L'antanairesi e la teoria armonica greca

Abstract

Antanairesis, literally «successive subtractions», is the method to-day known as Euclidean algorithm. In the Elements it is applied to numbers, for computing the GCD, and also to generic magnitudes, to determine if they are commensurable or not. The oldest example of an antanairetic procedure, described in a fragment of Philolaus, regards the construction of the musical intervals in the Pythagorean scale. These intervals ― fourth, tone, diesis, comma ― are in fact obtained from octave and fifth by successive subtractions. Since octave and fifth are incommensurable, such antanairesis is infinite; therefore does not exist an interval by which all the others can be measured (a passage from Plato's Republic is possibly a reference to that). In order to approximate this interval, the musical antanairesis has been forced to stop, an operation which corresponds to finding a convergent of a continued fraction. Those, as the Phytagoreans, who did not accept this kind of intervals, solved the problem by using two incommensurable measures. This suggests an alternative usage of the cut of an antanairesis, that we analyze in the second part of the paper.