28edo: Difference between revisions
Wikispaces>phylingual **Imported revision 327653320 - Original comment: ** |
Wikispaces>guest **Imported revision 328779090 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:guest|guest]] and made on <tt>2012-05-02 13:35:07 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>328779090</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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=Basic properties= | =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. | 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= | =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. | 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. | Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39. | ||
=Table of intervals= | =Table of intervals= | ||
The following table compares it to potentially useful nearby [[just intervals]]. | 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 || Just Interval || Just Cents || Difference (ET minus Just) || | ||
| Line 54: | Line 54: | ||
||~ Periods | ||~ Periods | ||
per octave ||~ Generator ||~ Temperaments || | per octave ||~ Generator ||~ Temperaments || | ||
|| 1 || 3\28 || [[Negri]] || | || 1 || 3\28 || [[xenharmonic/Negri|Negri]] || | ||
|| 1 || 5\28 || [[Machine]] || | || 1 || 5\28 || [[xenharmonic/Machine|Machine]]/[[Chromatic pairs#Antikythera|Antikythera]] || | ||
|| 1 || 9\28 || [[Würschmidt family#Worschmidt|Worschmidt]] || | || 1 || 9\28 || [[xenharmonic/Würschmidt family#Worschmidt|Worschmidt]] || | ||
|| 1 || 11\28 || || | || 1 || 11\28 || || | ||
|| 1 || 13\28 || <span style="background-color: #ffffff;">[[Thuja]]</span> || | || 1 || 13\28 || <span style="background-color: #ffffff;">[[xenharmonic/Thuja|Thuja]]</span> || | ||
|| || || || | || || || || | ||
|| 4 || 1\28 || || | || 4 || 1\28 || || | ||
|| 4 || 2\28 || [[Diminished#Demolished|Demolished]] || | || 4 || 2\28 || [[xenharmonic/Diminished#Demolished|Demolished]] || | ||
|| 4 || 3\28 || || | || 4 || 3\28 || || | ||
|| 7 || 1\28 || [[xenharmonic/Apotome family|Whitewood]] || | || 7 || 1\28 || [[xenharmonic/Apotome family|Whitewood]] || | ||
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=Commas= | =Commas= | ||
28 EDO tempers out the following [[comma]]s. (Note: This assumes the val < 28 44 65 79 97 104 |.) | 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 || | ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 || | ||
||= 2187/2048 || | -11 7 > ||> 113.69 ||= Apotome ||= || | ||= 2187/2048 || | -11 7 > ||> 113.69 ||= Apotome ||= || | ||
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=Some scales= | =Some scales= | ||
[[machine5]] | [[xenharmonic/machine5|machine5]] | ||
[[machine6]] | [[xenharmonic/machine6|machine6]] | ||
[[machine11]] | [[xenharmonic/machine11|machine11]] | ||
=Compositions= | =Compositions= | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Basic properties"></a><!-- ws:end:WikiTextHeadingRule:0 -->Basic properties</h1> | <!-- 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="/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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Subgroups"></a><!-- ws:end:WikiTextHeadingRule:2 -->Subgroups</h1> | <!-- 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="/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 /> | 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 /> | <br /> | ||
Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39.<br /> | Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39.<br /> | ||
<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> | <!-- 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="/just%20intervals">just intervals</a>.<br /> | 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 /> | <br /> | ||
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<td>3\28<br /> | <td>3\28<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Negri">Negri</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Negri">Negri</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 497: | Line 497: | ||
<td>5\28<br /> | <td>5\28<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Machine">Machine</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Machine">Machine</a>/<a class="wiki_link" href="/Chromatic%20pairs#Antikythera">Antikythera</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>9\28<br /> | <td>9\28<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/W%C3%BCrschmidt%20family#Worschmidt">Worschmidt</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/W%C3%BCrschmidt%20family#Worschmidt">Worschmidt</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>13\28<br /> | <td>13\28<br /> | ||
</td> | </td> | ||
<td><span style="background-color: #ffffff;"><a class="wiki_link" href="/Thuja">Thuja</a></span><br /> | <td><span style="background-color: #ffffff;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Thuja">Thuja</a></span><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td>2\28<br /> | <td>2\28<br /> | ||
</td> | </td> | ||
<td><a class="wiki_link" href="/Diminished#Demolished">Demolished</a><br /> | <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Diminished#Demolished">Demolished</a><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:8 -->Commas</h1> | <!-- 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="/comma">comma</a>s. (Note: This assumes the val &lt; 28 44 65 79 97 104 |.)<br /> | 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 /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Some scales"></a><!-- ws:end:WikiTextHeadingRule:10 -->Some scales</h1> | <!-- 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="/machine5">machine5</a><br /> | <a class="wiki_link" href="http://xenharmonic.wikispaces.com/machine5">machine5</a><br /> | ||
<a class="wiki_link" href="/machine6">machine6</a><br /> | <a class="wiki_link" href="http://xenharmonic.wikispaces.com/machine6">machine6</a><br /> | ||
<a class="wiki_link" href="/machine11">machine11</a><br /> | <a class="wiki_link" href="http://xenharmonic.wikispaces.com/machine11">machine11</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:12 -->Compositions</h1> | <!-- 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></pre></div> | <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=26UpCbrb3mE" rel="nofollow">28 tone Prelude</a> by Kosmorksy</body></html></pre></div> | ||