Physical Distinction between Noise and Music—A Musical
Tone Produced by Periodic, Noise Produced by Unperiodic,
Impulses—Production of Musical Sounds by Taps—Production
of Musical Sounds by Puffs—Definition of Pitch in
Music—Vibrations of a Tuning-Fork; their Graphic
Representation on Smoked Glass—Optical Expression of the
Vibrations of a Tuning-Fork—Description of the Siren—Limits
of the Ear; Highest and Deepest Tones—Rapidity of Vibration
Determined by the Siren—Determination of the Lengths
of Sonorous Waves—Wave-Lengths of the Voice in Man and
Woman—Transmission of Musical Sounds through Liquids and Solids
In our last chapter we considered the propagation through air of a
sound of momentary duration. We have to-day to consider continuous
sounds, and to make ourselves in the first place acquainted with the
physical distinction between noise and music. As far as sensation goes,
everybody knows the difference between these two things. But we have
now to inquire into the causes of sensation, and to make ourselves
acquainted with the condition of the external air which in one case
resolves itself into music and in another into noise.
We have already learned that what is loudness in our sensations is
outside of us nothing more than width of swing, or _amplitude_, of the
vibrating air-particles. Every other real sonorous impression of which
we are conscious has its correlative without, as a mere form or state
of the atmosphere. Were our organs sharp enough to see the motions
of the air through which an agreeable voice is passing, we might see
stamped upon that air the conditions of motion on which the sweetness
of the voice depends. In ordinary conversation, also, the physical
precedes and arouses the psychical; the spoken language, which is to
give us pleasure or pain, which is to rouse us to anger or soothe us
to peace, existing for a time, between us and the speaker, as a purely
mechanical condition of the intervening air.
Noise affects us as an irregular succession of shocks. We are conscious
while listening to it of a jolting and jarring of the auditory
nerve, while a musical sound flows smoothly and without asperity
or irregularity. How is this smoothness secured? _By rendering the
impulses received by the tympanic membrane perfectly periodic._ A
periodic motion is one that repeats itself. The motion of a common
pendulum, for example, is periodic, but its vibrations are far too
sluggish to excite sonorous waves. To produce a musical tone we
must have a body which vibrates with the unerring regularity of the
pendulum, but which can impart much sharper and quicker shocks to the
air.
Imagine the first of a series of pulses following each other at regular
intervals, impinging upon the tympanic membrane. It is shaken by the
shock; and a body once shaken cannot come instantaneously to rest. The
human ear, indeed, is so constructed that the sonorous motion vanishes
with extreme rapidity, but its disappearance is not instantaneous; and
if the motion imparted to the auditory nerve by each individual pulse
of our series continues until the arrival of its successor, the sound
will not cease at all. The effect of every shock will be renewed before
it vanishes, and the recurrent impulses will link themselves together
to a continuous musical sound. The pulses, on the contrary, which
produce noise, are of irregular strength and recurrence. The action of
noise upon the ear has been well compared to that of a flickering light
upon the eye, both being painful through the sudden and abrupt changes
which they impose upon their respective nerves.
The only condition necessary to the production of a musical sound is
that the pulses should succeed each other in the same interval of time.
No matter what its origin may be, if this condition be fulfilled the
sound becomes musical. If a watch, for example, could be caused to
tick with sufficient rapidity—say one hundred times a second—the ticks
would lose their individuality and blend to a musical tone. And if the
strokes of a pigeon’s wings could be accomplished at the same rate, the
progress of the bird through the air would be accompanied by music.
In the humming-bird the necessary rapidity is attained; and when we
pass on from birds to insects, where the vibrations are more rapid,
we have a musical note as the ordinary accompaniment of the insects’
flight.[24] The puffs of a locomotive at starting follow each other
slowly at first, but they soon increase so rapidly as to be almost
incapable of being counted. If this increase could continue up to fifty
or sixty puffs a second, the approach of the engine would be heralded
by an organ-peal of tremendous power.
§ 2. _Musical Sounds produced by Taps_
Galileo produced a musical sound by passing a knife over the edge of
a piastre. The minute serration of the coin indicated the periodic
character of the motion, which consisted of a succession of taps quick
enough to produce sonorous continuity. Every schoolboy knows how to
produce a note with his slate-pencil. I will not call it musical,
because this term is usually associated with pleasure, and the sound of
the pencil is not pleasant.
[Illustration: FIG. 15.]
The production of a musical sound by taps is usually effected by
causing the teeth of a rotating wheel to strike in quick succession
against a card. This was first illustrated by the celebrated Robert
Hooke,[25] and nearer our own day by the eminent French experimenter
Savart. We will confine ourselves to homelier modes of illustration.
This gyroscope is an instrument consisting mainly of a heavy brass
ring, _d_, Fig. 15, loading the circumference of a disk, through which
and at right angles to its surface, passes a steel axis, delicately
supported at its two ends. By coiling a string round the axis, and
drawing it vigorously out, the ring is caused to spin rapidly; and
along with it rotates a small-toothed wheel, w. On touching this wheel
with the edge of a card _c_, a musical sound of exceeding shrillness is
produced. I place my thumb for a moment against the ring; the rapidity
of its rotation is thereby diminished, and this is instantly announced
by a lowering of the pitch of the note. By checking the motion still
more, the pitch is lowered still further. We are here made acquainted
with the important fact that the pitch of a note depends upon the
rapidity of its pulses.[26] At the end of the experiment you hear the
separate taps of the teeth against the card, their succession not being
quick enough to produce that continuous flow of sound which is the
essence of music. A screw with a milled head attached to a whirling
table, and caused to rotate, produces by its taps against a card a note
almost as clear and pure as that obtained from the toothed wheel of the
gyroscope.
