# Limma

In musical harmony, a**limma**is the interval whose ratio is 256:243 = 2

^{8}/3

^{5}. 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

**Limma**can also refer to the ratio 135:128, which may occur in a 5-limit tuning system.