1) The science which treats of the nature and laws of
sound.
2) The sensation of sound consists in the communication of a
vibratory motion to the tympanic membrane of the ear, through
slight and rapid changes in the pressure of the air on its outer
surface.
3) The mode of propagation of sound in air may be explained in
the following manner. Suppose a small particle of fulminating
silver to be exploded in free air; the air particles immediately
contiguous are driven outwards in all directions by the explosion,
their motion is almost instantaneously communicated to the
adjacent ones, those first agitated coming at the same time to
rest; the adjacent ones pass on the impulse in the same way to
those at a greater distance, and so on; thus the explosion gives
rise to what may be looked on as a rapidly expanding shell or
constant thickness, containing at any instant between its exterior
and interior surfaces a stratum of agitated air particles each one
of which performs a single vibration to and fro during the passage
of the shell over it; in other words the exterior and interior
surfaces of the shell are at any time the loci of all those points
at which the particles at that instant come under the influence of
the impulse, and are left at rest by it respectively, so that its
thickness depends both on the rapidity oftheir vibration and the
rate at which they pass on the impulse, one to another. Let us
suppose now that immediately after the first explosion a second
were to take place; then, in exactly the same way there would be a
second pulse propagated in all directions. If a series of
explosions at regular intervals were to take place, there would be
a regular series of expanding shells; and if the intervals were
sufficiently small, the alternate changes of pressure, due to the
successive collisions of the air particles against the tympanic
membrane of an ear in the neighborhood of the explosions would
convey to the brain a sensation of a continuous note. Exactly the
same thing occurs if, for a series of explosions, are substituted
the vibrations of an elastic body; and it is, in general, by the
latter means that all sounds, and especially musical ones, are
produced. The motion of a sound wave must not be confounded
with the motion of the particles which transmit the wave.
In the passage of a single wave each particle over which it passes
makes only a small excursion to and fro, the semi-length of which
is called the amplitude of the vibration, the time occupied during
one vibration being called its period.
4) The intensity of a sound is proportional to the square of
the maximum velocity of the vibrating particle. It also
approximately varies inversely as the square of the distance from
the origin of the sound; for, supposing the latter to be produced
at a uniform loudness, the same amount of energy has to be
communicated to the particles contained within the external and
internal surfaces of shells of the same thickness but of different
radii. For example, if we take a shell of air whose internal
radius is one foot, one of the same thickness whose radius is two
feet will contain four times the quantity of matter; one whose
radius is three feet, nine times the quantity, and so on. Thus the
amount of matter over which a given quantity of energy has to be
distribute augments as the square of the distance from the origin
of sound, and therefore the amount of energy, or, what comes to
the same thing, the intensity of the sound, diminishes in the same
ratio.
5) At a temperature of zero Centigrade sound is propagated at
the rate of about 1090 feet per second, and this speed augments
about two feet per second for every additional degree of
temperature; thus at 15° C. the rate of propagation would be
about 1120 feet per second. The velocity of sound in air depends
on the elasticity of the air in relation to its density. It is
also directly proportional to the square root of the elasticity,
and inversely proportional to the square root of the density. Now
for a constant temperature the elasticity varies as the density,
hence in this case they neutralise one another, and the velocity
of the sound is independent of the density of the air.
6) One sound differs from another not only in quantity, but
also in quality and pitch. The pitch of a sound depends on the
number of vibrations per second by which it is caused; the greater
this number is the higher is the sound, and vice verse; thus pitch
is a more or less relative term, and it is therefore necessary to
have some standard to which different sounds may be referred. This
standard is so choosen that the middle C of the pianoforte shall
be produced by 264 vibrations per second. (That is, according to
German musical pitch, Western pitch is set at 244 cps.)
