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LIBRARY
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THE
THEORY OF SOUND
IN ITS
RELATION TO MUSIC.
BY
PROFESSOR PIETRO BLASERNA,
OF THE ROYAL UNIVERSITY OF ROME.
WITH NUMEROUS WOODCUTS.
NEW YORK:
D. APPLETON AND COMPANY,
549 & 551 BROADWAY.
1876.
Music
ML
3805
3643
PREFACE.
:
travefer to
Musie
12-22-65
THE present treatise neither pretends to nor aims
at giving a complete description or demonstration
of the phenomena of Sound, or of the history and
laws of Music. Those who wish to study these
subjects thoroughly, whether from their scientific
or artistic aspect, must have recourse to special
works and special study.
Following the example given by Helmholtz in
his now classical book, "Die Lehre von den Ton-
empfindungen,” I have tried to bring together in
a plain and simple form two subjects which have
hitherto been treated of separately. The student
of physics did not go much into the study of
musical arguments, and our artists do not suf
vi
PREFACE.
+
ficiently understand the very important, bearing
that the laws of sound have upon many musical
questions.
Science has latterly made notable progress in
this respect. She has come to regard the history
of the development of Music from a single point of
view, and is on the road to furnish a larger and
firmer basis to musical criticism.
To expound briefly the fundamental principles
of such a science, and to point out its most im-
portant applications, has been the object and
scope of this treatise. I hope, therefore, that it
will be received with some interest both by lovers
of science and by lovers of art.
PIETRO BLASERNA.
April 1875.
NOTE TO THE ENGLISH EDITION.
A FEW of the woodcuts in this work were de-
signed by Professor TYNDALL for his book on
Sound. They are here reproduced by his kind
permission.
CONTENTS.
CHAPTER I.
1. Periodic movements, vibration-2. Sonorous vibration.—3.
Vibration of a bell-4. Vibration of a tuning-fork, graphic
method-5. Vibration of a string-6. Of plates and mem-
branes-7. Vibration of air in a sounding pipe-8. Method
by the manometric flame-9. Conclusion
PAGE
1
CHAPTER II.
1
1. Transmission of sound-2. Propagation in air-3. In water
and other bodies-4. Velocity of sound in air-5. In water
and other bodies-6. Reflection of sound-7. Echo
28
CHAPTER III.
1. Characteristics of sound, and difference between musical
sound and noise-2. Loudness of sound, and the various
causes on which it depends-3. Principle of the super-
position of sounds-4. Sounding-boards and resonators
•
45
X
CONTENTS.
CHAPTER IV.
PAGE
1. Measure of the number of vibrations; the graphic method-
2. Cagniard de la Tour's siren-3. Pitch of sounds; limit
of audible sounds, of musical sounds and of the human
voice-4. The "normal pitch”—5. Laws of the vibrations
of a string, and of harmonics
59
CHAPTER V.
1. Musical sounds—2. Låw of simple ratio—3. Unison, inter-
ference-4. Beats-5. Their explanation-6. Resultant
notes-7. Octaves and other harmonics-8. Consonant
chords and their limits-9. The major fifth, fourth, sixth,
and third; the minor third and sixth-10. The seventh
harmonic
74
CHAPTER VI.
1. Helmholtz's double siren-2. Application of the law of simple
ratio to three or more notes-3. Perfect major and minor
chords, their nature-4. Their inversion
. 96
CHAPTER VII.
1. Discords-2 and 3. The nature of music and musical scales-
4. Ancient music—5. Greek scale-6. Scale of Pythagoras
7. Its decay-8. Ambrosian and Gregorian chants-9.
Polyphonic music, harmony, the Protestant Reformation,
Palestrina-10. Change of the musical scale, the tonic, and
fundamental chord-11. The major scale, musical intervals
CONTENTS.
xi
-12. The minor scale-13. Key and transposition-14 and
15. Sharps and flats-16. The temperate scale, its inaccu-
racy-17. Desirability of abandoning it.
PAGE
. 109
CHAPTER VIII.
1. Quality or timbre of musical sounds-2. Forms assumed by
the vibrations, optical method-3. Another optical method
—4. Phonautographic method-5. Laws of harmonics.— 6.
Quality or timbre of strings and of instruments-7. General
laws of chords-8. Noises accompanying musical sounds—
9. Quality or timbre of vocal musical sounds
146
CHAPTER IX.
1. Difference between science and art-2. Italian and German
music-3 and 4. Separation of the two schools-5. Influ-
ence of Paris-6. Conclusion
. 175
THE THEORY OF SOUND.
CHAPTER I.
1. PERIODIC MOVEMENTS, VIBRATION-2. SONOROUS VIBRATION-3. VIBRA-
TION OF A BELL-4. VIBRATION OF A TUNING-FORK, GRAPHIC METHOD—
5. VIBRATION of a string-6. OF PLATES AND MEMBRANES-7. VIBRA-
TION OF AIR IN A SOUNDING PIPE-8. METHOD BY THE MANOMETRIC
FLAME.-9. CONCLUSION.
1. AMONGST the innumerable forms of motion which exist
in nature, the science of Physics pays especial attention
to some, to which it assigns great importance. These are
those forms of motion in which a body, or part of a body,
arrives at an extreme point, remains at rest for a moment,
retraces its steps, again takes the road it has already passed
over, and continues thus, making regular to-and-fro move-
ments in a determinate line.
The pendulum offers us the most simple example of
such a periodic movement. Its laws have been determined
by Galileo, who discovered that the movement is isochron-
ous-that is to say, that the time in which the to-and-fro
movement is executed, is always the same for the same
pendulum, be its oscillations large or not.
2
THE THEORY OF SOUND.
In other words, if we give to a pendulum at rest a
slight impulse, or a strong impulse, the oscillations will
be respectively small or large; but for the same pendulum
the duration of each oscillation will be always the same;
which may be expressed as follows, that the duration of
the oscillations is independent of their extent.
The law of the isochronism of the pendulum is a very
general law in nature. Although it may not be mathe-
matically exact, still it is sufficiently so for the majority
of cases which we shall consider. Every periodic to-and-
fro movement comparable with that of the pendulum is
called an oscillation, and if it be smaller and quicker, it
is also termed a vibration. For greater clearness we will
give the name of simple vibrations to those which exactly
follow the laws of the pendulum, which, by the way, are
the most simple of all. On the other hand, we will give the
name of compound vibrations to those which follow more
complex laws.
An example will show how vibrations can be more
complex. The movement of the pendulum may be thus
summed up: When it has arrived at the end of its path,
it remains at rest for a moment, and comes back with a
constantly-increasing velocity, which becomes a maximum
in the vertical position, and then decreases during the
second half of its path. For the pendulum, then, the two
extreme points of the oscillation correspond to a velocity
zero, the middle point to the maximum velocity.
An example of compound periodic movement is obtained
COMPOUND PERIODIC MOVEMENTS.
3
by adding to the already existing oscillation of the pen-
dulum some other oscillatory movement. Suppose, for
example, that the pendulum rod be flexible and elastic,
and that it oscillates on its own account, and let us
further suppose the lower heavy part of the pendulum
to be an elastic ball, which, being violently impelled
when at rest, vibrates like a ball on a billiard-table-that
is to say, exhibits successive compressions and expansions.
We shall then have three vibratory movements united in
the pendulum, which will give a compound movement
obviously more complex than the first.
Another example of a compound movement is furnished
by the ball-player, who throws a ball vertically up in the
air, and then sends it up again without allowing it to fall
to the ground. Here the movement is different from that
of the pendulum: The ball goes up with a velocity de-
creasing nearly at the same rate as that of the pendulum
does (but the velocity decreases according to a different
law), comes to rest, and then falls with a constantly-
increasing velocity, and is suddenly stopped and thrown
up again by the muscular force of the player, whilst its
velocity is fairly high and still increasing, according to the
laws which regulate the falling of heavy bodies. In this
case, then, contrary to that which takes place in that of the
pendulum, the two extreme points of the path correspond
—the one to a velocity of zero, the other to a fairly high
velocity-in fact, to the highest velocity possible under
the circumstances. There exist in nature a very great
4
THE THEORY OF SOUND.
number of vibrations of various forms, and that these
have been carefully studied and their importance fully
understood, constitutes one of the most pregnant strides
which Physics has made in this century.
Of all these vibratory movements I will treat of one
group, which merits special attention on account of the
great facility which it offers for its study, and for the
great importance which its application has exercised on
the history of human culture.
2. I wish first of all to demonstrate that sound is formed
by the vibrations of the particles of bodies. To under-
stand these vibrations we are not obliged to know the
intimate structure of the bodies themselves; it is enough
to know that the body can be subdivided into little par-
ticles, and that these particles can move away from one
another, at least within certain limits, without thereby
causing the rupture or disaggregation of the body. This
is the result of everyday observation, and in regard
to our present study, I have no need to go farther
into the matter or to formulate a more or less hypothe-
tical opinion on the intimate structure of the bodies
themselves. We must, however, add to this conception
yet another-that of the elasticity of bodies. A body is
called elastic in which a particle, moved from its natural
position of equilibrium, has a tendency to return to its
first position as soon as the external cause which had
displaced it has ceased.
When a particle is under the circumstances here con-
VIBRATION OF A PARTICLE OF AN ELASTIC BODY.
LO
5
templated, it does what the pendulum does. The instant
it is free to move, it returns towards the position it
originally occupied; at first with small velocity, afterwards
with constantly-increasing velocity. Arrived at its position
of natural equilibrium, it continues for a certain space by
its momentum the movement which it has acquired, and
finally stops and retraces its steps. It oscillates then
about its position of natural equilibrium, precisely as the
pendulum oscillates on one side and the other of its
vertical position. Mathematical investigation shows that
in this case the vibration is simple, like that of the pen-
dulum. But in the study of the vibrations to which a
body, or part of a body, may be subjected, it is not enough
to consider the movement of one particle only. The body
is formed of a vast number of particles, and as each
one vibrates, it is important to know whether they influ-
ence each other in their respective movements. In this
respect any case is possible, according to the special
conditions under which the vibrations take place, and
the cause which excites them. It often happens that each
single particle vibrates on its own account, as if the others
were not in existence. Those vibrations which take place
irregularly in all possible directions, acquire a great im-
portance in respect to the phenomena of heat, and for
others besides, but have no direct bearing upon sound.
In order that the vibration may be sonorous, it is necessary
that the particles should execute their movements together
and with a certain regularity. The vibrations then acquire
в
THE THEORY OF SOUND.
a general and regular character. They may be compared
to the compact manœuvres of a company of soldiers, while
the thermal vibrations rather resemble the altogether
irregular movements of an undisciplined crowd.
3. It can be demonstrated by a certain number of
examples that sound is always accompanied by vibration
of the sounding body. A metal bell is taken, mouth
upwards, firmly attached to a foot A (fig. 1). A very light
a
A
Fig. 1.
pendulum a touches the bell to indicate the movements
that it may make at a given moment. If this bell be
rubbed by a violin bow, a very marked sound is obtained,
and immediately the pendulum is driven away, falls back
against the bell, and is again driven away, and so on; the
movement of the pendulum lasts for a certain time, and
VIBRATION OF A TUNING-FORK.
7
grows less as the sound dies away, and shows that the
bell, so long as it sounds, is in a state of vibration in all
its parts.
4. Another illustration is given by a sort of steel fork
D, which, if it be held by its foot, can be easily set in
vibration (fig. 2). A fork of this kind is called a tuning-
fork. If the tuning-fork be struck upon the table, or if
it be rubbed at the extremities of its branches with a
violin bow, a very faint and scarcely audible musical
sound is obtained. This sound is observably strengthened
if the foot of the tuning-fork be placed in contact with
the table, or, better still, with a hollow box. The sound
can then be clearly heard, and by this means it may be
shown that the sound really exists. This being so, it is
easy to believe that the two branches of the tuning-fork,
when it sounds, are in a state of continual vibratory move-
ment. The movement is very rapid and the eye cannot
follow it, but the outlines of the branches no longer have
a sharp and well-defined form, which clearly shows the
movement of the tuning-fork. This vibratory movement
is made very perceptible by touching the fork itself with
the finger. If the two branches be touched, the move-
ment ceases, and with it the sound. Sound and move-
ment are so correlated that one is strong when the other
is strong, one diminishes when the other diminishes, and
the one stops when the other stops.
But the vibrations of the tuning-fork can be made
visible by the following graphic method:-
8
THE THEORY OF SOUND.
A plate of glass L (fig. 2), coated with lamp-black by
a petroleum flame, and which slides easily in the frame
G, is taken, and a point P having been attached to one of
G
L
D
P
Fig. 2.
the branches of the vibrating tuning-fork, this point is
placed against it, and the plate is drawn along rapidly in
such a way that the point slides continuously on it.
Or, to make the experiment still more certain and more
elegant, a cylinder of brass is used, on which is stretched
a sheet of paper blackened by a petroleum flame. This
cylinder can be turned by means of a screw handle A
moved by hand or by mechanism (fig. 3). The tuning-
fork is now brought close to the cylinder, so that the point
D slightly scores the paper, and is fixed firmly by means
of a vice. If the tuning-fork remains at rest and the
cylinder turns, the point traces a straight line on the
paper, or, as the movement of the cylinder is spiral, it
traces a curve differing very little from a straight line.
If, on the other hand, the cylinder remains at rest and
the tuning-fork vibrates, its point traces a short line per-
pendicular to the first. If, lastly, the tuning-fork vibrates
and the cylinder turns, a regular and very characteristic
GRAPHIC METHOD.
9
undulating line is obtained on the paper, which represents
fairly well the vibratory movement of the tuning-fork.
When the tuning-fork gives a loud sound, the vibrations
D
Bomm
Fig. 3.
traced on the paper are very wide; later on, when the
sound is already weakened, the vibrations begin to dim-
inish in width; finally, when the sound is on the point of
ceasing, they become almost invisible, and are sensibly
confounded with the straight line.
5. The vibration of a string can also be very easily
demonstrated. For this purpose a metal string is used,
stretched over a wooden box (fig. 4). Two bridges, A and
10
THE THEORY OF SOUND.
B, on which the string rests, give it the exact length of
ore metre, and a scale placed beneath it enables the length
of any part to be determined at will. The wire is fixed
d
A
B
Fig. 4.
and kept stretched by means of pegs, exactly as in a
pianoforte. This instrument, which has been known since.
the days of the ancient Greeks, and which is called a
sonometer or monochord, is of great importance both in the
history of music and in the study of those phenomena in
which we are now interested. If the string be rubbed in
the middle with a violin bow, a note is obtained, the pitch
of which depends on many circumstances, as, for instance,
on the length, thickness, and density of the string, and on
its tension. If the string be rubbed lightly, the sound is
feeble; if, on the contrary, it be rubbed with some force,
the sound is loud; and, generally speaking, the intensity
or loudness of the sound depends on the greater or less
amount of force that produces it. The vibration of this
string can be demonstrated in the following manner :—
Simple observation has already shown that the string
VIBRATION OF A STRING.
i
11
At the
when rubbed is in a state of rapid vibration.
extremities, where it rests on the two bridges, the string
appears to be at rest, but when the middle part is exam-
ined, it is found that the string loses its sharpness of
outline. It appears sensibly thickened, and this thicken-
ing reaches its maximum at about the middle of the
string, which proves that each particle of the string per-
forms a to-and-fro movement in a direction perpendicular
to the length of the string. Vibrations of this sort are
called transverse, to distinguish them from longitudinal
vibrations, in which each particle vibrates in the direc-
tion of the string itself.
In practical music no use is made of the longitudinal
vibration of strings; the transverse vibrations only need
here be treated of. In order better to demonstrate their
existence, some little slips of paper, doubled in the middle
like riders, may be placed on the string; when it vibrates,
these riders are thrown up on account of their lightness
and fall back again on the string, thus indicating when it
is in a state of vibration and when it is at rest. The most
simple form of vibration is that in which the whole string
performs simultaneously one single vibration. This effect
can easily be obtained by leaving the string quite free,
and rubbing it with the bow close to one of its ends. In
that case all the riders are thrown up--first those in the
middle, where the movement is greatest, and afterwards
the others in succession. This shows that, with the ex-
ception of the two fixed points of the string, there is no
}
12
THE THEORY OF SOUND.
point of it that does not vibrate, or, in other terms, that
the whole string vibrates in one single vibration. The
note which is thus obtained from the string is the lowest
note corresponding to it, and it is for this reason that it is
called its fundamental note.
