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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
- <math> {2^3 \over (3/2)^5} = {2^3 \cdot 2^5 \over 3^5} = {2^8 \over 3^5} = {256 \over 243}. <math>
Limma can also refer to the ratio 135:128, which may occur in a 5-limit tuning system.
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