Vibration of Strings—How employed in Music—Influence of
Sound-Boards—Laws of Vibrating Strings—Combination of
Direct and Reflected Pulses—Stationary and Progressive
Waves—Nodes and Ventral Segments—Application of Results to the
Vibrations of Musical Strings—Experiments of Melde—Strings
set in Vibration by Tuning-Forks—Laws of Vibration thus
demonstrated—Harmonic Tones of Strings—Definitions of Timbre
or Quality, or Overtones and Clang—Abolition of Special
Harmonics—Conditions which affect the Intensity of the
Harmonic Tones—Optical Examination of the Vibrations of a
Piano-Wire.
§ 1. _Vibrations of Strings: Use of Sound-Boards_
We have to begin our studies to-day with the vibrations of strings
or wires; to learn how bodies of this form are rendered available
as sources of musical sounds, and to investigate the laws of their
vibrations.
To enable a musical string to vibrate _transversely_, or at right
angles to its length, it must be stretched between two rigid points.
Before you, Fig. 31 (next page), is an instrument employed to stretch
strings, and to render their vibrations audible. From the pin _p_,
to which one end of it is firmly attached, a string passes across
the two bridges B and B′, being afterward carried over the wheel H,
which moves with great freedom. The string is finally stretched by a
weight W, of 28 lbs., attached to its extremity. The bridges B and B′,
which constitute the real ends of the string, are fastened on to the
long wooden box M N. The whole instrument is called a monochord, or
sonometer.
[Illustration: FIG. 31.]
Taking hold of the stretched string B B′ at its middle and plucking it
aside, it springs back to its first position, passes it, returns, and
thus vibrates for a time to and fro across its position of equilibrium.
You hear a sound, but the sonorous waves which at present strike
your ears do not proceed immediately from the string. The amount of
wave-motion generated by so thin a body is too small to be sensible at
any distance. But the string is drawn tightly over the two bridges B
B′; and when it vibrates, its tremors are communicated through these
bridges to the entire mass of the box M N, and to the air within the
box, which thus become the real sounding bodies.
[Illustration: FIG. 32.]
That the vibrations of the string alone are not sufficient to produce
the sound may be thus experimentally demonstrated: A B, Fig. 32 (next
page), is a piece of wood placed across an iron bracket C. From each
end of the piece of wood depends a rope ending in a loop, while
stretching across from loop to loop is an iron bar _m n_. From the
middle of the iron bar hangs a steel wire _s s′_, stretched by a weight
W, of 28 lbs. By this arrangement the wire is detached from all large
surfaces to which it could impart its vibrations. Plucking the wire _s
s′_, it vibrates vigorously, but even those nearest to it do not hear
any sound. The agitation imparted to the air is too inconsiderable to
affect the auditory nerve at any distance. A second wire _t t′_, Fig.
33 (next page), of the same length, thickness, and material as _s s′_,
has one of its ends attached to the wooden tray A B. This wire also
carries a weight W, of 28 lbs. Finally, passing over the bridges B B′
of the sonometer, Fig. 31, is our third wire, in every respect like
the two former, and, like them, stretched by a weight W, of 28 lbs.
When the wire _t t′_, Fig. 33, is caused to vibrate, you hear its sound
distinctly. Though one end only of the wire is connected with the tray
A B, the vibrations transmitted to it are sufficient to convert the
tray into a sounding body. Finally, when the wire of the sonometer
M N, Fig. 31, is plucked, the sound is loud and full, because the
instrument is specially constructed to take up the vibrations of the
wire.
The importance of employing proper sounding apparatus in stringed
instruments is rendered manifest by these experiments. It is not the
strings of a harp, or a lute, or a piano, or a violin, that throw the
air into sonorous vibrations. It is the large surfaces with which the
strings are associated, and the air inclosed by these surfaces. The
goodness of such instruments depends almost wholly upon the quality and
disposition of their sound-boards.[33]
[Illustration: FIG. 33.]
Take the violin as an example. It is, or ought to be, formed of wood
of the most perfect elasticity. Imperfectly elastic wood expends
the motion imparted to it in the friction of its own molecules; the
motion is converted into heat, instead of sound. The strings of the
violin pass from the “tail-piece” of the instrument over the “bridge,”
being thence carried to the “pegs,” the turning of which regulates
the tension of the strings. The bow is drawn across at a point about
one-tenth of the length of the string from the bridge. The two “feet”
of the bridge rest upon the most yielding portion of the “belly” of
the violin, that is, the portion that lies between the two _f_-shaped
orifices. One foot is fixed over a short rod, the “sound post,” which
runs from belly to back through the interior of the violin. This foot
of the bridge is thereby rendered rigid, and it is mainly through
the other foot, which is not thus supported, that the vibrations are
conveyed to the wood of the instrument, and thence to the air within
and without. The sonorous quality of the wood of a violin is mellowed
by age. The very act of playing also has a beneficial influence,
apparently constraining the molecules of the wood, which in the first
instance might be refractory, to conform at last to the requirements of
the vibrating strings.
[Illustration: FIG. 34.]
This is the place to make the promised reference (page 38) to Prof.
Stokes’s explanation of the action of sound-boards. Although the
amplitude of the vibrating board may be very small, still its larger
area renders the abolition of the condensations and rarefactions
difficult. The air cannot move away in front nor slip in behind before
it is sensibly condensed and rarefied. Hence with such vibrating bodies
sound-waves may be generated, and loud tones produced, while the thin
strings that set them in vibration, acting alone, are quite inaudible.
The increase of sound, produced by the stoppage of lateral motion, has
been experimentally illustrated by Prof. Stokes. Let the two black
rectangles in Fig. 34 represent the section of a tuning-fork. After
it has been made to vibrate, place a sheet of paper, or the blade of
a broad knife, with its edge parallel to the axis of the fork, and as
near to the fork as may be without touching. If the obstacle be so
placed that the section of it is A or B, no effect is produced; but
if it be placed at C, so as to prevent the reciprocating to-and-fro
movement of the air, which tends to abolish the condensations and
rarefactions, the sound becomes much stronger.