But this is not the only note which can be obtained
from the string. If it be touched at its middle with the
finger, or, better still, with a feather (fig. 4), a note is
obtained which is observably higher-a note which the
musical ear easily distinguishes, and which practical
musicians call the octave of the fundamental note. The
string in this case vibrates in two parts in such a way
that the point touched remains at rest. This fixed point
is called a node of the vibrating string, and such a node
has been produced artificially by touching the string
at the point indicated. In fact, if the riders be now
placed on the string, it will be observed in this case
that the rider nearest to the finger does not move,
whilst all the others are thrown off. The rider by re-
maining at rest thus indicates the presence of the node.
Successively higher and higher notes can be obtained
from the string by touching it at a third, a fourth, and
a fifth of its length, &c. An experiment made by means
of riders shows that in each case of this sort the string
subdivides itself into a certain number of parts, invariably
equal-in the first case into three, in the second into four,
and in the third into five, &c.- and the riders that remain
on the string will indicate the equidistant nodes formed
1
VIBRATION OF A STRING.
13
in the string. Thus, for example, if the string be touched
at one-fifth of its length, it divides itself into five equal
parts, and four nodes are formed at distances of 1, 2, 3, 4
of the string's length, whilst at the intermediate points is
4
4
A
Y
Fig. 5.
found the maximum vibratory movement (fig. 5). The
parts of the string between the nodes which contain these
points of maximum movement are called ventral segments.
Fig. 6.
Fig. 6 represents in somewhat exaggerated dimensions
the different modes of vibration which a string assumes in
different cases, when it vibrates as a whole, or is divided
f
2
14
THE THEORY OF SOUND.
!
into 2, 3, 4, &c., parts.
formed, in the others we have 1, 2, 3, &c., nodes. It is
necessary to observe that these different cases correspond
to different, and successively higher and higher, notes of
the same string.
In the first case no node is
6. Another interesting example of vibration is that
offered by a plate or membrane. This case is somewhat
more complicated than that of a string, but the explana-
tion is almost the same. In fact, a plate may be con-
sidered as an assemblage of strings rigidly fastened to-
gether. As we obtained nodal points in the string, we
ought therefore to have nodal lines in the plate, formed
by the junction of the different nodal points.
Chladni, who first observed these nodal lines, indicated
a very simple means of demonstrating them, on which
account the figures produced by these lines are called
"Chladni's figures." Fig. 7 shows some plates, such as are
Fig. 7.
generally used, fixed in the middle to a solid foot, which
rests on a common bench. By sprinkling a little sand on
C
CHLADNI'S FIGURES.
15
the plate and rubbing it with a bow, musical sounds are
obtained, unpleasant because too shrill, but always quite
clear; and the moment the sound is produced, the sand
dances about and collects together on certain straight
or curved lines, which indicate the places where the
vibratory movement does not exist: these are Chladni's
nodal lines.
From the same plate very varied figures can be obtained
Fig. 8.
by applying the finger to suitable points, in order thus to
induce a nodal point, and thence a nodal line. Fig. 8
:
16
THE THEORY OF SOUND.
contains a very beautiful series of figures, which can be
obtained, according to Savart, from a square plate of
sufficient size. The number of these figures is rather large,
and those here drawn represent only a small part of those
which can be obtained, especially when the plate is large.
Membranes also vibrate in an analogous way. The
forms of their nodal lines are generally even more com-
plicated than those of plates, and are observed in an an-
alogous way. The rule is this, that to a determinate note
belongs a determinate figure for the same membrane or
plate, and that the figure is more complicated as the note
produced is higher.
No relation, however, between these phenomena, or
rather no law that will make known the relation between
the figure obtained and the corresponding note, has yet
been discovered.
7. We have up to this point only studied the case of
the vibration of solid bodies, but liquid and gaseous bodies
can also produce sound by vibrating. The best-understood
case is that of a sounding pipe, of which under many
different forms much use is made in practical music. It
is divided into two large categories-that of “flue" or
"mouth" pipes, and that of "reed" pipes. In both the
sound is produced either by breaking up the air which is
blown into them, or by causing it to enter in puffs. In
the first and most important—that is, in the flue-pipe-
this effect is obtained by a special arrangement, which is
called the embouchure of the pipe.

FLUE PIPES.
17
Fig. 9 represents the most common form of flue-pipe.
By blowing through the open tube a, whether by means
of the mouth or by connecting the pipe to a bellows,
a musical sound is obtained. The pipe is hollow within,
and open or closed at the top, according to circumstances,
and it has its embouchure at m and l. Fig. 10 shows better
R
M
a
R
Fig. 9.
Fig. 10.
in section the arrangement of such an embouchure, which
consists of an upper lip 7 and a lower lip m, which are
bevelled to an edge. The air enters by the tube a, passes
into the box b, and breaks through a narrow fissure against

18
THE THEORY OF SOUND.
the upper lip 7. It partly enters the pipe, and induces
vibrations, and produces a very clear and agreeable sound.
Fig. 11 gives an example of a reed pipe. The air, which
enters by r, is obliged, in order to get to the larger tube R,
to pass through a special apparatus, the reed, of which fig. 12
R
B
a
h
Fig. 11.
a
Fig. 12.
gives the detailed arrangement. The box acb is closed in
the middle by a small, thin, elastic, metallic plate d, called
the tongue. When this is lifted up, the air passes through
the fissure a; when the tongue springs back by its own
·
REED PIPES.
•
19
elasticity, it closes the passage. The vibration of the
tongue induces a rapid opening and closing of this fissure;
the air penetrates at intervals, in regular puffs, and thus
a sound is obtained. Whether the first or second kind of
pipe be used, if we blow hard the sound is loud; it be-
comes feeble if we blow gently, and in this last case the
note is the lowest that the pipe can give, for which
reason it is called the fundamental note of the pipe.
This note depends on the dimensions, and, above all, on
the length of the pipe, and also on the nature of the gas
that enters it; so that to determinate dimensions and to
a determinate mode of blowing corresponds a determinate
fundamental note. When the blowing is stronger, it may
happen that the pipe will give a different note from the
fundamental note. This happens especially if the pipe is
very narrow for its length; it is easy to prevent this by
giving to the tube a sectional area proportioned to its
length, according to certain rules suggested by practice.
A pipe, then, may be constructed at pleasure which
will give by preference the fundamental note, or one
which will give by preference higher notes. Sometimes
the one case and sometimes the other iş utilised in prac-
tical music.
In fact there are instruments, both with embouchures
and with reeds, in which each pipe is intended only to give
its fundamental note-as, for example, the organ, in all the
various and complicated forms which it can assume. Many
wind instruments-as the trumpet, trombone, &c., and also
20
THE THEORY OF SOUND.
the flute-are pipes, each one of which is intended to give
a series of notes. This is attained by giving the instru-
ments great length in proportion to their other dimensions
and turning and twisting the pipe when too long in order
to give it a more convenient form. Like a vibrating string,
a sounding pipe thus gives a series of notes successively
higher and higher. It is enough for this to strengthen
the current of air; but a better and quicker effect is ob-
tained by opening a communication with the external air
at suitable points by means of pistons in the cornet, and
by means of the fingers and of the keys in the flute. The
effect may be compared with that obtained when a point
in a vibrating string is touched by the finger; a node is
established, and the vibrating column of air subdivides
into a certain number of parts, according to simple rules,
which vary, however, with the nature of the pipe.
The flute and organ pipes for the most part are flue-
pipes, but in the first the embouchure is formed both by
the instrument itself and the lip of the player. The
clarionet, the oböe, and all the trumpet class, are reed
instruments; these last have a little funnel against which
the lips are placed, which by vibrating act as a reed.
All the various different-shaped pipes are divided into
open and stopped. Besides certain very characteristic
differences, the note produced by the two kinds of pipes
is, cæteris paribus, different in pitch. Two pipes of the
same shape and dimensions, the one stopped and the other
open, give two fundamental notes which are to each other
VIBRATION OF AIR IN A PIPE.
21
as the fundamental note and its octave. If therefore an
open pipe be stopped, its note descends to the octave
below; and if a stopped pipe be opened, its note rises to
the octave above.
Many different methods may be used to demonstrate
the vibration of air in a pipe. One
method is to introduce into an open
pipe, one wall of which is of glass,
by its upper end a light, well-
stretched membrane of paper m on
which some sand is sprinkled (fig.
13). The note is slightly altered
by the introduction of this ex-
traneous body, but nevertheless
continues to exist, and through the
glass wall it may be observed how
the sand is thrown up with some
noise, because the vibration of the
air is communicated to the paper
membrane and thence to the sand.
At the middle point of the pipe
the movement of the sand ceases,
which shows that at this point
there is a true node, and also that
the dancing of the sand is not
Fig. 13.
produced by air in a stream, but only by air in a state of
vibration.
Another simple mode of demonstrating the vibration

22
THE THEORY OF SOUND.
of the air is the following: A tube rather long for its
sectional area is taken, in which one wall is made of very
thin flexible wood. By blowing into it with some violence,
a note is obtained much higher than the fundamental
note of the pipe. The air in vibrating communicates its
vibrations to the flexible side of the tube. Therefore, if
the pipe be held horizontally, and sand be sprinkled on
the flexible side, the sand will dance about and accumu-
late in certain lines which are true nodal lines, and indi-
cate exactly the way in which the air is vibrating in the
interior of the pipe.
Another method of some importance has been pointed
out by Kundt. Take a tube of glass of sufficient sectional
area and about two metres long. Some light powder, such
as lycopodium or cork dust, is poured into the interior of
the tube, and distributed over it fairly evenly. The tube
is then stopped at the two extremities with corks; it is
then held firmly by its middle with one hand, whilst it is
rubbed with a slightly wetted cloth with the other. A
very sharp clear note is then formed; the vibrations of
Fig. 14.
the tube are transmitted to the enclosed air, and the light
powder distributes itself regularly in the way indicated in
fig. 14, where a portion of the tube is represented. The
i
KUNDT'S METHOD.
23
circles are nodes, and between them the powder shows
real transverse lines.
The form of the figure, and especially the distribution
of the nodes, depends on various circumstances-on the
dimensions of the tube, on the note which can consequently
be obtained from it, and on the gas which is enclosed in
the tube. In this respect Kundt's method is susceptible
of great accuracy, and is widely applicable, enabling us
to deduce the velocity with which sound is propagated
through different bodies.
8. The mechanician König has recently devised a very
elegant novel method of demonstrating the vibrations of
the air in sounding pipes-viz., that by means of the
manometric flame. Fig. 15 gives a sufficiently exact
idea of the apparatus contrived by him. One or more
pipes, equal or unequal (in the annexed drawing they are
equal A and B), are mounted on a small box b, which acts
as an air-chest. The indiarubber tube a puts it in com-
munication with a blower. The valves v serve to set in
action at will either pipe or both together. In the pipe
itself a hole is made and closed again by a capsule e,
under which is a flexible elastic membrane which serves
to separate the interior of the pipe from the interior of the
capsule, which is put in communication on one side with a
gas-tube c by means of the small tube d, and on the other
with the small tube f, which ends in a gas-burner. The
gas then enters the capsule, fills it, and passes through the
small tube and the burner. Lighting it, then, at the burner,
*

24
THE THEORY OF SOUND.
a small flame is obtained. If the pipe is not sounding, the
gas passes quietly through the capsule, and gives rise to a
quiet and normal flame. If, on the other hand, the pipe
B
S
Fig. 15.
produces its note, the movement of the vibrating air is
communicated to the membrane, from it to the gas, and
thence to the flame. When the pipe produces this note,
it will be seen that the flame lengthens, grows restless
and more blue, and by all its behaviour indicates some-
thing abnormal. This proceeds from the fact that the
flame takes part in the vibrations of the air in the pipe.

THE MANOMETRIC FLAME.
25
It rises and falls rapidly, and as this movement, on account
of its rapidity, cannot be followed by the eye, only a com-
plete image of the flame is seen, an image caused by the
superposition of the partial short and long flames.
In order readily to observe the vibrations of the flame,
recourse is had to a means often made use of in physics.
Behind the flames is placed a four-sided box of looking-
glass S, which can be rapidly turned round a vertical axis
by means of a handle and a system of toothed wheels.
When the flame burns steadily, a continuous luminous
band is formed in the turning mirror, because to each
position which the mirror takes up in its rotation, there
corresponds a similar image of the flame. If, on the other
hand, the flame vibrates-that is to say, if it is sometimes
short and sometimes long-there will correspond to cer-
tain positions of the mirror long, and to others short
images, and there will be seen in the turning mirror long
and short images succeeding each other (fig. 16, a).
The short flame is confounded sensibly with the lumin-
ous mass, because the lower part
of the flame does not take much
part in the vibrations. But the
long flames are seen clearly sepa-
rated one from the other; whence
the phenomenon presents the ap-
pearance of separate and equal
Fig. 16.
r
flames, as is shown in fig. 16. This method gives a clear
idea of what happens in the interior of the pipe, and if
26
THE THEORY OF SOUND.
care be taken to darken the room in order to remove all
other reflections from the mirror, the experiment becomes
perfectly visible even to a very numerous audience. A
smaller pipe may now be used, which gives a note exactly
an octave higher than that of the first.
When the apparatus is set in action in the same way as
at first, it will be observed that the vibrating flames are
considerably nearer together. In general terms, the higher
the note, for an equal speed of the mirror, the nearer
together are the flames-that is to say, the more rapid are
the vibrations of the air in the pipe. This is a very
important law, which will require further study later on,
but which it is useful to have already demonstrated by this
elegant experiment. We may even go farther, and deter-
mine the ratio between the number of vibrations per
second of the two pipes. Setting them both in action
independently of each other, two series of flames are
obtained one over the other (fig. 16, a b). And with a
little attention it will be seen that with whatever velocity
the mirror be turned, two images of the lower line cor-
respond to one image of the upper line. This conclusion
may then be arrived at, that the octave is always composed
of a number of vibrations per second double that of the
fundamental note.
9. Similar demonstrations might be continued, for, in
truth, examples are not wanting. A few only of the more
important have been cited, and these will suffice to demon-
strate the point to be settled in this chapter. Wherever
SOUND THE RESULT OF VIBRATION.
27
there is sound, there is always vibration; whence it may
be concluded that sound and vibration are concomitant
phenomena. The vibration may come from a solid, a
liquid, or a gaseous body; but there is no known case of
sound, without vibration of material bodies. It does not
necessarily follow that all vibrations must produce sound.
In order that this may happen, they must satisfy certain
special conditions, which will be considered later on. But
up to this point we may say, that wherever there is sound
there is vibration. But vibration is objective: it exists in
sounding bodies independently of the listener.
:
Sound, on the other hand, is produced in our ears, and is
therefore subjective. To a deaf man the vibration exists,
but the sound does not. He would be able to study the
`vibration, though entirely ignorant of the fact that it pro-
duces a special sensation on our normal organisation.
From this we may conclude that vibration is the cause and
sound the effect produced on our ears, or in other words,
that sound is the result of certain forms of vibration of
bodies.
CHAPTER II.
1. TRANSMISSION OF SOUND-2. PROPAGATION IN AIR-3. IN WATER AND
OTHER BODIES-4. VELOCITY OF SOUND IN AIR-5. IN WATER AND
OTHER BODIES-6. REFLECTION OF SOUND-7. ECHO.
1. WE possess in our organisation a special instrument,
the ear, adapted for perceiving sounds. But if the vibra-
tions of the sounding body are the cause of sound, the
question arises, How do these vibrations ultimately reach
our ears, so as to produce the sensation of sound? A
vibration evidently could not be propagated if there
were no medium fit to propagate it. This medium is
generally air; but it may also be any other solid, liquid,
or gaseous body, so long as it be elastic. Elasticity of the
body is a necessary condition, not only for the formation
of sound, but also for its transmission: since a vibration
can only be propagated by transmitting its own move-
ment to the layers of the medium nearest the vibrating
body; these layers communicate it to fresh layers in their
neighbourhood, and so on.
