Law of Vibratory Motions in Water and Air—Superposition of
Vibrations—Interference of Sonorous Waves—Destruction of
Sound by Sound—Combined Action of Two Sounds nearly in Unison
with each other—Theory of Beats—Optical Illustration of the
Principle of Interference—Augmentation of Intensity by Partial
Extinction of Vibrations—Resultant Tones—Conditions of their
Production—Experimental Illustrations—Difference-Tones and
Summation-Tones—Theories of Young and Helmholtz
§ 1. _Interference of Water-Waves_
From a boat in Cowes Harbor, in moderate weather, I have often watched
the masts and ropes of the ships, as mirrored in the water. The images
of the ropes revealed the condition of the surface, indicating by
long and wide protuberances the passage of the larger rollers, and,
by smaller indentations, the ripples which crept like parasites over
the sides of the larger waves. The sea was able to accommodate itself
to the requirements of all its undulations, great and small. When the
surface was touched with an oar, or when drops were permitted to fall
from the oar into the water, there was also room for the tiny wavelets
thus generated. This carving of the surface by waves and ripples had
its limit only in my powers of observation; every wave and every ripple
asserted its right of place, and retained its individual existence,
amid the crowd of other motions which agitated the water.
The law that rules this chasing of the sea, this crossing and
intermingling of innumerable small waves, is _that the resultant
motion of every particle of water is the sum of the individual motions
imparted to it_. If a particle be acted on at the same moment by two
impulses, both of which tend to raise it, it will be lifted by a force
equal to the sum of both. If acted upon by two impulses, one of which
tends to raise it, and the other to depress it, it will be acted upon
by a force equal to the difference of both. When, therefore, the sum
of the motions is spoken of, the _algebraic sum_ is meant—the motions
which tend to raise the particle being regarded as positive, and those
which tend to depress it as negative.
When two stones are cast into smooth water, 20 or 30 feet apart,
round each stone is formed a series of expanding circular waves,
every one of which consists of a ridge and a furrow. The waves touch,
cross each other, and carve the surface into little eminences and
depressions. Where ridge coincides with ridge, we have the water raised
to a double height; where furrow coincides with furrow, we have it
depressed to a double depth; where ridge coincides with furrow, we
have the water reduced to its average level. The resultant motion of
the water at every point is, as above stated, the algebraic sum of the
motions impressed upon that point. And if, instead of two sources of
disturbance, we had ten, or a hundred, or a thousand, the consequence
would be the same; the actual result might transcend our powers of
observation, but the law above enunciated would still hold good.
Instead of the intersection of waves from two distinct centres of
disturbance, we may cause direct and reflected waves, from the same
centre, to cross each other. Many of you know the beauty of the effects
produced when light is reflected from ripples of water. When mercury
is employed the effect is more brilliant still. Here, by a proper
mode of agitation, direct and reflected waves may be caused to cross
and interlace, and by the most wonderful self-analysis to untie their
knotted scrolls. The adjacent figure (Fig. 149), which is copied from
the excellent “Wellenlehre” of the brothers Weber, will give some idea
of the beauty of these effects. It represents the chasing produced by
the intersection of direct and reflected water-waves in a circular
vessel, the point of disturbance (marked by the smallest circle in the
figure) being midway between the centre and the circumference.
[Illustration: FIG. 149.]
This power of water to accept and transmit multitudinous impulses is
shared by air, which concedes the right of space and motion to any
number of sonorous waves. The same air is competent to accept and
transmit the vibrations of a thousand instruments at the same time.
When we try to visualize the motion of that air—to present to the eye
of the mind the battling of the pulses direct and reverberated—the
imagination retires baffled from the attempt. Still, amid all the
complexity, the law above enunciated holds good, every particle of air
being animated by a resultant motion, which is the algebraic sum of all
the individual motions imparted to it. And the most wonderful thing of
all is, that the human ear, though acted on only by a cylinder of that
air, which does not exceed the thickness of a quill, can detect the
components of the motion, and, by an act of attention, can even isolate
from the aërial entanglement any particular sound.
§ 2. _Interference of Sound_
When two unisonant tuning-forks are sounded together, it is easy to see
that the forks may so vibrate that the condensations of the one shall
coincide with the condensations of the other, and the rarefactions
of the one with the rarefactions of the other. If this be the case,
the two forks will assist each other. The condensations will, in
fact, become more condensed, the rarefactions more rarefied; and as
it is upon the difference of density between the condensations and
rarefactions that _loudness_ depends, the two vibrating forks, thus
supporting each other, will produce a sound of greater intensity than
that of either of them vibrating alone.
It is, however, also easy to see that the two forks may be so related
to each other that one of them shall require a condensation at the
place where the other requires a rarefaction; that the one fork shall
urge the air-particles forward, while the other urges them backward.