§ 2. _Laws of Vibrating Strings_
Having thus learned how the vibrations of strings are rendered
available in music, we have next to investigate the laws of such
vibrations. I pluck at its middle point the string B B′, Fig. 31. The
sound heard is the fundamental or lowest note of the string, to produce
which it swings, as a whole, to and fro. By placing a movable bridge
under the middle of the string, and pressing the string against the
bridge, it is divided into two equal parts. Plucking either of those
at its centre, a musical note is obtained, which many of you recognize
as the octave of the fundamental note. In all cases, and with all
instruments, the octave of a note is produced by doubling the number of
its vibrations. It can, moreover, be proved, both by theory and by the
siren, that this half string vibrates with exactly twice the rapidity
of the whole. In the same way it can be proved that one-third of the
string vibrates with three times the rapidity, producing a note a fifth
above the octave, while one-fourth of the string vibrates with four
times the rapidity, producing the double octave of the whole string. In
general terms, _the number of vibrations is inversely proportional to
the length of the string_.
Again, the more tightly a string is stretched the more rapid is its
vibration. When this comparatively slack string is caused to vibrate,
you hear its low fundamental note. By turning a peg, round which one
end of it is coiled, the string is tightened, and the pitch rendered
higher. Taking hold with my left hand of the weight w, attached to the
wire B B′ of our sonometer, and plucking the wire with the fingers of
my right, I alternately press upon the weight and lift it. The quick
variations of tension are expressed by a varying wailing tone. Now,
the number of vibrations executed in the unit of time bears a definite
relation to the stretching force. Applying different weights to the end
of the wire B B′, and determining in each case the number of vibrations
executed in a second, we find the numbers thus obtained to be
_proportional to the square roots of the stretching weights_. A string,
for example, stretched by a weight of one pound, executes a certain
number of vibrations per second; if we wish to double this number, we
must stretch it by a weight of four pounds; if we wish to treble the
number, we must apply a weight of nine pounds, and so on.
The vibrations of a string also depend upon its thickness. Preserving
the stretching weight, the length, and the material of the string
constant, _the number of vibrations varies inversely as the thickness
of the string_. If, therefore, of two strings of the same material,
equally long and equally stretched, the one has twice the diameter
of the other, the thinner string will execute double the number of
vibrations of its fellow in the same time. If one string be three times
as thick as another, the latter will execute three times the number of
vibrations, and so on.
Finally, the vibrations of a string depend upon the density of the
matter of which it is composed. A platinum wire and an iron wire,
for example, of the same length and thickness, stretched by the same
weight, will not vibrate with the same rapidity. For, while the
specific gravity of iron, or in other words its density, is 7·8, that
of platinum is 21·5. All other conditions remaining the same, _the
number of vibrations is inversely proportional to the square root of
the density of the string_. If the density of one string, therefore,
be one-fourth that of another of the same length, thickness, and
tension, it will execute its vibrations twice as rapidly; if its
density be one-ninth that of the other, it will vibrate with three
times the rapidity, and so on. The last two laws, taken together, may
be expressed thus: _The number of vibrations is inversely proportional
to the square root of the weight of the string_.
In the violin and other stringed instruments we avail ourselves of
thickness instead of length to obtain the deeper tones. In the piano
we not only augment the thickness of the wires intended to produce
the bass notes, but we load them by coiling round them an extraneous
substance. They resemble horses heavily jockeyed, and move more slowly
on account of the greater weight imposed upon the force of tension.
§ 3. _Mechanical Illustrations of Vibrations. Progressive and
Stationary Waves. Ventral Segments and Nodes_
These, then, are the four laws which regulate the _transverse_
vibrations of strings. We now turn to certain allied phenomena, which,
though they involve mechanical considerations of a rather complicated
kind, may be completely mastered by an average amount of attention.
And they _must_ be mastered if we would thoroughly comprehend the
philosophy of stringed instruments.
[Illustration: FIG. 35.]
[Illustration: FIG. 36.]
From the ceiling _c_, Fig. 35, of this room hangs an India-rubber
tube twenty-eight feet long. The tube is filled with sand to render
its motions slow and more easily followed by the eye. I take hold of
its free end _a_, stretch the tube a little, and by properly timing
my impulses cause it to swing to and fro as a whole, as shown in
the figure. It has its definite period of vibration dependent on
its length, weight, thickness, and tension, and my impulses must
synchronize with that period.
I now stop the motion, and by a sudden jerk raise a hump upon the tube,
which runs along it as a pulse toward its fixed end; here the hump
reverses itself, and runs back to my hand. At the fixed end of the
tube, in obedience to the law of reflection, the pulse reversed both
its position and the direction of its motion. Supposing _c_, Fig. 36,
to be the fixed end of the tube, and _a_ the end held in the hand:
if the pulse on reaching _c_ have the position shown in (1), after
reflection it will have the position shown in (2). The arrows mark
the direction of progression. The time required for the pulse to pass
from the hand to the fixed end and back is exactly that required to
accomplish one complete vibration of the tube as a whole. It is indeed
the addition of such impulses which causes the tube to continue to
vibrate as a whole.
[Illustration: FIG. 37.]
If, instead of a single jerk, a succession of jerks be imparted,
thereby sending a series of pulses along the tube, every one of them
will be reflected above, and we have now to inquire how the direct and
reflected pulses behave toward each other.
Let the time required by the pulse to pass from my hand to the fixed
end be one second; at the end of half a second it occupies the position
_a b_ (1), Fig. 37, its foremost point having reached the middle of
the tube. At the end of a whole second it would have the position _b
c_ (2), its foremost point having reached the fixed end _c_ of the
tube. At the moment when reflection begins at _c_, let another jerk
be imparted at _a_. The reflected pulse from _c_ moving with the same
velocity as this direct one from _a_, the foremost points of both will
arrive at the centre _b_ (3) at the same moment. What must occur? The
hump _a b_ wishes to move on to _c_, and to do so must move the point
_b_ to the right. The hump _c b_ wishes to move toward _a_, and to do
so must move the point _b_ to the left. The point _b_, urged by equal
forces in two opposite directions at the same time, will not move in
either direction. Under these circumstances, the two halves, _a b_,
_b c_ of the tube will oscillate as if they were independent of each
other (4). Thus by the combination of two _progressive pulses_, the one
direct and the other reflected, we produce two _stationary pulses_ on
the tube _a c_.
The vibrating parts _a b_ and _b c_ are called _ventral segments_; the
point of no vibration _b_ is called a _node_.
The term “pulse” is here used advisedly, instead of the more usual term
_wave_. For a wave embraces two of these pulses. It embraces both the
hump and the depression which follows the hump. The length of a wave,
therefore, is twice that of a ventral segment.
[Illustration: FIG. 38.]
