h 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.

Thus Pythagoras brought order into the music as well as
into the lives of people. But whereas it ennobled the
people, it killed the music, the one vent in life through
which unbounded utterance is possible; its essence is so
interwoven with spirituality that to tear it away and fetter
it with human mathematics is to lower it to the level of mere
utilitarianism. And so it was with Gree