28edo: Difference between revisions

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<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:~~hstraub~~|~~hstraub~~]] and made on <tt>2011-06-~~22 05~~:22:12 UTC</tt>.<br>

: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-30 16:18:29 UTC</tt>.<br>

: The original revision id was <tt>~~238134791~~</tt>.<br>

: The original revision id was <tt>239550315</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>

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=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.

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.

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.

The following table compares it to potentially useful nearby [[just intervals]].

|| Step # || ET Cents || Just Interval || Just Cents || Difference (ET minus Just) ||

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|| 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 |.)

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 ||= ||

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<br />

<h1 id="toc0"><a name="Basic properties"></a>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 />

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 />

<h1 id="toc1"><a name="Subgroups"></a>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 />

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 />

<h1 id="toc2"><a name="Table of intervals"></a>Table of intervals</h1>

The following table compares it to potentially useful nearby just intervals.<br />

The following table compares it to potentially useful nearby <a class="wiki_link" href="/just%20intervals">just intervals</a>.<br />

<br />

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<h1 id="toc3"><a name="Commas"></a>Commas</h1>

28 EDO tempers out the following ~~commas~~. (Note: This assumes the val &lt; 28 44 65 79 97 104 |.)<br />

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 />

@@ 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:hstraub|hstraub]] and made on <tt>2011-06-22 05:22:12 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-30 16:18:29 UTC</tt>.<br>
-: The original revision id was <tt>238134791</tt>.<br>
+: The original revision id was <tt>239550315</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 10: / Line 10: @@
 =Basic properties=
-edo, 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.
+edo, 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=
-edo 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.
+edo 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.
+The following table compares it to potentially useful nearby [[just intervals]].
 || Step # || ET Cents || Just Interval || Just Cents || Difference (ET minus Just) ||
@@ Line 49: / Line 49: @@
 || 28 || 1200 || 2:1 || 1200 || 0 ||
 =Commas=
-EDO tempers out the following commas. (Note: This assumes the val &lt; 28 44 65 79 97 104 |.)
+EDO tempers out the following [[comma]]s. (Note: This assumes the val &lt; 28 44 65 79 97 104 |.)
 ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
 ||= 2187/2048 || | -11 7 &gt; ||&gt; 113.69 ||= Apotome ||=   ||
@@ Line 74: / Line 74: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Basic properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Basic properties&lt;/h1&gt;
-edo, 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.&lt;br /&gt;
+edo, a multiple of both &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; and &lt;a class="wiki_link" href="/14edo"&gt;14edo&lt;/a&gt; (and of course &lt;a class="wiki_link" href="/2edo"&gt;2edo&lt;/a&gt; and &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt;), has a step size of 42.857 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s. It shares three intervals with &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the &lt;a class="wiki_link" href="/greater%20diesis"&gt;greater diesis&lt;/a&gt; &lt;a class="wiki_link" href="/648_625"&gt;648:625&lt;/a&gt;. It does not however temper out the &lt;a class="wiki_link" href="/128_125"&gt;128:125&lt;/a&gt; &lt;a class="wiki_link" href="/lesser%20diesis"&gt;lesser diesis&lt;/a&gt;, 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 &lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt; and its inversion &lt;a class="wiki_link" href="/14_9"&gt;14/9&lt;/a&gt; are also found in 14edo.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Subgroups"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Subgroups&lt;/h1&gt;
-edo 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 &lt;a class="wiki_link" href="/Semicomma%20family"&gt;orwell temperament&lt;/a&gt; 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 &lt;a class="wiki_link" href="/augmented%20triad"&gt;augmented triad&lt;/a&gt; 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.&lt;br /&gt;
+edo can approximate the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to &lt;a class="wiki_link" href="/Semicomma%20family"&gt;orwell temperament&lt;/a&gt; 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 &lt;a class="wiki_link" href="/augmented%20triad"&gt;augmented triad&lt;/a&gt; has a very low complexity, so many of them appear in the &lt;a class="wiki_link" href="/MOS%20scales"&gt;MOS scales&lt;/a&gt; for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Table of intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Table of intervals&lt;/h1&gt;
-  The following table compares it to potentially useful nearby just intervals.&lt;br /&gt;
+  The following table compares it to potentially useful nearby &lt;a class="wiki_link" href="/just%20intervals"&gt;just intervals&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 448: / Line 448: @@
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Commas&lt;/h1&gt;
-EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 28 44 65 79 97 104 |.)&lt;br /&gt;
+EDO tempers out the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the val &amp;lt; 28 44 65 79 97 104 |.)&lt;br /&gt;