(Gk.: ‘remainder’).
In ancient Greek music theory the interval that remains when two whole-tone intervals are subtracted from a perfect 4th. The whole tone is the difference between a perfect 5th and a perfect 4th. The limma is also the difference between three octaves and five perfect 5ths, in other words, the diatonic semitone. Several ancient writers defined the limma, among them Ptolemy (Harmonics, i.10), Theon of Smyrna (On Mathematics Useful for the Understanding of Plato, ed. E. Hiller, Leipzig, 1878/R, 66–70), who relied on Adrastus, Gaudentius (Harmonic Introduction, ed. C. von Jan, Musici scriptores graeci, Leipzig, 1895/R, 342.7ff) and Boethius (De institutione musica, ii.28). Another, generic name for the limma in ancient Greek music was diesis (‘passing through’), although it was used to refer to a variety of smaller intervals as well (see Diesis (ii)).
In the Pythagorean theory of ratios and proportions the limma is represented by 256:243, the difference between a perfect 4th (4:3) and two whole tones (9:8 + 9:8 = 81:64). By referring to the excess of a 4th over two whole tones as a remainder and not as a semitone, the term ‘limma’ reveals its Pythagorean as opposed to its Aristoxenian nature. Aristoxenian music theory divides the octave into exactly six whole tones, the 5th equalling three and a half whole tones and the 4th two and a half. ‘Limma’ may also indicate the Greek musico-theoretical procedure of continuous subtraction (antanairēsis): an octave (2:1) minus a 5th (3:2) produces a 4th; a 5th minus a 4th produces a whole tone (an interval that Nichomachus also called a limma: Manual of Harmonics, ed. Jan, op. cit., 250.11); a 4th minus two whole tones produces the limma; the whole tone minus the limma produces the apotomē (‘cutting off’) or large semitone (2187:2048); the apotomē minus the limma produces the Pythagorean comma (531,441:524,288), the interval by which six whole tones exceed an octave (the whole tone minus two limmas also produces the comma, as Boethius noted in De institutione musica, iii.6–7).
Aristides Quintilianus (On Music, i.18) also defined the shortest rest as one limma.
ANDRÉ BARBERA