If the opposing forces be equal, particles so solicited will move
neither backward nor forward, the aërial rest which corresponds to
silence being the result. Thus it is possible, by adding the sound of
one fork to that of another, to abolish the sounds of both. We have
here a phenomenon which, above all others, characterizes wave-motion.
It was this phenomenon, as manifested in optics, that led to the
undulatory theory of light, the most cogent proof of that theory being
based upon the fact that, by adding light to light, we may produce
darkness, just as we can produce silence by adding sound to sound.
[Illustration: FIG. 150.]
During the vibration of a tuning-fork the distance between the two
prongs is alternately increased and diminished. Let us call the motion
which increases the distance the _outward swing_, and that which
diminishes the distance the _inward swing_ of the fork. And let us
suppose that our two forks, A and B, Fig. 150, reach the limits of
their outward swing and their inward swing at the same moment. In this
case the _phases_ of their motion, to use the technical term, are
the same. For the sake of simplicity we will confine our attention
to the right-hand prongs, A and B, of the two forks, neglecting the
other two prongs; and now let us ask what must be the distance between
the prongs A and B, when the condensations and rarefactions of both,
indicated respectively by the dark and light shading, coincide? A
little reflection will make it clear that if the distance from B to A
be equal to the length of a whole sonorous wave, coincidence between
the two systems of waves must follow. The same would evidently occur
were the distance between A and B two wave-lengths, three wave-lengths,
four wave-lengths—in short, any number of whole wavelengths. In all
such cases we shall have coincidence of the two systems of waves, and
consequently a reinforcement of the sound of the one fork by that of
the other. Both the condensations and rarefactions between A and C are,
in this case, more pronounced than they would be if either of the forks
were suppressed.
[Illustration: FIG. 151.]
But if the prong B be only half the length of a wave behind A, what
must occur? Manifestly the rarefactions of one of the systems of waves
will then coincide with the condensations of the other system, the
air to the right of A being reduced to quiescence. This is shown in
Fig. 151, where the uniformity of shading indicates an absence both
of condensations and rarefactions. When B is two half wave-lengths
behind A, the waves, as already explained, support each other; when
they are three half wave-lengths apart, they destroy each other. Or
expressed generally, we have augmentation or destruction according as
the distance between the two prongs amounts to an even or an odd number
of semi-undulations. Precisely the same is true of the waves of light.
If through any cause one system of ethereal waves be any _even_ number
of semi-undulations behind another system, the two systems support each
other when they coalesce, and we have more light. If the one system be
any _odd_ number of semi-undulations behind the other, they oppose each
other, and a destruction of light is the result of their coalescence.
The action here referred to, both as regards sound and light is called
_Interference_.
§ 3. _Experimental Illustrations_
[Illustration: FIG. 152.]
Sir John Herschel was the first to propose to divide a stream of
sound into two branches, of different lengths, causing the branches
afterward to reunite, and interfere with each other. This idea has
been recently followed out with success by M. Quincke; and it has
been still further improved upon by M. König. The principle of these
experiments will be at once evident from Fig. 152. The tube _o f_
divides into two branches at _f_, the one branch being carried round
_n_, and the other round _m_. The two branches are caused to reunite
at _g_, and to end in a common canal, _g p_. The portion _b n_ of
the tube which slides over _a b_ can be drawn out as shown in the
figure, and thus the sound-waves can be caused to pass over different
distances in the two branches. Placing a vibrating tuning-fork at _o_,
and the ear at _p_, when the two branches are of the same length, the
waves through both reach the ear together, and the sound of the fork
is heard. Drawing _n b_ out, a point is at length obtained where the
sound of the fork is extinguished. This occurs when the distance _a
b_ is one-fourth of a wave-length; or, in other words, when the whole
right-hand branch is half a wave-length longer than the left-hand one.
Drawing _b n_ still further out, the sound is again heard; and when
twice the distance _a b_ amounts to a whole wave-length, it reaches a
maximum. Thus, according as the difference of both branches amounts to
half a wave-length, or to a whole wave-length, we have reinforcement
or destruction of the two series of sonorous waves. In practice, the
tube _o f_ ought to be prolonged until the direct sound of the fork is
unheard, the attention of the ear being then wholly concentrated on the
sounds that reach it through the tube.
It is quite plain that the wave-length of any simple tone may be
readily found by this instrument. It is only necessary to ascertain the
difference of path which produces complete interference. Twice this
difference is the wave-length; and if the rate of vibration be at the
same time known, we can immediately calculate the velocity of sound in
air.
Each of the two forks now before you executes exactly 256 vibrations
in a second. Sounded together, they are in unison. Loading one of
them with a bit of wax, it vibrates a little more slowly than its
neighbor. The wax, say, reduces the number of vibrations to 255
in a second; how must their waves affect each other? If they start
at the same moment, condensation coinciding with condensation, and
rarefaction with rarefaction, it is quite manifest that this state of
things cannot continue. At the 128th vibration their phases are in
complete opposition, one of them having gained half a vibration on
the other. Here the one fork generates a condensation where the other
generates a rarefaction; and the consequence is, that the two forks,
at this particular point, completely neutralize each other. From this
point onward, however, the forks support each other more and more,
until, at the end of a second, when the one has completed its 255th,
and the other its 256th vibration, condensation again coincides with
condensation, and rarefaction with rarefaction, the full effect of both
sounds being produced upon the ear.