Supposing the jerks to be so timed as to cause each hump to be
one-third of the tube’s length. At the end of one-third of a second
from starting the pulse will be in the position _a b_ (1), Fig. 38. In
two-thirds of a second it will have reached the position _b b′_ (2),
Fig. 38. At this moment let a new pulse be started at _a_; after the
lapse of an entire second from the commencement we shall have two humps
upon the tube, one occupying the position _a b_ (3), the other the
position _b′ c_ (3). It is here manifest that the end of the reflected
pulse from _c_, and the end of the direct one from _a_, will reach
the point _b′_ at the same moment. We shall therefore have the state
of things represented in (4), where _b b′_ wishes to move upward, and
_c b′_ to move downward. The action of both upon the point _b′_ being
in opposite directions, that point will remain fixed. _And from it,
as if it were a fixed point, the pulse b b′ will be reflected, while
the segment b′ c will oscillate as an independent string._ Supposing
that at the moment _b b′_ (4) begins to be reflected at _b′_ we start
another pulse from _a_, it will reach _b_ at the same moment the pulse
reflected from _b′_ reaches it. The pulses will neutralize each other
at _b_, and we shall have there a second node. Thus, by properly timing
our jerks, we divide the rope into three ventral segments, separated
from each other by two nodal points. As long as the agitation continues
the tube will vibrate as in (6).
There is no theoretic limit to the number of nodes and ventral
segments that may be thus produced. By the quickening of the impulses,
the tube is divided into four ventral segments separated by three
nodes; quickening still more we have five ventral segments and four
nodes. With this particular tube the hand may be caused to vibrate
sufficiently quick to produce ten ventral segments, as shown in Fig.
38 (7). When the stretching force is constant, the number of ventral
segments is proportional to the rapidity of the hand’s vibration. To
produce 2, 3, 4, 10 ventral segments requires twice, three times, four
times, ten times the rapidity of vibration necessary to make the tube
swing as a whole. When the vibration is very rapid the ventral segments
appear like a series of shadowy spindles, separated from each other by
dark motionless nodes. The experiment is a beautiful one, and it is
easily performed.
If, instead of moving the hand to-and fro, it be caused to describe a
small circle, the ventral segments become “surfaces of revolution.”
Instead of the hand, moreover, we may employ a hook turned by a
whirling-table. Before you is a cord more rigid than the India-rubber
tube, 25 feet long, with one of its ends attached to a freely-moving
swivel fixed in the ceiling of the room. By turning the whirling-table
to which the other end is attached, this cord may be divided into
as many as 20 ventral segments, separated from each other by their
appropriate nodes. In another arrangement a string of catgut 12 feet
long, with silvered beads strung along it, is stretched horizontally
between a vertical wheel and a free swivel fixed in a rigid stand. On
turning the wheel, and properly regulating both the tension and the
rapidity of rotation, the beaded cord may be caused to rotate as a
whole, and to divide itself successively into 2, 3, 4, or 5 ventral
segments. When we envelop the cord in a luminous beam, every spot of
light on every bead describes a brilliant circle, and a very beautiful
experiment is the result.
§ 4. _Mechanical Illustrations of Damping Various Points of Vibrating
Cord_
The subject of _stationary waves_ was first experimentally treated by
the Messrs. Weber, in their excellent researches on wave-motion. It is
a subject which will well repay your attention by rendering many of the
most difficult phenomena of musical strings perfectly intelligible. It
will make the connection of both classes of vibrations more obvious if
we vary our last experiments. Before you is a piece of India-rubber
tubing, 10 or 12 feet long, stretched from _c_ to _a_, Fig. 39, and
made fast to two pins at _c_ and _a_. The tube is blackened, and behind
it is placed a surface of white paper, to render its motions more
visible. Encircling the tube at its centre _b_ (1) by the thumb and
forefinger of my left hand, and taking the middle of the lower half
_b a_ of the tube in my right, I pluck it aside. Not only does the
lower half swing, but the upper half also is thrown into vibration.
Withdrawing the hands wholly from the tube, its two halves _a b_ and _b
c_ continue to vibrate, being separated from each other by a node _b_
at the centre (2).
[Illustration: FIG. 39.]
I now encircle the tube at a point _b_ (3) one-third of its length
from its lower end _a_, and, taking hold of _a b_ at its centre, pluck
it aside; the length _b c_ above my hand instantly divides into two
vibrating segments. Withdrawing the hands wholly, you see the entire
tube divided into three ventral segments, separated from each other
by two motionless nodes, _b_ and _b′_ (4). I pass on to the point _b_
(5), which marks off one-fourth of the length of the tube, encircle
it, and pluck the shorter segment aside. The longer segment above my
hand divides itself immediately into three vibrating parts. So that, on
withdrawing the hand, the whole tube appears before you divided into
four ventral segments, separated from each other by three nodes _b b′
b″_ (6). In precisely the same way the tube may be divided into five
vibrating segments with four nodes.
This sudden division of the long upper segment of the tube, without any
apparent cause, is very surprising; but if you grant me your attention
for a moment, you will find that these experiments are essentially
similar to those which illustrated the coalescence of direct and
reflected undulations. Reverting for a moment to the latter, you
observed that the to-and-fro motion of the hand through the space of
a single inch was sufficient to make the middle points of the ventral
segments vibrate through a foot or eighteen inches. By being properly
timed the impulses accumulated, until the amplitude of the vibrating
segments exceeded immensely that of the hand which produced them. The
hand, in fact, constituted a nodal point, so small was its comparative
motion. Indeed, it is usual, and correct, to regard the ends of the
tube also as nodal points.
Consider now the case represented in (1), Fig. 39, where the tube was
encircled at its middle, the lower segment _a b_ being thrown into the
vibration corresponding to its length and tension. The circle formed
by the finger and thumb permitted the tube to oscillate at the point
_b_ through the space of an inch; and the vibrations at that point
acted upon the upper half _b c_ exactly as my hand acted when it caused
the tube suspended from the ceiling to swing as a whole, as in Fig.
35. Instead of the timid vibrations of the hand, we have now the timid
vibrations of the lower half of the tube; and these, though narrowed to
an inch at the place clasped by the finger and thumb, soon accumulate,
and finally produce an amplitude, in the upper half, far exceeding
their own. The same reasoning applies to all the other cases of
subdivision. If, instead of encircling a point by the finger and thumb,
and plucking the portion of the tube below it aside, that same point
were taken hold of by the hand and agitated in the period proper to the
lower segment of the tube, precisely the same effect would be produced.
We thus reduce both effects to one and the same cause; namely, the
combination of direct and reflected undulations.
And here let me add that, when the tube was divided by the timid
impulses of the hand, not one of its nodes was, strictly speaking, a
point of no motion; for were the nodes not capable of vibrating through
a very small amplitude, the motion of the various segments of the tube
could not be maintained.
