Limma

The Limma or limma (Greek λεῖμμα, remnant ') referred to in the music since the Pythagorean diatonic semitone Euclid, the corresponding chromatic semitone is called apotome. The Limma is the difference between the perfect fourth and the Pythagorean major third ( Ditonus ). Its frequency ratio is the ratio of the frequency ratio 4/3 of the fourth and the frequency ratio 81/64 of Ditonus:

The fourth is the difference of an octave ( frequency ratio 2/1) and fifth ( frequency ratio 3/2), the Ditonus the sum of two whole tones ( frequency ratio 9/8), the whole tone, finally, the difference between two fifths and one octave and so the Limma also be considered as a difference of three octaves and five fifths. Example of a bottom chain according to the Pythagorean tuning: F - C - G - D - A - E - H

The E is a Limma under F and H a Limma under C.

Limma (EF) and apotome (F- sharp) add up to a greater whole tone (E - F # ). The apotome is a Pythagorean comma larger than the Limma.