Vibration seems, then, to travel from layer to layer, and
when circumstances permit, in every direction. The vibra-
tory movement is not possible if each particle of the
transmitting medium be not able to vibrate on its own
1
TRANSMISSION OF VIBRATION.
29
account-that is to say, if the medium be not elastic.
And thus the power that a body possesses of transmitting
sound constitutes one of the surest criteria of its elasticity.
A very clear idea of the transmission of a vibratory
movement may be obtained by observing a large surface
of water at rest. On throwing a stone into it, a series of
concentric waves is seen to start from the point struck by
the stone, which as they grow larger, become less distinct,
and end by becoming imperceptible. It would be a mis-
take to suppose that the water itself moved from point to
point. Each particle remains, so to speak, at its post, and
only executes a vibration perpendicular to the direction
of the wave; having accomplished which, it is at the very
same post as before. It is easy to demonstrate that this
is so, by throwing on the water some sawdust or other
floating body, when it will be observed that it is lifted by
the vibratory movement which passes under it, without
being sensibly displaced. It is, then, only the vibratory
movement which moves from one point to another, and
not the body itself or any part of it.
The case of several vibratory movements, which coming
from different points strike against or across each other,
is more complicated, and at the same time more interest-
ing. If two or more stones be thrown into still water at
different points, two or more systems of waves are formed,
which, as they grow larger, join together. Experience
shows that at each point common to two waves moving
in opposite directions, there are special phenomena, called
30
THE THEORY OF SOUND.
interference. But beyond these points, each wave is pro-
pagated exactly as if the other did not exist, and never
had existed.
This is the great principle of the coexistence of vibratory
movements, a principle discovered by experiment and
proved to demonstration by mathematical analysis. It is
applicable to all cases, whatever be the elastic body
examined, and whatever be the nature of its vibrations.
Applied to the case of sonorous vibrations and air, it leads
to this conclusion: that twenty, thirty, a hundred different
sounds may be transmitted in every direction without
reciprocal disturbance.
Ha
2. In order to demonstrate that air is really able to trans-
mit sound, use is made of a glass globe, into which a rod
of brass penetrates, carrying at its lower
extremity a little bell c (fig. 17) attached
by means of an inelastic cotton thread.
The globe is furnished with a stopcock
a in its neck, which enables it to be
opened or closed. If the air be ex-
hausted from the globe as thoroughly
as possible by means of an air-pump,
the globe, and thence the bell, may be
shaken with any degree of violence with-
out any sound being heard. By holding the ear, how-
ever, against the globe, a very feeble sound is heard,
the reason of which is, that the air is not completely
exhausted, and that the cotton thread by which the bell
Fig. 17.
ALL ELASTIC BODIES TRANSMIT SOUND.
31
is suspended is not completely devoid of elasticity, and
therefore transmits the sound, though but slightly. But
the phenomenon is very faint, and is imperceptible at a
very short distance. If the stopcock be opened for an
instant and again closed, a little air will enter the globe,
and the sound of the bell will begin to be heard. Its
vibrations now find an elastic medium, which, although
very rarefied, is able to transmit them at last to the
glass envelope of the globe. The glass, which is very
elastic, transmits them to the external air, and thence
to the ear of the observer, and therefore the sound
is heard, although feebly. If the stopcock be again
opened, and left open so that the air can enter freely,
the sound will grow louder, and when the air in the globe
has reached the same density as the external air, the bell
will be heard with its full loudness. This experiment
shows that air is able to transmit sound, that in this
case it was necessary for such transmission, and that it
transmits sound better as its density is greater.
1
3. Not only air, but all solid, liquid, and gaseous
bodies, are able, if elastic, to transmit sonorous vibrations.
It is a well-known fact that if, when bathing in the sea,
the head be plunged under water, or even the ears only,
the noise produced by the water striking against the
rocks is distinctly heard. And it is an equally well-
known fact, that to hear a distant noise produced by the
passage of men or animals, the ear must be held to the
ground; which shows not only that the earth transmits
32
THE THEORY OF SOUND.
1
sound, but that in some cases it transmits it even better
than air.
Almost all known bodies are able to transmit sound,
and metals are best of all adapted for this purpose. This
transmission succeeds best when the sound is circum-
scribed and obliged to travel in one direction only. This
is not the case for a bell sounding in the open air; the
sound is transmitted in every direction, and soon grows
feeble. But if, on the other hand, the transmission takes
place in one direction only, a sound, although feeble, may
be heard at a great distance. It is on this principle that
are founded the acoustic pipes, or speaking-tubes, in common
use. These are cylindrical tubes, generally of gutta-percha,
which are arranged from point to point as may be required,
with this condition, however, that they should not have
sharp curves. If words be spoken at one end, the sound
is transmitted from layer to layer of the enclosed air, and
easily reaches the other end; communication can thus be
made between two distant parts of a building. Theoreti-
cally, there is no limit to such transmission in cylindrical
tubes; in practice, however, the sound grows gradually
weaker in long tubes, because the vibrating air loses a
part of its vibratory movement by friction with the sides.
of the tube. Very long distances may nevertheless be
attained.
An elegant experiment on the transmission of sound
is that described by Wheatstone. A rod of wood several
yards in length passes from one room to another-for
VELOCITY OF SOUND.
33
example, from one room to the room on the floor below.
In order to preserve it from contact with other bodies, it
is surrounded by a tin tube and by indiarubber, but the
two extremities are left free. One extremity is put in
communication with the sounding-board of a pianoforte
or other musical instrument, and transmits all its notes
to the other extremity. To make these perceptible, it
need only be fixed to some other instrument—a violin,
harp, or pianoforte. The effect is astonishing; a piece
of music played in the other room or other floor is
perfectly heard.
4. This being so, the following questions now present
themselves: With what velocity is sound transmitted in
different bodies? Is it great or small? Is it the same
for all bodies?
By velocity is meant the space passed over in a second
of time; as an example, let us investigate the space passed
over by a sonorous vibration in one second of time in air.
It is a well-known fact that this velocity is not great.
In fact, when a man some distance off strikes an anvil
with a hammer, the movement of the hammer is first seen
and then the sound is heard; and if the distance is rather
great, the lapse of time between the moments at which
the blow is seen and heard becomes very considerable.
The firing of a cannon a long distance off is announced
first by the flame produced by the explosion of the powder,
and only some time afterwards by the report. Similar
examples are very numerous. They demonstrate that
34
THE THEORY OF SOUND.
sound is transmitted much more slowly than light, and
that the velocity of sound in each case cannot be great.
The method of determining the velocity of sound is in
itself very simple. It may be done by merely placing
two cannon at two different stations as far apart as pos-
sible, and having exactly measured the distance between
them, firing them at previously-arranged moments, and
observing by means of a stop-watch the moment at which
the report from the first station reaches the second, and
vice versa. The time which the sound takes to pass over
the space comprised between the two stations is thus
ascertained, and further, the distance between the two
stations being also known, the required velocity is found
by dividing the latter by the former. Such experiments
must be made at night, so as not to be disturbed by other
noises. They ought, further, to be made on calm nights
when there is no wind, because wind, which is merely a
change of place of a great mass of air, increases or
diminishes the velocity of sound according as it is favour-
able or unfavourable—that is to say, according to whether
its direction is the same as, or contrary to, the direction in
which the sound is transmitted. But as it is impossible
to be quite free from wind, the cannon are fired first from
one station and then from the other, because in this case
the wind will be favourable to the transmission of the
one sound, and unfavourable to the transmission of the
other. One of the two velocities will then be too great,
the other too small, and the mean will represent a very
VELOCITY OF SOUND.
35
close approximation to the value which would have been
found had there been no wind.
Experiments of this kind have often been performed.
We may note especially those made by members of the
French Academy in the year 1822 between Monthléry and
Villejuif, those made by Moll and Van der Beck, and lastly,
those lately made by Regnault with much more perfect
means.
The result of these experiments is, that the velocity of
sound in air, at a temperature of 0° C. (32° F.), may be
fixed in round numbers at 330 metres* in a second
of time; and that this velocity increases regularly with
the temperature, so that at a temperature of 16° C.
(60·8° F.) it is about 340 metres per second.
It was
not at first known that it is rather greater for loud
sounds than for feeble ones. However, this difference
observed by Regnault is very small, and may be neglected
in most cases. To confirm the influence of temperature
on the velocity of sound, we may here notice the
experiment made by Captain Parry in Melville Island,
lying in the middle of the group of islands near North
America, from which it results that for the very low
temperature of 38.5° below zero C. (37.3° below 0° F.), the
least velocity was 309 metres per second.
Another question is this, Are low and high notes pro-
pagated with the same velocity? Listening to a military
band playing at a distance, it will be observed that the
* One metre is equal to 3.28 feet, i.e., to about 3 feet 3 inches.
36
THE THEORY OF SOUND.
Y
piece of music which is being executed completely pre-
serves its rhythmic movement.
The notes reach the
observer enfeebled by a great distance, but they maintain
exactly the same consecutive order. This, however, would
not be possible if the different notes, whether high or low,
had not the same velocity. Biot endeavoured to make
more accurate experiments on this point by causing a
known and very simple melody to be played on a flute in
such a way that the sound entered one of the water-
pipes at Paris. On listening at the other end of this
very long pipe, he found the rhythm of the melody un-
altered. It is not at all impossible that still more accurate
experiments may reveal some such small difference in
this respect as was found for loud and feeble sounds in
certain of Regnault's experiments, but the difference
would be certainly very small, and might in most cases
be neglected.
5. The velocity of sound in water has been determined
by Colladon and Sturm in the Lake of Geneva. A bell
was suspended under water and sounded at known mo-
ments; at a long distance from this a tube was led into
the water from the boat, in which was the observer. The
lower end of the tube was much enlarged, like a colossal
ear, and was closed by means of an elastic membrane,
which was completely under water. The sonorous vibra-
tions from the bell were propagated through the water to
the membrane, and from it to the air in the tube. The
observer holding his ear to the tube, distinctly perceived
VELOCITY OF SOUND.
37
the sound. Whence, by measuring the distance from
the bell to the observer, and the time occupied by the sound
in traversing the whole distance, Colladon and Sturm found
a velocity of 1435 metres per second. The velocity of
sound through water is therefore appreciably greater than
through air.
Many other experiments have been tried in order to
arrive at the velocity of sound in different bodies. It
would be impossible to enter into further details on these
points without overstepping the limits I have laid down,
the more so as the methods employed in these researches
are very various, and require a somewhat profound know-
ledge of the theory of sound. I wish therefore to limit
myself to the statement, that the velocity of sound is
small in gaseous bodies, such as air, and is smaller in
proportion as the gas is more dense; that it is therefore
the smallest possible in carbonic anhydride [262 metres
per second], which is a gas one and a half times denser
than air, and the greatest possible in hydrogen [1269
metres per second], a gas which is fourteen times less
dense than air. In gas an increase of temperature.
considerably increases the velocity.
In liquids the velocity is, generally speaking, sensibly
greater than in gas [with the exception of hydrogen]. In
solids it is found to be still greater, especially in metals,
in which it rises as high as twenty times the velocity in
air. But increase of temperature generally considerably
diminishes the velocity, except in iron, in which at first
3
38
THE THEORY OF SOUND.
the velocity increases with the temperature up to 100°
C. (212° F.), and then rapidly diminishes.
These differences and these anomalies arise from the
intimate structure of the various bodies, and from the way
in which this structure varies with temperature. The
velocity of sound depends on two quantities-on the elas-
ticity and on the density of the body; it increases when
the first increases and when the second diminishes. But
as the laws according to which elasticity and density
vary with the temperature are very various, especially in
solid bodies, it follows that the variations of the velocity
of sound in solids must also follow complicated laws.
For different kinds of wood there are very different
values, according to the direction of the fibre and of the
rings. The following table contains some results obtained
in this respect, and will serve to render the above explana-
tion more clear :-
Air
Oxygen
Hydrogen
VELOCITY OF SOUND IN VARIOUS BODIES.
at
0° C.
0° C. (32° F.), { according to aundry
F.),
0° C. (32° F.), according to Dulong.
Metres per
Second.
330
99
0° C. (32° F.)
317
1269
""
Carbonic
anhydride
}
"
0° C.
(32° F.)
262
99
""
Common gas
0° C.
""
(32° F.)
314
وو
Seine water
15° C.
(59° F.), according to Wertheim 1437
Sea-water
20° C.
(68° F.)
1453
""
99
Absolute alcohol
"9
20° C.
(68° F.)
1160
"9
"9
Ethyl æther.
0° C.
""
(32° F.)
1159
99
""
Lead
20° C.
99
(68° F.)
1228
""
""
"9
99
100° C.
(212° F.)
""
33
1204
REFLECTION OF SOUND.
39
Metres per
Second.
Gold
at 20° C.
(68° F.), according to Wertheim 1743
""
""
100° C.
(212° F.)
1719
""
""
•
""
Silver
""
200° C.
(392° F.)
1634
""
99
•
99
20° C.
(68° F.)
2707
""
""
""
""
100° C.
(212° F.)
2639
""
99
""
200° C.
(392° F.)
2477
99
Copper
""
20° C.
(68° F.)
3556
""
""
""
"
100° C.
(212° F.)
3292
دو
""
""
Iron
""
200° C.
(392° F.)
2954
""
99
وو
20° C.
(68° F.)
5127
""
"
""
100° C.
(212° F.)
5299
""
""
200° C.
(392° F.)
4719
""
99
Cast steel.
""
20° C.
(68° F.)
4986
""
""
""
""
100° C. (212° F.)
4925
"9
""
""
200° C. (392° F.)
4788
""
Acacia wood along the fibre
4714
""
99
across the rings
with the rings
""
across the rings.
""
with the rings
1475
1352
Pine in the direction of the fibre
3322
1405
•
794
6. When a sonorous wave strikes against an obstacle,
it exhibits the same phenomena as an elastic body striking
an elastic wall. The sonorous wave is reflected in such a
way that the angle of incidence is equal to the angle of
reflection. The angle formed by the ray of sound which
impinges
on the wall, with the perpendicular to the wall
drawn from the point at which it impinges, is called the
angle of incidence. The angle formed by this perpendi-
cular with the reflected ray of sound is called the angle of
reflection. By means of this law the direction taken by
a ray of sound after its reflection can always be perfectly
determined.
40
THE THEORY OF SOUND.
An innumerable quantity of phenomena of reflection
exist. The two most distinct forms are resonance and
echo. When a sound is produced in a closed chamber,
the sonorous waves are propagated in every direction,
strike against the walls of the chamber, and are sent back
from them by reflection, and can be repeated several times
from one wall to another. An observer within the cham-
ber will hear not only the sound which comes direct from
the sounding body, but will also receive the vibrations
which come by reflection from all parts of the chamber.
The sound is thus remarkably strengthened, and this is
the reason why it is easier to hear and to make one's self
heard in a closed room than in an open space.
Evidently in such a case the sound will not only return
strengthened, but even altered; because the reflections
from the walls, on account of the low velocity of sound,
require some time, and prolong the sound more or less
considerably. If the chamber be small, this prolongation
is not considerable and can be neglected; but when the
chamber assumes large proportions-as, for example, in a
theatre-each note spoken, sung, or played, may be con-
siderably prolonged: it is confounded with the next note;
and this phenomenon of resonance may become extremely
troublesome unless it be remedied. This happens in all
large, enclosed, empty places, where reflection takes place
freely. There is only one way to prevent it, which con-
sists in breaking up the large walls. The seats of a theatre,
the decorations between them, the galleries, even the
REFLECTION OF SOUND.
41
hangings, serve not merely for the accommodation of the
spectators and for the internal beauty of the theatre, but
also fill an even more important office-viz., that of pre-
venting the disagreeable resonance of the place. It is one
of the most difficult problems for an architect to construct
a room on proper acoustic principles-that is to say, a room
in which sound shall be considerably strengthened without
degenerating into resonance, and it may be said that up
to the present time this problem has been solved in very
few theatres in a satisfactory way.
The reflection of sound has been utilised in various
ways; nature and art have combined to solve some
problems not deficient in interest. The celebrated "ear of
Dionysius" is well known; it is a sort of hole excavated
in the rocks near Syracuse, where the least sound is trans-
formed into a deafening roar. The great dome of St Paul's
Cathedral in London is so constructed that two persons at
opposite points of the internal gallery, placed in the drum
of the dome, can talk together in a mere whisper. The
sound is transmitted from one to the other by successive
reflections along the curve of the dome. Similar phe-
nomena are often met with under the large arches of
bridges, viaducts, &c.; and there was a period when
problems of this nature were much sought after, and
often solved, by architects. It is for this reason that
whispering - galleries, speaking - pipes, &c., are so often
met with in old houses.
