Harmonics.
In considering harmonics, the names of two of the greatest violin players the world has ever seen force themselves to one's mind:—Paganini and Spohr; the exponents of two schools of violin playing as diametrically opposed to each other as darkness is to light. Paganini the weird, fiery Italian, astonishing the world with hitherto undreamt of effects, not the least marvellous in the eyes of critics and multitude alike, being his wonderful command over every possible form of harmonic playing. Spohr with his solid, classical, German temperament, attempting nothing out of the established limits of real, solid playing, countenancing nothing which the "great in music" before him had not accepted and stamped with their hall-mark. Considering this, and also that Spohr may have been annoyed at the allegiance which nearly the whole music-loving public were only too eager to pay to the Italian violinist, it is not to be wondered at that he should find it necessary to denounce the whole art of harmonic playing as trick playing and unworthy of a great artist. One can hardly forgive Spohr's description of harmonic tones as "foreign and childish"; they certainly are entirely different to the tone produced by stopped notes, but this very difference, instead of condemning them, should rather recommend them to the instrumentalist as another means of adding variety, that essence of life, to his playing. It is really surprising what an electrical effect on an audience has a well executed passage in harmonics; "harmonics excite wonder"! true, but if well played they also excite enthusiasm.
Spohr is to be praised for his recommendations to young violinists not to neglect that which is useful, in the prosecution of the study of harmonics; young violoncellists please attend to the advice! yet every player should be thoroughly conversant with the science of harmonics even if he has to defer perfection in the art of their production until a later period. "Harmonics" are described as "the accessory sounds generated with the predominant and apparently simple tones of any vibrating string or column of air." Science teaches us that a single note is impossible; immediately a note is sounded, certain tones more or less related to the fundamental note, are generated. These overtones may be distinctly heard if one of the open strings of a good old violoncello is vigorously sounded; as the fundamental note decreases in power, the harmonic over-tones will be easily heard in their order of production—first the octave, then the fifth to the octave, then the major third to the octave above. Another method of hearing these harmonics is by causing a note in unison with one of the open strings of the violoncello to be sung, or played upon some other instrument, the string in unison with the note sounded, through sympathetic vibration will give out the overtones only, as previously described. However it is not these fleeting overtones which demand our attention, although they form the natural basis to the whole matter, it is the production of harmonic tones in the form of independent or primary notes. To accomplish this on the violoncello the string must be touched lightly with the finger at certain places, not as for the production of a stopped note, by pressing the string firmly against the fingerboard, but by allowing one finger to lightly rest upon it with sufficient "touch" to divert the vibrations. It will be found that only at certain places are harmonic notes possible. These places are called nodes or nodal points; they are to be found at the mathematical divisions of the strings into halves, thirds, quarters, etc. The class of harmonics produced in this manner are termed "Natural Harmonics." As each string gives out the same notes relative to the pitch of the open string, one description will suffice.
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.
