256/243

Interval information

Ratio	256/243
Factorization	28 × 3-5
Monzo	[8 -5⟩
Size in cents	90.225¢
Names	Pythagorean limma, Pythagorean diatonic semitone, blackwood comma
Color name	sw2, sawa 2nd
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
Comma size	medium
S-expression	S7⋅S82
[sound info]
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Semitone #Pythagorean tuning

The interval 256/243, the Pythagorean limma, or Pythagorean diatonic semitone factors as 2⁸/3⁵, is about 90.2 cents in size, and 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.

Approximation

This interval is well approximated by any tuning generated with accurate octaves and fifths. For example, 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. 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 use the cycle of fifths and fourths along with seven note names, 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.

In musical notations that use the cycle of fifths and fourths with seven note names, such as the ups and downs notation, the limma is an important interval. The scale is structured with the following step pattern:

   A to B: whole tone
   B to C: limma
   C to D: whole tone
   D to E: whole tone
   E to F: limma
   F to G: whole tone
   G to A: whole tone

This pattern highlights the placement of the limma intervals between B and C, and E and F, distinguishing them from the whole tones that occur between the other note pairs.

See also

• 243/128 – its octave complement
• 729/512 – its fifth complement
• 16/15 – the classic (5-limit) diatonic semitone
• Gallery of just intervals
• Medium comma
• Pythagorean tuning