256/243: Difference between revisions

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{{interwiki

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| en = 256/243

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{{Infobox Interval

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

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~~The interval~~ '''256/243''', the '''Pythagorean limma''', or '''Pythagorean diatonic semitone''' factors as 28/35, is about 90.2 [[cent]]s in size~~, and is the [[diatonic semitone]] in [[Pythagorean tuning]]~~. It can be generated by stacking five [[4/3]] just perfect fourths and [[~~Octave~~ reduction|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 28/35, 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 ==

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== 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'''. 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.

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 ~~use~~ the ~~cycle~~ of fifths ~~and fourths along with seven note names~~, 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.

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 ==

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[[Category:Semitone]]

[[Category:Blackwood]]

[[Category:Commas named after composers]]

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+{{interwiki
+| de = 256/243
+| en = 256/243
+| es =
+| ja =
+}}
 {{Infobox Interval
 | Name = Pythagorean limma, Pythagorean diatonic semitone, blackwood comma
@@ Line 7: / Line 13: @@
 {{Wikipedia| Semitone #Pythagorean tuning }}
-The interval '''256/243''', the '''Pythagorean limma''', or '''Pythagorean diatonic semitone''' factors as 2<sup>8</sup>/3<sup>5</sup>, is about 90.2 [[cent]]s in size, and is the [[diatonic semitone]] in [[Pythagorean tuning]]. It can be generated by stacking five [[4/3]] just perfect fourths and [[Octave reduction|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]].
 == Approximation ==
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 == 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'''. 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.
+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 use the cycle of fifths and fourths along with seven note names, 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.
+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 ==
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 [[Category:Semitone]]
 [[Category:Blackwood]]
+[[Category:Commas named after composers]]