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.
[Illustration: Fig 5]
[Illustration: Fig 6]
[Illustration: Fig 7]
ARTIFICIAL HARMONICS.
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.