An elegant mode of demonstrating the reflection of
42
THE THEORY OF SOUND.
sound is by the use of two parabolic reflectors MN M'N'
(fig. 18), placed one opposite the other in such a way that
their centres shall be on the straight line AA'. Placing a
sounding body at a particular point F, called the focus of the
reflector MN, the sound-wave strikes against the reflector,
is driven back thence by reflection on to the second
M
M'
A
KURENCONTIN
N
F
A
N
Fig. 18.
reflector, and by it is concentrated on to its own focus F';
that is to say, the ray FC is reflected in the direction CC',
and by a second reflection along CF. Each of the
remaining rays is reflected in the same way, and they all
ECHO.
43
concentrate at F. An ear placed at this point distinctly
perceives a very slight sound made at F, by means of
the reflectors and of the double reflection which they
give, whilst without the mirrors it would only be possible
to perceive the ray FF', which is much too feeble by itself
to excite a sufficiently strong sensation.
Another case of multiple reflection is met with in
the famous Baptistry at Pisa, a building surmounted by
a narrow cupola of peculiar form. Placing one's self
under the cupola inside the Baptistry and singing a note,
the sound is prolonged for a very considerable time; there-
fore by singing three or four notes in cadence, by the effect
of the reflections, a most beautiful chord is heard, as if
from an organ, which is considerably prolonged.
7. The best understood of all the cases of reflection is
that which is called echo. In order that an echo may be
produced, it is necessary that there should be, at some
distance from the observer, a large vertical wall, or some
other object—as, for example, a rock-which roughly re-
sembles a wall. A sound sent by the observer towards
the wall returns from it by reflection, and if the distance
passed over by the sound be sufficiently great, the reflected
sound will be clearly separated from the sound uttered.
The velocity of sound being, at our ordinary temperature,
about 340 metres per second, the tenth of this is
34 metres. But experiment shows that about five
syllables are pronounced in one second, therefore the
time necessary to pronounce one syllable is one-fifth of a
44
THE THEORY OF SOUND.
second.
68 metres.
1
In this time sound passes over twice 34, or
It follows that, if the reflecting wall be at a distance
of 34 metres from the observer, one syllable when pro-
nounced would take one-tenth of a second to be trans-
mitted to the wall, and another tenth of a second to
return to the observer; in all, therefore, one-fifth of a
second. Therefore the echo would reach the ear of the
observer after the syllable had been pronounced, and
therefore separate and distinct. In this case the echo is
called monosyllabic; it is called dissyllabic when two syl-
lables can reach the observer distinctly. This happens
when the wall is at twice the distance that is to
say, at a distance of about 68 metres. At a triple distance
an echo may be trisyllabic, and so on.
An echo may
also be multiple, when the sound is reflected from two
parallel walls, placed at a sufficient distance from each
other. The most interesting case of this sort is certainly
that of Simonetta, near Milan, a villa with two lateral
wings. The report of a pistol is repeated as often as
thirty-two times.
Examples of echoes are found almost everywhere.
Their explanation is always easy; it seems therefore use-
less to dwell longer on the point.
CHAPTER III.
1. CHARACTERISTICS OF SOUND, AND DIFFERENCE BETWEEN MUSICAL SOUND
AND NOISE-2. LOUDNESS OF SOUND, AND THE VARIOUS CAUSES ON
WHICH IT DEPENDS-3. PRINCIPLE OF THE SUPERPOSITION OF SOUNDS
-4. SOUNDING-BOARDS AND RESONATORS.
1. ALL the different musical sounds in nature, whatever
may be their origin, and by whatever means they may
be propagated, may be distinguished from each other by
three different qualities.
First By the greater or less energy by which they are
produced, or by their loudness.
Secondly, By their pitch.
Thirdly, By a certain characteristic difference, by which
even an almost unpractised ear easily distinguishes the
sound of the violin from that of the flute, that of the
pianoforte from that of the human voice, &c., even though
these sounds are all of the same loudness and the same
pitch. This characteristic difference is called "quality,"
'tone," or "timbre."
<<
We ought, then, to examine on what these three different
characteristics of sound depend. But before entering into
this important matter, it is necessary to explain what is
really meant by sound, when its qualities are spoken of
46
THE THEORY OF SOUND.
A distinction is generally made in physics between sound
and noise. Sound is the result of very regular vibrations
which follow a law, complicated perhaps, but still a law.
When the vibrations assume the simplest possible form-
viz., that offered by the oscillations of the pendulum-the
resulting sound is called simple, or a simple note; if the
law be more complex, the sound is called compound, or a
compound note. Noise, on the contrary, is a mixture of
sounds collected together under no law, or under some
law so complicated that the ear neither understands nor
feels it. It follows that in most cases it is easy to dis-
tinguish the one from the other, but the limit between
sound and noise is not always so clearly drawn. That which
is a sound to one, is a noise to another, and vice versa. The
confused sound produced by the movement of the waves
of the sea is generally considered to be a noise; but an
attentive and practised ear distinguishes determinate
musical sounds, and finds a musical meaning. Thus the
poets speak often, and not without reason, of the harmony
of the waves. An orchestra, when the individual instru-
mentalists are tuning their instruments and preparing to
play, produces a noise which may perhaps be considered as
the line of demarcation between musical sound and noise.
In fact, there really is a considerable amount of music in
it, although perhaps somewhat irregular, and the general
impression produced is by no means disagreeable.
A fine or practised ear is able to pick out a determinate
note from the midst of a confused noise. Often those who
LOUDNESS OF MUSICAL SOUNDS.
47
have not the habit are not aware of the presence of a more
marked note in the midst of so many others; but with
very little attention it becomes easy to recognise it.
In order to demonstrate this fact, use is made of
a series of eight small boards, which are all of the same
length and breadth, and which differ only in thickness.
If one of these boards be allowed to fall on a bench, most
people would be unable to distinguish any note in the
noise of the blow. But a very marked note is there. To
make it, however, perfectly evident, the eight boards may
be allowed to fall one after another. They are tuned so as
to produce the musical scale, which will be perceived very
distinctly. It follows that in the confused noise, produced
by the fall of each board, there is a note, which at first is
not easily perceived, but which is nevertheless sufficiently
clear and distinct.
In the study which we have now commenced, I shall
always consider musical sounds, or notes, and not noises,
because the attempt to determine the quality of a noise
would be meaningless. It has no definite pitch, loudness,
or timbre.
2. This being so, let us now investigate what may be
the causes on which the modification of the loudness of
musical sounds depends, or by which it is produced. The
loudness depends, in the first place, on the greater or
less energy by which the sound is produced. Now, all
the experiments described in the earlier part of this work
show that greater energy produces a more marked vibra-
48
THE THEORY OF SOUND.
tory movement in the particles of the sonorous body, in
the sense that each vibrating particle traverses a longer
space. The law of isochronism of vibrations shows that
the duration is independent of the space passed over,
within a certain approximation, which is generally con-
sidered sufficient. We will call the greatest space
passed over by each particle the amplitude of its vibration,
therefore we may say that the greater or less energy by
which a sound is produced only influences the amplitude
of the vibrations, and not their duration. In other words,
the loudness of a sound is represented by the amplitude of the
vibrations causing it.
The loudness of a sound depends also on the nature and
density of the body which is to transmit it. In fact, a
sounding body is heard in different degrees of loudness,
according as the sound is transmitted by air, some other
gas, water, or some other liquid or solid body. As to the
density, it is enough for me to refer to the experiment
described in the second chapter, of a bell under a glass
receiver. When the air is completely exhausted, the sound
can scarcely be heard, and the sound becomes stronger and
stronger as the air is gradually allowed to enter the receiver.
The loudness depends, again, on the distance of the
sounding body. It is a general law of nature, confirmed
by numerous experiments and by theory, that all those
phenomena, whatever they may be, which have the
property of being transmitted equally in all directions,
must follow the inverse ratio of the square of the distance.
SYMPATHETIC VIBRATION.
49
Sound belongs to precisely this class of phenomena—
in fact, under like conditions, it is transmitted equally in
every direction. It follows that its loudness must vary
inversely as the square of the distance; which means that
a sound that has a given loudness at a certain distance
is found at double the distance to have a loudness four
times less-in other words, its loudness is reduced to one-
fourth. At three times the distance the loudness would
be one-ninth, and for a distance twenty times greater, the
loudness would be of the original loudness.
.
1
400
3. The loudness depends, again, on the presence of other
bodies, capable of vibrating together with the principal
body. We have already seen that sound is stronger in an
enclosed place than in an open one. This arises from
the multiple reflection in the interior of the place, by
which the vibrations which exist within it are not able
to disperse, and therefore come in greater number to the
ear of the observer. This is only a particular case, treat-
ing rather of the conservation of existing vibrations than
of the creation of new ones.
But experiment shows that whenever a body vibrates,
other bodies placed near it are able to enter into a state
of vibration, on this condition only, that such bodies shall
be capable by themselves of producing the same note. This
interesting fact, which deserves a moment's consideration,
may be demonstrated in many ways.
Take a sonometer on which two equal strings are
stretched, tuned to give the same note. In order to show
50
THE THEORY OF SOUND.
whether they are in a state of vibration in any given case
or no, place paper riders on the two strings, as described
in the first chapter [5]. If one of the two strings be
rubbed with the bow so that it may give its fundamental
note, all the riders placed on this string will be thrown up
into the air. But it will be observed at the same time,
that the other string, which has not even been touched,
also exhibits the same phenomenon, although more feebly:
its riders will also after a little hesitation be thrown off.
If the riders be replaced on the two strings, and one of
the strings be touched at its middle point and rubbed with
the bow, a node is set up in the middle, and a higher
note is produced. The second string begins to vibrate of
its own accord in the same way; all the riders are thrown
off except one, which is the one corresponding to the
middle node. This means that the second string vibrates
in the same way as the first.
This may be continued: the first string being made to
vibrate in any way whatever, the riders on the second
string will show that it immediately begins to vibrate in
the same way. The vibrations of the first string are trans-
mitted to the wooden bridge on which it rests, and thence
to the second string. They are also transmitted from the
first to the second string by means of the air, and the
vibratory movement is the same in both strings.
But the vibratory movement of the second string no
longer takes place, if it is unable by vibrating alone to
give the same note as the first.
SYMPATHETIC VIBRATION.
51
To demonstrate this, let one of the strings be stretched
a little more, so that there may be a sensible difference of
note between the two strings-for example, a semi-tone.
The first string may then be rubbed how and to what-
ever extent you please, but no movement is now observed
in the second. It was not therefore the purely mechan-
ical action of the blow or shock given to the instrument
which produced in the previous instance the beautiful
phenomenon observed.
The following is another experiment, tending to show
the same law: Take a tuning-fork, mounted, as is usual,
on a wooden box. Being rubbed with a bow, it gives a
very clear pure note. Now take an organ-pipe, which
itself would give the same note. Scarcely is it made to
sound near the tuning-fork, without, however, touching it,
than the tuning-fork is heard to reproduce the same note.
But the phenomenon no longer takes place, when instead
of the first pipe one is used which gives a different note
from that of the tuning-fork.
Two equal tuning-forks exhibit this phenomenon in a very
marked manner. Even when placed at a great distance
from each other, the one sounds directly the other sounds.
This no longer happens if the tuning-forks do not give the
same note. A convincing proof of this may be obtained
by taking two different tuning-forks, or even by slightly
altering the note of one of the two former tuning-forks by
fastening, by means of some wax, a small coin to one of its
branches. It will not now sound.
52
THE THEORY OF SOUND.
The following is a third method of demonstrating the
same law (fig. 19): Take a cylindrical glass jar a, and
Fig. 10.
make a tuning-fork b vibrate over it. The sound of the
tuning-fork is not in the least reinforced. By pouring
water into the jar, however, the volume of the air enclosed
in it is gradually diminished. By pouring in more and
more, a point is arrived at where the sound is considerably
reinforced. If more be poured in, the phenomenon ceases.
The quantity of water which must be put into the jar
in order to obtain the greatest possible reinforcement, can
thus be determined by a few trials. This point being
SAVART'S BELL.
53
found, let us next look for the cause of this reinforcement
of the sound.
Take the jar and blow gently across the upper edge; a
feeble note is produced by the vibration of the air, like
that of an organ-pipe, and this note is exactly that of the
tuning-fork. If, on the other hand, the water be poured
away, or more be added, notes may be obtained by blow-
ing in the same manner, but they are no longer the same
as the note given by the tuning-fork.
The same conclusion is arrived at by means of Savart's
bell (fig. 20). A large bell a when rubbed by a bow pro-
b
α
Fig. 20.
duces a powerful note. A hollow cylinder of wood, with
a movable bottom b, is so constructed that it can be
brought near it. By altering the position of the movable
bottom, and thus modifying the internal dimensions of the
cylinder (the open end of which is turned towards the
54
THE THEORY OF SOUND.
bell), the point at which the reinforcement of the sound
The effect obtained is con-
is greatest is easily found.
siderable when the cylinder is brought near.
When the sound of the bell is still strong, the reinforce-
ment produced by the cylinder is very sensible. The
effect is still more remarkable when the sound of the bell
is allowed to diminish so that it can scarcely be heard; on
bringing the cylinder nearer, it becomes very marked.
4. These experiments demonstrate, then, that the rein-
forcement of a sound only takes place when there are
C
மா
A
B
Fig. 21.
other bodies in the neighbourhood of the sounding body
themselves capable of giving the same note. This im-
portant law of resonance has been applied to many cases.
The sounding-board is founded on this law. In fact,
tuning-forks give a very feeble sound by themselves.
They are therefore often mounted on wooden boxes, as in
HELMHOLTZ'S RESONATOR.
55
fig. 21, where the tuning-fork AC is attached to the box
that supports it by means of the foot B.
The boxes have different dimensions, according to the
dimensions of their tuning-forks, and enclose a quantity
of air determined for each note. They considerably rein-
force the sound of the tuning-fork, provided that their
dimensions have been well chosen.
An interesting form of sounding-board, which has ac-
quired a great importance of late years, is that called Helm-
holtz's resonator. These resonators are hollow metallic
spheres or even cylinders of different sizes, furnished with
two apertures. One a, the larger, only serves to maintain
a communication between the external air and that in the
sphere; the other b, the smaller, has the form of an orifice.
with an elongated neck, and is intended to be inserted in
the ear [figs. 22 and 23].
บ
a
Fig. 22.
For use, it is necessary to have a series of these reso-
nators of different sizes. Each of them, according to the
volume of air that it contains, reinforces one single note;
56
THE THEORY OF SOUND.
the larger ones serve for the low, the smaller for the high
notes.
The spherical resonators are really the best, and give
the clearest phenomena. Nevertheless, a cylindrical and
a
Fig. 23.
even conical form is sometimes adopted, because they are
more readily held in the hand, and are therefore more
convenient to manage.
It is easy to show that these resonators reinforce musical
sounds, and each one only one particular note. If a series
of tuning-forks, which give notes corresponding to those
of the resonators, be taken, the sound of each fork is rein-
forced by its corresponding resonator. This effect may be
still better observed if the point of the resonator be intro-
duced into one ear and the other be closed with the hand.
It is to be noticed that no resonator will produce this
effect unless it be combined with its corresponding tuning-
fork.
Let us suppose that there are a number of notes mixed
together, our ears then separate them with difficulty.
But if an observer wishes to know whether amongst all
these there is some one particular note, he need only take
the corresponding resonator and hold it to his ear. If the
note in question is there, it will be reinforced, and thus he
will easily be able to distinguish it amongst all the others.
1
RESONATORS AND SOUNDING-BOARDS.
57
An example of this kind is easily to be found. If a
number of suitable tuning - forks be sounded together,
a very agreeable harmony will be produced, in which,
however, an unpractised ear would not perhaps be able to
distinguish the individual notes composing it. By means
of resonators it is quite easy to do so. To take another
example, the human voice is very rich in notes, and
even when merely speaking, we modulate the voice much
more than is generally believed. If a resonator be taken
and held to the ear while the observer speaks in his
natural voice, every now and then he will distinctly
perceive in the resonator the note to which it corresponds,
which signifies that amongst the many notes which he
uses in speaking, there is the particular one to which the
resonator corresponds. He could thus with a little patience
analyse successively all the notes used by a person whilst
speaking.
