt;span style="background-color: #ffffff;"&gt;154°17'9"&lt;/span&gt; || 4:3 || 498.04
+°24&lt;span style="background-color: #ffffff;"&gt;'49"&lt;/span&gt; || 16.25
+°52'20" ||
+|| 13 || 557.14
+°&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; || 11:8 || 551.32
+°&lt;span style="background-color: #ffffff;"&gt;23'43"&lt;/span&gt; || 5.82
+°45'9" ||
+|| 14 || 600
+&lt;span style="background-color: #ffffff;"&gt;180°&lt;/span&gt; || 7:5 10:7 || 582.51 617.49
+°45'13" &lt;span style="background-color: #ffffff;"&gt;185°14'47"&lt;/span&gt; || ±17.49
+±&lt;span style="background-color: #ffffff;"&gt;5°14'47"&lt;/span&gt; ||
+|| 15 || 642.86
+&lt;span style="background-color: #ffffff;"&gt;192°51'26"&lt;/span&gt; || 16:11 || 648.68
+&lt;span style="background-color: #ffffff;"&gt;194°36'35"&lt;/span&gt; || -5.82
+-1°45'9" ||
+|| 16 || 685.71
+&lt;span style="background-color: #ffffff;"&gt;205°42'51"&lt;/span&gt; || 3:2 || 701.96
+&lt;span style="background-color: #ffffff;"&gt;210°35'11"&lt;/span&gt; || -16.25
+-4°52'20" ||
+|| 17 || 728.57
+°&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; || 32:21 || 729.22
+°45'57" || -0.65
+-11'40" ||
+|| 18 || 771.43
+&lt;span style="background-color: #ffffff;"&gt;231°25'43"&lt;/span&gt; || 14:9 || 764.92
+°28'30" || 6.51
+°57'13" ||
+|| 19 || 814.29
+°&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; || 5:8 || 813.68
+°&lt;span style="background-color: #ffffff;"&gt;6'21"&lt;/span&gt; || 0.61
+'48" ||
+|| 20 || 857.14
+&lt;span style="background-color: #ffffff;"&gt;257°8'34"&lt;/span&gt; || 18:11 || 852.59
+&lt;span style="background-color: #ffffff;"&gt;258°30'29"&lt;/span&gt; || 4.55
+°21'55" ||
+|| 21 || 900
+° || 5:3 || 884.36
+°18'27" || 15.64
+°41'33" ||
+|| 22 || 942.86
+&lt;span style="background-color: #ffffff;"&gt;282°51'26"&lt;/span&gt; || 12:7 || 933.13
+&lt;span style="background-color: #ffffff;"&gt;285°46'33"&lt;/span&gt; || 9.73
+°55'7" ||
+|| 23 || 985.71
+&lt;span style="background-color: #ffffff;"&gt;295°42'51"&lt;/span&gt; || 30:17 || 983.31
+&lt;span style="background-color: #ffffff;"&gt;296°26'4"&lt;/span&gt; || 2.40
+'13" ||
+|| 24 || 1028.57
+&lt;span style="background-color: #ffffff;"&gt;308°34'17"&lt;/span&gt; || 20:11 || 1035.00
+°29'55" || -6.43
+-1°55'38° ||
+|| 25 || 1071.42
+°&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; || 13:7 || 1071.70
+°&lt;span style="background-color: #ffffff;"&gt;30'38"&lt;/span&gt; || -0.27
+-4'55" ||
+|| 26 || 1114.29
+°17'9" || 40:21 || 1115.53
+°39'35" || -1.24
+-22'26" ||
+|| 27 || 1157.14
+°&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; ||   ||   ||   ||
+|| 28 || 1200
+&lt;span style="background-color: #ffffff;"&gt;360°&lt;/span&gt; || 2:1 || 1200 || 0 ||
 =&lt;span style="background-color: #ffffff;"&gt;Rank two temperaments&lt;/span&gt;=
@@ Line 115: / Line 194: @@
 &lt;/td&gt;
          &lt;td&gt;ET Cents&lt;br /&gt;
+DMS&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Just Interval&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Just Cents&lt;br /&gt;
+DMS&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Difference (ET minus Just)&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 139: / Line 208: @@
 &lt;/td&gt;
          &lt;td&gt;42.86&lt;br /&gt;
+°&lt;span style="background-color: #ffffff;"&gt;51'26&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 151: / Line 221: @@
 &lt;/td&gt;
          &lt;td&gt;85.71&lt;br /&gt;
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          &lt;td&gt;21:20&lt;br /&gt;
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          &lt;td&gt;84.47&lt;br /&gt;
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@@ Line 163: / Line 236: @@
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      &lt;/tr&gt;
@@ Line 175: / Line 251: @@
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          &lt;td&gt;165.00&lt;br /&gt;
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@@ Line 187: / Line 266: @@
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@@ Line 199: / Line 281: @@
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@@ Line 211: / Line 296: @@
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@@ Line 223: / Line 311: @@
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@@ Line 235: / Line 326: @@
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@@ Line 247: / Line 341: @@
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@@ Line 259: / Line 356: @@
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-         &lt;td&gt;7:5&lt;br /&gt;
+         &lt;td&gt;7:5 10:7&lt;br /&gt;
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-         &lt;td&gt;582.51&lt;br /&gt;
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@@ Line 355: / Line 476: @@
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@@ Line 463: / Line 609: @@
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          &lt;td&gt;2:1&lt;br /&gt;

28edo: Difference between revisions

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