It is quite manifest that under these circumstances we cannot have the
continuous flow of perfect unison. We have, on the contrary, alternate
reinforcements and diminutions of the sound. We obtain, in fact, the
effect known to musicians by the name of _beats_, which, as here
explained, are a result of interference.
I now load this fork still more heavily, by attaching a fourpenny-piece
to the wax; the coincidences and interferences follow each other more
rapidly than before; we have a quicker succession of beats. In our
last experiment, the one fork accomplished one vibration more than the
other in a second, and we had a single beat in the same time. In the
present case, one fork vibrates 250 times, while the other vibrates
256 times in a second, and the number of beats per second is 6. A
little reflection will make it plain that in the interval required
by the one fork to execute one vibration more than the other, a beat
must occur; and inasmuch as, in the case now before us, there are
six such intervals in a second, there must be six beats in the same
time. In short, _the number of beats per second is always equal to the
difference between the two rates of vibration_.
§ 4. _Interference of Waves from Organ-pipes_
[Illustration: FIG. 153.]
Beats may be produced by all sonorous bodies. These two tall
organ-pipes, for example, when sounded together, give powerful
beats, one of them being slightly longer than the other. Here are
two other pipes, which are now in perfect unison, being exactly of
the same length. But it is only necessary to bring the finger near
the embouchure of one of the pipes, Fig. 153, to lower its rate of
vibration, and produce loud and rapid beats. The placing of the
hand over the open top of one of the pipes also lowers its rate of
vibration, and produces beats, which follow each other with augmented
rapidity as the top of the pipe is closed more and more. By a stronger
blast the first two harmonics of the pipes are brought out. These
higher notes also interfere, and you have these quicker beats.
[Illustration: FIG. 154.]
No more beautiful illustration of this phenomenon can be adduced than
that furnished by two sounding-flames. Two such flames are now before
you, the tube surrounding one of them being provided with a telescopic
slider, Fig. 154. There are, at present, no beats, because the tubes
are not sufficiently near unison. I gradually lengthen the shorter tube
by raising its slider. Rapid beats are now heard; now they are slower;
now slower still; and now both flames sing together in perfect unison.
Continuing the upward motion of the slider, I make the tube too long;
the beats begin again, and quicken, until finally their sequence is
so rapid as to appeal only as roughness to the ear. The flames, you
observe, dance within their tubes in time to the beats. As already
stated, these beats cause a silent flame within a tube to quiver when
the voice is thrown to a proper pitch, and when the position of the
flame is rightly chosen, the beats set it singing. With the flames of
large rose-burners, and with tin tubes from 3 to 9 feet long, we obtain
beats of exceeding power.
[Illustration: FIG. 155.]
You have just heard the beats produced by two tall organ-pipes nearly
in unison with each other. Two other pipes are now mounted on our
wind-chest, Fig. 155, each of which, however, is provided at its centre
with a membrane intended to act upon a flame.[70] Two small tubes lead
from the spaces closed by the membranes, and unite afterward, the
membranes of both the organ-pipes being thus connected with the same
flame. By means of the sliders, _s s′_, near the summits of the pipes,
they are either brought into unison or thrown out of it at pleasure.
They are not at present in unison, and the beats they produce follow
each other with great rapidity. The flame connected with the central
membranes dances in time to the beats. When brought nearer to unison,
the beats are slower, and the flame at successive intervals withdraws
its light and appears to exhale it. A process which reminds you of the
inspiration and expiration of the breath is thus carried on by the
flame. If the mirror, M, be now turned, the flame produces a luminous
band—continuous at certain places, but for the most part broken into
distinct images of the flame. The continuous parts correspond to the
intervals of interference where the two sets of vibrations abolish each
other.
Instead of permitting both pipes to act upon the same flame, we may
associate a flame with each of them. The deportment of the flames is
then very instructive. Imagine both flames to be in the same vertical
line, the one of them being exactly under the other. Bringing the
pipes into unison, and turning the mirror, we resolve each flame into
a chain of images, but we notice that the images of the one occupy the
spaces _between_ the images of the other. The periods of extinction of
the one flame, therefore, correspond to the periods of kindling of the
other. The experiment proves that, when two unisonant pipes are placed
thus close to each other, their vibrations are in opposite phases. The
consequence of this is, that the two sets of vibrations permanently
neutralize each other, so that at a little distance from the pipes you
fail to hear the fundamental tone of either. For this reason we cannot,
with any advantage, place close to each other in an organ several pipes
of the same pitch.