In a later chapter, following the example of Helmholtz,
I will show what use can be made of the employment of
these resonators. I will show [chapter ix.] how one of the
most important and most delicate laws may be observed
and followed by this means.
The case of resonators and of the sounding-boards
described above must not be confounded with that of
the sounding-boards forming part of certain musical
instruments. The sonometer, which has been described
in an earlier chapter, the violin and other stringed instru-
ments, the pianoforte, &c., have sounding-boards intended
58
THE THEORY OF SOUND.
༣
to reinforce not only one note alone, but all the notes in
turn, and as far as possible to a uniform extent. It
would be a very bad musical instrument in which the
different notes had not the same loudness, when the
method of producing them is the same.
The theory
of these sounding - boards is much more complicated,
and is not easy to follow. I will confine myself to
saying, that in order to obtain this effect it is necessary
that the sounding-board be relatively very large, and that
it should have a particular shape determined by experience.
In this case the sounding-board corresponds to a very low
note, and is subject, like a vibrating string, to such laws
that it corresponds not only to the lower note, but also
to many successively higher notes.
If the lowest note be very low, it can reinforce so many
notes that their number may be considered infinite.
This takes place especially in the case of plates, mem-
branes, and large vibrating boards, and practice shows that
all that is wanted in the way of reinforcing notes can be
obtained.
CHAPTER IV.
1. MEASURE OF THE NUMBER OF VIBRATIONS; THE GRAPHIC METHOD-
2. CAGNIARD DE LA TOUR'S SIREN-3. PITCH OF SOUNDS; LIMIT OF
AUDIBLE SOUNDS, OF MUSICAL SOUNDS AND OF THE HUMAN VOICE-
4. THE "NORMAL PITCH"-5. LAWS OF THE VIBRATIONS OF A STRING,
AND OF HARMONICS.
1. THE second characteristic quality of musical sounds
is their pitch. Every ear, however little practised, dis-
tinguishes a high note from a low one, even when the
interval is not large. I propose to demonstrate that the
pitch depends on the number of vibrations that a sounding
body makes in each second of time, in such a way that the
low notes are characterised by the small number, the high
notes by the large number of their vibrations per second.
In order to solve this problem, we must first solve
another: how the number of vibrations is to be deter-
mined? There are many methods in physics used for this
purpose.
One method I have already to a great extent indicated—
the graphic method, by means of which the vibrations of
a tuning-fork were traced in the first chapter. In that
experiment use was made of a cylinder, turned by hand.
Naturally the motion could not be very regular, but if in-
1
60
THE THEORY OF SOUND.
पै
stead of the hand, use is made of one of the many mechani-
cal equivalents, a perfectly regular motion can easily be
obtained; and further, the velocity of this motion may be
determined. Suppose, for example, that the cylinder has a
velocity of one turn per second; the tuning-fork will then
record its vibrations, and to know their number per second
we have only to count how many there are in one complete.
turn of the cylinder. The calculation is equally simple if the
cylinder makes any other number of turns in a second. If,
for example, the cylinder only makes five, we have only to
count the vibrations in five turns, and the determination
will be accurate, if the number of turns that the cylinder
makes per second is accurately determined. This is often
possible, and I will describe later on a very simple counter,
by which the number of turns of a rotating apparatus is
measured. The problem may then be solved by this
means, as far as the vibrations of a tuning-fork are
concerned.
2. At this point it becomes necessary to make the reader
acquainted with another instrument, which answers this pur-
pose to perfection, and which offers the advantage over the
turning cylinder of not requiring for its use a slight change
of the note, for a point has to be attached to the vibrating
tuning-fork to trace its vibrations. This instrument is
Cagniard de la Tour's Siren. Figs. 24, 25, and 26 show the
arrangement of the instrument. It consists of a hollow
empty cylindrical box BBB, which by means of its neck
can be put in communication with a blower capable of
រ

CAGNIARD DE LA TOUR'S SIREN.
61
furnishing a constant current of air. In the upper end of
the cylinder are a certain number of equidistant holes,
disposed on the periphery of a circle concentric to the
outline of the rim itself. These holes are all oblique, so
B
B
Fig. 24.
Fig. 25.
R
Fig. 26.
as to form an angle of about 45° with the vertical line.
Over these holes is a metal disc C, which covers them
entirely, and which is able to turn rapidly on a vertical
axis A. This disc carries an equal number of holes,
corresponding exactly in position and size with the holes
in the disc below. These holes are also oblique, but in a
different direction, and form an angle of about 45° with
the vertical, but an angle of 90° with the direction of the
holes beneath. Fig. 25 shows the section of the two discs,
the fixed one and the movable one, as well as the arrange-
ment of the holes. The holes p slope in one direction and
the holes p' in the other. Fig. 26 is a drawing in section of
the whole apparatus, in which is shown the hollow cylindri-
4
62
THE THEORY OF SOUND.
cal box B of the siren, the movable disc CF, and the axis Aa
on which it turns. When a current of air is forced into the
cylinder by means of the blower, this current passes, it is
true, through the holes; but on account of their obliquity, it
strikes against their sides. The movable disc thus receives
a series of impulses, all in the same direction, and therefore
begins to turn. It turns quickly when the current is strong,
and slowly when the current is weak; by suitably control-
ling the blower, the force of the current can be regulated at
will, and through it the velocity of rotation of the movable
disc.
But the current of air that enters the cylinder in a
regular jet, can only pass out through the holes inter-
mittently as soon as the movable disc has begun to
turn. The reason for this is very simple. The air only
passes out at the moments when the holes in the movable
disc coincide with the holes in the fixed disc; it is, on the
other hand, intercepted when the holes in the movable
disc are over the unperforated parts between the holes
in the fixed disc. It follows that the air must come out
of the siren in the form of little puffs, which will be more
frequent as the number of holes in the two discs is greater,
and the velocity of rotation of the movable disc is greater.
Suppose, for example, that each disc has twenty-five holes,
as is the case in the siren I am now describing [twelve
are shown in the drawing, but the number is arbitrary].
Suppose further that the velocity of rotation is one turn
per second of time. In this case each hole in the upper
1
THE COUNTER.
63
disc will give out twenty-five puffs of air in a second of
time. If, however, the movable disc makes 2, 3, 4, &c.,
turns in a second, I must multiply the number of holes
-i.e., 25-by 2, 3, 4, &c., or generally by the number
of turns, to find the number of puffs of air produced by
each hole per second.
When the siren is set in action, a very pure note is
formed, low at first when the disc turns slowly, but higher
as the velocity of the disc increases. The note is produced
because the external air over the instrument is struck
regularly by the puffs of air from the siren. These
periodical blows produce vibrations in the external air,
the number of which per second evidently corresponds
to the number of blows received per second. We can
therefore produce at will, by controlling the blower, any
note we please, be it high or low; we can at the same
time calculate the number of vibrations per second
corresponding to it, as there is a means of determining
the number of turns that the instrument makes in a
second of time.
The very simple counter, which is placed at the top of
the instrument, and which is represented in fig. 26,
answers this purpose. To the movable disc CF is
fixed a vertical steel arbor Aa, which carries at its upper
end a few turns S of what is called an endless screw.
The teeth of a toothed wheel EH gear into this screw, SO
that it moves forward one tooth for one turn of the arbor
and movable disc. To the axis of the toothed wheel is
64
THE THEORY OF SOUND.
fastened a hand, like that of a clock [see fig. 24]; a
graduated circle enables its movements, and by its means
those of the toothed wheel, to be observed. Each division
on the circle corresponds to one tooth of the wheel, or one
revolution of the movable disc of the siren. The toothed
wheel has 100 teeth, therefore the divisions under the
hand are hundredth parts, arranged round an internal
periphery. A second toothed wheel, with a second hand
and a second set of divisions, is so arranged that for each
revolution of the first wheel it moves forward one tooth
and its hand one division. Its divisions show, therefore,
each hundred turns, whilst those of the first show the
single turns of the movable disc of the siren.
This system enables us easily to determine the number
of turns made by the siren in a given time, even though
it may be very high.
To facilitate the counting, the toothed wheel can be
brought near to the main arbor and moved away from it
at will by means of the button a (fig. 24), which allows of
a small displacement; the wheel is thus put in action for
as long as may be desired, and may be stopped at will.
The instrument is thus used. Suppose we wish to
determine the number of vibrations per second which
corresponds to the note of a tuning-fork. We have:
(1.) To reproduce with the siren the same note in such
a way as to keep it constant for a certain length of
time.
(2.) This being obtained, to set in action a stop-
1
THE SIREN.
65
watch and the counter of the siren, and thus to determine
the number of turns per second.
These two operations are easily performed. The first
result is obtained by suitably loading the blower. The
note of the siren is at first very low, then rises slowly,
and after some time remains constant. The reason of
this constancy is this, that the current of air is just able
to overcome the friction of the instrument for this given
velocity. But say the note of the tuning-fork is higher:
then in order to get up to it, the blower must be loaded
more. The note rises suddenly, and after a few trials
the weight is found which must be put on the blower in
order to reproduce the note of the tuning-fork and keep it
steady. The note keeps constant for a certain length of
time, during which the second operation can be easily
completed. This being arranged, the exact position of
the pointers is observed, the stop-watch is set going, and so
afterwards, at a chosen moment, is the counter of the siren,
which is left in action for ten seconds. It is always
better to operate for a somewhat long time rather than
for one second only, because in the former case a small
error committed in setting the counter going is less felt
than in the latter.
When the ten seconds have elapsed, the action of the
counter is stopped, and from the position of the pointers
the number of turns made by the instrument in ten
seconds is seen.
In this case say there are 358 turns in ten seconds-
1
66
THE THEORY OF SOUND.
that is to say, that 35-8 turns correspond to one second.
To know now the number of vibrations per second, since
the disc of the siren has 25 holes, 35.8 must be multiplied
by 25, which gives 895 as the number of vibrations per
second. The tuning-fork in question, therefore, produces
a note of 895 vibrations per second of time. As we have
operated in this case, so we can operate in any other case
whatsoever. The siren lends itself admirably to researches
of this nature. It only requires a practised ear to be able
to reproduce exactly a particular note. Whether we wish
to have recourse to the graphic method, or whether we
wish to make use of the siren, we have acquired the power
of determining the number of vibrations of very many,
or, I might say, of all notes.
3. Let us now consider the more important results
which have been arrived at on this point, by a minute
and accurate examination of the facts.
Does our ear
What is the limit of audible sounds?
perceive, as a note, any number of vibrations whatever,
or is our perception confined between certain limits?
That there is a lower limit may easily be demonstrated
by means of the siren. When the siren is set in action,
and at first turns very slowly, the single puffs of air are
heard singly, but no note is perceived. A very low note,
however, begins when the siren turns a little faster. By
more exact experiments it is found that there must be at
least sixteen vibrations in a second of time in order to
produce a note; and this limit is only reached by using a
1
LIMIT OF AUDIBLE SOUNDS.
67
very powerful instrument-that is to say, an instrument
able to give a somewhat loud note. In other cases—as,
for instance, in the case of the common siren-twenty or
twenty-five vibrations must take place per second, in
order to produce an appreciable note.
It is more difficult to fix a high limit for sound. If
the blower be successively loaded, the siren turns faster
and faster, the note grows sharper and sharper, and at
last becomes shrill and disagreeable. But with an ordi-
nary siren it would not be possible to obtain a velocity
above a certain limit, because the friction would prevent
a very high velocity. To solve the problem, Despretz
made use of smaller and smaller tuning-forks, and finally
succeeded in demonstrating that there is an upper limit
for sound, beyond which our ear perceives nothing.
This limit was fixed by him at very nearly 38,000
vibrations in a second, a figure that has been finally con-
firmed by Helmholtz; but it is probable that it differs in
different individuals. We may conclude that sonorous
vibrations lie between the limits 16 and 38,000 per
second.
But all the notes comprised between these extreme
limits are not musical notes, properly so called—that is to
say, notes advantage of which is taken in practical music.
The notes that are too low are badly heard; those that are
too high are unpleasant.
In the modern pianoforte of seven complete octaves, the
base A corresponds to about 271, the highest A to 3480
68
THE THEORY OF SOUND.
vibrations per second. Therefore, taking into account the
differences of tuning, it may be said that the notes of the
pianoforte range from 27 to 3500 vibrations per second.
In the violin, the fourth open string [the lowest note]
corresponds to about 193 vibrations; the highest note
may be fixed at about 3500.
This number is not, however, the highest. Some piano-
fortes go up to the seventh C, which corresponds to about
4200; and with the piccolo 4700 and more vibrations per
second are reached. But the real gain that music has
realised from so great an extension is very doubtful.
Notes that are too high are shrill, and lose entirely that
full, sweet quality which constitutes the principal char-
acteristic of musical notes. It may be concluded, without
exaggeration, that musical notes are comprised between
27 and 4000 vibrations per second.
The question of the human voice, and of the limits
between which it acts, is also interesting. In consider-
ing it, we must distinguish between the voice of men
and of women. The latter is represented by about twice
as many vibrations per second as that of men. Subdivi-
sions are made for musical purposes in each of these
classes of voice: thus there are, for men, bass, baritone,
and tenor voices; for women, contralto, mezzo-soprano,
and soprano voices. The following table shows the limits
of each of these voices for a normal case, as they may
reasonably be expected from a good and practised singer.
The figures written in brackets represent cases of excep-
EXTENT AND LIMITS OF VOICES.
69
tional voices which the stage has produced up to the
present time:
Extent and Limits of the Human Voice.
Bass
[B 61]
E
82
D 293 [F
348]
Baritone
(D
73) F =
87 .
F
370 [G
392]
Tenor
(G
98)
A
109
435 (C#
544)*
Contralto.
(C = 110)
E
164
F
696 (A
870)
A
870 (B
976)
C1044
1044 (E = 1305)
Mezzo-soprano (E=164) F = 174
(G - 196) A 218
Soprano
The well-developed voice of a single singer embraces
about two octaves, in the case of women a little more.
The extreme limits of the human voice [man's and
woman's combined] may be fixed within four octaves,
from C=65 up to C1044, certain extreme cases not
included.†
4. A question of some practical importance has latterly
been raised and solved: that of establishing a uniform
pitch for all countries, so as to make it possible to tune
instruments uniformly. For the purpose of tuning musi-
cal instruments, a small tuning-fork is generally used,
which gives the A which corresponds to the second open
string of the violin, and is in a seven-octave pianoforte
the fifth A, counting from the lowest note. All the
* Tamberlik's overpraised C#.
+ Certain marvellously gifted voices have had more extended limits;
the voices of Cruvelli, Catalani, Patti, and Nilsson will always be cele-
brated for this. The highest voice seems to be that of Bastardella, whom
Mozart heard at Parma in 1770, which had three and a half octaves, and
went up almost to 2000 vibrations. Also the voice of eunuchs, and
especially that of the celebrated Farinelli, have a very great range.
70
THE THEORY OF SOUND.
different theatres of Italy and Europe have adopted
pitches differing from each other; and even in the same
theatre the A goes on gradually rising. At Paris, in
1700, it was 405; later on, 425; in 1855, 440; and in
1857, 448 vibrations per second. This last number has
since remained steady at the Berlin Theatre; but the
pitch of the Scala at Milan corresponds to 4511, and that
of Covent Garden Theatre at London to 455 vibrations
per second.
This state of things was very unpleasant to singers,
for whom it was no easy matter to satisfy requirements
differing so sensibly in different countries, especially
when it is remembered that modern music, to increase
the effect, is written very much on the extreme notes,
and especially on the high ones, and therefore makes a
great call upon the singers. To this may be added the
tendency of the manufacturers of musical instruments,
and especially of brass instruments, to raise the pitch
continually, in order to give a greater brilliancy of tone to
their instruments.
As may be seen from the example given of the rise of
pitch at Paris, it thus came about that from the last
century until now the pitch had been rising considerably
everywhere, and had a tendency to rise still higher. It
was therefore necessary to find a remedy for so grave an
inconvenience, and an international commission fixed as
the normal pitch (usually called the diapason normal) a
tuning-fork giving 435 vibrations per second of time.
