256/243: Difference between revisions

(6 intermediate revisions by 5 users not shown)

Line 1:

{{interwiki

| de = 256/243

| en = 256/243

| es =

| ja =

}}

{{Infobox Interval

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

Line 7:

Line 13:

~~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 ==

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''', 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 ==

Line 39:

Line 45:

[[Category:Semitone]]

[[Category:Blackwood]]

~~[[Category:Commas named after polymaths]]~~

~~[[Category:Commas named after controversial individuals]]~~

[[Category:Commas named after composers]]

@@ Line 1: / Line 1: @@
+{{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''', 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 ==
@@ Line 39: / Line 45: @@
 [[Category:Semitone]]
 [[Category:Blackwood]]
-[[Category:Commas named after polymaths]]
-[[Category:Commas named after controversial individuals]]
 [[Category:Commas named after composers]]