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.
The original Pythagorean limma, 256/243, discussed at Pythagorean intervals.
The 5-limit limma, 128/125, the amount by which three just major thirds fall short of an octave.
Although closer in size to the Pythagorean apotome than to the limma, it has been so called because of its function as a diatonic semitone rather than a chromatic one.
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