§ 5. _Lissajous’s Illustration of Beats of Two Tuning-forks_
[Illustration: FIG. 156.]
In the case of beats, the amplitude of the oscillating air reaches
a maximum and a minimum periodically. By the beautiful method of M.
Lissajous we can illustrate optically this alternate augmentation and
diminution of amplitude. Placing a large tuning-fork, T′, Fig. 156,
in front of the lamp L, a luminous beam is received upon the mirror
attached to the fork. This is reflected back to the mirror of a second
fork, T, and by it thrown on to the screen, where it forms a luminous
disk. When the bow is drawn over the fork T′, the beam, as in the
experiments described in the second chapter, is tilted up and down, the
disk upon the screen stretching to a luminous band three feet long.
If, in drawing the bow over this second fork, the vibrations of both
coincide in phase, the band will be lengthened; if the phases are in
opposition, total or partial neutralization of the one fork by the
other will be the result. It so happens that in the present instance
the second fork adds something to the action of the first, the band
of light being now four feet long. These forks have been tuned as
perfectly as possible. Each of them executes exactly 64 vibrations in
a second; the initial relation of their phases remains, therefore,
constant, and hence you notice a gradual shortening of the luminous
band, like that observed during the subsidence of the vibrations of a
single fork. The band at length dwindles to the original disk, which
remains motionless upon the screen.
By attaching, with wax, a threepenny-piece to the prong of one of
these forks, its rate of vibration is lowered. The phases of the
two forks cannot now retain a constant relation to each other. One
fork incessantly gains upon the other, and the consequence is that
sometimes the phases of both coincide, and at other times they are in
opposition. Observe the result. At the present moment the two forks
conspire, and we have a luminous band four feet long upon the screen.
This slowly contracts, drawing itself up to a mere disk; but the action
halts here only during the moment of opposition. That passed, the
forks begin again to assist each other, and the disk once more slowly
stretches into a band. The action here is very slow; but it may be
quickened by attaching a sixpence to the loaded fork. The band of light
now stretches and contracts in perfect rhythm. The action, rendered
thus optically evident, is impressed upon the air of this room; its
particles alternately vibrate and come to rest, and, as a consequence,
beats are heard in synchronism with the changes of the figure upon the
screen.
The time which elapses from maximum to maximum, or from minimum to
minimum, is that required for the one fork to perform one vibration
more than the other. At present this time is about two seconds. In two
seconds, therefore, one beat occurs. When we augment the dissonance
by increasing the load, the rhythmic lengthening and shortening of
the band is more rapid, while the intermittent hum of the forks is
more audible. There are now six elongations and shortenings in the
interval taken up a moment ago by one; the beats at the same time being
heard at the rate of three a second. By loading the forks still more,
the alternations may be caused to succeed each other so rapidly that
they can no longer be followed by the eye, while the beats, at the
same time, cease to be individually distinct, and appeal as a kind of
roughness to the ear.
[Illustration: FIG. 157.]
In the experiments with a single tuning-fork, already described (Fig.
22, Chapter II.), the beam reflected from the fork was received on
a looking-glass, and, by turning the glass, the band of light on
the screen was caused to stretch out into a long wavy line. It was
explained at the time that the loudness of the sound depended on the
depth of the indentations. Hence, if the band of light of varying
length now before us on the screen be drawn out in a sinuous line,
the indentations ought to be at some places deep, while at others
they ought to vanish altogether. This is the case. By a little tact
the mirror of the fork T (Fig. 156) is caused to turn through a small
angle, a sinuous line composed of swellings and contractions (Fig. 157)
being drawn upon the screen. The swellings correspond to the periods
of sound, and the contractions to those of silence.[71]
Two vibrating bodies, then, each of which separately produces a musical
sound, can, when acting together, neutralize each other. Hence, by
quenching the vibrations of one of them, we may give sonorous effect to
the other. It often happens, for instance, that when two tuning-forks,
on their resonant cases, are vibrating in unison, the stoppage of one
of them is accompanied by an augmentation of the sound. This point
may be further illustrated by the vibrating bell, already described
(Fig. 78, Chapter IV.) Placing its resonant tube in front of one of
its nodes, a sound is heard, but nothing like what is heard when the
tube is opposed to a ventral segment. The reason of this is that the
vibrations of a bell on the opposite sides of a nodal line are in
opposite directions, and they therefore interfere with each other.
By introducing a glass plate between the bell and the tube, the
vibrations on one side of the nodal line may be intercepted; an instant
augmentation of the sound is the consequence.
§ 6. _Interference of Waves from a Vibrating Disk. Hopkins’s and
Lissajous’s Illustrations_
In a vibrating disk every two adjacent sectors move at the same time
in opposite directions. When the one sector rises the other falls, the
nodal line marking the limit between them. Hence, at the moment when
any sector produces a condensation in the air above it, the adjacent
sector produces a rarefaction in the same air. A partial destruction
of the sound of one sector by the other is the result. You will now
understand the instrument by which the late William Hopkins illustrated
the principle of interference. The tube A B, Fig. 158, divides at B
into two branches. The end A of the tube is closed by a membrane.