The production of a musical sound by taps may also be pleasantly
illustrated in the following way: In this vise are fixed vertically
two pieces of sheet-lead, with their horizontal edges a quarter of an
inch apart. I lay a bar of brass across them, permitting it to rest
upon the edges, and, tilting the bar a little, set it in oscillation
like a see-saw. After a time, if left to itself, it comes to rest. But
suppose the bar on touching the lead to be always tilted upward by a
force issuing from the lead itself, it is plain that the vibrations
would then be rendered permanent. Now such a force is brought into
play _when the bar is heated_. On its then touching the lead the heat
is communicated, a sudden jutting upward of the lead at the point of
contact being the result. Hence an incessant tilting of the bar from
side to side, so long as it continues sufficiently hot. Substituting
for the brass bar the heated fire-shovel shown in Fig. 16, the same
effect is produced.
[Illustration: FIG. 16.]
In its descent upon the lead the bar taps it gently, the taps being so
slow that you may readily count them. But a mass of metal differently
shaped may be caused to vibrate more briskly, and the taps to succeed
each other more rapidly. When such a heated _rocker_, Fig. 17, is
placed upon a block of lead, the taps hasten to a loud rattle. When,
with the point of a file, the rocker is pressed against the lead,
the vibrations are rendered more rapid, and the taps link themselves
together to a deep musical tone. A second rocker, which oscillates
more quickly than the last, produces music without any other pressure
than that due to its own weight. Pressing it, however, with the file,
the pitch rises, until a note of singular force and purity fills the
room. Relaxing the pressure, the pitch instantly falls; resuming the
pressure, it again rises; and thus by the alternation of the pressure
we obtain great variations of tone. Nor are such rockers essential.
Allowing one face of the clean, square end of a heated poker to rest
upon the block of lead, a rattle is heard; causing another face to rest
upon the block, a clear musical note is obtained. The two faces have
been bevelled differently by a file, so as to secure different rates
of vibration.[27] This curious effect was discovered by Schwartz and
Trevelyan.
[Illustration: FIG. 17.]
§ 3. _Musical Sounds produced by Puffs_
Prof. Robison was the first to produce a musical sound by a quick
succession of _puffs_ of air. His device was the first form of an
instrument which will soon be introduced to you under the name of the
_siren_. Robison describes his experiment in the following words: “A
stop-cock was so constructed that it opened and shut the passage of a
pipe 720 times in a second. The apparatus was fitted to the pipe of a
conduit leading from the bellows to the wind-chest of an organ. The air
was simply allowed to pass gently along this pipe by the opening of the
cock. When this was repeated 720 times in a second, the sound _g in
alt_ was most smoothly uttered, equal in sweetness to a clear female
voice. When the frequency was reduced to 360, the sound was that of a
clear but rather a harsh man’s voice. The cock was now altered in such
a manner that it never shut the hole entirely, but left about one-third
of it open. When this was repeated 720 times in a second, the sound was
uncommonly smooth and sweet. When reduced to 360, the sound was more
mellow than any man’s voice of the same pitch.”
[Illustration: FIG. 18.]
But the difficulty of obtaining the necessary speed renders another
form of the experiment preferable. A disk of Bristol board, B, Fig.
18, twelve inches in diameter, is perforated at equal intervals along
a circle near its circumference. The disk, being strengthened by
a backing of tin, can be attached to a whirling table, and caused
to rotate rapidly. The individual holes then disappear, blending
themselves into a continuous shaded circle. Immediately over this
circle is placed a bent tube, _m_, connected with a pair of acoustic
bellows. The disk is now motionless, the lower end of the tube being
immediately over one of the perforations of the disk. If, therefore,
the bellows be worked, the wind will pass from _m_ through the hole
underneath. But if the disk be turned a little, an unperforated portion
of the disk comes under the tube, the current of air being then
intercepted. As the disk is slowly turned, successive perforations are
brought under the tube, and whenever this occurs a puff of air gets
through. On rendering the rotation rapid, the puffs succeed each other
in very quick succession, producing pulses in the air which blend to a
continuous musical note, audible to you all. Mark how the note varies.
When the whirling table is turned rapidly the sound is shrill; when its
motion is slackened the pitch immediately falls. If instead of a single
glass tube there were two of them, as far apart as two of our orifices,
so that whenever the one tube stood over an orifice, the other should
stand over another, it is plain that if both tubes were blown through,
we should, on turning the disk, get a puff through two holes at the
same time. The intensity of the sound would be thereby augmented, but
the pitch would remain unchanged. The two puffs issuing at the same
instant would act in concert, and produce a greater effect than one
upon the ear. And if instead of two tubes we had ten of them, or better
still, if we had a tube for every orifice in the disk, the puffs from
the entire series would all issue, and would be all cut off at the
same time. These puffs would produce a note of far greater intensity
than that obtained by the alternate escape and interruption of the air
from a single tube. In the arrangement now before you, Fig. 19, there
are nine tubes through which the air is urged—through nine apertures,
therefore, puffs escape at once. On turning the whirling table, and
alternately increasing and relaxing its speed, the sound rises and
falls like the loud wail of a changing wind.
[Illustration: FIG. 19.]