§ 5. _Stationary Water-waves_
What is true of the undulations of an India-rubber tube applies to all
undulations whatsoever. Water-waves, for example, obey the same laws,
and the coalescence of direct and reflected waves exhibits similar
phenomena. This long and narrow vessel with glass sides, Fig. 40, is
a copy of the wave-canal of the brothers Weber. It is filled to the
level A B with colored water. By tilting the end A suddenly, a wave
is generated, which moves on to B, and is there reflected. By sending
forth fresh waves at the proper intervals, the surface is divided into
two stationary undulations. Making the succession of impulses more
rapid we can subdivide the surface into three, four (shown in the
figure), or more stationary undulations, separated from each other by
nodes. The step of a water-carrier is sometimes so timed as to throw
the surface of the water in his vessel into stationary waves, which may
augment in height until the water splashes over the brim. Practice has
taught the water-carrier what to do; he changes his step, alters the
period of his impulses, and thus stops the accumulation of the motion.
[Illustration: FIG. 40.]
In travelling recently in the coupé of a French railway carriage, I had
occasion to place a bottle half filled with water on one of the little
coupé tables. It was interesting to observe it. At times it would be
quite still; at times it would oscillate violently. To the passenger
within the carriage there was no sensible change in the motion of the
train to which the difference could be ascribed. But in the one case
the tremor of the carriage contained no vibrations synchronous with
the oscillating period of the water, while in the other case such
vibrations were present. Out of the confused assemblage of tremors the
water selected the particular constituent which belonged to itself,
and declared its presence when the traveller was utterly unconscious of
its introduction.
§ 6. _Application of Mechanical Illustrations to Musical Strings_
From these comparatively gross, but by no means unbeautiful, mechanical
vibrations, we pass to those of a sounding string. In the experiments
with our monochord, when the wire was to be shortened, a movable bridge
was employed, against which the wire was pressed so as to deprive the
point resting on the bridge of all possibility of motion. This strong
pressure, however, is not necessary. Placing the feather-end of a
goose-quill lightly against the middle of the string, and drawing a
violin-bow over one of its halves, the string yields the octave of the
note yielded by the whole string. The mere _damping_ of the string at
the centre, by the light touch of the feather, is sufficient to cause
the string to divide into two vibrating segments. Nor is it necessary
to hold the feather there throughout the experiment: after having
drawn the bow, the feather may be removed; the string will continue
to vibrate, emitting the same note as before. We have here a case
exactly analogous to that in which the central point of our stretched
India-rubber tube was damped, by encircling it with the finger and
thumb as in Fig. 39 (1). Not only did the half plucked aside vibrate,
but the other half vibrated also. We can, in fact, reproduce, with
the vibrating string, every effect obtained with the tube. This,
however, is a point of such importance as to demand full experimental
illustration.
To prove that when the centre is damped, and the bow drawn across one
of the halves of the string, the other half vibrates, I place across
the middle of the untouched half a little rider of red paper. Damping
the centre and drawing the bow, the string shivers, and the rider is
overthrown, Fig. 41.
[Illustration: FIG. 41.]
[Illustration: FIG. 42.]
When the string is damped at a point which cuts off one-third of its
length, and the bow drawn across the shorter section, not only is this
section thereby thrown into vibration, but the longer section divides
itself into two ventral segments with a node between them. This is
proved by placing small riders of red paper on the ventral segments,
and a rider of blue paper at the node. Passing the bow across the short
segment you observe a fluttering of the red riders, and now they are
completely tossed off, while the blue rider which crosses the node is
undisturbed, Fig. 42.
[Illustration: FIG. 43.]
Damping the string at the end of one-fourth of its length, the bow is
drawn across the shorter section; the remaining three-fourths divide
themselves into three ventral segments, with two nodes between them.
This is proved by the unhorsing of the three riders placed astride
the ventral segments, the two at the nodes keeping their places
undisturbed, Fig. 43.
[Illustration: FIG. 44.]
Finally, damping the string at the end of one-fifth of its length,
and arranging, as before, the red riders on the ventral segments and
the blue ones on the nodes, by a single sweep of the bow the four red
riders are unhorsed, and the three blue ones left undisturbed, Fig.
44. In this way we perform with a sounding string the same series of
experiments that were formerly executed with a stretched India-rubber
tube, the results in both cases being identical.[34]
To make, if possible, this identity still more evident to you, a stout
steel wire 28 feet in length is stretched behind the table from side
to side of the room. I take the central point of this wire between
my finger and thumb, and allow my assistant to pluck one-half of it
aside. It vibrates, and the vibrations transmitted to the other half
are sufficiently powerful to toss into the air a large sheet of paper
placed astride the wire. With this long wire, and with riders not of
one-eighth of a square inch, but of 30, 40, or 50 square inches in
area, we may repeat all the experiments which you have witnessed with
the musical string. The sheets of paper placed across the nodes remain
always in their places, while those placed astride the ventral segments
are tossed simultaneously into the air when the shorter segment of
the wire is set in vibration. In this case, when close to it, you can
actually see the division of the wire.
§ 7. _Melde’s Experiments_
It is now time to introduce to your notice some recent experiments
on vibrating strings, which appeal to the eye with a beauty and a
delicacy far surpassing anything attainable with our monochord. To M.
Melde, of Marburg, we are indebted for this new method of exhibiting
the vibrations of strings. The scale of the experiments will be here
modified so as to suit our circumstances.
[Illustration: FIG. 45.]
First, then, you observe here a large tuning-fork T, Fig. 45, with a
small screw fixed into the top of one of its prongs, by which a silk
string can be firmly attached to the prong. From the fork the string
passes round a distant peg P, by turning which it may be stretched
to any required extent. When the bow is drawn across the fork, an
irregular flutter of the string is the only result. On tightening it,
however, when at the proper tension it expands into a beautiful gauzy
spindle six feet long, more than six inches across at its widest part,
and shining with a kind of pearly lustre. The stretching force at the
present moment is such that the string swings to and fro as a whole,
its vibrations being executed in a vertical plane.
Relaxing the string gradually, when the proper tension has been
reached, it suddenly divides into two ventral segments, separated from
each other by a sharply-defined and apparently motionless node.
While the fork continues vibrating, if the string be relaxed still
further, it divides into three vibrating parts. Slackening it still
more, it divides into four vibrating parts. And thus we might continue
to subdivide the string into ten, or even twenty ventral segments,
separated from each other by the appropriate number of nodes.
[Illustration: FIG. 46.]