Scattering sand upon this membrane, and holding the ends of the
branches over _adjacent_ sectors of a vibrating disk, no motion (or, at
least, an extremely feeble motion) of the sand is perceived. Placing
the ends of the two branches over _alternate_ sectors of the disk, the
sand is tossed from the membrane, proving that in this case we have
coincidence of vibration on the part of the two sectors.
[Illustration: FIG. 158.]
[Illustration: FIG. 159.]
We are now prepared for a very instructive experiment, which we owe to
M. Lissajous. Drawing a bow over the edge of a brass disk, I divide it
into six vibrating sectors. When the palm of the hand is brought over
any one of them, the sound, instead of being diminished, is augmented.
When two hands are placed over two _adjacent_ sectors, you notice
no increase of the sound; but when they are placed over _alternate_
sectors, as in Fig. 159, a striking augmentation of the sound is
the consequence. By simply lowering and raising the hands, marked
variations of intensity are produced. By the approach of the hands
the vibrations of the two sectors are intercepted; their interference
right and left being thus abolished, the remaining sectors sound more
loudly. Passing the single hand to and fro along the surface, you also
hear a rise and fall of the sound. It rises when the hand is over a
vibrating sector; it falls when the hand is over a nodal line. Thus,
by sacrificing a portion of the vibrations, we make the residue more
effectual. Experiments similar to these may be made with light and
radiant heat. If of two beams of the former, which destroy each other
by interference, one be removed, light takes the place of darkness;
and if of two interfering beams of the latter one be intercepted, heat
takes the place of cold.
§ 7. _Quenching the Sound of one Prong of a Tuning-fork by that of the
other_
You have remarked the almost total absence of sound on the part of a
vibrating tuning-fork when held free in the hand. The feebleness of the
fork as a sounding body arises in part from interference. The prongs
always vibrate in opposite directions, one producing a condensation
where the other produces a rarefaction, a destruction of sound being
the consequence. By simply passing a pasteboard tube over one of the
prongs of the fork, its vibrations are in part intercepted, and an
augmentation of the sound is the result. The single prong is thus
proved to be more effectual than the two prongs. There are positions in
which the destruction of the sound of one prong by that of the other
is _total_. These positions are easily found by striking the fork and
turning it round before the ear. When the back of the prong is parallel
to the ear, the sound is heard; when the side surfaces of both prongs
are parallel to the ear, the sound is also heard; but when the _corner_
of a prong is carefully presented to the ear, the sound is utterly
destroyed. During one complete rotation of the fork we find four
positions where the sound is thus obliterated.
[Illustration: FIG. 160.]
Let _s s_ (Fig. 160) represent the two ends of the tuning-fork, looked
down upon as it stands upright. When the ear is placed at _a_ or _b_,
or at _c_ or _d_, the sound is heard. Along the four dotted lines,
on the contrary, the waves generated by the two prongs completely
neutralize each other, and nothing is there heard. These lines have
been proved by Weber to be hyperbolic curves; and this must be their
character according to the principle of interference.
This remarkable case of interference, which was first noticed by Dr.
Thomas Young, and thoroughly investigated by the brothers Weber, may
be rendered audible by means of resonance. Bringing a vibrating fork
over a jar which resounds to it, and causing the fork to rotate slowly,
in four positions we have a loud resonance; in four others absolute
silence, alternate risings and fallings of the sound accompanying
the fork’s rotation. While the fork is over the jar with its corner
downward and the sound entirely extinguished, let a pasteboard tube
be passed over one of its prongs, as in Fig. 161, a loud resonance
announces the withdrawal of the vibrations of that prong. To obtain
this effect, the fork must be held over the centre of the jar, so that
the air shall be symmetrically distributed on both sides of it. Moving
the fork from the centre toward one of the sides, without altering its
inclination in the least, we obtain a forcible sound. Interference,
however, is also possible near the side of the jar. Holding the fork,
not with its corner downward, but with both its prongs in the same
horizontal plane, a position is soon found near the side of the jar
where the sound is extinguished. In passing completely from side to
side over the mouth of the jar, two such places of interference are
discoverable.
[Illustration: FIG. 161.]
[Illustration: FIG. 162.]
A variety of experiments will suggest themselves to the reflecting
mind, by which the effect of interference may be illustrated. It is
easy, for example, to find a jar which resounds to a vibrating plate.