THE HARMONIC SERIES.
71
5. I will close this chapter by demonstrating an impor-
tant law, at which we arrive by studying the number of
the vibrations per second of a string. When the whole
string vibrates in one vibration, it gives its lowest note,
which I have called the fundamental note. If the string
be divided, by touching it with the finger or a feather,
into two, three, four, &c., parts, higher and higher notes
are obtained, which form that which is called an harmonic
series. The notes of this harmonic series are not notes
taken at random. They are very agreeable to the ear in
relation to the fundamental note, and have great import-
ance, as we shall see in the sequel, in the theory of music
and of musical instruments. It may be asked, then, if
there is a simple law to regulate these notes, as the
method of their production is so simple.
To answer this question, all that is needed is to deter-
mine the number of vibrations of the string for the funda-
mental note, and for the successive harmonic notes.
Carefully repeated experiments show that simple
relations exist between all these notes. Let us suppose,
for example, that the fundamental note makes 128 vibra-
tions per second; the second harmonic, which is obtained
by dividing the string into two parts, then makes twice
128 vibrations, or 256 per second; the third harmonic,
which is obtained by dividing the string into three parts,
makes three times 128, or 384 vibrations per second; the
fourth harmonic, which arises from the division of the
string into four parts, makes four times 128, or 512
72
THE THEORY OF SOUND.
4
vibrations per second, and so on. Therefore, calling the
fundamental note 1, the harmonic notes will be exactly
represented, in respect of their vibrations per second, by
the whole numbers, 2, 3, 4, &c. By considering the mode
of formation of these notes, the following two laws are
arrived at:-
(1.) The harmonics increase, in respect of the number of
their vibrations per second, as the whole numbers.
(2.) The number of vibrations per second of a string
always varies inversely as its length.
This second law holds for all cases, even when a string
is shortened in any manner whatever; on the sonometer
this shortening is accomplished in a very simple manner.
Besides the two fixed bridges on which the string rests,
there is a third movable bridge, by means of which the
string can be shortened at will. A scale of centimetres
and millimetres allows the length of the effective part of
the string to be measured in any case. A sonometer
constructed in this manner affords the simplest and
shortest means of determining the number of vibrations
of a note. The operation is performed as follows: The
string of the sonometer is stretched so that when vibrating
in its whole length (one metre), it gives a known, constant
note-for example, one of 128 vibrations per second. When
the string is thus tuned, the sonometer is ready for imme-
diate use.
If it be desired to know the number of vibra-
tions of a given musical sound, that note is exactly
reproduced by sliding the bridge along, and so shortening
USE OF THE SONOMETER.
73
the string; the scale under the string gives its new length.
Let this be, for example, 432 millimetres; then, as the
number of vibrations varies inversely as the length of the
string, we arrive at the following proportion:-
432: 1000 :: 128: x
Whence x
128 x 1000
432
= 296
Therefore the note makes 296 vibrations per second.
This method of determining the number of the vibra-
tions is the simplest of all. It is capable of giving results
of sufficient accuracy, and may be adopted directly the
laws of the vibrations of strings have been established.
CHAPTER V.
1. MUSICAL SOUNDS-2. LAW OF SIMPLE RATIO-3. UNISON, INTERFERENCE
-4. BEATS-5. THEIR EXPLANATION—6. RESULTANT NOTES-7.
OCTAVES AND OTHER HARMONICS-8. CONSONANT CHORDS AND THEIR
LIMITS-9. THE MAJOR FIFTH, FOURTH, SIXTH, AND THIRD; THE MINOR
THIRD AND SIXTH-10. THE SEVENTH HARMONIC.
1. IT has been shown in the last chapter that all the
sounds in nature are not musical sounds, properly so
called. In order that a sound may acquire a musical
character, it must satisfy the essential condition of being
agreeable to the ear. It is on this account that all the
sounds produced by imperfect instruments must be re-
jected, whatever may be their pitch. All those, also,
which are too high or too low must be rejected as either
disagreeable or insignificant. There remain, therefore, the
notes comprised between about 27 and 4000 vibrations
per second, which form an interval of a little more than
seven octaves, between which limits the music of all
countries and all nations is written.
But it would be a grave error to suppose that between
the limits hinted at, all the notes can be used arbitrarily
or at hazard. Experience shows that any one of these
notes may be chosen in executing or beginning a piece
MUSICAL SOUNDS.
75
of music. But when once this note is selected, all the
others that are to follow or accompany it are limited, and
we move in a very restricted circle. This is not only the
case in our modern music, but also holds good for the
music of every epoch. There is no instance known of a
musical system, however barbarous it may be, in which
the choice of the notes is left to the fancy of the com-
poser or performer. The history of music, on the con-
trary, teaches us that it has always been sought to select,
from the enormous number of possible notes, an infinitely
more restricted number, according to certain established
rules, in which musical instinct was at times influenced
by scientific theories of greater or less value, giving the
preference to one and sometimes to another of such
theories. We will consider later on the different concep-
tions which instinctively or rationally have guided different
nations in the historical development of music. For the
present I will content myself by saying that in our
modern music, art has outstripped science with rapid
strides, and it is only quite recently that the latter has
been able to give a complete and rational explanation of
what the former has effected by means of delicate
æsthetic feeling.
2. It may be established as one of the fundamental
principles of our music, that the ear can only endure
notes, be they simultaneous or successive, on this condi-
tion—namely, that they should bear simple ratios to each
other in respect of the number of their vibrations per second;
1
76
THE THEORY OF SOUND.
*
that is to say, that the ratio of the number of vibrations
per second of the notes should be expressed by low
numbers. All the bearings of this simple principle will
be pointed out in a later chapter, and since, thanks to the
great researches of Helmholtz, it has acquired of late years,
notwithstanding its simplicity, an even simpler and wider
significance, I will for the present content myself with
indicating its more important consequences.
It is not without some hesitation that I enter upon
such a subject. I shall have to go through a series of
figures, and indeed to argue entirely upon figures. The
road is rather a rough and thorny one, but I trust that,
like the traveller who courageously climbs the steep and
rugged sides of a mountain in order to enjoy at last a vast
and magnificent panorama, so from the highest peak of this
argument a vast horizon will open out before the reader,
in which he will discover the synthesis of one of the
grandest creations of the imagination-a creation that in
itself forms one of the most brilliant pages in the
history of human culture.
3. The most simple ratio that can be imagined between
the vibrations per second of two notes is that in which
both are represented by the same number of vibrations.
The two notes are then said to be in unison. If they be
sounded one after the other, they only form one more pro-
longed note; if they be sounded together, they only give
one note of double loudness. It sometimes happens,
however, that two equal notes, instead of supporting each
1
INTERFERENCE.
77
other, are enfeebled in their effects. Cases of this kind
are due to what is called interference. This happens
whenever the vibrations of the two notes are made in the
reverse way—that is to say, when the vibrating body of
the first note makes a movement in one given direction,
whilst the other makes a precisely contrary movement.
It is evident that such opposing vibratory movements
must destroy each other's effect, when superposed in the
air in which they are propagated; as a particle of air
which ought to move at the same time and with the
same force in two opposite directions, not being able to
follow either, remains at rest.
Fig. 27.
The apparatus represented in fig. 27 enables us to pro-
duce interference at will. It is composed of a vibrating
plate in which a Chladni's figure is formed by the
vibrating segments A B, A' B'. The vibrations in two
78
THE THEORY OF SOUND.
contiguous segments, as A' and B', are contrary, or reverse,
inasmuch as when the particles at A' fall, those at B'
rise, and vice versa; they are similar in two opposite
segments, as at A and A'. DCE is a bifurcated pipe
which gives by itself the same note as the plate, and is
closed at the top by a paper membrane, which serves,
when sprinkled with sand, to indicate the vibrations
in the pipe. If now the plate be caused to vibrate,
and some sand be sprinkled on it to indicate its mode
of vibration, two points having similar vibrations, as
A and A', may be selected, and the branches of the
pipe placed over them, without, however, allowing them
to touch; the sand on the membrane will then dance
about and dispose itself regularly, which shows that the
air in the pipe vibrates, because the vibrations of A and A'
support each other in producing this effect. Again, two
points having an opposite motion, as A' and B', may be
selected and the branches of the pipe placed over them;
the effect will then be nil, and the sand will not move.
From all this we may conclude that when two equal and
simultaneous vibratory movements are superimposed, they
support each other, but that, on the other hand, their effect is
destroyed if they are equal and opposite.
4. The question is interesting as to what happens when
the two notes produced are almost though not quite
identical, and have not therefore quite the same number
of vibrations per second. A new phenomenon then appears,
known by the name of beats.
:
BEATS.
79
In order to show what these beats really are, the follow-
ing experiment may be adopted. Two large equal organ-
pipes are taken which give two low, strong, identical
notes. These being attached to a blower, and sounded
together, we obtain the same note, only of double loud-
ness, which is obtained by sounding either pipe without
the other. But the pipes are so made that it is easy
slightly to alter the note of either.
For this purpose
there is an aperture in the upper part of one of the walls
of each pipe closed by a movable plate, by lowering
which more or less the aperture in the pipe can be opened
to a greater or less extent; the effect thus produced is
similar to that obtained by shortening the pipe. The
note is slightly raised, and by means of the movable plate
it can be regulated at will.
Now let the note of one of the two pipes be slightly
raised. The difference between the notes of the two pipes
is so small that even a practised ear can scarcely perceive
it for such low notes. But if the two pipes be sounded
together, a sound is obtained of varying loudness, now
strong and now feeble, and very marked jerks or shocks.
are perceived. These shocks are the beats. If the dif
ference between the notes of the two pipes be very small,
the beats will be very slow, not more perhaps than one
in the second; but if, on the other hand, the difference
be increased by raising the note of the first pipe, they
will become more frequent. By suitably regulating the
note of one of the pipes by means of the movable plate,
ļ
80
THE THEORY OF SOUND.
靠
​and leaving the other untouched, 2, 3, 4, 5, or 10 beats
per second may be obtained. In the last case, however,
it is difficult to count them; but they are distinctly
audible, and remain so up to 20 or even 25 per second,
beyond that the ear is no longer able to distinguish
them.
Beats of this kind are very common. They are more
especially heard in instruments with fixed strong notes, as,
for example, in the organ. They are a sure sign that the
instrument is not well tuned, and afford a very simple and
correct method of bringing two slightly differing notes
into unison. All that need be done is to tune them until
the beats cease.
In the sound of bells the phenomenon of beats is very
common. In fact, it is not possible to cast a large bell so
that it may present at every point a perfect homogeneity
and an equal density and elasticity. The bell, therefore,
easily divides into two not perfectly equal parts, which by
vibrating somewhat differently produce beats.
5. It is easy to understand how these beats are produced.
If two notes make exactly the same number of vibrations
per second, except in the special case of interference,
which will not be here taken into account, the vibrations
of the two bodies coincide, and a note is produced of
double loudness; but if the two notes be not of exactly the
same pitch, the phenomenon is more complicated. Let it
be supposed, for example, that the first note makes 100
vibrations in a second, and the other 101.
1
BEATS.
81
If they be sounded together, the first vibrations will
almost correspond with each other, their effects will be
added together, and a louder note will be produced. But
at the fiftieth vibration of the first, the second note will
have accomplished fifty and a half vibrations. But as a
vibration always has one half in which the vibrating body
moves in one direction, and another half in which it
moves in the opposite direction, it follows that at the
fiftieth vibration of the first note, and the corresponding
half vibration of the second, the movements will be
contrary, and the note will therefore be sensibly deadened,
or at least more or less considerably enfeebled. To the
hundredth vibration of the first note corresponds the one
hundred and first vibration of the second, and from this
point they again reinforce each other, and so on. It
is thus seen that for each difference of one vibration
there must be one strengthening and one enfeebling of
the note—that is to say, one beat per second. There-
fore, if there be two notes, which differ by 2, 3, 4, 5,
10 vibrations per second, there will be 2, 3, 4, 5, 10
beats per second; and the number of beats, which is
easily observed, gives a very exact measure of the differ-
ence between the number of vibrations per second of the
two notes. This is a very safe, practical method for
determinations of this sort, because it is independent, so
to speak, of the ear, or, at all events, does not require a
very delicate ear.
The phenomenon of beats is not only observed when
82
THE THEORY OF SOUND.
two notes are almost in unison, but even more when the
two notes are to each other, in respect to their number of
vibrations per second, almost, but not quite, in some
other simple proportion. Let, for example, the number of
their vibrations per second be as one to two, then, if the
proportion is exact, there will be no beats; if, on the
other hand, the proportion be not exact, the beats are at
once heard. In order to show this, take two pipes giving
the fundamental note and its octave, and which can be
slightly altered at will. If they be perfectly tuned and
sounded together, there are no beats, and the harmony is
agreeable, like one clearer and fuller note; but if one of
the two notes be ever so slightly altered, immediately
unpleasant beats appear, which spoil the harmony. It
is easy, by an analogous reasoning to the foregoing, to
explain this phenomenon. It is enough, however, here to
draw this conclusion, that the beats are the simplest and
most sure means of observing that two notes are not so
tuned that their vibrations may be represented by a simple
ratio. But as simple ratio is a necessary condition in
order that harmonies may be produced which shall be
agreeable to the ear, it follows that the presence of beats
is a sure proof that an instrument is not properly tuned.
6. In strict relation to the phenomenon of beats, and as
a necessary consequence to the combination of two notes,
are those notes whose discovery, made towards the
middle of the last century, is generally attributed to the
celebrated violinist Tartini, and to which the name of
}
RESULTANT NOTES.
83
resultant notes, or sometimes difference notes, is usually
given. The theory of these notes is not easy to give.
Hitherto it has been held that when the beats become
very rapid, so as to be more than 16 per second, they
generate on their own account a very low note, which is
the resultant note. If there be two notes—one which
makes 100, and the other 125 vibrations per second-they
will give 25 beats a second, which will generate a note
of 25 vibrations a second. There are thus three notes
—the original two of 100 and 125, and the resultant note
of 25 vibrations per second.
But this explanation, however simple it may appear,
and however well it may correspond to the results, is open
to certain serious objections on which it is impossible to
dwell. The true theory of resultant notes can only be
given by means of mathematical calculation.
All that need here be said is, that resultant notes are
really difference notes in this sense, that the number of
their vibrations per second really corresponds to the
difference of the vibrations per second of. the two com-
bined notes. Thus in the example given above, when two
notes are combined together, one of 100 and the other of
125 vibrations per second, a resultant note is obtained
which really corresponds to 25 vibrations per second.
These resultant notes may be experimentally observed
by means of two organ-pipes, one making 200 and the
other 250 vibrations per second, giving a harmony; which
is represented by the ratio §, and which, as will be seen
84
THE THEORY OF SOUND.
later on, is called a major third. When they are sounded
together, besides these two notes, a low note is very clearly
heard, which corresponds to 50 vibrations per second,
a number which is the fourth part of 200, or the half of
the half. It will be seen later on that the half means
the lower octave of a note, therefore the half of the half
signifies two octaves below this same note. It follows
that the resultant note which is formed ought to be the
second octave below of the note of 200 vibrations a
second, which with a little attention is found to be the
fact.
Resultant notes are always present, whenever two
different notes are combined; and there is a very simple
rule for determining them: the number of vibrations per
second of the resultant note is always equal to the difference
between the number of vibrations per second of the notes that
are combined. But as it is more important in the theory of
sound to know the ratios of the numbers of vibrations per
second of the different notes to each other than to know
the absolute number of their vibrations per second, the
different notes are expressed by whole numbers; such
being the case, the resultant note will also be expressed
by whole numbers.
In the example given above it may be said that the
notes 4 and 5 have been combined, since the ratio is
the same as that between 200 and 250. The resultant
note is then represented by the difference 1.
The resultant notes have great importance in the theory
1
THE OCTAVE.
85
of music, as will be shown later on in this chapter. As
they are frequently very loud, it is necessary to take them
into account, and also their ratios to other notes. If, then,
several notes be combined together, it is not enough to
select those which by themselves will give an agreeable
harmony; it is necessary further to examine the resultant
notes, and to see how these will behave in relation to the
combined notes.
