256/243: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES<~~/~~h2>~~

{{interwiki

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

| de = 256/243

~~: This revision was by author [[User:spt3125~~|~~spt3125]] and made on <tt>2014-06-30 20:59:28 UTC</tt>.<br>~~

| en = 256/243

~~: The original revision id was <tt>515317390</tt>.<br>~~

| es =

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

| ja =

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

}}

~~<h4>Original Wikitext content:</h4>~~

{{Infobox Interval

~~<div style~~="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">**256/243**

| Name = Pythagorean limma, Pythagorean diatonic semitone, blackwood comma

|~~8 -5>~~

| Color name = sw2, sawa 2nd

~~90.225 cents~~

| Sound = jid_256_243_pluck_adu_dr220.mp3

~~[[media type~~=~~"file" key="jid_256_243_pluck_adu_dr220.mp3"]] [[file:xenharmonic/~~jid_256_243_pluck_adu_dr220.mp3|~~sound sample]]~~

| Comma = yes

}}

~~The~~ Pythagorean limma, or Pythagorean semitone, is the ~~interval of size 256/243 =~~ 2^8/3^5 (about 90.~~2¢), which is the diatonic semitone~~ in ~~Pythagorean tuning~~. It can be generated by stacking five [[~~4_3|~~4/3]] just perfect fourths and octave-reducing the resulting interval.

'''256/243''', the '''Pythagorean limma''' or '''Pythagorean diatonic semitone''', is the [[diatonic semitone]] in [[Pythagorean tuning]]. In other words, it is the [[3-limit]] minor second. It factors as 2<sup>8</sup>/3<sup>5</sup>, and is about 90.2 [[cent]]s in size. It can be generated by stacking five [[4/3]] just perfect fourths and [[octave reduction|octave-reducing]] the resulting interval, or equivalently by decreasing 4/3 by two [[9/8]] major seconds. Unlike the situation in [[meantone]] tunings, it is smaller, not larger, than the corresponding [[chromatic semitone]], which is the Pythagorean augmented unison of [[2187/2048]].

~~See: [[Gallery of Just Intervals]]~~, [[~~comma~~]]~~</pre></div>~~

== Approximation ==

~~<h4>Original HTML content:</h4>~~

This interval is well approximated by any tuning generated with accurate octaves and fifths. For example, [[53edo|4\53]] is a very good approximation.

~~<div style~~=~~"width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style~~=~~"margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class~~=~~"old-revision-html"><html><head><title>256_243</title></head><body><strong>256/243</strong><br />~~

~~|8 -~~5~~&gt;<br />~~

== Temperaments ==

~~90.225 cents<br />~~

When this ratio is taken as a comma to be tempered in the [[5-limit]], it produces the [[blackwood]] temperament, and it may be called the '''blackwood comma''', named after [[Easley Blackwood Jr]]. Edos tempering it out include [[5edo]], [[10edo]], [[15edo]], [[20edo]], [[25edo]] and [[30edo]]. See [[Limmic temperaments]] for a number of other temperaments where it is tempered out.

~~<!~~-~~- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www~~.~~wikispaces~~.~~com/site/embedthumbnail/file-audio/jid_256_243_pluck_adu_dr220~~.~~mp3?h~~=~~20&amp;w~~=~~240&quot; class~~=~~&quot;WikiMedia WikiMediaFile&quot; id~~=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;jid_256_243_pluck_adu_dr220.mp3&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --~~><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x~~-~~shockwave~~-~~flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic~~.wikispaces.com%252Ffile%252Fview%252Fjid_256_243_pluck_adu_dr220.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed> <a href="http~~://~~xenharmonic.wikispaces.com~~/~~file/view/jid_256_243_pluck_adu_dr220.mp3/515315614/jid_256_243_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http~~:/~~/xenharmonic.wikispaces.com/file/view/jid_256_243_pluck_adu_dr220.mp3/515315614/jid_256_243_pluck_adu_dr220.mp3');">sound sample</a><br />

~~<br~~ /~~>~~

== Notation ==

~~The Pythagorean~~ limma, ~~or Pythagorean semitone, is~~ the ~~interval of size 256~~/~~243 = 2^~~8~~/3^5 (about 90.2¢), which is~~ the ~~diatonic semitone in Pythagorean tuning~~. ~~It can be generated by stacking five <a class~~=~~"wiki_link" href~~="/~~4_3">4/3</a> just perfect fourths and~~ octave~~-reducing the resulting interval.<br~~ /~~>~~

In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the limma is represented by the distances between B and C, as well as between E and F.

~~<br~~ /~~>~~

~~See: <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">~~Gallery of ~~Just Intervals</a>, <a class="wiki_link" href="/~~comma~~">comma</a></body></html></pre></div>~~

The scale is structured with the following step pattern:

* A to B: [[9/8|whole tone]]

* B to C: [[256/243|limma]]

* C to D: [[9/8|whole tone]]

* D to E: [[9/8|whole tone]]

* E to F: [[256/243|limma]]

* F to G: [[9/8|whole tone]]

* G to A: [[9/8|whole tone]]

This pattern highlights the placement of the limma intervals between the note pairs above, distinguishing them from the [[9/8|whole tone]] that occur between the other note pairs.

