1.  Fifths too flat.

    2.  Fifths not flat enough.

    3.  Some fifths correctly tempered and others not.

    4.  Some fifths sharper instead of flatter than perfect; a
    condition that must be watched with vigilance.

    5.  Some or all of the strings tuned fall from the pitch at which
    they were left.

From a little reflection upon these causes, it is seen that the last trial may prove a correct fifth and yet the temperament be imperfect.  If this is the case, it will be necessary to go all over the temperament again.  Generally, however, after you have had a little experience, you will find the trouble in one of the first two causes above, unless it be a piano wherein, the strings fall as in Cause 5.  This latter cause can be ascertained in cases only where you have started from a tuning pipe or fork.  Sometimes you may find that the temperament may be corrected by the alteration of but two or three tones; so it is always well to stop and examine carefully before attempting the correction.  A haphazard attempt might cause much extra work.

In temperament setting by our system, if the fifths are properly tempered and the octaves are left perfect, the other intervals will need no attention, and will be found beautifully correct when used in testing.

The mistuned or tempered intervals are as follows:—­

INTERVALS FLATTENED.     INTERVALS SHARPENED.
The Fifth, slightly.      The Fourth, slightly. 
The Minor Third,         The Major Third,
considerably.             greatly. 
The Minor Sixth,         The Major Sixth,
considerably.             greatly.

Tuning the Treble.—­In tuning the treble, which is always tuned by exact octaves, from their corresponding tones within the temperament, the ear will often accept an octave as true before its pitch has been sufficiently raised.  Especially is this true in the upper octaves.  After tuning a string in the treble by its octave in the temperament, test it as a fifth.  For instance, after tuning your first string beyond the temperament, 3C[#], test it as a fifth to 2F[#].  If you are yet uncertain, try it as a major third in the chord of A. The beats will serve you as a guide in testing by fifths, up to about an octave and a half above the highest tone of the temperament; but beyond this point they become so rapid as to be only discernible as degrees of roughness.  The beats will serve as a guide in tuning octaves higher in the treble than the point at which the beats of the fifth become unavailable; and in tuning unisons, the beats are discernible almost to the last tone.

The best method to follow in tuning the treble may be summed up as follows:  Tune the first octave with the beats as guides both in the octave and in testing it by the fifth.  If yet uncertain, test by chords.  Above this octave, rely somewhat upon the beats in the octave, still use the fifth for testing, but listen for the pitch in the extreme upper tones and not so much for the beats except in bringing up unisons, in which the beats are more prominent.