When white-silk strings vibrate thus, the nodes appear perfectly fixed,
while the ventral segments form spindles of the most delicate beauty.
Every protuberance of the twisted string, moreover, writes its motion
in a more or less luminous line on the surface of the aërial gauze. The
four nodes of vibration just illustrated are represented in Fig. 46, 1,
2, 3, 4.[35]
When the synchronism between fork and string is perfect, the vibrations
of the string are steady and long-continued. A slight departure
from synchronism, however, introduces unsteadiness, and the ventral
segments, though they may show themselves for a time, quickly disappear.
[Illustration: FIG. 47.]
In the experiments just executed the fork vibrated in the direction of
the length of the string. Every forward stroke of the fork raised a
protuberance, which ran to the fixed end of the string, and was there
reflected; so that when the _longitudinal_ impulses were properly timed
they produced a _transverse_ vibration. Let us consider this further.
One end of this heavy cord is attached to a hook A, Fig. 47, fixed in
the wall. Laying hold of the other end I stretch the cord horizontally,
and then move my hand to and fro in the direction of the cord. It
swings as a whole, and you may notice that always, when the cord is
at the limit of its swing, the hand is in its most forward position.
If it vibrate in a vertical plane, the hand, in order to time the
impulses properly, must be at its forward limit at the moment the cord
reaches the upper boundary, and also at the moment it reaches the lower
boundary of its excursion. A little reflection will make it plain
that, in order to accomplish this, the hand must execute a complete
vibration while the cord executes a semi-vibration; in other words, the
vibrations of the hand must be twice as rapid as those of the cord.
Precisely the same is true of our tuning-fork. When the fork vibrates
in the direction of the string, the number of vibrations which it
executes in a certain time is twice the number executed by the string
itself. And if, while arranged thus, a fork and string vibrate with
sufficient rapidity to produce musical notes, the note of the fork will
be an octave above that of the string.
But if, instead of the hand being moved to and fro in the direction of
this heavy cord, it is moved at right angles to that direction, then
every upward movement of the hand coincides with an upward movement of
the cord; every downward movement of the hand with a downward movement
of the cord. In fact, the vibrations of hand and string, in this case,
synchronize perfectly; and if the hand could emit a musical note, the
cord would, emit a note of the same pitch. The same holds good when a
vibrating fork is substituted for the vibrating hand.
Hence, if the string vibrate as a whole when the vibrations of the
fork are _along_ it, it will divide into two ventral segments when the
vibrations are _across_ it; or, more generally expressed, preserving
the tension constant, whatever be the number of ventral segments
produced by the fork when its vibrations are in the direction of the
string, twice that number will be produced when the vibrations are
transverse to the string. The string A B, for example, Figs. 48 and
49, passing over a pulley B, is stretched by a definite weight (not
shown in the figure). When the tuning-fork vibrates _along_ it, as in
Fig. 48, the string divides into two equal ventral segments. When the
fork is turned so that it shall vibrate at right angles to the string,
the number of ventral segments is four, Fig. 49, or double the former
number. Attaching two strings of the same length to the same fork, the
one parallel and the other perpendicular to the direction of vibration,
and stretching both with equal weights, when the fork is caused to
vibrate, one of them divides itself into twice the number of ventral
segments exhibited by the other.
[Illustration: FIG. 48.]
[Illustration: FIG. 49.]
A number of exquisite effects may be obtained with these vibrating
cords. The path described by any point of any one of them may be
studied, after the manner of Dr. Young, by illuminating that point,
and watching the line of light which it describes. This is well
illustrated by a flat burnished silver wire, twisted so as to form a
spiral surface, from which, at regular intervals, the light flashes
when the wire is illuminated. When the vibration is steady, the
luminous spots describe straight lines of sunlike brilliancy. On
slackening the wire, but not so much as to produce its next higher
subdivision, upon the larger motion of the wire are superposed a
host of minor motions, the combination of all producing scrolls of
marvellous complication and of indescribable splendor.
In reflecting on the best means of rendering these effects visible,
the thought occurred to me of employing a fine platinum wire heated
to redness by an electric current. Such a wire now stretches from a
tuning-fork over a bridge of copper, and then passes round a peg. The
copper bridge on the one hand and the tuning-fork on the other are the
poles of a voltaic battery, from which a current passes through the
wire and causes it to glow. On drawing the bow across the fork, the
wire vibrates as a whole; its two ends are brilliant, while its middle
is dark, being chilled by its rapid passage through the air. Thus you
have a shading off of incandescence from the ends to the centre of the
wire. On relaxing the tension, the wire divides itself into two ventral
segments; on relaxing still further, we obtain three; still further,
and the wire divides into four ventral segments, separated from each
other by three brilliant nodes. Right and left from every node the
incandescence shades away until it disappears. You notice also, when
the wire settles into steady vibration, that the nodes shine out with
greater brilliancy than that possessed by the wire before the vibration
commenced. The reason is this. Electricity passes more freely along a
cold wire than along a hot one. When, therefore, the vibrating segments
are chilled by their swift passage through the air, their conductivity
is improved, more electricity passes through the vibrating than through
the motionless wire, and hence the augmented glow of the nodes. If,
previous to the agitation of the fork, the wire be at a bright-red
heat, when it vibrates its nodes may be raised to the temperature of
fusion.
§ 8. _New Mode of determining the Laws of Vibration_
We may extend the experiments of M. Melde to the establishment of all
the laws of vibrating strings. Here are four tuning-forks, which we may
call _a_, _b_, _c_, _d_, whose rates of vibration are to each other as
the numbers 1, 2, 4, 8. To the largest fork is attached a string, _a_,
stretched by a weight, which causes it to vibrate as a whole. Keeping
the stretching weight the same, I determine the lengths of the same
string, which, when attached to the other three forks, _b_, _c_, _d_,
swing as a whole. The lengths in the four respective cases are as the
numbers 8, 4, 2, 1.
From this follows the first law of vibration, already established (p.
126) by another method; viz., _the length of the string is inversely
proportional to the rapidity of vibration_.[36]
In this case the longest string vibrates as a whole when attached to
the fork _a_. I now transfer the string to _b_, still keeping it
stretched by the same weight. It vibrates when _b_ vibrates; but how?
By dividing into two equal ventral segments. In this way alone can it
accommodate itself to the swifter vibrating period of _b_. Attached
to _c_, the same string separates into four, while when attached
to _d_, it divides into eight ventral segments. The number of the
ventral segments is proportional to the rapidity of vibration. It is
evident that we have here, in a more delicate form, a result which we
have already established in the case of our India-rubber tube set in
motion by the hand. It is also plain that this result might be deduced
theoretically from our first law.