§ 4. _Musical Sounds produced by a Tuning-fork_
Various other means may be employed to throw the air into a state of
periodic motion. A stretched string pulled aside and suddenly liberated
imparts vibrations to the air which succeed each other in perfectly
regular intervals. A tuning-fork does the same. When a bow is drawn
across the prongs of this tuning-fork, Fig. 20, the resin of the bow
enables the hairs to grip the prong, which is thus pulled aside. But
the resistance of the prong soon becomes too strong, and it starts
suddenly back; it is, however, immediately laid hold of again by the
bow, to start back once more as soon as its resistance becomes great
enough. This rhythmic process, continually repeated during the passage
of the bow, finally throws the fork into a state of intense vibration,
and the result is a musical note. A person close at hand could see the
fork vibrating; a deaf person bringing his hand sufficiently near would
feel the shivering of the air. Or causing its vibrating prong to touch
a card, taps against the card link themselves, as in the case of the
gyroscope, to a musical sound, the fork coming rapidly to rest. What we
call silence expresses this absence of motion.
[Illustration: FIG. 20.]
When the tuning-fork is first excited the sound issues from it with
maximum loudness, becoming gradually feebler as the fork continues to
vibrate. A person close to the fork can notice at the same time that
the amplitude, or space through which the prongs oscillate, becomes
gradually less and less. But the most expert ear in this assembly
can detect no change in the pitch of the note. The lowering of the
intensity of a note does not therefore imply the lowering of its pitch.
In fact, though the amplitude changes, the rate of vibration remains
the same. Pitch and intensity must therefore be held distinctly apart;
the latter depends solely upon the amplitude, the former solely upon
the rapidity of vibration.
This tuning-fork may be caused to write the story of its own motion.
Attached to the side of one of its prongs, F, Fig. 21, is a thin
strip of sheet-copper which tapers to a point. When the tuning-fork
is excited it vibrates, and the strip of metal accompanies it in its
vibration. The point of the strip being brought gently down upon a
piece of smoked glass, it moves to and fro over the smoked surface,
leaving a clear line behind. As long as the hand is kept motionless,
the point merely passes to and fro over the same line; but it is plain
that we have only to draw the fork along the glass to produce a sinuous
line, Fig. 21.
[Illustration: FIG. 21.]
When this process is repeated without exciting the fork afresh, the
depth of the indentations diminishes. The sinuous line approximates
more and more to a straight one. This is the visual expression of
decreasing amplitude. When the sinuosities entirely disappear, the
amplitude has become zero, and the sound, which depends upon the
amplitude, ceases altogether.
[Illustration: FIG. 22.]
To M. Lissajous we are indebted for a very beautiful method of giving
optical expression to the vibrations of a tuning-fork. Attached to one
of the prongs of a very large fork is a small metallic mirror, F, Fig.
22, the other prong being loaded with a piece of metal to establish
equilibrium. Permitting a slender beam of intense light to fall upon
the mirror, the beam is thrown back by reflection. In my hands is held
a small looking-glass, which receives the reflected beam, and from
which it is again reflected to the screen, forming a small luminous
disk upon the white surface. The disk is perfectly motionless, but
the moment the fork is set in vibration the reflected beam is tilted
rapidly up and down, the disk describing a band of light three feet
long. The length of the band depends on the amplitude of the vibration,
and you see it gradually shorten as the motion of the fork is expended.
It remains, however, a straight line as long as the glass is held in
a fixed position. But on suddenly turning the glass so as to make
the beam travel from left to right over the screen, you observe the
straight line instantly resolved into a beautiful luminous ripple _m
n_. A luminous impression once made upon the retina lingers there
for the tenth of a second; if then the time required to transfer the
elongated image from side to side of the screen be less than the tenth
of a second, the wavy line of light will occupy for a moment the whole
width of the screen. Instead of permitting the beam from the lamp to
issue through a single aperture, it may be caused to issue through two
apertures, about half an inch asunder, thus projecting two disks of
light, one _above_ the other, upon the screen. When the fork is excited
and the mirror turned, we have a brilliant double sinuous line running
over the dark surface, Fig. 23. turning the diaphragm so as to place
the two disks _beside_ each other, on exciting the fork and moving the
mirror we obtain a beautiful interlacing of the two sinuous lines, Fig.
24.
[Illustration: FIG. 23.]
[Illustration: FIG. 24.]
§ 5. _The Waves of Sound_
How are we to picture to ourselves the condition of the air through
which this musical sound is passing? Imagine one of the prongs of the
vibrating fork swiftly advancing; it compresses the air immediately in
front of it, and when it retreats it leaves a partial vacuum behind,
the process being repeated by every subsequent advance and retreat.
The whole function of the tuning-fork is to carve the air into these
condensations and rarefactions, and they, as they are formed, propagate
themselves in succession through the air. A condensation with its
associated rarefaction constitutes, as already stated, a sonorous
wave. In water the length of a wave is measured from crest to crest;
while, in the case of sound, the _wave-length_ is the distance between
two successive condensations. The condensation of the sound-wave
corresponds to the crest, while the rarefaction of the sound-wave
corresponds to the _sinus_, or depression, of the water-wave. Let the
dark spaces, _a_, _b_, _c_, _d_, Fig. 25, represent the condensations,
and the light ones, _a′_, _b′_, _c′_, _d′_, the rarefactions of the
waves issuing from the fork A B: the wave-length would then be measured
from _a_ to _b_, from _b_ to _c_, or from _c_ to _d_.
[Illustration: FIG. 25.]
§ 6. _Definition of Pitch: Determination of Rates of Vibration_
When two notes from two distinct sources are of the same pitch, their
rates of vibration are the same. If, for example, a string yield the
same note as a tuning-fork, it is because they vibrate with the same
rapidity; and if a fork yield the same note as the pipe of an organ or
the tongue of a concertina, it is because the vibrations of the fork in
the one case are executed in precisely the same time as the vibrations
of the column of air, or of the tongue, in the other. The same holds
good for the human voice. If a string and a voice yield the same note,
it is because the vocal chords of the singer vibrate in the same time
as the string vibrates. Is there any way of determining the actual
number of vibrations corresponding to a musical note? Can we infer from
the pitch of a string, of an organ-pipe, of a tuning-fork, or of the
human voice, the number of waves which it sends forth in a second? This
very beautiful problem is capable of the most complete solution.