7) Knowing the velocity of sound in air we can estimate the
different wave lengths corresponding to notes of different pitch
in the following manner. The wave length is the distance through
which the sound travels while any particle over which it passes
describes a complete vibration; hence, if we know the number of
vibrations the particle performs per second, the required wave
length will be found by dividing the number of feet over which the
sound travels per second, by that number. Now, by means of an
instrument invented by Cagniard de la Tour, and by him named
syren, the number of vibrations corresponding to a note of any
given pitch can be determined very exactly. For a detailed account
of this instrument and of its improvements by Helmholtz, the
reader is referred to Tyndall's Lectures on Sound, p. 64; but to
describe it shortly may be said in its original form to consist of
two equal disks, one forming the top of a hollow fixed cylinder,
into which air can be driven, the other capable of revolving
concentrically upon it with the smallest possible amount of
friction. A circle of small holes equidistant from each other is
bored upon each disk and concentric with it; those in the upper
disk being inclined slantwise to its plane, those in the lower
being slantwise also but in the opposite direction; there are also
arrangements both for driving a constant supply of air into the
hollow cylinder, and for regestering the number of revolutions the
upper disk performs in a minute; thus, when the upper disk is so
turned that its holes coincide with those of the lower, and air is
forced into the cylinder, it will pass out through the
perforations, and by reason of their obliquity will cause the
moveable disk to revolve with a rapidity corresponding to the
pressure; and each time that the holes of the former coincide with
those of the latter a number of little puffs of air get through
simultaneously, giving rise to an agitation in the surrounding
atmosphere which spreads round in all directions in the way before
described, and if the pressure of the air in the cylinder is
sufficient, the series of impulses thus given will link themselves
together, forming a continuous note. (It should be remarked that
the pitch of the sound would be exactly the same if there were
only one perforation in the revolving disk, the number of holes
merely serving to increase its intensity; if the number of holes
in the revolving disk is less than the number in the lower one,
those of the former must be situated so as all to coincide
simultaneously with an equal number of the latter. (It should also
be remarked that this syren is an old invention. Today there are
any number of electronic and computer instruments that are far
more accurate and precise than this mechanial device.)
Hence to determine the number of vibrations per second,
corresponding to a sound of a given pitch, we have only to
maintain such a pressure of air in the syren as will cause it to
produce the same sound for the space of a minute, and note the
number of revolutions registered in that time. Now, for every
revolution of the upper disk, the same number of sound waves are
propagated around as there are perforations, hence the whole
number propagated in a second will be the product of the number of
holes and number of revolutions per minute divided by 60; and this
result will evidently be the required number of vibrations per
second caused by the given sound.
To apply this to find the wave length corresponding to the
note given by the open C string of the violoncello, we should
adjust the supply of air to the syren until it gives a note of the
same pitch. Supposing the number of holes in each disk to be 18,
the number of revolutions per minute would be found to be 220.
Hence the number of vibrations per second of the string, and
therefore of the surrounding particles of air, would be 220 x 18/
60 = 66. Supposing the temperature were 16° C the velocity of
sound would be about 1122 feet per second, and the number by 66
gives the wave length correspnding to that number of vibrations
per second; that is, just 17 feet; the sound then will travel
through this distance during the time the string takes to perform
one complete vibration.
8) If the number of vibrations per second be increased, the
pitch of the sound caused by them is raised, and vice versa, as
can easily be illustrated by driving more or less air into the
syren, and observing the sound it produces. Dr. Wollaston has
shown (Phil. Trans. 1820, p. 336) that if the number be increased,
beyond a certain limit the sound becomes inaudible, although this
limit is not the same for all ears, some persons being perfectly
sensible to sounds inaudible to others. In general it is probable
that no sound is heard when the number of vibrations per second
exceeds 40,000; while on the other hand the perception of pitch
appears to begin when the number of vibrations is somewhere
between 8 and 32, the wave length being in the former case about
0.03 of an inch - in the latter ranging from 140 feet to 35
feet.
9) Sounds are primarily divided into two classes, musical and
unmusical; the former being defined as those produced by regular
or periodic vibrations, the latter by such as are irregular or
non-periodic. These definitions require some explanation, since,
by sounding together a sufficient number of notes sufficiently
near in pitch, it is plain that we could produce as unmusical a
sound as we pleased, although the components would be themselves
due to periodic vibrations, and would therefore musical. The
answer to this is found in the fact that when two or more sets of
sound waves impinge on the ear at the same instant, since each one
cannot impress its own particular vibration on the tympanum
contemporaneously with those of the others, the motion of the
latter membrane must be in some way [be] the sum of all
the different motions which the different sets of waves would have
separately caused it to follow; and this is what in fact does
happen, i.e., the vibrations due to each set combine and throw the
tympanum into a complicated state of vibration, causing the
sensation of the consonance or combination of the different sounds
from which the sets of sound waves proceed.