We may extend the experiment. Here are two tuning-forks separated from
each other by the musical interval called a fifth. Attaching a string
to one of the forks, I stretch the string until it divides into two
ventral segments: attached to the other fork, and stretched by the
same weight, it divides instantly into three segments when the fork is
set in vibration. Now, to form the interval of a fifth, the vibrations
of the one fork must be to those of the other in the ratio of 2:3.
The division of the string, therefore, declares the interval. In, the
same way the division of the string in relation to all other musical
intervals may be illustrated.[37]
Again. Here are two tuning-forks, _a_ and _b_, one of which (_a_)
vibrates twice as rapidly as the other. A string of silk is attached to
_a_, and stretched until it synchronizes with the fork, and vibrates as
a whole. Here is a second string of the same length, formed by laying
four strands of the first one side by side. I attach this compound
thread to _b_, and, keeping the tension the same as in the last
experiment, set _b_ in vibration. The compound thread synchronizes with
_b_, and swings as a whole. Hence, as the fork _b_ vibrates with half
the rapidity of _a_, by quadrupling the weight of the string we halved
its rapidity of vibration. In the same simple way it might be proved
that by augmenting the weight of the string nine times we reduce the
number of its vibrations to one-third. We thus demonstrate the law:
_The rapidity of vibration is inversely proportional to the square root
of the weight of the string._
An instructive confirmation of this result is thus obtained: Attached
to this tuning-fork is a silk string six feet long. Two feet of the
string are composed of four strands of the single thread, placed side
by side; the remaining four feet are a single thread. A stretching
force is applied, which causes the string to divide into two ventral
segments. But how does it divide? Not at its centre, as is the case
when the string is of uniform thickness throughout, but at the precise
point where the thick string terminates. This thick segment, two feet
long, is now vibrating at the same rate as the thin segment four feet
long, a result which follows by direct deduction from the two laws
already established.
Here again are two strings of the same length and thickness. One of
them is attached to the fork _a_, the other to the fork _b_, which
vibrates with twice the rapidity of _a_. Stretched by a weight of 20
grains, the string attached to _b_ vibrates as a whole. Substituting
_b_ for _a_, a weight of 80 grains causes the string to vibrate as a
whole. Hence, to double the rapidity of vibration, we must quadruple
the stretching weight. In the same way it might be proved that to
treble the rapidity of vibration we should have to make the stretching
weight ninefold. Hence our third law:
_The rapidity of vibration is proportional to the square root of the
tension._
[Illustration: FIG. 50.]
Let us vary this experiment. This silk cord is carried from the
tuning-fork over the pulley, and stretched by a weight of 80 grains.
The string vibrates as a whole as at A, Fig. 50. By diminishing the
weight the string is relaxed, and finally divides sharply into two
ventral segments, as at B, Fig. 50. What is now the stretching
weight?—20 grains, or one-fourth of the first. With a stretching
weight of almost exactly 9 grains it divides into three segments,
as at C; while with a stretching weight of 5 grains it divides into
four segments, as at D. Thus then, a tension of one-fourth doubles, a
tension of one-ninth trebles, and a tension of one-sixteenth quadruples
the number of ventral segments. In general terms, the number of
segments is inversely proportional to the square root of the tension.
This result may be deduced by reasoning from our first and third laws,
and its realization here confirms their correctness.
Thus, by a series of reasonings and experiments totally different from
those formerly employed, we arrive at the self-same laws. In science,
different lines of reasoning often converge upon the same truth; and
if we only follow them faithfully, we are sure to reach that truth at
last. We may emerge, and often do emerge, from our reasoning with a
contradiction in our hands; but on retracing our steps, we infallibly
find the cause of the contradiction to be due, not to any lack of
constancy in Nature, but of accuracy in man. It is the millions of
experiences of this kind which science furnishes that give us our
present faith in the stability of Nature.
HARMONIC SOUNDS OR OVERTONES
§ 9. _Timbre; Klangfarbe; Clang-tint_
We now approach a portion of our subject which will subsequently
prove to be of the very highest importance. It has been shown by the
most varied experiments that a stretched string can either vibrate
as a whole, or divide itself into a number of equal parts, each of
which vibrates as an independent string. Now it is not possible to
sound the string as a whole without at the same time causing, to a
greater or less extent, its subdivision; that is to say, superposed
upon the vibrations of the whole string we have always, in a greater
or less degree, the vibrations of its aliquot parts. The higher notes
produced by these latter vibrations are called the _harmonics_ of the
string. And so it is with other sounding bodies; we have in all cases
a coexistence of vibrations. Higher tones mingle with the fundamental
one, and it is their intermixture which determines what, for want of
a better term, we call the _quality_ of the sound. The French call it
_timbre_, and the Germans call it _Klangfarbe_.[38] It is this union
of high and low tones that enables us to distinguish one musical
instrument from another. A clarinet and a violin, for example, though
tuned to the same fundamental note, are not confounded; the auxiliary
tones of the one are different from those of the other, and these
latter tones, uniting themselves to the fundamental tones of the two
instruments, destroy the identity of the sounds.
All bodies and instruments, then, employed for producing musical sounds
emit, besides their fundamental tones, others due to higher orders of
vibration. The Germans embrace all such sounds under the general term
_Obertöne_. I think it will be an advantage if we in England adopt the
term _overtones_ as the equivalent of the term employed in Germany. One
has occasion to envy the power of the German language to adapt itself
to requirements of this nature. The term _Klangfarbe_, for example,
employed by Helmholtz is exceedingly expressive, and we need its
equivalent also. Color depends upon rapidity of vibration, blue light
bearing to red the same relation that a high tone does to a low one. A
simple color has but one rate of vibration, and it may be regarded as
the analogue of a simple tone in music. A _tone_, then, may be defined
as the product of a vibration which cannot be decomposed into more
simple ones. A compound color, on the contrary, is produced by the
admixture of two or more simple ones, and an assemblage of tones, such
as we obtain when the fundamental tone and the harmonics of a string
sound together, is called by the Germans a _Klang_. May we not employ
the English word _clang_ to denote the same thing, and thus give the
term a precise scientific meaning akin to its popular one? And may we
not, like Helmholtz, add the word _color_ or _tint_, to denote the
character of the clang, using the term _clang-tint_ as the equivalent
of Klangfarbe?
