Limma

In musical harmony, a limma is the interval whose ratio is 256:243 = 2⁸/3⁵. It is the semitone of the
3-limit Pythagorean diatonic scale (see Pythagorean tuning).

It is derived by moving up three octaves, then moving down five perfect fifths. An octave has 12 semitones, and a perfect fifth has 7 semitones, so moving up three octaves equals moving up 3x12 = 36 semitones, and moving down five fifth equals moving down 5x7 = 35 semitones. Moving up three octaves and moving down five fifths equals 36 − 35 = 1 semitone.

Now consider the intervals as ratios of frequencies. The octave has ratio 2:1, and the perfect fifth has ratio 3:2 (especially in a Pythagorean scale). Therefore a semitone equals

${2^{3} \over (3/2)^{5}}={2^{3}\cdot 2^{5} \over 3^{5}}={2^{8} \over 3^{5}}={256 \over 243}.$