§ 7. _The Siren: Analysis of the Instrument_
By the rotation of a perforated pasteboard disk, it has been proved to
you that a musical sound is produced by a quick succession of puffs.
Had we any means of registering the number of revolutions accomplished
by that disk in a minute, we should have in it a means of determining
the number of puffs per minute due to a note of any determinate pitch.
The disk, however, is but a cheap substitute for a far more perfect
apparatus, which requires no whirling table, and which registers its
own rotations with the most perfect accuracy.
I will take the instrument asunder, so that you may see its various
parts. A brass tube, _t_, Fig. 26, leads into a round box, C, closed
at the top by a brass plate _a b_. This plate is perforated with four
series of holes, placed along four concentric circles. The innermost
series contains 8, the next 10, the next 12, and the outermost 16
orifices. When we blow into the tube _t_, the air escapes through the
orifices, and the problem now before us is to convert these continuous
currents into discontinuous puffs. This is accomplished by means of
a brass disk _d e_, also perforated with 8, 10, 12, and 16 holes, at
the same distances from the centre and with the same intervals between
them as those in the top of the box C. Through the centre of the disk
passes a steel axis, the two ends of which are smoothly bevelled off
to points at _p_ and _p′_. My object now is to cause this perforated
disk to rotate over the perforated top _a b_ of the box C. You will
understand how this is done by observing how the instrument is put
together.
[Illustration: FIG. 26.]
[Illustration: FIG. 27.]
In the centre of _a b_, Fig. 26, is a depression _x_ sunk in steel,
smoothly polished and intended to receive the end _p′_ of the axis. I
place the end _p′_ in this depression, and, holding the axis upright,
bring down upon its upper end _p_ a steel cap, finely polished within,
which holds the axis at the top, the pressure both at top and bottom
being so gentle, and the polish of the touching surfaces so perfect,
that the disk can rotate with an exceedingly small amount of friction.
At _c_, Fig. 27, is the cap which fits on to the upper end of the
axis _p p′_. In this figure the disk _d e_ is shown covering the top
of the cylinder C. You may neglect for the present the wheel-work of
the figure. Turning the disk _d e_ slowly round, its perforations
may be caused to coincide or not coincide with those of the cylinder
underneath. As the disk turns, its orifices come alternately over
the perforations of the cylinder and over the spaces between the
perforations. Hence it is plain that if air were urged into C, and
if the disk could be caused to rotate at the same time, we should
accomplish our object, and carve into puffs the streams of air. In
this beautiful instrument the disk is caused to rotate by the very air
currents which it renders intermittent. This is done by the simple
device of causing the perforations to pass _obliquely_ through the top
of the cylinder C, and also obliquely, but oppositely inclined, through
the rotating disk _d e_. The air is thus caused to issue from C, not
vertically, but in side currents, which impinge against the disk and
drive it round. In this way, by its passage through the siren, the air
is molded into sonorous waves.
Another moment will make you acquainted with the recording portion of
the instrument. At the upper part of the steel axis _p p′_, Fig. 27,
is a screw _s_, working into a pair of toothed wheels (seen when the
back of the instrument is turned toward you). As the disk and its
axis turn, these wheels rotate. In front you simply see two graduated
dials, Fig. 28, each furnished with an index like the hand of a clock.
These indexes record the number of revolutions executed by the disk
in any given time. By pushing the button _a_ or _b_ the wheel-work is
thrown into or out of action, thus starting or suspending, in a moment,
the process of recording. Finally, by the pins _m_, _n_, _o_, _p_,
Fig. 27, any series of orifices in the top of the cylinder C can be
opened or closed at pleasure. By pressing _m_, one series is opened; by
pressing _n_, another. By pressing two keys, two series of orifices are
opened; by pressing three keys, three series; and by pressing all the
keys, puffs are caused to issue from the four series simultaneously.
The perfect instrument is now before you, and your knowledge of it is
complete.
[Illustration: FIG. 28.]
This instrument received the name of siren from its inventor, Cagniard
de la Tour. The one now before you is the siren as greatly improved by
Dove. The pasteboard siren, whose performance you have already heard,
was devised by Seebeck, who gave the instrument various interesting
forms, and executed with it many important experiments. Let us now make
the siren sing. By pressing the key _m_, the outer series of apertures
in the cylinder C is opened, and by working the bellows, the air is
caused to impinge against the disk. It begins to rotate, and you hear a
succession of puffs which follow each other so slowly that they may be
counted. But as the motion augments, the puffs succeed each other with
increasing rapidity, and at length you hear a deep musical note. As the
velocity of rotation increases the note rises in pitch; it is now very
clear and full, and as the air is urged more vigorously, it becomes so
shrill as to be painful. Here we have a further illustration of the
dependence of pitch on rapidity of vibration. I touch the side of the
disk and lower its speed; the pitch falls instantly. Continuing the
pressure the tone continues to sink, ending in the discontinuous puffs
with which it began.
Were the blast sufficiently powerful and the siren sufficiently free
from friction, it might be urged to higher and higher notes, until
finally its sound would become inaudible to human ears. This, however,
would not prove the absence of vibratory motion in the air; but would
rather show that our auditory apparatus is incompetent to take up and
translate into sound vibrations whose rapidity exceeds a certain limit.
