Epimoric Ratios and Greek Musical Theory

Abstract

The discovery to which the historian of mathematics Eric Temple Bell refers consisted in observing that the three fundamental kinds of consonances, the octave (for instance, the interval C-c),the fifth (as the interval C-G) and the fourth (as the interval C-F) were obtained by plunking two sections of a string whose lengths were in the ratio of 2 to 1 for the octave, of 3 to 2 for the fifth and of 4 to 3 for the fourth. So, the three most natural consonances could be expressed by means of the numbers 1, 2, 3 and 4 (the numbers which form the tetraktys of the decad!), and the following figure (in which 6, 8, 9, 12 are the smallest numbers producing these ratios) became the manifesto of the Greek music theory. (We observe that, if we set the numbers in incresing order and we consider them as representing the lenghts of the strings, then the pitches of the corresponding notes are decreasing. And actually the Greek scales were descending.)

If one landmark overtops all others in the evolution of science, it is the discovery by Pythagoras of the connection between musical harmonies and numbers.