False waves will occur in an annoying degree when the tuner sets a mute on a nodal point in the string; it will cause the muted string to sound a real harmonic tone.  This does not happen in the upright, as the mutes are set so near the end of the string as to preclude this possibility.  In the square, however, it very frequently happens, as there are so many nodes between the dampers and the bridge, where the tuner sets his mutes.  If, for instance, he is tuning an octave and has his mute set precisely in the middle of the vibrating segment, in place of muting the string it sounds its own octave, which will disturb the ear in listening for the tone from the one free string.  Move the mute either way until it is found to mute the string entirely.

QUESTIONS ON LESSON XIV.

    1.  Explain the cause of the beats.

    2.  How many beats per second in a unison of two strings, one
    tuned to 100, the other to 101 vibrations per second?

    3.  How many beats per second in an octave, the lower tone of which
    is tuned to 100, the upper to 201 vibrations per second?

    4.  How many beats per second in a fifth, the fundamental of which
    is tuned to 100, the fifth to 151?

    5.  The fifth, 2F-3C, when properly tempered, should beat 1-3/4
    times per second.  How often should a fifth, an octave higher,
    beat?

LESSON XV.

MISCELLANEOUS TOPICS PERTAINING TO THE PRACTICAL WORK OF TUNING, REGULATING, AND REPAIRING.

Comparison of the Different Systems.—­Up to this time, we have given no account of any system of tuning except the one recommended.  For the purpose of making the student more thoroughly informed we detail here several different systems which have been devised and practiced by other tuners.  It is a matter of history that artisans in this profession and leaders in musical science have endeavored to devise a system of temperament having all the desirable qualifications.

The aims of many have been to invent a system which uses the fewest number of tones; working under the impression that the fewer the tones used in the temperament, the easier the tuner’s work.  These have reduced the compass of the temperament to the twelve semi-tones from middle C to B above; or from F below, to E above middle C. This system requires the tuner to make use of both fourths and fifths.  Not only does he have to use these two kinds of intervals in tuning, but he has to tune by fourths up and fourths down, and, likewise, by fifths up and fifths down.  When tuning a fifth upward, he flattens it; and when tuning a fifth downward he sharpens the lower tone; when tuning a fourth upward, he sharpens it; when tuning a fourth downward, he flattens the lower tone.

It is readily seen that by a system of this kind the tuner’s mind is constantly on a strain to know how to temper the interval he is tuning, and how much to temper it, as fourths require a different degree of tempering from the fifths; and he is constantly changing from an interval upward to one downward; so, this system must be stamped as tedious and complicated, to say the least.  Yet this system is much followed in factories for rough tuning, and also by many old professional tuners.