The ear, as we shall immediately learn, is in this respect similar to
the eye.
By means of this siren we can determine with extreme accuracy the
rapidity of vibration of any sonorous body. It may be a vibrating
string, an organ-pipe, a reed, or the human voice. Operating
delicately, we might even determine from the hum of an insect the
number of times it flaps its wings in a second. I will illustrate the
subject by determining in your presence a tuning-fork’s rapidity of
vibration. From the acoustic bellows I urge the air through the siren,
and, at the same time, draw my bow across the fork. Both now sound
together, the tuning-fork yielding at present the highest note. But the
pitch of the siren gradually rises, and at length you hear the “beats”
so well known to musicians, which indicate that the two notes are not
wide apart in pitch. These beats become slower and slower; now they
entirely vanish, both notes blending as it were to a single stream of
sound.
All this time the clockwork of the siren has remained out of action.
As the second-hand of a watch crosses the number 60, the clockwork is
set going by pushing the button _a_. We will allow the disk to continue
its rotation for a minute, the tuning-fork being excited from time
to time to assure you that the unison is preserved. The second-hand
again approaches 60; as it passes that number the clockwork is stopped
by pushing the button _b_; and then, recorded on the dials, we have
the exact number of revolutions performed by the disk. The number is
1,440. But the series of holes open during the experiment numbers 16;
for every revolution, therefore, we had 16 puffs of air, or 16 waves
of sound. Multiplying 1,440 by 16, we obtain 23,040 as the number of
vibrations executed by the tuning-fork in a minute. Dividing this by
60, we find the number of vibrations executed in a second to be 384.
§ 8. _Determination of Wave-lengths: Time of Vibration_
Having determined the rapidity of vibration, the length of the
corresponding sonorous wave is found with the utmost facility. Imagine
a tuning-fork vibrating in free air. At the end of a second from the
time it commenced its vibrations the foremost wave would have reached
a distance of 1,090 feet in air of the freezing temperature. In the
air of a room which has a temperature of about 15° C., it would reach
a distance of 1,120 in a second. In this distance, therefore, are
embraced 384 sonorous waves. Dividing 1,120 by 384, we find the length
of each wave to be nearly 3 feet. Determining in this way the rates of
vibration of the four tuning-forks now before you, we find them to be
256, 320, 384, and 512; these numbers corresponding to wave-lengths
of 4 feet 4 inches, 3 feet 6 inches, 2 feet 11 inches, and 2 feet 2
inches respectively. The waves generated by a man’s voice in common
conversation are from 8 to 12 feet, those of a woman’s voice are from 2
to 4 feet in length. Hence a woman’s ordinary pitch in the lower sounds
of conversation is more than an octave above a man’s; in the higher
sounds it is two octaves.
And here it is important to note that by the term vibrations is meant
_complete ones_; and by the term sonorous wave is meant a condensation
and its associated rarefaction. By a vibration an excursion _to and
fro_ of the vibrating body is to be understood. Every wave generated
by such a vibration bends the tympanic membrane once in and once
out. These are the definitions of a vibration and of a sonorous wave
employed in England and Germany. In France, however, a vibration
consists of an excursion of the vibrating body _in one direction_,
whether to or fro. The French vibrations, therefore, are only the
halves of ours, and we therefore call them semi-vibrations. In all
cases throughout these chapters, when the word vibration is employed
without qualification, it refers to complete vibrations.
During the time required by each of those sonorous waves to pass
entirely over a particle of air, that particle accomplishes one
complete vibration. It is at one moment pushed forward into the
condensation, while at the next moment it is urged back into the
rarefaction. The time required by the particle to execute a complete
oscillation is, therefore, that required by the sonorous wave _to move
through a distance equal to its own length_. Supposing the length of
the wave to be eight feet, and the velocity of sound in air of our
present temperature to be 1,120 feet a second, the wave in question
will pass over its own length of air in, 1/140th of a second: this is
the time required by every air-particle that it passes to complete an
oscillation.
In air of a definite density and elasticity a certain length of wave
always corresponds to the same pitch. But supposing the density or
elasticity not to be uniform; supposing, for example, the sonorous
waves from one of our tuning-forks to pass from cold to hot air: an
instant augmentation of the wave-length would occur, without any change
of pitch, for we should have no change in the rapidity with which the
waves would reach the ear. Conversely with the same length of wave
the pitch would be higher in hot air than in cold, for the succession
of the waves would be quicker. In an atmosphere of hydrogen, waves of
a certain length would produce a note nearly two octaves higher than
waves of the same length in air; for, in consequence of the greater
rapidity of propagation, the number of impulses received in a given
time in the one case would be nearly four times the number received in
the other.
§ 9. _Definition of an Octave_
Opening the innermost and outermost series of the orifices of our
siren, and sounding both of them, either together or in succession, the
musical ears present at once detect the relationship of the two sounds.
They notice immediately that the sound which issues from the circle of
sixteen orifices is the octave of that which issues from the circle of
eight. But for every wave sent forth by the latter, two waves are sent
forth by the former. In this way we prove that the physical meaning
of the term “octave” is, that it is a note produced by double the
number of vibrations of its fundamental. By multiplying the vibrations
of the octave by two, we obtain _its_ octave, and by a continued
multiplication of this kind we obtain a series of numbers answering to
a series of octaves. Starting, for example, from a fundamental note of
100 vibrations, we should find, by this continual multiplication, that
a note five octaves above it would be produced by 3,200 vibrations.
Thus:
100 Fundamental note.
2
————
200 1st octave.
2
————
400 2d octave.
2
————
800 3d octave.
2
————
1600 4th octave.