Such a jar, placed over a vibrating segment of the plate, produces a
powerful resonance. Placed over a nodal line, the resonance is entirely
absent; but if a piece of pasteboard be interposed between the jar and
plate, so as to cut off the vibrations on one side of the nodal line,
the jar instantly resounds to the vibrations of the other. Again,
holding two forks, which vibrate with the same rapidity, over two
resonant jars, the sound of both flows forth in unison. When a bit of
wax is attached to one of the forks, powerful beats are heard. Removing
the wax, the unison is restored. When one of these unisonant forks is
placed in the flame of a spirit-lamp its elasticity is changed, and it
produces long loud beats with its unwarmed fellow.[72] If while one
of the forks is sounding on its resonant case the other be excited
and brought near the mouth of the case, as in Fig. 162, loud beats
declare the absence of unison. Dividing a jar by a vertical diaphragm,
and bringing one of the forks over one of its halves, and the other
fork over the other; the two semi-cylinders of air produce beats by
their interference. But the diaphragm is not necessary; on removing
it, the beats continue as before, one-half of the same column of air
interfering with the other.[73]
The intermittent sound of certain bells, heard more especially when
their tones are subsiding, is an effect of interference. The bell,
through lack of symmetry, as explained in the fourth chapter, vibrates
in one direction a little more rapidly than in the other, and beats
are the consequence of the coalescence of the two different rates of
vibration.
RESULTANT TONES
We have now to turn from this question of interference to the
consideration of a new class of musical sounds, of which the beats were
long considered to be the progenitors. The sounds here referred to
require for their production the union of two distinct musical tones.
Where such union is effected, under the proper conditions, _resultant
tones_ are generated, which are quite distinct from the primaries
concerned in their production. They were discovered, in 1745, by a
German organist named Sorge, but the publication of the fact attracted
little attention. They were discovered independently, in 1754, by the
celebrated Italian violinist Tartini, and after him have been called
Tartini’s tones.
To produce them it is desirable, if not necessary, to have the two
primary tones of considerable intensity. Helmholtz prefers the siren
to all other means of exciting them, and with this instrument they
are very readily obtained. It requires some attention at first, on
the part of the listener, to single out the resultant tone from the
general mass of sound; but, with a little practice, this is readily
accomplished; and though the unpracticed ear may fail, in the first
instance, thus to analyze the sound, the clang-tint is influenced in an
unmistakable manner by the admixture of resultant tones. I set Dove’s
siren in rotation, and open two series of holes at the same time; with
the utmost strain of attention, I am as yet unable to hear the least
symptom of a resultant tone. Urging the instrument to greater rapidity,
a dull, low droning mingles with the two primary sounds. Raising the
speed of rotation, the low, resultant tone rises rapidly in pitch, and
now, to those who stand close to the instrument, it is very audible.
The two series of holes here open number 8 and 12 respectively. The
resultant tone is in this case an octave below the deepest of the two
primaries. Opening two other series of orifices, numbering 12 and
16 respectively, the resultant tone is quite audible. Its rate of
vibration is one-third of the rate of the deepest of the two primaries.
In all cases, _the resultant tone is that which corresponds to a rate
of vibration equal to the difference of the rates of the two primaries_.
The resultant tone here spoken of is that actually heard in the
experiment. But with finer methods of experiment other resultant tones
are proved to exist. Those on which we have now fixed our attention
are, however, the most important. They are called _difference-tones_ by
Helmholtz, in consequence of the law just mentioned.
To bring these resultant tones audibly forth, the primaries must,
as already stated, be forcible. When they are feeble the resultants
are unheard. I am acquainted with no method of exciting these tones
more simple and effectual than a pair of suitable singing-flames.
Two such flames may be caused to emit powerful notes—self-created,
self-sustained, and requiring no muscular effort on the part of the
observer to keep them going. Here are two of them. The length of the
shorter of the two tubes surrounding these flames is 10-3/8 inches,
that of the other is 11·4 inches. I hearken to the sound, and in the
midst of the shrillness detect a very deep resultant tone. The reason
of its depth is manifest: the two tubes being so nearly alike in
length, the difference between their vibrations is small, and the note
corresponding to this difference, therefore, low in pitch. Lengthening
one of the tubes by means of its slider, the resultant tone rises
gradually, and now it swells surprisingly. When the tube is shortened
the resultant tone falls, and thus, by alternately raising and lowering
the slider, the resultant tone is caused to rise and sink in accordance
with the law which makes the number of its vibrations the difference
between the number of its two primaries.
We can determine, with ease, the actual number of vibrations
corresponding to any one of those resultant tones. The sound of
the flame is that of the open tube which surrounds it, and we have
already learned (Chapter III.) that the length of such a tube is half
that of the sonorous wave it produces. The wave-length, therefore,
corresponding to our 10-3/8-inch tube is 20-3/4 inches. The velocity
of sound in air of the present temperature is 1,120 feet a second.
Bringing these feet to inches, and dividing by 20-3/4, we find the
number of vibrations corresponding to a length of 10-3/8 inches to be
648 per second.
