The Scientific Basis of Harmonics—Some Peculiar Laws which Govern a Vibrating String—"Natural" and "Artificial" Harmonics—The Manner of Bowing Harmonics.

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.

The name "artificial" is used merely in contrast to "natural," they are only artificial in the sense that they are produced on an artificial or "made" basis, instead of being in the key of the open string.

If one was restricted to the use of natural harmonics it would be impossible to play scale passages, or any passages foreign to the key of the four open strings. However by causing the string to be shortened by the employment of a stopped note the pitch of the fundamental note may be raised to any desired height, the harmonics being produced exactly as before, on the fresh root thus formed; the harmonic being in the key of the shortened string. In playing artificial harmonics on the violoncello it is usual to employ the thumb for the stopped note, the third finger should then be caused to touch the string lightly at one of its nodal points, the distances of course being now calculated from the thumb to the bridge. Owing to the sometimes great length of string between the stopped note and the bridge it is often found impossible to use any but the smaller divisions, the form most often met with being harmonics produced by fourths, that is, the third finger touching the string at the interval of one fourth (stopped note) above the thumb: the harmonic thus produced will be two octaves above the pitch of the note stopped by the thumb. Whole passages are written for this form of artificial harmonics. To the student who has thoroughly mastered octave passages in stopped notes the technical difficulties presented ought to be easily conquered, as the same technique really suffices for both; the only difference being that whilst octaves are played across two strings—artificial harmonics by fourths are played with the same fingering and the same intervals on one string.

With respect to the bowing of harmonics, the peculiar manner in which the string vibrates during the production of harmonic notes must be taken into consideration. It is important to notice that the vibration of the string near the nodal points is the least intense (see double lines in diagram), whereas the part mid-way between the nodes vibrates with the greatest intensity; if the bow is caused to pass near one of these nodes it may possibly give a greater vibration to that portion of the string than is required, which would have the effect of displacing the natural division, and placing of the nodes, thus causing confusion or perhaps another harmonic than the one intended to be produced. It is general to give an all round rule that in playing harmonics the bow must always be drawn close to the bridge, the student will comprehend that the smaller the division of the string used to produce the harmonic, the greater need will there be to attend to this rule. The progression of harmonic notes should be clearly defined; in passages composed wholly of harmonics it may sometimes be found advisable to use a down bow-stroke for each note, slightly striking the commencement of each harmonic; this will give the necessary attack, and will cause each note to ring out clear as a bell, providing the fingering is correctly managed.

Harmonics are indicated in various ways, sometimes causing much confusion and indecision as to what is really required; thus in a single composition it is possible to find the real notes which are to be produced being given with the word "harmonique" or "flageolet" or sometimes the sign O (usually employed for natural harmonics) added, the player to produce the harmonics as he pleases; again the position of the node which produces the harmonic will be indicated by a blank note, the pitch of the harmonic to be produced being left to fate and the performer. To indicate artificial harmonics it is usual to write the stopped notes as an ordinary passage, then with the aid of blank notes the positions where the string has to be lightly touched are shown.