Now the unassisted ear is only able to distinguish the
separate notes out of a number sounded at once up to a certain
point; beyond this it fails to distinguish them individually, and
is conscious only of a confused mixture of sounds which approaches
the more nearly to the character of noise the more components
there are, or the nearer they lie to one another. A noise, then,
may be defined as a sound so complicated that the ear is unable to
resolve or analyse it into its original constituents.
10) As the character of a sound depends upon that of the
vibrations by which it is caused, it is important to know of what
kind the latter must be in order that they may give the sensation
of a perfectly simple tone, i.e., one which the ear cannot resolve
into any others. Such a vibration is perhaps best realised by
comparison with that of the pendulum of a clock when it is
swinging only a little to and fro. Under these circumstances it is
performing what are called harmonic vibrations, and when the air
particles in the neighborhood of the ear are caused by any means
to vibrate according to the same law as that which the pendulum
follows, and also with sufficient rapidity, a perfect simple tone
is the result. Such a tone is, however, rarely heard except when
produced by means specially contrived for the purpose. If a note
on the pianoforte is struck, the impact of the hammer on the
string throws it into a state of vibration, which, though
periodic, is not really harmonic; consequently we do not hear a
perfectly simple tone, but one which is in reality a mixture of
several higher simple tones with that one which corresponds to the
actual length of the string. The former are, however, generally
faint, and become associated by habit with the latter, appearing
to form with it a single note of determinate pitch. These higher
tones are harmonics of the string, and are produced by vibrations
whose numbers per second are respectively twice, three times, four
times, etc., as great as those of the fundamental tone of the
string. The same may be said of the notes of all instruments,
including the human voice, which are usually employed for the
production of musical sounds.
11) Since the consonance of two or more such simple tones
always gives a more or less musical sound, and since also the ear
is always more or less capable of resolving the latter into its
components, the question naturally arises whether all sounds are
not, theoretically at least, resolvable into simple tones. The
answer to this is contained in a celebrated theorem due to the
French mathematician Fourier. He has shown that any periodic
vibration is the result of combining together a certain number of
simple harmonic vibrations whose periods are aliquot parts of that
of the former; and we have conclusive reasons for supposing that,
in the same way as a compound periodic vibration gives rise to a
compound sound, so the simple tones into which the ear resolves
the latter are respectively due to the simple harmonic vibrations
which, as the above mentioned theorem proves, make up the
former.
12) The theorem of Fourier referred to in the preceeding
article is of such great importance in all questions connected
with acoustics that a few words illustrative of it may not be out
of place.
If a peg is fixed into the rim of a wheel capable of revolving
about a fixed center, and at right angles to the plane of the
wheel, and if the latter is caused to rotate uniformly and is
looked at edgeways the peg will appear to move up and down in a
straight line, its velocity being the greatest at the middle of
its course, and diminishing as it approaches each end. Under these
circumstance the peg appears to perform harmonic vibrations.
Now suppose a second wheel, also furnished with a peg in its
rim, is made to revolve about the peg of the first as an axis. If
the latter is at rest the peg of the second will appear, looked at
as above, to perform harmonic vibrations; but if the former is
also caused to revolve these vibrations are no longer harmonic,
but are the result of adding together the separate harmonic
vibrations of the two pegs, in other words of superposing the
harmonic vibrations which the second peg performs ifthe first
wheel is at rest, upon those which the first peg performs when it
is itself in motion. Now it is evident that by continuing this
process indefinitely, and by giving the wheels different radii,
and different uniform velocities of rotation, the final motion of
the last peg looked at sideways as before, would be an exceedingly
complicated one, and that an indefinite number of different
vibrations could be produced by varying the number, position at
starting, radii, and velocities of the wheels, though it could not
be assumed without proof that every possible variety could be so
produced. This however is what Fourier's theorem asserts, provided
that the velocities of rotation of the several wheels of the
series are in the proportion of 1, 2, 3, 4, 5, etc. In other
words, every periodic vibration is the resultant of a certain
number of harmonic vibrations whose periods are one-half,
one-third, one-fourth, etc., that of the former.
