28edo
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2011-03-28 23:14:59 UTC.
- The original revision id was 214918260.
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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 || || || || || 12 || 514.29 || 4:3 || 498.04 || 16.25 || || 13 || 557.14 || || || || || 14 || 600 || 7:5 || 582.51 || 17.49 || || 15 || 642.86 || || || || || 16 || 685.71 || 3:2 || 701.96 || -16.25 || || 17 || 728.57 || || || || || 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:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- 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 />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><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:<h1> --><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:<h1> --><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><br />
</td>
<td><br />
</td>
<td><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><br />
</td>
<td><br />
</td>
<td><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><br />
</td>
<td><br />
</td>
<td><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><br />
</td>
<td><br />
</td>
<td><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>