It may be added that these resultant notes are notes
that really exist. It follows from this that they can
combine with each other and produce new resultant notes,
which are called resultant notes of the second order.
There are thus resultant notes of the third, fourth order,
&c. But as these are so very feeble that even a
practised ear cannot succeed in distinguishing them, in
most cases it is not worth while to take them into
account.
7. The next most simple ratio that can be imagined
after unison is that of 1:2. This is the ratio called that
of the octave. That note is called the octave of the funda-
mental note that makes twice the number of vibrations
per second. Doubling the number of vibrations of a note
means raising it to its octave above, and vice versa. So,
too, reducing the number of vibrations per second of ja
note to one-half, means descending to its octave below.
The octave of the octave is represented by a number of
vibrations per second four times greater, the third
octave by a number eight times greater; the second,
5
86
THE THEORY OF SOUND.
third, &c., octaves below are expressed by 1,, &c., of
the vibrations per second of the fundamental note.
The harmony of the octave with the fundamental note
is very consonant. When the two notes are perfectly
in tune, which is recognised by the complete absence of
beats, the ear does not distinguish two notes. One
single, open, clear note, as it were, is heard, with a some-
what modified timbre.
The Greeks, who did not use harmony, properly so
called, in their music, nevertheless admitted singing in
octaves; which is easily understood, when it is remem-
bered that the voices of women and boys are an octave
higher than those of adult men; therefore a chorus,
singing all together, must produce an accompaniment
in octaves.
The resultant note produced by the combination of
the notes 1 and 2 is again 1, which means, that in the
harmony of the fundamental note and its octave the
resultant note serves to reinforce the fundamental note.
Other simple relations are furnished by the fundamental
note 1 united to one of the notes of the harmonic series
2, 3, 4, 5, &c. The note 2 represents, as has been seen,
the octave; the note 3 is the twelfth, or as it may also
be called, for reasons which will be seen later on, the fifth
of the octave; the note 4 is the octave of the octave, &c.
All these notes form agreeable harmonies with the
fundamental note. Their only defect, musically speaking,
is that the intervals between them and the fundamental
1
ļ
NOTES NOT IN THE HARMONIC SERIES.
•
87
note are very great. These harmonies, however, are
certainly poor, but not unpleasant, and are especially
used on the violin and other stringed instruments. The
fundamental character of these harmonies is, that the
resultant notes arising from them also belong to the
harmonic series. Thus, for example, the resultant note
of 1 and 3 is 2, that of 1 and 4 is 3, and so on; and
the resultant notes of the second order are found, when
we come to analyse them, to strengthen the fundamental
note.
A
8. But music would be extremely poor if it were wished
to limit it to these few notes, although they are the most
natural ones. Certain brass instruments, indeed, have no
other notes at their disposition, as, for example, the
primitive keyless trumpet; but the melodies played on
such instruments are very restricted and monotonous.
Practical musicians, therefore, have been compelled to
go farther into the matter, and to see if they could not find
other ratios which, although more complicated than the
first, would be still simple enough to be acceptable. But
it naturally follows, from the principle laid down at the
head of this chapter, that the less complicated the ratios,
the more perfect are the harmonies. The introduction
into music of more and more complicated harmonies has
therefore been made slowly and gradually. This must be
considered as progress in the sense that it has increased
musical resources, but it is progress made at the expense
of primitive purity.
!
88
THE THEORY OF SOUND.
Starting from this principle, let us see how far this has
been done up to the present time, and how much farther
we can reasonably go. What has been said is of itself
sufficient apology for the opinion of those who maintain
that music is not the result of absolute æsthetic prin-
ciples, but that it is rather the result of successive musical
education, an education which evidently primarily depends
on the æsthetic aspirations of different nations, and on the
state of their culture. In fact, history shows that all bold
musical innovations have had to contend with immense
opposition, and it is convenient, though not consistent
with truth, to look upon such resistance as only rancour
or personal envy. The true reason is, that there is no
mathematical expression by which to define with certainty
when a ratio is simple and when not; and it is equally
difficult to establish when a sound ceases to be agreeable.
Whether it be more or less simple, more or less compli-
cated, more or less agreeable, it depends on the habit of
the ear how far it will follow a bold innovation. In truth,
certain harmonies, which are now considered perfectly
admissible, were not so considered in past centuries,
especially in the early stages of music.
9. The jump from the fundamental note 1 to its octave
2 is very great, and leads to the inquiry, whether it be
not possible to introduce other notes in this interval. The
study of the harmonic series here offers a good precedent.
In fact we have seen that the ratios 1:2, 1:3, 1: 4,
&c., are consonant. It may then be asked, if the ratios
•
THE FIFTH.
89
which result from taking another note of the harmonic
series as a point of departure such as 2:3, 2:4, 3:4
-are not also consonant; which signifies, in other terms,
that the notes of the harmonic series are consonant not
only with the fundamental note, but also with each other.
This question may be approached in a different way.
Given that the interval between 1 and 2 can or ought to
be filled up with other notes, it may be asked, what will
be the notes which will present the simplest possible
ratios? It is evident that these notes will be expressed
by the figures 12, 13, 13, 14, 14, &c., or by the ratios 3, 4,
15, 11, 1, &c.
3
5 7
The simplest note is that of, which corresponds to the
ratio 2:3. It signifies that the new note makes three
vibrations in the same time in which the fundamental
note makes two, and represents also the harmony of the
second and third harmonic. This ratio was recognised as
consonant even by the ancient Greeks, who made it with
scientific exaggeration the starting-point of their music
and of the formation of the musical scale.
If the harmony
It is the harmony of the fifth.
2:3 be taken, its resultant note is 1-that is to say,
the octave below the fundamental note, seeing that this is
equal to two. This resultant note contributes considerably
to the improvement of the harmony of the fifth.
Another simple ratio is, which may also be written
3:4. This ratio was also known to and admitted by the
ancient Greeks. It is to a certain extent a consequence
90
THE THEORY OF SOUND.
3
of the fifth, and can be derived from it; for if the funda
mental note be 1, the fifth below it is evidently, and its
octave is obtained by doubling its value, hence we have
This ratio is called in music the fourth, whence the
fourth is the octave of the fifth below the fundamental
note. In the harmonic series it represents the harmony
of the third and fourth harmonic. Writing down the
ratio 3: 4, it is seen that the resultant note is 1, which
does not correspond to any lower octave of the fundamental
note, but is instead the second octave below the note 4-
that is to say, of the fourth itself. The harmony of the
fundamental note with the fourth presents, then, this some-
what strange character, that the resultant note, which
arises from it, does not reinforce the fundamental note,
but the fourth, and makes it to a certain extent more
important than the fundamental note.
Another fairly simple ratio is that expressed by; it
corresponds to the major sixth in music. This ratio was
unknown to the Greeks; it is, in fact, more complicated
than the preceding, and it was some time before it was
adopted. Indeed it presents this hitherto new character,
that the resultant note reinforces neither the one or the
other of the two notes, but is an altogether new note.
Writing the ratio 3: 5, the resultant note is 2, which is
the fifth below the fundamental note 3.
Another important ratio is that furnished by the fourth
and fifth harmonics, and expressed by or 4:5; it is
called the major third. Written in the second manner, it
1
MODERN AND PYTHAGOREAN THIRD.
91
gives as a resultant note 1-that is to say, the second
octave below the fundamental note 4. It is a very im-
portant ratio, which was unknown to the ancient Greeks, and
was introduced into modern music in the fifteenth and six-
teenth centuries. The Greeks had in its stead the not widely
different, but obviously dissonant harmony, 81, formed
from the fundamental note 1 by four successive fifths.
3
27
1, 2, 3 × 2 = 2, 3 × 1 × 1 = 4
81
× × × = }}
T 6
This note, lowered by two octaves in order to bring it
near to the note 1, and to keep it in the same octave,
becomes, which is precisely the Greek third, also called
the Pythagorean third, from the name of its inventor.
It may be said without exaggeration that the substitu-
tion for it of the consonant and harmonic third, con-
stitutes the most remarkable and most decisive progress of
our scale as compared with that of the Greeks. The third
5
T
5
also comes into the category of consonant harmonies
through the consideration that its resultant note being the
second octave below, reinforces the fundamental note.
Another harmony, which was introduced into music, is
the minor third. It is expressed by the ratio g, or also
5: 6. It was only adopted in the seventeenth century
with many reservations, together with the harmony of the
sixth, from which it can be easily derived. In fact, it is
only the octave of the sixth inverted. As late as the middle
of the last century, even in the compositions of Mozart,
this harmony was considered as imperfect, and was
92
THE THEORY OF SOUND.
avoided as far as possible as the final chord of a piece.
The resultant note is very low, and does not reinforce
either of the notes of the harmony. Writing it
56, the resultant note is 1, and is in respect to the
note 6 of the harmony the second octave below its fifth.
This note is very low, and not dissonant with the notes
5 and 6. But in the further combinations to which the
harmony of the minor third becomes subjected, the result-
ant note, as will be seen later on, becomes obviously
dissonant.
There is one other harmony, which is now considered
consonant, although imperfectly so that of the minor
sixth, or also 5: 8. It was the last with that of the
minor third to be adopted. The resultant note is 3-
that is to say, the major sixth below the fundamental
note 5-a new note which is not dissonant in itself, but
which becomes so in the fuller chords to which the minor
sixth gives rise.
10. This harmony of the minor sixth is evidently on the
limit of dissonant notes. It may, however, be asked, if it
would not be possible to push on farther in this direction,
so as to enrich music with other fairly consonant harmonies.
This is a question of high musical art, one which has latterly
been much discussed, and one which would merit even
deeper study. It is, however, perhaps too delicate a question
to be touched on here. To treat it properly it would be
necessary to enter into a series of most minute details,
and to consider the harmonies in their relations to three
SEVENTH HARMONIC.
8
93
7 7 8
or four different notes. Such an investigation would go
far beyond the limits of this treatise. It may, however,
be observed that, in order to enlarge the field of music
in this direction, it would be proper to have recourse to
the seventh harmonic, and to consider the ratios I
&c., in which the seventh harmonic has a decisive impor-
tance. Some of these ratios-as,, -are undoubtedly
&
dissonant to our ears. Their resultant notes are so also,
or are at great intervals, and therefore insignificant, and I
do not believe that future generations will ever be taught
to think otherwise. The same cannot be said a prior.
of the harmony. But it is a strange phenomenon to
observe how the seventh harmonic can be entirely
banished from music even as a dissonant note, notwith-
standing that much more complicated, and therefore
much more dissonant, ratios are adopted, as, for ex-
ample,,, &c., of which we shall speak hereafter. To
an ear accustomed to our music, as it is, the seventh
harmonic may appear like an unpleasant note; but an
unprejudiced examination, according to the opinion of
some an opinion with which I entirely agree-shows
that it is rather strange than unpleasant; that in
certain special cases it affords very good discords and
passing chords, and that the strangeness arises rather from
our want of familiarity with it than from its inherent
nature.
It need not, however, be a matter of surprise that this
note should be thus banished from practical music. The
94
THE THEORY OF SOUND.
reason is, scientifically speaking, that the number 7 is not
sufficiently small for a consonant harmony, and that being
great, it has the fault of being a prime number. Since, even
in discords, it is of the greatest importance not to make
too much use of such numbers, numbers greater than 7,
but divisible by 2, 3, 4, or 5, have a great numerical
And this is the true and principal
advantage over it.*
reason why no use is made of it in music.
Without wishing to push too far forward, and to pro-
phesy what will happen in the future, it may be observed
that the systematic introduction of the seventh harmonic
into music would produce in it a very deep and almost
incalculable revolution—a revolution which does not seem
justifiable, because for our magnificent musical system
another would be substituted, perhaps as magnificent, but
certainly not better, and probably worse, and at any rate
more artificial. This does not, however, exclude the pos-
sibility that a secondary part in the musical system will
hereafter be assigned to the seventh harmonic. To certain
chords-for example, the chord of the diminished seventh
-and discords it is well suited, and may sometimes be
substituted with advantage for some in present use.
However this may be, it is certain that for us the
seventh harmonic represents the great line of demarcation
between consonant harmonies and discords. Below it is
* Euler had remarked the importance of the numbers 2, 3, and 5,
and established upon them a rule for the development of our musical
system.
CONSONANT AND DISSONANT INTERVALS.
95
ļ
consonance, and above it dissonance, and between these
is a great hiatus. We have thus the following harmonies
in the interval of one octave:-
Perfectly consonant
Consonant
Imperfectly consonant
5p
5
Hiatus formed by the seventh harmonic.
Dissonant
+100 20+ copo
៖
,
10 &c.
This demonstration may be terminated by represent-
ing in musical notation the harmonies examined above,
with the resultant notes of the first order which arise
from them. The harmonies will be found on the upper
line, in the treble clef; the resultant notes on the lower
line, in the bass clef.
This mode of illustration will serve to make what has
been explained above more clear.
8ve. | 5th.
4th. 6th.
maj. maj. maj.
6th. 3d.
min,
3d.
min. min.
6th.
6th.
be
CHAPTER VI.
1. HELMHOLTZ'S DOUBLE SIREN-2. APPLICATION OF THE LAW OF SIMPLE
RATIO TO THREE OR MORE NOTES-3. PERFECT MAJOR AND MINOR
CHORDS, THEIR NATURE-4. THEIR INVERSION.
1. THE laws explained in the preceding chapter can be
demonstrated by means of the siren constructed by
Helmholtz, which is called the double siren (fig. 28).
It is composed of two complete sirens, do, a, placed
one over the other, so that their revolving discs face each
other. These are attached to the same arbor k, and there-
fore turn together with the same velocity. In the middle
of this arbor is the counter (not represented in the figure),
which is intended to measure the number of turns when
absolute measurement is required. Each disc carries four
concentric circles of holes, according to an idea already
realised by Dove; and by means of four buttons i any one
of the circles of holes, or all simultaneously, can be set in
action. There are thus eight notes, which can be pro-
duced at will, at the disposition of the experimenter.
A strong current of air, which can be caused to enter
either siren at go or g₁, produces, as in the simple siren,
the rotation of the discs and the formation of the
notes.

hi
g1
21
ho
k
Co
Fig. 28.
ho
98
THE THEORY OF SOUND.
In the upper disc the circles have successively 9, 12,
15, 16, in the lower disc, 8, 10, 12, 18 holes. There are
thus many possible combinations of notes, the vibrations.
of which have simple ratios to each other.
The revolving discs are covered by cylindrical boxes
ho, ho, h₁, hi, which serve to reinforce the notes pro-
duced, and to make them clearer.
•
For the study of beats there is an arrangement by
which these ratios can be slightly altered. In fact, the
box of the upper siren can be turned independently of the
rotatory movement of the discs. This is attained by means
of a handle d, which turns the toothed wheel e, which
gears into one on the upper siren, and is so arranged that
for each three turns there is one complete revolution of
the siren itself. Under the handle is a graduated circle
divided into sixteenths, whence each division corresponds
to of a complete turn of the siren. So when the siren
is turned, it follows the holes in the revolving disc, or
moves in a contrary direction to them, according to the
direction of the rotation.
1
48
The effect in the first case is to somewhat lower the
note, in the second to raise it. The reason of this is, that
the note depends on the number of puffs of air which are
produced each time that the holes of the revolving disc
coincide with the holes of the siren; these puffs will
be more or fewer in number, according as the siren is
caused to move in a direction contrary to, or the same
as that of the revolving disc. Very varied experiments
1
HELMHOLTZ'S DOUBLE SIREN.
99
may be made by means of this double siren. The most
important and the most interesting of them may now
be described. If the twelve holes of both upper and
lower siren be set in action, two identical notes are
obtained. Their effects are added together, and they
therefore give a reinforced note, when the two sirens
are so placed that the puffs of air from both are made
at the same instant.
But, on the other hand, they enfeeble each other when
the two sirens are so placed that their puffs are alter-
nate. If the handle d be slowly and regularly turned,
the notes sometimes strengthen, sometimes weaken
each other; beats are thus obtained, the number of which
per second corresponds to the difference of the number
of vibrations per second of the two notes produced by the
sirens.
If the twelve holes of the two discs be kept in action,
and the blower be more loaded, the siren will turn
quicker, and the two notes will rise in pitch but will
still remain in unison, which demonstrates that unison
is independent of the absolute number of vibrations
per second.
