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 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 understand what has to be done, go about it systematically, and have plenty of patience.

In this lesson we give you our system of setting the temperament; that is, the succession in which the different tones of the temperament are tuned.  We advise, however, that you do not attempt to set a temperament until after studying Lesson IX, which enters into the theory of temperament, testing, etc.