With your permission I shall henceforth employ these terms; and now it
becomes our duty to look a little more closely than we have hitherto
done into the subdivision of a string into its harmonic segments. Our
monochord with its stretched wire is before you. The scale of the
instrument is divided into 100 equal parts. At the middle point of the
wire stands the number 50; at a point almost exactly one-third of its
length from its end stands the number 33; while at distances equal to
one-fourth and one-fifth of its length from its end stand the numbers
25 and 20 respectively. These numbers are sufficient for our present
purpose. When the wire is plucked at 50 you hear its clang, rather
hollow and dull. When plucked at 33, the clang is different. When
plucked at 25, the clang is different from either of the former. As
we retreat from the centre of the string, the clang-tint becomes more
“brilliant,” the sound more brisk and sharp. What is the reason of
these differences in the sound of the same wire?
The celebrated Thomas Young, once professor in this Institution,
enables us to solve the question. He proved that when any point of a
string is plucked, all the higher tones _which require that point for
a node_ vanish from the clang. Let me illustrate this experimentally.
I pluck the point 50, and permit the string to sound. It may be
proved that the first overtone, which corresponds to a division of
the string into two vibrating parts, is now absent from the clang.
If it were present, the damping of the point 50 would not interfere
with it, for this point would be its node. But on damping the point
50 the fundamental tone is quenched, and no octave of that tone is
heard. Along with the octave its whole progeny of overtones, with rates
of vibration four times, six times, eight times—all even numbers of
times—the rate of the fundamental tone, disappear from the clang. All
these tones require that a node should exist at the centre, where,
according to the principle of Young, it cannot now be formed. Let us
pluck some other point, say 25, and damp 50 as before. The fundamental
tone is now gone, but its octave, clear and full, rings in your ears.
The point 50 in this case not being the one plucked, a node can form
there; it _has_ formed, and the two halves of the string continue
to vibrate after the vibrations of the string as a whole have been
extinguished. Plucking the point 33, the second harmonic or overtone
is absent from the clang. This is proved by damping the point 33. If
the second harmonic were on the string this would not affect it, for 33
is its node. The fundamental is quenched, but no tone corresponding to
a division of the string into three vibrating parts is now heard. The
tone is not heard because it was never there.
All the overtones which depend on this division, those with six times,
nine times, twelve times the rate of vibration of the fundamental one,
are also withdrawn from the clang. Let us now pluck 20, damping 33 as
before. The second harmonic is not extinguished, but continues to sound
clearly and fully after the extinction of the fundamental tone. In
this case the point 33 not being that plucked, a node can form there,
and the string can divide itself into three parts accordingly. In like
manner, if 25 be plucked and then damped, the third harmonic is not
heard; but when a point between 25 and the end of the wire is plucked,
and the point 25 damped, the third harmonic is plainly heard. And thus
we might proceed, the general rule enunciated by Young, and illustrated
by these experiments, being, that when any point of a string is plucked
or struck, or, as Helmholtz adds, agitated with a bow, the harmonic
which requires that point for a node vanishes from the general clang of
the string.
§ 10. _Mingling of Overtones with Fundamental. The Æolian Harp_
You are now in a condition to estimate the influence which these
higher vibrations must have upon the quality of the tone emitted by
the string. The sounds which ring in your ears so plainly after the
fundamental tone is quenched mingled with that note before it was
extinguished. It seems strange that tones of such power could be
so masked by the fundamental one that even the disciplined ear of a
musician is unable to separate the one from the other. But Helmholtz
has shown that this is due to want of practice and attention. The
musician’s faculties were never exercised in this direction. There
are numerous effects which the musician can distinguish, because his
art demands the habit of distinguishing them. But it is no necessity
of his art to resolve the clang of an instrument into its constituent
tones. By attention, however, even the unaided ear can accomplish this,
particularly if the mind be informed beforehand what the ear has to
bend itself to find.
And this reminds me of an occurrence which took place in this room at
the beginning of my acquaintance with Faraday. I wished to show him a
peculiar action of an electro-magnet upon a crystal. Everything was
arranged, when just before the magnet was excited he laid his hand
upon my arm and asked, “What am I to look for?” Amid the assemblage
of impressions connected with an experiment, even this prince of
experimenters felt the advantage of having his attention directed to
the special point to be illustrated. Such help is the more needed
when we attempt to resolve into its constituent parts an effect so
intimately blended as the composite tones of a clang. When we desire
to isolate a particular tone, one way of helping the attention is to
sound that tone feebly on a string of the proper length. Thus prepared,
the ear glides more readily from the single tone to that of the same
pitch in a composite clang, and detaches it more readily from its
companions. In the experiments executed a moment ago, where our aim
in each respective case was to bring out the higher tone of the string
in all its power, we entirely extinguished its fundamental tone. It
may, however, be enfeebled without being destroyed. I pluck this string
at 33, and lay the feather lightly for a moment on the string at 50.
The fundamental tone is thereby so much lowered that its octave can
make itself plainly heard. By again touching the string at 50, the
fundamental tone is lowered still more; so that now its first harmonic
is more powerful than itself. You hear the sound of both, and you
might have heard them in the first instance by a sufficient stretch of
attention.
The harmonics of a string may be augmented or subdued within wide
limits. They may, as we have seen, be masked by the fundamental
tone, and they may also effectually mask it. A stroke with a hard
body is favorable, while a stroke with a soft body is unfavorable
to their development. They depend, moreover, on the promptness with
which the body striking the string retreats after striking. Thus they
are influenced by the weight and elasticity of the hammers in the
pianoforte. They also depend upon the place at which the shock is
imparted. When, for example, a string is struck in the centre, the
harmonics are less powerful than when it is struck near one end.
Helmholtz, who is equally eminent as a mathematician and as an
experimental philosopher, has calculated the theoretic intensity of
the harmonics developed in various ways; that is to say, the actual
_vis viva_ or energy of the vibration, irrespective of its effects
upon the ear. A single example given by him will suffice to illustrate
this subject. Calling the intensity of the fundamental tone, in each
case, 100, that of the second harmonic, when the string was simply
pulled aside at a point one-seventh of its length from its end and then
liberated, was found to be 56·1, or a little better than one-half. When
the string was struck with the hammer of a pianoforte, whose contact
with the string endured for three-sevenths of the period of vibration
of the fundamental tone, the intensity of the same tone was 9. In
this case the second harmonic was nearly quenched. When, however, the
duration of contact was diminished to three-twentieths of the period of
the fundamental, the intensity of the harmonic rose to 357; while, when
the string was sharply struck with a very hard hammer, the intensity
mounted to 505, or to more than quintuple that of the fundamental
tone.[39] Pianoforte manufacturers have found that the most pleasing
tone is excited by the middle strings of their instruments, when the
point against which the hammer strikes is from one-seventh to one-ninth
of the length of the wire from its extremity.
