256/243: Difference between revisions

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**Imported revision 282665078 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-12-05 21:36:19 UTC</tt>.<br>
| en = 256/243
: The original revision id was <tt>282665078</tt>.<br>
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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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<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 (about 90.225¢), 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.
| Name = Pythagorean limma, Pythagorean diatonic semitone, blackwood comma
| Color name = sw2, sawa 2nd
| Sound = jid_256_243_pluck_adu_dr220.mp3
| Comma = yes
}}
{{Wikipedia| Semitone #Pythagorean tuning }}


See: [[Gallery of Just Intervals]], [[comma]]</pre></div>
'''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]].
<h4>Original HTML content:</h4>
 
<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 (about 90.225¢), 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;
== Approximation ==
&lt;br /&gt;
This interval is well approximated by any tuning generated with accurate octaves and fifths. For example, [[53edo|4\53]] is a very good approximation.
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>
 
== 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]]

Latest revision as of 16:36, 27 November 2025

Interval information
Ratio 256/243
Factorization 28 × 3-5
Monzo [8 -5⟩
Size in cents 90.225¢
Names Pythagorean limma,
Pythagorean diatonic semitone,
blackwood comma
Color name sw2, sawa 2nd
FJS name [math]\displaystyle{ \text{m2} }[/math]
Special properties reduced,
reduced subharmonic
Tenney norm (log2 nd) 15.9248
Weil norm (log2 max(n, d)) 16
Wilson norm (sopfr(nd)) 31
Comma size medium
S-expression S7⋅S82

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

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 28/35, and is about 90.2 cents in size. It can be generated by stacking five 4/3 just perfect fourths and 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, 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 diatonic 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:

This pattern highlights the placement of the limma intervals between the note pairs above, distinguishing them from the whole tone that occur between the other note pairs.

See also

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