correspond in the two strings and then antagonize. This is known in
physics as "interference of sound-waves."

~The Octave.~--When perfectly tuned, the upper tone of the octave has
exactly double the number of vibrations of the lower. If the lower
tone vibrates 1000 per second, the upper will vibrate 2000. Of course,
the ear cannot ascertain in any way the number of vibrations per
second; we use these figures for scientific demonstration only.
However, there is an instrument called the Siren which is constructed
for the purpose of ascertaining the number of vibrations per second of
any given tone, and which is delicately accurate in its work. By its
assistance we know, definitely, a great many things regarding our
musical scale of which we would otherwise be ignorant. But, while we
cannot, by the ear, ascertain these numbers, we can, by the
"interference of sound-waves" above referred to, ascertain, to the
most delicate point, when the relative vibration of two strings is
mathematically exact, if they are tuned to a unison, octave, fifth,
etc.

Practice now on tuning the octave. Find an octave in which the upper
tone is flat. Mute all but one string in the lower tone to make sure
of getting a pure tone, then select one string (the middle one if a
piano has three strings) of the upper octave and proceed to pull it up
gradually until all beats disappear. This being done, bring up the
unisons.

~The Fifth.~--In our system, when we speak of a fifth, we mean a fifth
upward. The fifth to C is G, to G is D, and so on.

The vibration of the fifth is one and a half times that of its