256/243: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = Pythagorean limma, Pythagorean diatonic semitone, blackwood comma | |||
| Color name = sw2, sawa 2nd | |||
| Name = Pythagorean limma, | |||
| Sound = jid_256_243_pluck_adu_dr220.mp3 | | Sound = jid_256_243_pluck_adu_dr220.mp3 | ||
| | | Comma = yes | ||
}} | }} | ||
{{Wikipedia| Semitone #Pythagorean tuning }} | |||
'''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 == | |||
This interval is well approximated by any tuning generated with accurate octaves and fifths. For example, [[53edo|4\53]] is a very good approximation. | |||
== 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. | |||
== Notation == | |||
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 == | == See also == | ||
* [[243/128]] – its [[octave complement]] | * [[243/128]] – its [[octave complement]] | ||
* [[729/512]] – its [[fifth complement]] | * [[729/512]] – its [[fifth complement]] | ||
* [[16/15]] – the classic (5-limit) diatonic semitone | |||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[Medium comma]] | * [[Medium comma]] | ||
* [[Pythagorean tuning]] | * [[Pythagorean tuning]] | ||
[[Category: | [[Category:Second]] | ||
[[Category: | [[Category:Semitone]] | ||
[[Category:Blackwood]] | |||
[[Category: | [[Category:Commas named after composers]] | ||
[[Category: | |||