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 chain-of-fifths notation 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.

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 tone 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