He gave the Greeks their first scientific analysis of sound.  The legend runs that, passing a blacksmith’s shop and hearing the different sounds of the hammering, he conceived the idea that sounds could be measured by some such means as weight is measured by scales, or distance by the foot rule.  By weighing the different hammers, so the story goes, he obtained the knowledge of harmonics or overtones, namely, the fundamental, octave, fifth, third, etc.  This legend, which is stated seriously in many histories of music, is absurd, for, as we know, the hammers would not have vibrated.  The anvils would have given the sound, but in order to produce the octave, fifth, etc., they would have had to be of enormous proportions.  On the other hand, the monochord, with which students in physics are familiar, was his invention; and the first mathematical demonstrations of the effect on musical pitch of length of cord and tension, as well as the length of pipes and force of breath, were his.

These mathematical divisions of the monochord, however, eventually did more to stifle music for a full thousand years than can easily be imagined.  This division of the string made what we call harmony impossible; for by it the major third became a larger interval than our modern one, and the minor third smaller.  Thus thirds did not sound well together, in fact were dissonances, the only intervals which did harmonize being the fourth, fifth, and octave.  This system of mathematically dividing tones into equal parts held good up to the middle of the sixteenth century, when Zarlino, who died in 1590, invented the system in use at the present time, called the tempered scale, which, however, did not come into general use until one hundred years later.

Aristoxenus, a pupil of Aristotle, who lived more than a century after Pythagoras, rejected the monochord as a means for gauging musical sounds, believing that the ear, not mathematical calculation, should be the judge as to which interval sounds “perfect.”  But he was unable to formulate a system that would bring the third (and naturally its inversion the sixth) among the harmonizing intervals or consonants.  Didymus (about 30 B.C.) first discovered that two different-sized whole tones were necessary in order to make the third consonant; and Ptolemy (120 A.D.) improved on this system somewhat.  But the new theory remained without any practical effect until nearly the seventeenth century, when the long respected theory of the perfection of mathematical calculation on the basis of natural phenomena was overthrown in favour of actual effect.  If Aristoxenus had had followers able to combat the crushing influence of Euclid and his school, music might have grown up with the other arts.  As it is, music is still in its infancy, and has hardly left its experimental stage.