256/243: Difference between revisions

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}}

{{Infobox Interval

| Name = Pythagorean limma, Pythagorean diatonic semitone~~, 3-limit minor 2nd~~, blackwood comma

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

| Color name = sw2, sawa 2nd

| Sound = jid_256_243_pluck_adu_dr220.mp3

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~~The interval~~ '''256/243''', the '''Pythagorean limma''' or '''Pythagorean diatonic semitone''' or the ~~'''~~3-limit minor ~~2nd'''~~ 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, or equivalently by decreasing 4/3 by two [[9/8]] major seconds.

'''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''', 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.

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 ==

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 }}
 {{Infobox Interval
-| Name = Pythagorean limma, Pythagorean diatonic semitone, 3-limit minor 2nd, blackwood comma
+| Name = Pythagorean limma, Pythagorean diatonic semitone, blackwood comma
 | Color name = sw2, sawa 2nd
 | Sound = jid_256_243_pluck_adu_dr220.mp3
@@ Line 13: / Line 13: @@
 {{Wikipedia| Semitone #Pythagorean tuning }}
-The interval '''256/243''', the '''Pythagorean limma''' or '''Pythagorean diatonic semitone''' or the '''3-limit minor 2nd''' 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, or equivalently by decreasing 4/3 by two [[9/8]] major seconds.
+'''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 19: / 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''', 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.
+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 ==