The following suggestions will serve you greatly in testing:  When a third sounds disagreeably sharp, one or more fifths have not been sufficiently flattened.[E] While it is true that thirds are tuned sharp, there is a limit beyond which we cannot go, and this excessive sharpness of the third is the thing that tuners always listen for.

[E] In making these suggestions, no calculation is made for the liability of the tones tuned to fall.  This often happens, in which case your first test will display a sharp third.  In cases like this it is best to go on through, taking pains to temper carefully, and go all over the temperament again, giving all the strings an equal chance to fall.  If the piano is very bad, you may have to bring up the unisons roughly, inuring this portion of the instrument to the increased tension, when you may again place your continuous mute and set your temperament with more certainty.

The fundamental sounds better to the ear when too sharp.  The reason for this is the same as has already been explained above; namely, if the fundamental is too sharp the third will be less sharp to it, and, therefore, nearer perfect.

After you have gone all over your temperament, test every member of the chromatic scale as a fundamental of a chord, as a third, and as a fifth.  For instance:  try middle C as fundamental in the chord of C (G-C-E or E-G-C or C-E-G).  Then try it as third in the chord A flat (E flat-A flat-C or C-E flat-A flat or A flat-C-E flat).  Then try it as fifth in the chord of F (C-F-A or A-C-F or F-A-C).  Take G likewise and try it as fundamental in the chord of G in its three positions, then try it as a third in the chord of E flat, then as fifth in the chord of C. In like manner try every tone in this way, and if there is a falsely tempered interval in the scale you will be sure to find it.

You now understand that the correctness of your temperament depends entirely upon your ability to judge the degree of flatness of your fifths; provided, of course, that the strings stand as tuned.  We have told you something about this, but you may not be able at once to judge with sufficient accuracy to insure a good temperament.  Now, we have said, let the fifths beat a little more slowly than once a second; but the question crops up, How am I to judge of a second of time?  The fact is that a second of time is quickly learned and more easily estimated, perhaps, than any other interval of time; however, we describe here a little device which will accustom one to estimate it very accurately in a short time.  The pendulum oscillates by an invariable law which says that a pendulum of a certain length will vibrate always in a corresponding period of time, whether it swings through a short arc or a long one.  A pendulum thirty-nine and a half inches long will vibrate seconds by a single swing; one nine and seven-eighths inches long will vibrate seconds at the double swing, or