rument 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
fundamental. If a certain F vibrates 100, the C, a fifth above, will
vibrate 150, if tuned so that no waves are heard; but for reasons
which will be fully explained later, the fifth cannot be tuned with
mathematical precision. On account of certain peculiarities in our
tempered scale, the fifth must always be left somewhat flatter than
perfect. This fact is always learned with some astonishment by
beginners.

In your practice on tuning the fifth, first tune it perfectly, so that
no waves are perceptible; then flat it so that there are very slow
waves; less than one per second. Some authorities say there should be
three beats in five seconds; but the tuner must learn to determine
this by his own judgment. The tempering of the fifth will be treated
exhaustively in subsequent lessons.

We advise that you confine your practice to the unison until you are
sure you have a clear conception of its peculiarities in all portions
of the key-board, except the extreme lower and upper octaves; do not
try these yet. Do not begin to practice on the octave until you are
very familiar with the beats in the unison. By gradual progress you
will avoid confusing the ear, each step being thoroughly mastered
before advancing to the next. Remember, there is nothing that is
extremely difficult in learning to tune if you but understa