But it must not be forgotten here that the air in which the vibrations
are actually executed is much more elastic than the surrounding
air. The flame heats the air of the tube, and the vibrations must,
therefore, be executed more rapidly than they would be in an ordinary
organ-pipe of the same length. To determine the actual number of
vibrations, we must fall back upon our siren; and with this instrument
it is found that the air within the 10-3/8 inch tube executes 717
vibrations in a second. The difference of 69 vibrations a second is
due to the heating of the aërial column. Carbonic acid and aqueous
vapor are, moreover, the product of the flame’s combustion, and their
presence must also affect the rapidity of the vibration.
Determining in the same way the rate of vibration of the 11·4-inch
tube, we find it to be 667 per second; the difference between
this number and 717 is 50, which expresses the rate of vibration
corresponding to the first deep resultant tone.
But this number does not mark the limit of audibility. Permitting the
11·4-inch tube to remain as before, and lengthening its neighbor, the
resultant tone sinks near the limit of hearing. When the shorter tube
measures 11 inches, the deep sound of the resultant tone is still
heard. The number of vibrations per second executed in this 11-inch
tube is 700. We have already found the number executed in the 11·4-inch
tube to be 667; hence 700-667=33, which is the number of vibrations
corresponding to the resultant tone now plainly heard when the
attention is converged upon it. We here come very near the limit which
Helmholtz has fixed as that of musical audibility. Combining the sound
of a tube 17-3/8 inches in length with that of a 10-3/8-inch tube, we
obtain a resultant tone of higher pitch than any previously heard. Now
the actual number of vibrations executed in the longer tube is 459; and
we have already found the vibrations of our 10-3/8-inch tube to be 717;
hence 717-459=258, which is the number corresponding to the resultant
tone now audible. This note is almost exactly that of one of our series
of tuning-forks, which vibrates 256 times in a second.
And now we will avail ourselves of a beautiful check which this result
suggests to us. The well-known fork which vibrates at the rate just
mentioned is here, mounted on its case, and I touch it with the bow so
lightly that the sound alone could hardly be heard; but it instantly
coalesces with the resultant tone, and the beats produced by their
combination are clearly audible. By loading the fork, and thus altering
its pitch, or by drawing up the paper slider, and thus altering the
pitch of the flame, the rate of these beats can be altered, exactly
as when we compare two primary tones together. By slightly varying
the size of the flame, the same effect is produced. We cannot fail to
observe how beautifully these results harmonize with each other.
Standing midway between the siren and a shrill singing-flame, and
gradually raising the pitch of the siren, the resultant tone soon makes
itself heard, sometimes swelling out with extraordinary power. When a
pitch-pipe is blown near the flame, the resultant tone is also heard,
seeming, in this case, to originate in the ear itself, or rather in the
brain. By gradually drawing out the stopper of the pipe, the pitch of
the resultant tone is caused to vary in accordance with the law already
enunciated.
The resultant tones produced by the combination of the ordinary
harmonic intervals[74] are given in the following table:
Interval Ratio of Difference The resultant tone is
vibrations deeper than the lowest
primary tone by
Octave 1 : 2 1 0
Fifth 2 : 3 1 an octave
Fourth 3 : 4 1 a twelfth
Major third 4 : 5 1 two octaves
Minor third 5 : 6 1 two octaves and a
major third
Major sixth 3 : 5 2 a fifth
Minor sixth 5 : 8 3 major sixth
The celebrated Thomas Young thought that these resultant tones were
due to the coalescence of rapid beats, which linked themselves
together like the periodic impulses of an ordinary musical note. This
explanation harmonized with the fact that the number of the beats,
like that of the vibrations of the resultant tone, is equal to the
difference between the two sets of vibrations. This explanation,
however, is insufficient. The beats tell more forcibly upon the ear
than any continuous sound. They can be plainly heard when each of the
two sounds that produce them has ceased to be audible. This depends
in part upon the sense of hearing, but it also depends upon the fact
that when two notes of the same intensity produce beats, the amplitude
of the vibrating air-particles is at times destroyed, and at times
doubled. But by doubling the amplitude we quadruple the intensity of
the sound. Hence, when two notes of the same intensity produce beats,
_the sound incessantly varies between silence and a tone of four times
the intensity of either of the interfering ones_.
If, therefore, the resultant tones were due to the beats of their
primaries, they ought to be heard, even when the primaries are feeble.
But they are not heard under these circumstances. When several sounds
traverse the same air, each particular sound passes through the air as
if it alone were present, each particular element of a composite sound
asserting its own individuality. Now, this is in strictness true only
when the amplitudes of the oscillating particles are infinitely small.
Guided by pure reasoning, the mathematician arrives at this result.
The law is also practically true when the disturbances are _extremely_
small; but it is _not_ true after they have passed a certain limit.
