28edo
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[[toc|flat]] ---- =Basic properties= 28edo, a multiple of both [[7edo]] and [[14edo]] (and of course [[2edo]] and [[4edo]]), has a step size of 42.857 [[cent]]s. It shares three intervals with [[12edo]]: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it [[tempering out|tempers out]] the [[greater diesis]] [[648_625|648:625]]. It does not however temper out the [[128_125|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|9/7]] and its inversion [[14_9|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. 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 [[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 [[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= [[machine5]] [[machine6]] [[machine11]]
Original HTML content:
<html><head><title>28edo</title></head><body><!-- ws:start:WikiTextTocRule:10:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- 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:<h1> --><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="/7edo">7edo</a> and <a class="wiki_link" href="/14edo">14edo</a> (and of course <a class="wiki_link" href="/2edo">2edo</a> and <a class="wiki_link" href="/4edo">4edo</a>), has a step size of 42.857 <a class="wiki_link" href="/cent">cent</a>s. It shares three intervals with <a class="wiki_link" href="/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="/tempering%20out">tempers out</a> the <a class="wiki_link" href="/greater%20diesis">greater diesis</a> <a class="wiki_link" href="/648_625">648:625</a>. It does not however temper out the <a class="wiki_link" href="/128_125">128:125</a> <a class="wiki_link" href="/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="/9_7">9/7</a> and its inversion <a class="wiki_link" href="/14_9">14/9</a> 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 <a class="wiki_link" href="/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="/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 <a class="wiki_link" href="/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:<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 <a class="wiki_link" href="/just%20intervals">just intervals</a>.<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:<h1> --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h1>
28 EDO tempers out the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val < 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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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:<h1> --><h1 id="toc4"><a name="Some scales"></a><!-- ws:end:WikiTextHeadingRule:8 -->Some scales</h1>
<a class="wiki_link" href="/machine5">machine5</a><br />
<a class="wiki_link" href="/machine6">machine6</a><br />
<a class="wiki_link" href="/machine11">machine11</a></body></html>