13) A harmonic scale is formed by taking a series of notes
produced by vibration whose numbers in a given time are
respectively as 1, 2, 3, 4, et.
If we take as fundamental tone the open C string of the
violoncello, the series of tones which with it forms a harmonic
scale will be as pictured:
The notes marked with an asterisk do not exactly represent the
corresponding tones; but are the nearest representives which the
modern notation supplies. All the notes of the harmonic scale can
theoretically be produced by either a single string, or by a
simple tube used as a trumpet. If we lightly touch the string of a
violin, without causing it to come in contact with the finger
baord, at any one of a series of points dividing it into a number
of a equal parts, and excite it by means of a bow, it no longer
vibrates as a whole, but separates into the number of equal
vibrating segments which is the least possible consistent with
that point forming one of their points of diversion; the latter
remain stationary, or very nearly so, and are called nodes, their
number being evidently just one less than that of the segments. It
is plain that if the point of appication of the bow be one of a
series of nodes, no sound will be produced, provided, of course,
the finger remains on any other of the same series, and this may
serve to explain why it is sometimes difficult to bring out the
higher harmonics of a violin, as the bow may, unconsciously to the
performer, be passing exactly over one of the corresponding nodes.
The first harmonic, as it is called, of the open string is
produced by touching it while on a state of vibration at its
middle point, and thereby dividing it into two equal portions,
both of which vibrate twice as fast as the whole, and accordingly
give the octave. The second harmonic, or the twelfth of the
fundamental, corresponds to a division of the string into three
equal portions, and so on. And generally, in order to produce the
nth harmonic the finger should touch the string at any one of the
series of points which divide it into n equal portions. In
practise, however, the finger should always touch the string at
the point of division adjacent to either end.
14) The harmonics of a simple tube used as a trumpet are the
same as those of a vibrating string, viz., the octave, twelfth,
fifteenth, etc., and are produced by modifications of the breath
and lips; but there is a great difference between the nature of
the vibrations which produce sound, in the case of strings and
pipes. In the former case the vibrations are executed at right
angles to the length of the string, that is, are lateral or
transverse, while in the latter they are in the direction of the
pipe, or longitudinal, and are the vibrations of the air itself
with it. Actually the string has all three primary modes of
vibration with the lateral or transverse predominating.
15) When an open organ pipe is sounding its fundamental tone,
the particles of the column of air within it are all, more or
less, in a state of vibration parallel to the length of the pipe,
of which the intensity is at its maximum at the two ends, growing
less and less towards the middle, where there is a node, that is,
a point of no disturbance. The harmonics of an open organ pipe
follow the same law as those of a simple trumpet, or vibrating
string.
The fundamental note of a stopped organ pipe is an octave
below the fundamental note of an open organ pipe of the same
length. When it is sounding this note there is no node, and the
first harmonic is a fifth above the octave, the second a major
sixth above the first, the third a diminished fifth above the
second, and so on. Or, more simply, the successive tones of the
harmonic scale of an open pipe are produced by vibrations which
are as 1, 2, 3, 4, etc., those of a stopped pipe by vibrations
which are as 1, 3, 5, 7, etc.
16) It was stated (§10) that the sound of a vibrating
string was in general compounded of a number of simple tones, and
a well trained ear can detect a considerable number of them. If it
were not for these harmonic components the tones of strings,
pipes, of the human voice, or in short, of every instrument most
generally used for the production of sound, would be flat and
uninteresting like pure water. Each harmonic component is by
itself a simple tone, and is due to the vibration of the
corresponding segment of the string superposed upon that of the
whole*. The same statement applies, mutatis mutandis to pipes,
whether open or stopped. That the harmonics of different
instruments greatly influence their several charcters is
observable in the differene of the tones of a flute, and clarinet.
A flute is an open pipe, a clarinet a stopped one end; in the
former, therefore, the harmonics follow the order of the natural
numbers 1, 2, 3, 4, and in the latter the order 1, 3, 5, 7; the
intermediate notes being supplied by opening the lateral orifices
of the instrument.
* Society is an accurate analogy to this statement: The whole
is made up of individualized whole tones of the individuals.
