If a string is lightly touched at its half length during vibration, the octave to the open string will be produced; at one-third or two-thirds its length, measuring either from the bridge to the nut or _vice versâ_, the fifth above the octave; at one-fourth or three-fourths the double octave; at one-fifth, two-fifths, three-fifths and four-fifths the major third above the second octave; at one-sixth and five-sixths the fifth above the second octave, and at one-eighth, three-eighths, five-eighths and seven-eighths a harmonic note three octaves above the open string will be produced. The difference between the vibration of a musical string during the production of a stopped note, and a harmonic note is of sufficient interest, and of enough importance to merit description. Most of my readers will be aware that when the string is pressed firmly against the fingerboard for a stopped note, the portion between the finger and the nut does not vibrate, the string is practically shortened to the dimensions of that portion which lies between the finger and the bridge; when a harmonic note is played however, the finger being lightly placed on the string merely diverts the vibration; the whole length of the string vibrates, the part between the finger and the nut assisting as actively in producing the note as the part between the finger and bridge. The following rough sketch will illustrate this more clearly than is possible in words alone; Fig. 5 represents a vibrating string; (_a_) is the nut, (_c_) the bridge, the string being stopped at its half length (_b_), the only portion which vibrates is that between (_b_) and (_c_). Fig. 6 represents a harmonic note produced at the half string (_b_) in this case the whole string vibrates yet the string is divided into two equal parts, the part between (_a_) and (_b_) vibrating in unison with that between (_b_) and (_c_). The student will here see the importance of keeping all the fingers quite clear of the string except of course the one producing the note, so that the vibrations may not be impeded. So far this seems quite logical, it is in proceeding further that one realizes the wonderful laws which govern the production of harmonics. Fig. 7 represents the string touched lightly at its fourth part (_d_) or (_e_) giving the harmonic note two octaves above the open string; the student will observe that it is quite immaterial whether the fourth be calculated from the bridge or from the nut; the vibrations in each case will be thus:--If the string is touched at (_d_) the portion between (_d_) and (_c_), that is between the finger and the bridge, will naturally divide itself into three equal parts, each part vibrating in unison with the part between (_a_) and (_d_); again, if the finger is placed at (_e_) the part behind the finger, that is, the portion of the string between (_e_) and (_a_) will divide itself into equal parts in like manner. One of the fourth parts is to be found at (_b_); how is it then that if the string is touched there a harmonic note of lower pitch than those given out at the first and third fourths is produced? The reason is that the portion of string at each side of (_b_) being of equal length, the string naturally divides itself into halves; we have found that this gives the octave to the open string. The student may work out for himself the reason why the fifth above the second octave is only playable at the one-sixth and five-sixth parts and not at the two-sixth, three-sixth and four-sixth, and why the third octave is not possible at the two-eighth, four-eighth, and six-eighth parts.