256/243: Difference between revisions

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{{interwiki

| de = 256/243

| en = 256/243

| es =

| ja =

}}

{{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.

In musical notations that use the cycle of fifths and fourths with seven note names, such as the ups and downs notation, the limma is an important interval. The scale is structured with the following step pattern:

*A to B: whole tone

*B to C: limma

*C to D: whole tone

*D to E: whole tone

*E to F: limma

*F to G: whole tone

*G to A: whole tone

This pattern highlights the placement of the limma intervals between ~~B and C, and E and F~~, distinguishing them from the whole ~~tones~~ that occur between the other note pairs.

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 ==
@@ Line 13: / Line 19: @@
 == 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.
-In musical notations that use the cycle of fifths and fourths with seven note names, such as the ups and downs notation, the limma is an important interval. The scale is structured with the following step pattern:
-*A to B: whole tone
-*B to C: limma
-*C to D: whole tone
-*D to E: whole tone
-*E to F: limma
-*F to G: whole tone
-*G to A: whole tone
-This pattern highlights the placement of the limma intervals between B and C, and E and F, distinguishing them from the whole tones that occur between the other note pairs.
+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]]