17) When two simple tones, that is (as explained above), notes
deprived of all the harmonic components which under ordinary
circumstances accompany them, are sounded together very nearly in
unison, there are heard what are called beats succeeding one
another at regular intervals, their rapidity depending inversely
on the smallness of the interval between the two tones. Their
origin may be explained thus: Suppose the tones to be produced by
vibrations numbering 500 and 501 per second respectively, then
every 500th sound wave of the former will strike on the tympanum
at exactly the same instant as every 501st of the latter and will
reinforce it; while at the 250th of the first the corresponding
wave of the other will be just half a period in front of it. Now a
sound wave consists of a condensed and rarified stratum of air
particles, and therefore the condensed portion of one wave here
coincides with the rarified portion of the other and neutralises
it. Thus there will be an alternate reinforcement and diminution
of sound, every second, from the maximum intensity when both waves
impinge on the tympanum at the same instant to the minimum when
they counteract each other as much as possible and vice
versa.
In the above case it was supposed that the number of
vibrations of one tone were only one more per second than those of
the other; but if the difference of the numbers had been two, for
instance, then in one second the first tone would have gained two
vibrations on the other, and there would have been two beats; and
in general the number of beats per second is always equal to the
difference between the two rates of vibrations per second.
18) In the preceding section, the cause of beats due to two
simple tones of nearly the same pitch was explained, and it was
seen that the number of beats per second was always equal to the
difference of the numbers of vibrations per second of each tone;
so that as the interval between them increased so would the number
of beats increase in a given time. Hence it is obvious that if the
interval became sufficiently large, the beats would succeed each
other so rapidly as to become undistinguished. For instance, in
the case of the fifth whose lower and upper tones are produced by
vibrations numbering 264 and 396 per second respectively, the
number of beats per second would be 132 and would therefore be
undistinguished &endash; and still more so supposing the upper
tone to have 397 or more vibrations per second; but, on the other
hand it is a well-known fact, that if an imperfect fifth, octave,
or any other tolerably simple interval is played on a violin or
vioncello, the beats are most distinctly heard succeeding each
other at perceptible intervals &endash; whereas according to what
was said above they should occur so rapidly as not to be heard at
all. Two explanations of this phenomenon have been given, of which
by far the most simple is due to Helmholtz &endash; and which here
follows. It appears that when the tones are simple and at a
sufficiently large interval the beats should occur too rapidly to
be heard, wheras when the interval is played on a violin they are
easily distinguishable. The reason of this fact is that in the
latter case the tones are no longer simple but compound -;
and the beats which are heard are not due to the fundamental tones
themselves but arise from two of their harmonic components which
are nearly unison. Suppose the ratio of the interval between the
fundamental tones to be m/n, that is, let m/n, be the fraction,
reduced to its lowest terms, which is formed by putting in the
numerator the number of vibrations per second of the upper tone,
and in the denominator those of the lower. Then it is plain that
the nth harmonic component of the tone m, will be of the same
pitch as the mth harmonic component of the tone n; for they will
each have exactly mn vibrations per second. Now let M/N be the
ratio, expressed in the same way, of another interval, nearly, but
not quite, equal to m/n; then the nth harmonic component of M will
have Nm vibrations per second, while the mth component of N will
have Nm. Now since M/N is nearly equal to m/n, the difference
between Mn and Nm will be a small number; and when the two notes
are sounded together the number of beats per second will be equal
to that difference.
For example, let m/n, be the ratio of a fifth, that is the
fraction 3/2, and let M/N represent very nearly the same interval,
say 397/264; then the difference between Mn and Nm, or 794 and
792, is 2; hence if two strings tuned apart at an interval
represented by 397/264 are sounded simultaneously there will be
two beats heard per second.
19) When the vibrations of the air due to a number of
different sounds which co-exist at the same time are infinitely
small, they are merely superposed one on another, so that each
separate sound passes through the air as if it alone were present;
and this Law of Supposition holds, though only approximately,
until the vibrations have increased up to a certain limit, beyond
which it is no longer true. Vibrations which give rise to a large
amount of disturbance produce secondary waves; and it is to these
that the phenomena of resultant tones are due.