Why should this be the case? Helmholtz has given the answer. Up to
the tones which require these points as nodes the overtones all form
chords with the fundamental; but the sixth and eighth overtones of
the wire do not enter into such chords; they are dissonant tones, and
hence the desirability of doing away with them. This is accomplished by
making the point at which a node is required that on which the hammer
falls. The possibility of the tone forming is thereby shut out, and its
injurious effect is avoided.
The strings of the Æolian harp are divided into harmonic parts by a
current of air passing over them. The instrument is usually placed in
a window between the sash and frame, so as to leave no way open to the
entrance of the air except over the strings. Sir Charles Wheatstone
recommends the stretching of a first violin-string at the bottom of a
door which does not closely fit. When the door is shut, the current
of air entering beneath sets the string in vibration, and when a fire
is in the room, the vibrations are so intense that a great variety
of sounds are simultaneously produced.[40] A gentleman in Basel once
constructed with iron wires a large instrument which he called the
weather-harp or giant-harp, and which, according to its maker, sounded
as the weather changed. Its sounds were also said to be evoked by
changes of terrestrial magnetism. Chladni pointed out the error of
these notions, and reduced the action of the instrument to that of the
wind upon its strings.
§ 11. _Young’s Optical Illustrations_
Finally, with regard to the vibrations of a wire, the experiments
of Dr. Young, who was the first to employ optical methods in such
experiments, must be mentioned. He allowed a sheet of sunlight to
cross a pianoforte-wire, and obtained thus a brilliant dot. Striking
the wire he caused it to vibrate, the dot described a luminous line
like that produced by the whirling of a burning coal in the air, and
the form of this line revealed the character of the vibration. It was
rendered manifest by these experiments that the oscillations of the
wire were not confined to a single plane, but that it described in its
vibrations curves of greater or less complexity. Superposed upon the
vibration of the whole string were partial vibrations, which revealed
themselves as loops and sinuosities. Some of the lines observed by Dr.
Young are given in Fig. 51. Every one of these figures corresponds to a
distinct impression made by the wire upon the surrounding air. The form
of the sonorous wave is affected by these superposed vibrations, and
thus they influence the clang-tint or quality of the sound.
[Illustration: FIG. 51.]
SUMMARY OF CHAPTER III
The amount of motion communicated by a vibrating string to the air is
too small to be perceived as sound, even at a small distance from the
string.
When a broad surface vibrates in air, condensations and rarefactions
are more readily formed than when the vibrating body is of small
dimensions like a string. Hence, when strings are employed as sources
of musical sounds, they are associated with surfaces of larger area
which take up their vibrations, and transfer them to the surrounding
air.
Thus the tone of a harp, a piano, a guitar, or a violin, depends mainly
upon the sound-board of the instrument.
The following four laws regulate the vibrations of strings: The rate
of vibrations is inversely proportional to the length; it is inversely
proportional to the diameter; it is directly proportional to the
square root of the stretching weight or tension; and it is inversely
proportional to the square root of the density of the string.
When strings of different diameters and densities are compared, the law
is, that the rate of vibration is inversely proportional to the square
root of the weight of the string.
When a stretched rope, or an India-rubber tube filled with sand, with
one of its ends attached to a fixed object, receives a jerk at the
other end, the protuberance raised upon the tube runs along it as a
pulse to the fixed end, and, being there reflected, returns to the hand
by which the jerk was imparted.
The time required for the pulse to travel from the hand to the fixed
end of the tube and back is that required by the whole tube to execute
a complete vibration.
When a series of pulses are sent in succession along the tube, the
direct and reflected pulses meet, and by their coalescence divide the
tube into a series of vibrating parts, called _ventral segments_, which
are separated from each other by points of apparent rest called _nodes_.
The number of ventral segments is directly proportional to the rate of
vibration at the free end of the tube.
The hand which produces these vibrations may move through less than an
inch of space; while by the accumulation of its impulses the amplitude
of the ventral segments may amount to several inches, or even to
several feet.
If an India-rubber tube, fixed at both ends, be encircled at its centre
by the finger and thumb, when either of its halves is pulled aside and
liberated, both halves are thrown into a state of vibration.
If the tube be encircled at a point one-third, one-fourth, or one-fifth
of its length from one of its ends, on pulling the shorter segment
aside and liberating it, the longer segment divides itself into two,
three, or four vibrating parts, separated from each other by nodes.
The number of vibrating segments depends upon the rate of vibration at
the point encircled by the finger and thumb.
Here also the amplitude of vibration at the place encircled by the
finger and thumb may not be more than a fraction of an inch, while the
amplitude of the ventral segments may amount to several inches.
A musical string damped by a feather at a point one-half, one-third,
one-fourth, one-fifth, etc., of its length from one of its ends, and
having its shorter segment agitated, divides itself exactly like the
India-rubber tube. Its division may be rendered apparent by placing
little paper riders across it. Those placed at the ventral segments are
thrown off, while those placed at the nodes retain their places.
The notes corresponding to the division of a string into its aliquot
parts are called the _harmonics_ of the string.
When a string vibrates as a whole, it usually divides at the same time
into its aliquot parts. Smaller vibrations are superposed upon the
larger; the tones corresponding to those smaller vibrations, and which
we have agreed to call overtones, mingling at the same time with the
fundamental tone of the string.
The addition of these overtones to the fundamental tone determines the
_timbre_ or _quality_ of the sound, or, as we have agreed to call it,
the _clang-tint_.
It is the addition of such overtones to fundamental tones of the
same pitch which enables us to distinguish the sound of a clarionet
from that of a flute, and the sound of a violin from both. Could the
pure fundamental tones of these instruments be detached, they would
be indistinguishable from each other; but the different admixture
of overtones in the different instruments renders their clang-tints
diverse, and therefore distinguishable.
Instead of the heavy India-rubber tube in the experiment above referred
to, we may employ light silk strings, and, instead of the vibrating
hand, we may employ vibrating tuning-forks, and cause the strings to
swing as a whole, or to divide themselves into any number of ventral
segments. Effects of great beauty are thus obtained, and by experiments
of this character all the laws of vibrating strings may be demonstrated.
When a stretched string is plucked aside or agitated by a bow, all the
overtones which require the agitated point for a node vanish from the
clang of the string.
The point struck by the hammer of the piano is from one-seventh to
one-ninth of the length of the string from its end: by striking this
point, the notes which require it as a node cannot be produced, a
source of dissonance being thus avoided.