2
————
3200 5th octave.
This result is more readily obtained by multiplying the vibrations of
the fundamental note by the fifth power of two. In a subsequent chapter
we shall return to this question of musical intervals. For our present
purpose it is only necessary to define an octave.
§ 10. _Limits of the Ear; and of Musical Sounds_
The ear’s range of hearing is limited in both directions. Savart fixed
the lower limit at eight complete vibrations a second; and to cause
these slowly recurring vibrations to link themselves together he was
obliged to employ shocks of great power. By means of a toothed wheel
and an associated counter, he fixed the upper limit of hearing at
24,000 vibrations a second. Helmholtz has recently fixed the lower
limit at 16 vibrations, and the higher at 38,000 vibrations, a second.
By employing very small tuning-forks, the late M. Depretz showed that
a sound corresponding to 38,000 vibrations a second is audible.[28]
Starting from the note 16, and multiplying continually by 2, or more
compendiously raising 2 to the 11th power, and multiplying this by 16,
we should find that at 11 octaves above the fundamental note the number
of vibrations would be 32,768. Taking, therefore, the limit assigned
by Helmholtz, the entire range of the human ear embraces about eleven
octaves. But all the notes comprised within these limits cannot be
employed in music. The practical range of musical sounds is comprised
between 40 and 4,000 vibrations a second, which amounts, in round
numbers, to seven octaves.[29]
The limits of hearing are different in different persons. While
endeavoring to estimate the pitch of certain sharp sounds, Dr.
Wollaston remarked in a friend a total insensibility to the sound of
a small organ-pipe, which, in respect to acuteness, was far within
the ordinary limits of hearing. The sense of hearing of this person
terminated at a note four octaves above the middle E of the pianoforte.
The squeak of the bat, the sound of a cricket, even the chirrup of the
common house-sparrow, are unheard by some people who for lower sounds
possess a sensitive ear. A difference of a single note is sometimes
sufficient to produce the change from sound to silence. “The suddenness
of the transition,” writes Wollaston, “from perfect hearing to total
want of perception, occasions a degree of surprise which renders an
experiment of this kind with a series of small pipes among several
persons rather amusing. It is curious to observe the change of feeling
manifested by various individuals of the party, in succession, as the
sounds approach and pass the limits of their hearing. Those who enjoy a
temporary triumph are often compelled, in their turn, to acknowledge to
how short a distance their little superiority extends.” “Nothing can be
more surprising,” writes Sir John Herschel, “than to see two persons,
neither of them deaf, the one complaining of the penetrating shrillness
of a sound, while the other maintains there is no sound at all. Thus,
while one person mentioned by Dr. Wollaston could but just hear a
note four octaves above the middle E of the pianoforte, others have
a distinct perception of sounds full two octaves higher. The chirrup
of the sparrow is about the former limit; the cry of the bat about an
octave above it; and that of some insects probably another octave.” In
“The Glaciers of the Alps” I have referred to a case of short auditory
range, noticed by myself in crossing the Wengern Alps in company with
a friend. The grass at each side of the path swarmed with insects,
which to me rent the air with their shrill chirruping. My friend heard
nothing of this, the insect-music lying beyond his limit of audition.
§ 11. _Drum of the Ear. The Eustachian Tube_
Behind the tympanic membrane exists a cavity—the drum of the ear—in
part crossed by a series of bones, and in part occupied by air. This
cavity communicates with the mouth by means of a duct called the
Eustachian tube. This tube is generally closed, the air-space behind
the tympanic membrane being thus shut off from the external air. If,
under these circumstances, the external air becomes denser, it will
press the tympanic membrane inward. If, on the other hand, the air
outside becomes rarer, while the Eustachian tube remains closed, the
membrane will be pressed outward. Pain is felt in both cases, and
partial deafness is experienced. I once crossed the Stelvio Pass by
night in company with a friend who complained of acute pain in the
ears. On swallowing his saliva the pain instantly disappeared. By the
act of swallowing, the Eustachian tube is opened, and thus equilibrium
is established between the external and internal pressure.
It is possible to quench the sense of hearing of low sounds by stopping
the nose and mouth, and trying to expand the chest, as in the act
of inspiration. This effort partially exhausts the space behind the
tympanic membrane, which is then thrown into a state of tension by
the pressure of the outward air. A similar deafness to low sounds is
produced when the nose and mouth are stopped, and a strong effort is
made to expire. In this case air is forced through the Eustachian tube
into the drum of the ear, the tympanic membrane being distended by the
pressure of the internal air. The experiment may be made in a railway
carriage, when the low rumble will vanish or be greatly enfeebled,
while the sharper sounds are heard with undiminished intensity. Dr.
Wollaston was expert in closing the Eustachian tube, and leaving the
space behind the tympanic membrane occupied by either compressed or
rarefied air. He was thus able to cause his deafness to continue
for any required time without effort on his part, always, however,
abolishing it by the act of swallowing. A sudden concussion may produce
deafness by forcing air either into or out of the drum of the ear, and
this _may_ account for a fact noticed by myself in one of my Alpine
rambles. In the summer of 1858, jumping from a cliff on to what was
supposed to be a deep snowdrift, I came into rude collision with a rock
which the snow barely covered. The sound of the wind, the rush of the
glacier-torrents, and all the other noises which a sunny day awakes
upon the mountains, instantly ceased. I could hardly hear the sound of
my guide’s voice. This deafness continued for half an hour; at the end
of which time the blowing of the nose opened, I suppose, the Eustachian
tube, and restored, with the quickness of magic, the innumerable
murmurs which filled the air around me.
