81/80: 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:

: This revision was by author [[User:~~Gedankenwelt~~|~~Gedankenwelt~~]] and made on <tt>~~2015~~-01-26 ~~18:07~~:35 UTC</tt>.

: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-12 04:26:04 UTC</tt>.

: The original revision id was <tt>~~538626006~~</tt>.

: The original revision id was <tt>589239734</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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21.506290 cents

The **syntonic** or **Didymus comma** (frequency ratio **81/80**) is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16_15|16/15]], [[625_624|625/624]], [[2401_2400|2401/2400]] and [[4096_4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10_9|10/9]] and [[9_8|9/8]], the product of which is the just major third, [[5_4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.

The **syntonic** or **Didymus comma** (frequency ratio **81/80**) is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16_15|16/15]], [[625_624|625/624]], [[2401_2400|2401/2400]] and [[4096_4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10_9|10/9]] and [[9_8|9/8]], the product of which is the just major third, [[5_4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. [[55edo]] tempers it out, while [[15edo]] does not.

Tempering out 81/80 gives a tuning for the [[tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]].

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21.506290 cents

The syntonic or Didymus comma (frequency ratio 81/80) is the smallest <a class="wiki_link" href="/superparticular">superparticular interval</a> which belongs to the <a class="wiki_link" href="/5-limit">5-limit</a>. Like <a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/625_624">625/624</a>, <a class="wiki_link" href="/2401_2400">2401/2400</a> and <a class="wiki_link" href="/4096_4095">4096/4095</a> it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between <a class="wiki_link" href="/10_9">10/9</a> and <a class="wiki_link" href="/9_8">9/8</a>, the product of which is the just major third, <a class="wiki_link" href="/5_4">5/4</a>. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.

The syntonic or Didymus comma (frequency ratio 81/80) is the smallest <a class="wiki_link" href="/superparticular">superparticular interval</a> which belongs to the <a class="wiki_link" href="/5-limit">5-limit</a>. Like <a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/625_624">625/624</a>, <a class="wiki_link" href="/2401_2400">2401/2400</a> and <a class="wiki_link" href="/4096_4095">4096/4095</a> it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between <a class="wiki_link" href="/10_9">10/9</a> and <a class="wiki_link" href="/9_8">9/8</a>, the product of which is the just major third, <a class="wiki_link" href="/5_4">5/4</a>. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. <a class="wiki_link" href="/55edo">55edo</a> tempers it out, while <a class="wiki_link" href="/15edo">15edo</a> does not.

Tempering out 81/80 gives a tuning for the <a class="wiki_link" href="/tone">whole tone</a> which is intermediate between 10/9 and 9/8, and leads to <a class="wiki_link" href="/Meantone%20family">meantone temperament</a>.

@@ 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:Gedankenwelt|Gedankenwelt]] and made on <tt>2015-01-26 18:07:35 UTC</tt>.<br>
+: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-12 04:26:04 UTC</tt>.<br>
-: The original revision id was <tt>538626006</tt>.<br>
+: The original revision id was <tt>589239734</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: @@
 .506290 cents
-The **syntonic** or **Didymus comma** (frequency ratio **81/80**) is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16_15|16/15]], [[625_624|625/624]], [[2401_2400|2401/2400]] and [[4096_4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10_9|10/9]] and [[9_8|9/8]], the product of which is the just major third, [[5_4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.
+The **syntonic** or **Didymus comma** (frequency ratio **81/80**) is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16_15|16/15]], [[625_624|625/624]], [[2401_2400|2401/2400]] and [[4096_4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10_9|10/9]] and [[9_8|9/8]], the product of which is the just major third, [[5_4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. [[55edo]] tempers it out, while [[15edo]] does not.
 Tempering out 81/80 gives a tuning for the [[tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]].
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 .506290 cents&lt;br /&gt;
 &lt;br /&gt;
-The &lt;strong&gt;syntonic&lt;/strong&gt; or &lt;strong&gt;Didymus comma&lt;/strong&gt; (frequency ratio &lt;strong&gt;81/80&lt;/strong&gt;) is the smallest &lt;a class="wiki_link" href="/superparticular"&gt;superparticular interval&lt;/a&gt; which belongs to the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;. Like &lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, &lt;a class="wiki_link" href="/625_624"&gt;625/624&lt;/a&gt;, &lt;a class="wiki_link" href="/2401_2400"&gt;2401/2400&lt;/a&gt; and &lt;a class="wiki_link" href="/4096_4095"&gt;4096/4095&lt;/a&gt; it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt; and &lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, the product of which is the just major third, &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.&lt;br /&gt;
+The &lt;strong&gt;syntonic&lt;/strong&gt; or &lt;strong&gt;Didymus comma&lt;/strong&gt; (frequency ratio &lt;strong&gt;81/80&lt;/strong&gt;) is the smallest &lt;a class="wiki_link" href="/superparticular"&gt;superparticular interval&lt;/a&gt; which belongs to the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;. Like &lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, &lt;a class="wiki_link" href="/625_624"&gt;625/624&lt;/a&gt;, &lt;a class="wiki_link" href="/2401_2400"&gt;2401/2400&lt;/a&gt; and &lt;a class="wiki_link" href="/4096_4095"&gt;4096/4095&lt;/a&gt; it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt; and &lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, the product of which is the just major third, &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. &lt;a class="wiki_link" href="/55edo"&gt;55edo&lt;/a&gt; tempers it out, while &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt; does not.&lt;br /&gt;
 &lt;br /&gt;
 Tempering out 81/80 gives a tuning for the &lt;a class="wiki_link" href="/tone"&gt;whole tone&lt;/a&gt; which is intermediate between 10/9 and 9/8, and leads to &lt;a class="wiki_link" href="/Meantone%20family"&gt;meantone temperament&lt;/a&gt;.&lt;br /&gt;