== See also ==

* [[243/128]] – its [[octave complement]]

* [[729/512]] – its [[fifth complement]]

* [[16/15]] – the classic (5-limit) diatonic semitone

* [[Gallery of just intervals]]

* [[Medium comma]]

* [[Pythagorean tuning]]

[[Category:Second]]

[[Category:Semitone]]

[[Category:Blackwood]]

[[Category:Commas named after composers]]

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{interwiki
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+| de = 256/243
-: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-06-30 20:59:28 UTC</tt>.<br>
+| en = 256/243
-: The original revision id was <tt>515317390</tt>.<br>
+| es =
-: The revision comment was: <tt></tt><br>
+| ja =
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
+}}
-<h4>Original Wikitext content:</h4>
+{{Infobox Interval
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">**256/243**
+| Name = Pythagorean limma, Pythagorean diatonic semitone, blackwood comma
-|8 -5&gt;
+| Color name = sw2, sawa 2nd
-.225 cents
+| Sound = jid_256_243_pluck_adu_dr220.mp3
-[[media type="file" key="jid_256_243_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_256_243_pluck_adu_dr220.mp3|sound sample]]
+| Comma = yes
+}}
+{{Wikipedia| Semitone #Pythagorean tuning }}
-The Pythagorean limma, or Pythagorean semitone, is the interval of size 256/243 = 2^8/3^5 (about 90.2¢), which is the diatonic semitone in Pythagorean tuning. It can be generated by stacking five [[4_3|4/3]] just perfect fourths and octave-reducing the resulting interval.
+'''256/243''', the '''Pythagorean limma''' or '''Pythagorean diatonic semitone''', is the [[diatonic semitone]] in [[Pythagorean tuning]]. In other words, it is the [[3-limit]] minor second. It factors as 2<sup>8</sup>/3<sup>5</sup>, and is about 90.2 [[cent]]s in size. It can be generated by stacking five [[4/3]] just perfect fourths and [[octave reduction|octave-reducing]] the resulting interval, or equivalently by decreasing 4/3 by two [[9/8]] major seconds. Unlike the situation in [[meantone]] tunings, it is smaller, not larger, than the corresponding [[chromatic semitone]], which is the Pythagorean augmented unison of [[2187/2048]].
-See: [[Gallery of Just Intervals]], [[comma]]</pre></div>
+== Approximation ==
-<h4>Original HTML content:</h4>
+This interval is well approximated by any tuning generated with accurate octaves and fifths. For example, [[53edo|4\53]] is a very good approximation.
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;256_243&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;256/243&lt;/strong&gt;&lt;br /&gt;
-|8 -5&amp;gt;&lt;br /&gt;
+== Temperaments ==
-.225 cents&lt;br /&gt;
+When this ratio is taken as a comma to be tempered in the [[5-limit]], it produces the [[blackwood]] temperament, and it may be called the '''blackwood comma''', named after [[Easley Blackwood Jr]]. Edos tempering it out include [[5edo]], [[10edo]], [[15edo]], [[20edo]], [[25edo]] and [[30edo]]. See [[Limmic temperaments]] for a number of other temperaments where it is tempered out.
-&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_256_243_pluck_adu_dr220.mp3?h=20&amp;amp;w=240&amp;quot; class=&amp;quot;WikiMedia WikiMediaFile&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;file&amp;amp;quot; key=&amp;amp;quot;jid_256_243_pluck_adu_dr220.mp3&amp;amp;quot;&amp;quot; title=&amp;quot;Local Media File&amp;quot;height=&amp;quot;20&amp;quot; width=&amp;quot;240&amp;quot;/&amp;gt; --&gt;&lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_256_243_pluck_adu_dr220.mp3?file_extension=mp3&amp;autostart=false&amp;repeat=false&amp;showdigits=true&amp;showfsbutton=false&amp;width=240&amp;height=20"&gt;&lt;/embed&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt; &lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_256_243_pluck_adu_dr220.mp3/515315614/jid_256_243_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_256_243_pluck_adu_dr220.mp3/515315614/jid_256_243_pluck_adu_dr220.mp3');"&gt;sound sample&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
+== Notation ==
-The Pythagorean limma, or Pythagorean semitone, is the interval of size 256/243 = 2^8/3^5 (about 90.2¢), which is the diatonic semitone in Pythagorean tuning. It can be generated by stacking five &lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt; just perfect fourths and octave-reducing the resulting interval.&lt;br /&gt;
+In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the limma is represented by the distances between B and C, as well as between E and F.
-&lt;br /&gt;
-See: &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Gallery of Just Intervals&lt;/a&gt;, &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+The scale is structured with the following step pattern:
+* A to B: [[9/8|whole tone]]
+* B to C: [[256/243|limma]]
+* C to D: [[9/8|whole tone]]
+* D to E: [[9/8|whole tone]]
+* E to F: [[256/243|limma]]
+* F to G: [[9/8|whole tone]]
+* G to A: [[9/8|whole tone]]
+This pattern highlights the placement of the limma intervals between the note pairs above, distinguishing them from the [[9/8|whole tone]] that occur between the other note pairs.
+== See also ==
+* [[243/128]] – its [[octave complement]]
+* [[729/512]] – its [[fifth complement]]
+* [[16/15]] – the classic (5-limit) diatonic semitone
+* [[Gallery of just intervals]]
+* [[Medium comma]]
+* [[Pythagorean tuning]]
+[[Category:Second]]
+[[Category:Semitone]]
+[[Category:Blackwood]]
+[[Category:Commas named after composers]]