Thus if two notes a fifth apart, for instance, are forcibly
sounded together, a third tone is heard an octave below the lower
of the two, and this ceases to be perceptible when the loudness of
the concord diminishes. In general the resultant tone of any
combination of two notes is produced by a number of vibrations per
second equal to the difference of the numbers per second of the
notes. This fact formerly led to the supposition that the
resultant tone was produced by the beats due to the consonance,
which, when they occurred with sufficient rapidity, linked
themselves together so as to form a continuous musical note. If
this were so it is clear that the resultant ought to be heard when
the original notes are sounded gently as well as forcibly; and it
was the failure of this condition that led Helmholtz to the
re-investigation of their origin. These resultant tones have been
named by him difference tones; he has also discovered the
existence of resultant tones formed by the sum of the numbers of
vibrations of the primaries. These summation tones as they are
called cannot be explained on the old theory.
20) The theory of beats explains the law that the smaller the
two numbers are, which express the ratio of their vibrations, the
smoother is the combination of any two tones. When two simple
tones are sounded together whose rates of vibration per second
differ by more than 32, the beats, according to Helmholtz, totally
disappear. As the difference grows less the beats become more and
more audible, the interval meanwhile growing proportionately
dissonant, till they number 33 per second, at which point the
dissonance of the interval is at its maximum.
This, however, depends upon the position of the interval as
regards its pitch. For it should be remembered that though the
ratio of any given interval remains the same whatever the absolute
pitch of its tones may be, yet the difference of the actual
numbers of their vibrations, and therefore the number of beats due
to their consonance, alters with it. And vice verse, if the
difference of the number of vibrations remains constant, the
interval must dimish as its pitch rises. For instance, either of
the following combinations would give rise to 33 beats per second,
since the numbers of vibrations of their tones per second, are
99-66, and 528-495, respectively. Now it is obvious that in the
latter case the dissonance would be far greater than in the
former.
The above explanation of the cause of dissonance is also due
to Helmholtz, and completely solves a question which had remained
unanswered since the time of Pythagoras, although that philosopher
made the important discovery that the simpler the ratio of the two
parts into which a vibrating string was divided, the more perfect
was the consonance of the two sounds.
21) The sound of the piano, violin, etc., is only in a small
measure due to the actual vibration of the strings themselves. The
latter communicate their own motion to the sound board of the
piano, and to the front, back, and enclosed air of the violin. In
the latter instrument communication is made to the surrounding air
from that within it by means of the holes.
If a string were merely stretched between two pegs firmly
fixed in a stone wall and caused to vibrate, scarcely any sound
would be heard at all, owing to the mass and rigidity of the wall,
which would refuse to be thrown into vibration by so small an
amount of energy as that which the string would possess. On the
other hand, the sound board of a piano readily answers to the
vibrations imposed on it when the string is struck, and having a
large surface in contact with the air, every point of which
originates a system of waves, it causes a full and powerful
sound.
22) The vibrations of straight rods may be either longitudinal
or transversal. The former have not been generally employed for
the production of musical sounds; the latter are such as take
place when a tuning fork is struck, or when a musical box or
triangle is played. In the case of a curved rod the vibrations are
more complicated, but there is one interesting case, namely, that
in which the curved rod takes the form of a circular ring. In this
case the fundamental tone is obtained by suspending it
horizontally by four strings attached at equidistant points in the
circumference, and by lightly tapping it midway between any two.
If the number of vibrations then given be 2n per second, those of
the successive harmonics are proportional to 3nsqrt6, 4nsqrt13, 5nsqrt22,
etc.
23) The nature of the vibrations of a bell may be partly
inferred from those of a ring, as the bell may be considered as
consisting of a connected series of rings of different diameters
all vibrating simultaneously; thus the fundamental tone of a bell
would cause it to divide itself longitudinally into four equal
segments, corresponding to the four quadrants into which the
suspended ring divides. The period of its vibrations could not,
however, be similarly inferred.
24) The vibration of plates is not, musically speaking, a
subject of much interest, as the only instruments which depend
upon it directly for the production of their sounds, are gongs and
cymbals, and the same may be said of membranes. Chladni was the
first to show the positions of the lines of nodes on a plate, by
clamping it horizontally in a vice, and causing it to vibrate by
passing a violin bow over one edge, having previously sprinkled it
with a little sand. The lines of nodes being those parts of the
plate which, like the nodes of a string (§13), are not thrown
into vibration, remain covered with the sand which collects there
from the vibrating portions, and in this way very curious and
interesting figures are produced.