Light, like sound, is excited by pulses or waves; and lights of
different colors, like sounds of different pitch, are excited by
different rates of vibration. But in its width of perception the ear
exceedingly transcends the eye; for while the former ranges over eleven
octaves, but little more than a single octave is possible to the
latter. The quickest vibrations which strike the eye, as light, have
only about twice the rapidity of the slowest;[30] whereas the quickest
vibrations which strike the ear, as a musical sound, have more than two
thousand times the rapidity of the slowest.
§ 12. _Helmholtz’s Double Siren_
Prof. Dove, as we have seen, extended the utility of the siren of
Cagniard de la Tour, by providing it with four series of orifices
instead of one. By doubling all its parts, Helmholtz has recently
added vastly to the power of the instrument. The double siren, as it
is called, is now before you, Fig. 29 (next page). It is composed
of two of Dove’s sirens, C and C′, one turned upside down. You will
recognize in the lower siren the instrument with which you are already
acquainted. The disks of the two sirens have a common axis, so that
when one disk rotates the other rotates with it. As in the former case,
the number of revolutions is recorded by clockwork (omitted in the
figure). When air is urged through the tube _t′_ the upper siren alone
sounds; when urged through _t_, the lower one only sounds; when it is
urged simultaneously through _t′_ and _t_, both the sirens sound. With
this instrument, therefore, we are able to introduce much more varied
combinations than with the former one. Helmholtz has also contrived a
means by which not only the disk of the upper siren, but the box C′
above the disk, can be caused to rotate. This is effected by a toothed
wheel and pinion, turned by a handle. Underneath the handle is a dial
with an index, the use of which will be subsequently illustrated.
[Illustration: FIG. 29.]
Let us direct our attention for the present to the upper siren. By
means of an India-rubber tube, the orifice _t′_ is connected with an
acoustic bellows, and air is urged into C′. Its disk turns round, and
we obtain with it all the results already obtained with Dove’s siren.
The pitch of the note is uniform. Turning the handle above, so as to
cause the orifices of the cylinder C′ to _meet_ those of the disk,
the two sets of apertures pass each other more rapidly than when the
cylinder stood still. An instant rise of pitch is the result. By
reversing the motion, the orifices are caused to pass each other more
slowly than when C′ is motionless, and in this case you notice an
instant fall of pitch when the handle is turned. Thus, by imparting
in quick alternation a right-handed and left-handed motion to the
handle, we obtain successive rises and falls of pitch. An extremely
instructive effect of this kind may be observed at any railway station
on the passage of a rapid train. During its approach the sonorous waves
emitted by the whistle are virtually shortened, a greater number of
them being crowded into the ear in a given time. During its retreat
we have a virtual lengthening of the sonorous waves. The consequence
is, that, when approaching, the whistle sounds a higher note, and when
retreating it sounds a lower note, than if the train were still. A fall
of pitch, therefore, is perceived as the train passes the station.[31]
This is the basis of Doppler’s theory of the colored stars. He supposes
that all stars are white, but that some of them are rapidly retreating
from us, thereby lengthening their luminiferous waves and becoming red.
Others are rapidly approaching us, thereby shortening their waves, and
becoming green or blue. The ingenuity of this theory is extreme, but
its correctness is more than doubtful.
§ 13. _Transmission of Musical Sounds by Liquids and Solids_
We have thus far occupied ourselves with the transmission of musical
sounds through air. They are also transmitted by liquids and solids.
When a tuning-fork screwed into a little wooden foot vibrates, nobody,
except the persons closest to it, hears its sound. On dipping the foot
into a glass of water a musical sound is audible: the vibrations having
been transmitted through the water to the air. The tube M N, Fig. 30,
three feet long, is set upright upon a wooden tray A B. The tube ends
in a funnel at the top, and is now filled with water to the brim. The
fork F is thrown into vibration, and on dipping its foot into the
funnel at the top of the tube, a musical sound swells out. I must so
far forestall matters as to remark that in this experiment the tray
is the real sounding body. It has been thrown into vibration by the
fork, but the vibrations have been conveyed to the tray _by the water_.
Through the same medium vibrations are communicated to the auditory
nerve, the terminal filaments of which are immersed in a liquid:
substituting mercury for water, a similar result is obtained.
[Illustration: FIG. 30.]
The siren has received its name from its capacity to sing under water.
A vessel now in front of the table is half filled with water, in which
a siren is wholly immersed. When a cock is turned, the water from the
pipes which supply the house forces itself through the instrument. Its
disk is now rotating, and a sound of rapidly augmenting pitch issues
from the vessel. The pitch rises thus rapidly because the heavy and
powerfully pressed water soon drives the disk up to its maximum speed
of rotation. When the supply is lessened, the motion relaxes and the
pitch falls. Thus, by alternately opening and closing the cock, the
song of the siren is caused to rise and fall in a wild and melancholy
manner. You would not consider such a sound likely to woo mariners to
their doom.
The transmission of musical sounds through solid bodies is also
capable of easy and agreeable illustration. Before you is a wooden
rod, thirty feet long, passing from the table through a window in
the ceiling, into the open air above. The lower end of the rod rests
upon a wooden tray, to which the musical vibrations of a body applied
to the upper end of the rod are to be transferred. An assistant is
above, with a tuning-fork in his hand. He strikes the fork against a
pad; it vibrates, but you hear nothing. He now applies the stem of
the fork to the end of the rod, and instantly the wooden tray upon
the table is rendered musical. The pitch of the sound, moreover, is
exactly that of the tuning-fork; the wood has been passive as regards
pitch, transmitting the precise vibrations imparted to it without any
alteration. With another fork a note of another pitch is obtained. Thus
fifty forks might be employed instead of two, and 300 feet of wood
instead of 30; the rod would transmit the precise vibrations imparted
to it, and no other.
