the to-and-fro swing.  You can easily make one by tying any little heavy article to a string of either of these lengths.  Measure from the center of such heavy article to the point of contact of the string at the top with some stationary object.  This is a sure guide.  Set the pendulum swinging and count the vibrations and you will soon become quite infallible.  Having acquired the ability to judge a second of time you can go to work with more confidence.

Now, as a matter of fact, in a scale which is equally tempered, no two fifths beat exactly alike, as the lower a fifth, the slower it should beat, and thus the fifths in the bass are hardly perceptibly flat, while those in the treble beat more rapidly.  For example, if a certain fifth beat once a second, the fifth an octave higher will beat twice a second, and one that is two octaves higher will beat four times a second, and so on, doubling the number of beats with each ascending octave.

In a subsequent lesson, in which we give the mathematics of the temperament, these various ratios will be found accurately figured out; but for the present let us notice the difference between the actual tempered scale and the exact mathematical scale in the point of the flattening of the fifth.  Take for example 1C, and for convenience of figuring, say it vibrates 128 per second.  The relation of a fundamental to its fifth is that of 2 to 3.  So if 128 is represented as 2, we think of it as 2 times 64.  Then with another 64 added, we have 192, which represents 3.  In other words, a fundamental has just two-thirds of the number of vibrations per second that its fifth has, in the exact scale.  This would mean a fifth in which there would be no beats.  Now in the tempered scale we find that G vibrates 191.78 instead of 192; so we can easily see how much variation from the mathematical standard there is in this portion of the instrument.  It is only about a fourth of a vibration.  This would mean that, in this fifth we would hear the beats a little slower than one per second.  Take the same fifth an octave higher and take 2C as fundamental, which has 256 for its vibration number.  The G, fifth above, should vibrate 384, but in the tempered scale it beats but 383.57, almost half a vibration flat.  This would give nearly 2 beats in 3 seconds.

These figures simply represent to the eye the ratios of these sounds, and it is not supposed that a tuner is to attain to such a degree of accuracy, but he should strive to arrive as near it as possible.

It is well for the student to practice temperament setting and regular tuning now if he can do so.  After getting a good temperament, proceed to tune by octaves upward, always testing the tone tuned as a fifth and third until his ear becomes sufficiently true on the octave that testing otherwise is unnecessary.  Tune the overstrung bass last and your work is finished.  If your first efforts are at all satisfactory you should be greatly encouraged and feel assured that accuracy will reward continued practice.