256/243: Difference between revisions

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

| Comma = yes

'''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]].  

 

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.

 

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.

 

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.

[[Category:Second]]

[[Category:Semitone]]

[[Category:Blackwood]]

[[Category:Commas named after composers]]