We are now prepared to appreciate an extremely beautiful experiment,
for which we are indebted to Sir Charles Wheatstone. In a room
underneath this, and separated from it by two floors, is a piano.
Through the two floors passes a tin tube 2-1/2 inches in diameter,
and along the axis of this tube passes a rod of deal, the end of
which emerges from the floor in front of the lecture-table. The rod
is clasped by India-rubber bands, which entirely close the tin tube.
The lower end of the rod rests upon the sound-board of the piano, its
upper end being exposed before you. An artist is at this moment engaged
at the instrument, but you hear no sound. When, however, a violin
is placed upon the end of the rod, the instrument becomes instantly
musical, not, however, with the vibrations of its own strings, but
with those of the piano. When the violin is removed, the sound ceases;
putting in its place a guitar, the music revives. For the violin and
guitar we may substitute a plain wooden tray, which is also rendered
musical. Here, finally, is a harp, against the sound-board of which
the end of the deal rod is caused to press; every note of the piano
is reproduced before you. On lifting the harp so as to break the
connection with the piano, the sound vanishes; but the moment the
sound-board is caused to press upon the rod the music is restored. The
sound of the piano so far resembles that of the harp that it is hard
to resist the impression that the music you hear is that of the latter
instrument. An uneducated person might well believe that witchcraft or
“spiritualism” is concerned in the production of this music.
What a curious transference of action is here presented to the mind!
At the command of the musician’s will, the fingers strike the keys;
the hammers strike the strings, by which the rude mechanical shock
is converted into tremors. The vibrations are communicated to the
sound-board of the piano. Upon that board rests the end of the deal
rod, thinned off to a sharp edge to make it fit more easily between the
wires. Through the edge, and afterward along the rod, are poured with
unfailing precision the entangled pulsations produced by the shocks of
those ten agile fingers. To the sound-board of the harp before you the
rod faithfully delivers up the vibrations of which it is the vehicle.
This second sound-board transfers the motion to the air, carving it and
chasing it into forms so transcendently complicated that confusion
alone could be anticipated from the shock and jostle of the sonorous
waves. But the marvellous human ear accepts every feature of the
motion, and all the strife and struggle and confusion melt finally into
music upon the brain.[32]
SUMMARY OF CHAPTER II
A musical sound is produced by sonorous shocks which follow each other
at regular intervals with a sufficient rapidity of succession.
Noise is produced by an irregular succession of sonorous shocks.
A musical sound may be produced by _taps_ which rapidly and regularly
succeed each other. The taps of a card against the cogs of a rotating
wheel are usually employed to illustrate this point.
A musical sound may also be produced by a succession of _puffs_. The
siren is an instrument by which such puffs are generated.
The pitch of a musical note depends solely on the number of vibrations
concerned in its production. The more rapid the vibrations, the higher
the pitch.
By means of the siren the rate of vibration of any sounding body may be
determined. It is only necessary to render the sound of the siren and
that of the body identical in pitch to maintain both sounds in unison
for a certain time, and to ascertain, by means of the counter of the
siren, how many puffs have issued from, the instrument in that time.
This number expresses the number of vibrations executed by the sounding
body.
When a body capable of emitting a musical sound—a tuning-fork, for
example—vibrates, it molds the surrounding air into sonorous waves,
each of which consists of a condensation and a rarefaction.
The length of the sonorous wave is measured from condensation to
condensation, or from rarefaction to rarefaction.
The wave-length is found by dividing the velocity of sound per second
by the number of vibrations executed by the sounding body in a second.
Thus a tuning-fork which vibrates 256 times in a second produces in air
of 15° C., where the velocity is 1,120 feet a second, waves 4 feet 4
inches long. While two other forks, vibrating respectively 320 and 384
times a second, generate waves 3 feet 6 inches, and 2 feet 11 inches
long.
A vibration, as defined in England and Germany, comprises a motion to
_and_ fro. It is a _complete_ vibration. In France, on the contrary, a
vibration comprises a movement to _or_ fro. The French vibrations are
with us semi-vibrations.
The time required by a particle of air over which a sonorous wave
passes to execute a complete vibration is that required by the wave to
move through a distance equal to its own length.
The higher the temperature of the air, the longer is the sonorous
wave corresponding to any particular rate of vibration. Given the
wave-length and the rate of vibration, we can readily deduce the
temperature of the air.
The human ear is limited in its range of hearing musical sounds. If
the vibrations number less than 16 a second, we are conscious only of
the separate shocks. If they exceed 38,000 a second, the consciousness
of sound ceases altogether. The range of the best ear covers about
11 octaves, but an auditory range limited to 6 or 7 octaves is not
uncommon.
The sounds available in music are produced by vibrations comprised
between the limits of 40 and 4,000 a second. They embrace 7 octaves.
The range of the ear far transcends that of the eye, which hardly
exceeds an octave.
By means of the Eustachian tube, which is opened in the act of
swallowing, the pressure of the air on both sides of the tympanic
membrane is equalized.
By either condensing or rarefying the air behind the tympanic membrane,
deafness to sounds of low pitch may be produced.
On the approach of a railway train the pitch of the whistle is higher,
on the retreat of the train the pitch is lower, than it would be if the
train were at rest.
Musical sounds are transmitted by liquids and solids. Such sounds may
be transferred from one room to another; from the ground-floor to the
garret of a house of many stories, for example, the sound being unheard
in the rooms intervening between both, and rendered audible only when
the vibrations are communicated to a suitable sound-board.