Vibrations which produce a large amount of disturbance give birth to
secondary waves, which appeal to the ear as resultant tones. This
has been proved by Helmholtz, and, having proved this, he inferred
further that there are also resultant tones formed by the _sum_ of
the primaries, as well as by their difference. He thus discovered the
_summation-tones_ before he had heard them; and bringing his result to
the test of experiment, he found that these tones had a real physical
existence. They are not at all to be explained by Young’s theory.
Another consequence of this departure from the law of superposition is,
that a single sounding body, which disturbs the air beyond the limits
of the law of superposition, also produces secondary waves, which
correspond to the harmonic tones of the vibrating body. For example,
the rate of vibration of the first overtone of a tuning-fork, as stated
in the fourth chapter, is 6-1/4 times the rate of the fundamental
tone. But Helmholtz shows that a tuning-fork, not excited by a bow, but
vigorously struck against a pad, emits the _octave_ of its fundamental
note, this octave being due to the secondary waves set up when the
limits of the law of superposition have been exceeded.
These considerations make it probably evident to you that a coalescence
of musical sounds is a far more complicated dynamical condition than
you have hitherto supposed it to be. In the music of an orchestra,
not only have we the fundamental tones of every pipe and of every
string, but we have the overtones of each, sometimes audible as far
as the sixteenth in the series. We have also resultant tones; both
difference-tones and summation-tones; all trembling through the
same air, all knocking at the self-same tympanic membrane. We have
fundamental tone interfering with fundamental tone; overtone with
overtone; resultant tone with resultant tone. And, besides this,
we have the members of each class interfering with the members of
every other class. The imagination retires baffled from any attempt
to realize the physical condition of the atmosphere through which
these sounds are passing. And, as we shall immediately learn, the
aim of music, through the centuries during which it has ministered
to the pleasure of man, has been to arrange matters empirically, so
that the ear shall not suffer from the discordance produced by this
multitudinous interference. The musicians engaged in this work knew
nothing of the physical facts and principles involved in their efforts;
they knew no more about it than the inventors of gunpowder knew about
the law of atomic proportions. They tried and tried till they obtained
a satisfactory result; and now, when the scientific mind is brought to
bear upon the subject, order is seen rising through the confusion, and
the results of pure empiricism are found to be in harmony with natural
law.
SUMMARY OF CHAPTER VIII
When several systems of waves proceeding from distinct centres of
disturbance pass through water or air, the motion of every particle is
the algebraic sum of the several motions impressed upon it.
In the case of water, when the crests of one system of waves coincide
with the crests of another system, higher waves will be the result of
the coalescence of the two systems. But when the crests of one system
coincide with the sinuses, or furrows, of the other system, the two
systems, in whole or in part, destroy each other.
This coalescence and destruction of two systems of waves is called
_interference_.
Similar remarks apply to sonorous waves. If in two systems of sonorous
waves condensation coincides with condensation, and rarefaction with
rarefaction, the sound produced by such coincidence is louder than that
produced by either system taken singly. But if the condensations of the
one system coincide with the rarefactions of the other, a destruction,
total or partial, of both systems is the consequence.
Thus, when two organ-pipes of the same pitch are placed near each other
on the same wind-chest and thrown into vibration, they so influence
each other that as the air enters the embouchure of the one it quits
that of the other. At the moment, therefore, the one pipe produces a
condensation the other produces a rarefaction. The sounds of two such
pipes mutually destroy each other.
When two musical sounds of nearly the same pitch are sounded together
the flow of the sound is disturbed by _beats_.
These beats are due to the alternate coincidence and interference
of the two systems of sonorous waves. If the two sounds be of the
same intensity, their coincidence produces a sound of four times the
intensity of either; while their opposition produces absolute silence.
The effect, then, of two such sounds, in combination, is a series of
shocks, which we have called “beats,” separated from each other by a
series of “pauses.”
The rate at which the beats succeed each other is equal to the
difference between the two rates of vibration.
When a bell or disk sounds, the vibrations on opposite sides of the
same nodal line partially neutralize each other; when a tuning-fork
sounds, the vibrations of its two prongs in part neutralize each other.
By cutting off a portion of the vibrations in these cases the sound may
be intensified.
When a luminous beam, reflected on to a screen from two tuning-forks
producing beats, is acted upon by the vibrations of both, the
intermittence of the sound is announced by the alternate lengthening
and shortening of the band of light upon the screen.
The law of the superposition of vibrations above enunciated is
strictly true only when the amplitudes are exceedingly small. When the
disturbance of the air by a sounding body is so violent that the law no
longer holds good, secondary waves are formed, which correspond to the
harmonic tones of the sounding body.
When two tones are rendered so intense as to exceed the limits of
the law of superposition, their secondary waves combine to produce
_resultant tones_.
Resultant tones are of two kinds; the one class corresponding to rates
of vibration equal to the difference of the rates of the two primaries;
the other class corresponding to rates of vibration equal to the sum of
the two primaries. The former are called _difference-tones_, the latter
_summation-tones_.