256/243: Difference between revisions

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The '''Pythagorean limma''', or '''Pythagorean diatonic semitone''', is the interval of size 256/243 = 2<sup>8</sup>/3<sup>5</sup> (about 90.2¢), which is the diatonic semitone in Pythagorean tuning. It can be generated by stacking five [[4/3]] just perfect fourths and octave-reducing the resulting interval.
The '''Pythagorean limma''', or '''Pythagorean diatonic semitone''', is the interval of size 256/243 = 2<sup>8</sup>/3<sup>5</sup> (about 90.2¢), which is the diatonic semitone in [[Pythagorean tuning]]. It can be generated by stacking five [[4/3]] just perfect fourths and octave-reducing the resulting interval.


When this ratio is taken as a comma to be tempered, it produces [[blackwood]] temperament.
== Temperament ==
When this ratio is taken as a comma to be tempered, it produces [[blackwood]] temperament. Edos tempering it out include [[5edo]], [[10edo]], [[15edo]], [[20edo]], [[25edo]] and [[30edo]].


== See also ==
== See also ==
 
* [[243/128]] – its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[729/512]] – its [[fifth complement]]
* [[Medium commas]]
* [[Gallery of just intervals]]
* [[5edo]], [[10edo]], [[15edo]], [[20edo]], [[25edo]] and [[30edo]], which temper it out.
* [[Medium comma]]
* [[Pythagorean tuning]]
* [[53edo|4\53]] is a very good approximation of the interval
* [[53edo|4\53]] is a very good approximation of the interval



Revision as of 04:34, 13 March 2021

Interval information
Ratio 256/243
Factorization 28 × 3-5
Monzo [8 -5
Size in cents 90.225¢
Names Pythagorean limma,
Pythagorean diatonic semitone
FJS name [math]\displaystyle{ \text{m2} }[/math]
Special properties reduced,
reduced subharmonic
Tenney norm (log2 nd) 15.9248
Weil norm (log2 max(n, d)) 16
Wilson norm (sopfr(nd)) 31

[sound info]
Open this interval in xen-calc

The Pythagorean limma, or Pythagorean diatonic semitone, is the interval of size 256/243 = 28/35 (about 90.2¢), which is the diatonic semitone in Pythagorean tuning. It can be generated by stacking five 4/3 just perfect fourths and octave-reducing the resulting interval.

Temperament

When this ratio is taken as a comma to be tempered, it produces blackwood temperament. Edos tempering it out include 5edo, 10edo, 15edo, 20edo, 25edo and 30edo.

See also