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:genewardsmith|genewardsmith~~]] ~~and made on~~ <tt>~~2011-08-08 21:56:24 UTC~~</tt>.<br>

| en = 256/243

~~: The original revision id was~~ <tt>~~244949379<~~/~~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">The Pythagorean limma~~, ~~or Pythagorean semitone~~, is the ~~interval of size 256/243 = 2^8/3^5~~, ~~which~~ is the ~~diatonic semitone in Pythagorean tuning. <~~/~~pre><~~/~~div>~~

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

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

| Color name = sw2, sawa 2nd

~~<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>The Pythagorean~~ limma, ~~or Pythagorean semitone, is~~ the ~~interval of size 256~~/243 ~~= 2^8~~/3^5~~, which is the~~ diatonic semitone in Pythagorean tuning~~.</body></html></pre></div>~~

| Sound = jid_256_243_pluck_adu_dr220.mp3

| Comma = yes

}}

'''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]].

== Approximation ==

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

== Temperaments ==

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.

== Notation ==

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.

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]]

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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:genewardsmith|genewardsmith]] and made on <tt>2011-08-08 21:56:24 UTC</tt>.<br>
+| en = 256/243
-: The original revision id was <tt>244949379</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">The Pythagorean limma, or Pythagorean semitone, is the interval of size 256/243 = 2^8/3^5, which is the diatonic semitone in Pythagorean tuning. </pre></div>
+| Name = Pythagorean limma, Pythagorean diatonic semitone, blackwood comma
-<h4>Original HTML content:</h4>
+| Color name = sw2, sawa 2nd
-<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;The Pythagorean limma, or Pythagorean semitone, is the interval of size 256/243 = 2^8/3^5, which is the diatonic semitone in Pythagorean tuning.&lt;/body&gt;&lt;/html&gt;</pre></div>
+| Sound = jid_256_243_pluck_adu_dr220.mp3
+| Comma = yes
+}}
+{{Wikipedia| Semitone #Pythagorean tuning }}
+'''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]].
+== Approximation ==
+This interval is well approximated by any tuning generated with accurate octaves and fifths. For example, [[53edo|4\53]] is a very good approximation.
+== Temperaments ==
+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.
+== Notation ==
+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.
+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]]