But if the eight and sixteen holes be set in action, a
note and its octave is obtained, since the vibrations in any
given time are as 1 to 2. The octave is quite perfect,
and remains so, however much the velocity of rotation of
the discs may be altered by loading the blower. The
notes change, but their ratio remains the same. Whatever,
100
THE THEORY OF SOUND.
then, may be the fundamental note, the octave always
makes double the number of vibrations in a second. If it
be desired to produce beats, the upper siren has but to be
turned. The phenomenon which results is easy to foresee
and to explain.
The ratio of the fifth can be produced in different ways.
by combining the numbers 8 and 12, 10 and 15, or 12 and
18, since in each case there is the ratio 2:3. The ratio
of the fourth is obtained by the holes 9 and 12, 12 and 16;
that of the major third by 8 and 10, 12 and 15, since they
have the ratio 4: 5.
The major and minor sixth are obtained by the combina-
tions 9 and 15, and 10 and 16.
Lastly, the ratio of the minor third is obtained by the
holes 10 and 12, 15 and 18.
There are many other combinations possible, but these
examples are enough for the present purpose. The
harmonies obtained by this means are mathematically
correct, and therefore preferable to those obtained by other
instruments. Experiments made with this instrument
show clearly that the absolute number of vibrations per
second has no influence on the harmonies, because the
ratios always remain the same; since in this instrument
the ratios are fixed by the number of holes in action, whilst
the absolute number of vibrations per second depends on
the velocity of rotation of the revolving discs, which can
be controlled at will. The fundamental note, then, may
be any note whatever; but when once a note has been
-
THEORY OF CONSONANT CHORDS.
101
decided on for this office, the number of vibrations per
second of all the others is fixed by the ratios pointed out
above.
Resultant notes also can be studied by means of this
instrument. The laws are thus arrived at, which were
set forth in the last chapter, by which the resultant note
of a combination of any two notes was defined as a true
difference note.
2. We must now treat of another question which is
intimately connected with that just studied. As yet only
the case of the combination of two notes has been con-
sidered, and the conditions under which they give a con-
sonant chord. But the question may be generalised. It may
be asked, Can three, four, or more notes be so combined as
to produce a consonant chord? In this case the following
rule holds good, which is nothing more than the generalisa-
tion of that explained and illustrated in the last chapter:
In order that a chord, produced by three or more notes, may
be consonant, it is necessary that the different notes that com-
pose it bear, in respect of the number per second of their vibra-
tions, simple ratios not only to the fundamental note but
also to each other. This rule shows that in order to
produce a consonant chord, it is not enough to take the
most simple ratios of octave, fifth, fourth, &c., but it is
necessary also to bear in mind that the different notes that
compose it must bear simple ratios to each other. Thus,
for example, a chord formed by the fundamental note, its
fourth, fifth, and octave, is clearly dissonant, notwith-
1
102
THE THEORY OF SOUND.
7
standing that these notes are represented by the most
simple ratios existing-
4
1, 1, 2, 2.
The reason is that the ratio of the fourth to the fifth is
too complex, and therefore dissonant. In fact, this ratio
is expressed by g, which ratio is not comprised amongst
those giving consonant harmonies.
3. In this respect a much better chord is obtained by
substituting the major third for the fourth, notwithstanding
the fact that it has a more complex ratio with the funda-
mental note. We thus obtain the notes-
1,,, 2,
whose ratios are—
which are all consonant.
56
0
T B
The above chord is the most consonant that exists in
music, and it is therefore called the perfect chord; to
which is also added the word major, because the major
third forms part of it, and to distinguish it from the
perfect minor chord, in which the minor third is substituted
for the major third.
The perfect major chord, as its name expresses, is the
most consonant chord that can be imagined. In its most
simple form it is composed of the fundamental note, major
third and fifth, and it is usual to add the octave. The ratios
of these notes to each other are a major third, a minor
third, and a fourth. The resultant notes which they form
PERFECT MAJOR AND MINOR CHORDS.
103
can be easily determined. In fact, writing the chord in
figures-
1,
5
1, 3, 2,
by combining each note with all the others, it is seen
that the resultant notes are the following:-
1, 1, 1, 1.
4
The first represents the second octave below the funda-
mental note; the second, the first octave below the
same note; the third, the octave below the fifth; and
the last, the fundamental note itself. Therefore the re-
sultant notes reinforce existing notes, more especially the
fundamental note, which contributes greatly to giving the
chord a firm, bold, and decided character. In this respect
the perfect minor chord is inferior to the first, although
its structure is not very different. In fact, if it be written
in the following manner:—
1, 1, 2, 2,
%,
it will be seen that the ratios of the successive notes to
each other are-
음​, 물​, 흉기
​5 4
which only differ from those of the major chord by the
order in which they follow each other. The ratios of the
notes in the perfect major chord to each other are the
intervals of a major third, a minor third and a fourth;
whilst in the minor chord they are a minor third, a major
third and a fourth. The difference is very small in itself,
and is not enough to explain the great fundamental
difference that exists between these two chords.
104
THE THEORY OF SOUND.
But a far more conclusive explanation is found by means
These are in the perfect minor
of the resultant notes.
chord-
3
, 10, 1, 4, 1.
The first represents the major third below the second
octave below the fundamental note, and is a new note in
the chord, which is dissonant with the fifth; the second
is the second octave below the minor third; the third is
the octave below the fundamental note; the fourth is the
major third below the fundamental note-that is to say,
the second octave above the first resultant note-which
thus adds to the existing dissonance of the chord; the last
finally reinforces the fundamental note.
Whilst, then, all the resultant notes in the perfect major
chord reinforce the existing harmony, certain of them in
the minor chord tend to disturb it. If these resultant
notes were stronger, they would suffice to make the chord
dissonant. Even as it is, they impart to it a disturbed
and undecided character.
These two perfect chords, the major and the minor,
form the keystone of our musical system. They are often
met with in compositions, and every piece of music must
end with one or the other.
They are, in truth, the fundamental chords, and impart
to the composition its special character. Pieces founded
on the perfect major chord have a cheerful, brisk, bold,
open character, and are therefore well adapted for the
expression of such conditions of the mind. On the other
靠
​PERFECT MAJOR AND MINOR CHORDS.
105
hand, those founded on the perfect minor chord are sad
and melancholy, or to express it more exactly, are dis-
turbed and undecided, and therefore express well condi-
tions of the mind in which uncertainty and indecision
play the greatest part. Thus it is seen that theory and
practice perfectly agree in defining these two fundamental
chords. It may be added that the history of music also
corroborates these views. Whilst the perfect major chord
has been accepted from the very birth of harmonised
music properly so called, the perfect minor chord was
considered for a long time rather as a slightly dissonant
passing chord than as a fundamental chord. Up to the
time of Sebastian Bach that is, up to the middle of
the last century-there were those who would hesitate
to conclude a piece of music with the minor chord, even
when its character demanded it. In Mozart we find a
certain reluctance to use it as a closing chord; his pre-
decessors willingly omitted the minor third, which
probably did not sound well enough to them. It
may almost be said that the most highly gifted musical
natures may have, as it were, felt beforehand that which
theory has since been able to explain in a simple and
conclusive way. To make this more clear with respect
to the resultant notes of the two perfect chords, these
notes are given below, written in the musical notation.
On the first line are the two chords, on the second the
resultant notes of the first order which they generate.
Those of the first order only are considered, because those
106
THE THEORY OF SOUND.
1
Y
of higher orders are only heard under exceptional circum-
stances, and have no great practical importance.
PERFECT CHORD.
MAJOR
2-2:
-b:
MINOR
70000
4. The perfect chord can, however, be used in different
ways. Carrying one or more of the notes an octave
higher or lower is called inverting the chord. The chord
then acquires a somewhat different character from that
which it had at first, and the resultant notes especially
are considerably modified. This is a point which can be
only indicated, as a fuller study of it would require wider
limits than those of this treatise. However, it may be
remarked that music is very rich in this respect, and that
inverted chords constituted the principal resource of
Palestrina, and of the composers of his school and time.
Chords which in a given position have consonant resultant
notes, can be transformed by invertion into others with
more or less dissonant resultant notes. This is especially
the case with the perfect minor chord, in which the
slight existing dissonance can be considerably reinforced
}
SECONDARY DISSONANCES.
107
by means of invertion, and can be transformed into a
much more marked dissonance. Palestrina, who had cer-
tainly one of the most finely gifted musical organisations,
made very great use of this musical process; and it is
wonderful to see how, without being guided by theory,
and only by the delicacy of his ear, he has been able to
feel and to appreciate such slight differences. In his music
he makes but very little use of true dissonances, but, on
the other hand, much use of those secondary dissonances
which are produced by the resultant notes by means of
invertions of the chords. This justifies the saying of a
great German thinker, who called Palestrina's music,
"music of angels, grieving indeed over things terrestrial,
but not disturbing themselves on that account in their
celestial serenity." As an example, a few inverted chords
MAJOR CHORD
MINOR CHORD
9
are given with their resultant notes, taking the simplest
case of three notes. The other cases are too complicated
to find room here.
Besides the perfect chords, major and minor, there
are also other simple chords. Amongst these may
108
THE THEORY OF SOUND.
1
be mentioned the chord formed by the fundamental
note, fourth, sixth, and octave, and that of the funda-
mental note with the third, sixth, and octave, which have
characters differing somewhat from each other, and
especially when they are compared with the perfect major
or minor chord. These chords can be studied by the
same rules as the perfect chords, and the conclusion is
easily arrived at, that they are nothing more than these
last suitably inverted, as is also seen by the example in
musical notation on the preceding page.
CHAPTER VII
1. DISCORDS-2 AND 3. THE NATURE OF MUSIC AND MUSICAL SCALES-
4. ANCIENT MUSIC-5. GREEK SCALE-6. SCALE OF PYTHAGORAS-
7. ITS DECAY-8. AMBROSIAN AND GREGORIAN CHANTS-9. POLY-
PHONIC MUSIC, HARMONY, THE PROTESTANT REFORMATION, Palestrina
-10. CHANGE OF THE MUSICAL SCALE, THE TONIC, AND FUNDAMENTAL
CHORD-11. THE MAJOR SCALE, MUSICAL INTERVALŠ—12. THE MINOR
SCALE-13. KEY AND TRANSPOSITION-14 AND 15. SHARPS AND FLATS
-16. THE TEMPERATE SCALE, ITS INACCURACY-17. DESIRABILITY OF
ABANDONING IT.
1. Up to this point only the case of consonant chords has
been considered; but music would be very poor if it were
limited to these, and to the few notes that compose them.
It may further be said that music formed only of conso-
nant chords would be extremely monotonous and quite
without vigour; it would be a sort of lullaby only intended
to catch the ear without touching the mind, and without
expressing anything.
To increase their resources, and to acquire greater
vigour and strength in the expression of their ideas,
musicians have been obliged to have recourse to dissonant
notes and chords.
Strictly speaking, much greater satisfaction is felt when
a dissonant chord is resolved into a consonant chord than
6
110
THE THEORY OF SOUND.
when nothing but consonant chords has been heard. It
is the force of contrast which produces these sensations.
in us, just as we doubly appreciate a calm after a storm.
This is exactly the idea which has unconsciously guided
music up to our time. Its strength lies in dissonances, if
they do not last too long, and they be at last resolved into
consonant chords.
The perfect chord being the most consonant of all, must
necessarily close the piece of music. There can be no
absolute rule for the admission of dissonance, and for
determining the limit up to which it may be used. All
this depends on the degree of musical culture and on habit.
Discords which now are perfectly permissible would have
appeared monstrosities in the time of Palestrina. On the
other hand, certain notes adopted by the Greeks at a period
of decline-as, for example, quarter tones-are decisively
rejected by us. It is therefore an error which many
commit, to think that music, and especially modern music,
has absolute character and values, and therefore to reject
every musical system which does not agree with ours.
There is nothing absolute in it but the laws of notes and
their combinations; but the application of these laws is
rather vague, and there remains a very wide and indeter-
minate field, which will be traversed in very different ways
by different nations at various historical epochs.
2. If the history of music be examined with attention,
even if all the evidence possible concerning the music of
barbarous nations be collected, this constant phenomenon
MUSICAL INTERVALS.
111
will be found, that music proceeds by notes clearly
separated from one another. Among the immense number
of notes adopted for musical purposes, there are only a few
that go to make up the various musical systems. A style
of music in which it would be necessary to pass from
one note to another, through all the intermediate notes,
would become almost intolerable. It is true that our
singers, and violin and violoncello players, sometimes
make use of this style with success. But the slide from
one note to another is only tolerated when it is used
sparingly; and it always remains doubtful whether it
would not be better to forbear from it entirely. Music
proceeds, then, by musical intervals, precisely as a man
walks with separate, firm, and decided steps. It seems
that it is in its movement by intervals and by rhythmic
steps, as also by the different shades of piano and forte,
crescendo and diminuendo, accelerando and rallantando, of
legato and staccato, which constitute musical accent, that
the secret of the great impression which music makes
upon the human heart resides. It has thus very varied
means of completely adapting itself to the psychological
movements which constitute any given state of mind,
because it is to be observed that music does not express
determinate sentiments; however, it is applicable to
certain states of mind from which a special sentiment may
arise. That this is the case is easily seen from instru-
mental music; the determinate sentiment is added by
means of words united with music. But if the words
112
THE THEORY OF SOUND.
be taken away, or modified in meaning, it will be seen
that the same melody and the same music may be adapted
to widely different sentiments.
Music is certainly the least material of all the fine arts.
There is no question in it, as in sculpture, of copying
idealised nature; nor, as in painting, of uniting to the
study of nature the geometrical idea of perspective, and
the optical idea of colours and their contrasts. Even
architecture has a larger basis in nature itself. The
trunks of trees and their branches, the grotto, the cavern,
have suggested to the architect the first principles of his
art, dictated to him by the wants of man and the condi-
tions of the strength of materials; but in music nature
offers scarcely anything. It is true that it abounds in
musical sounds, but the idea of musical interval is but
little suggested by the song of birds, and the idea of
simple ratios is almost entirely wanting, and without
these two ideas no music can exist. Man has therefore
been obliged to create for himself his own instrument,
and this is the reason why music has attained its full
development so much later than its sister arts. Music
resembles architecture more than any other art, as in it
also numerical relations are considered. In fact, the
height and width of a building or of a room, the height
and width of windows, the thickness and height of
columns-in a word, all dimensions are linked together
by.numerical relation. But these are only approximate
relations which allow of a certain amount of freedom,
MUSICAL SCALES.
113
whilst in music the relations must be exact, and nature
revenges itself by beats whenever this fundamental law is
departed from, however slightly.
3. In the music of all nations two unfailing characters
are found, rhythmic movement and procedure by deter-
minate intervals. The first appertains also to the speech
and other acts of man, as walking, swimming, dancing,
&c.; the second belongs exclusively to music.
All nations have selected notes to be used, have col-
lected together those intended to be together, and have
thus created one or more musical scales.
By musical scale is meant the collection of all the notes,
comprised between the fundamental note and its octave,
which succeed each other, and are intended to succeed
each other, with a certain pre-established regularity. The
study of the musical scale gives one of the most important
and concise means of judging of the musical state of a
nation. The examination of the musical scale is then
of the greatest assistance, and for this reason a few hints
will here be given on the most important musical systems
that history has noticed up to the present time.
It seems strange that a few notes put together in a
musical scale should be able to acquire a true importance
in the study of music. If it were a question of an as-
semblage of notes made at hazard or capriciously, the
matter would be of no importance; but the musical
scale is always the product of the musical activity of
many centuries. It is not established before music, but
114
THE THEORY OF SOUND.
is developed with it. A very perfect form of music
must have a very perfect scale; an imperfect and primi-
tive form of music, on the other hand, will have a scale
of little value.
In this respect, also, the comparison with archi-
tecture holds good.* In Greek architecture, the
distances between column and column, and wall
and wall, were small; the roofs were flat. Everything
therefore was reducible to vertical and horizontal lines,
and it is this great simplicity that constitutes one of
the most beautiful characteristics of this form of archi-
tecture. The ancient Etruscans invented the arch, which
allows of greater dimensions without impairing stability,
and from which comes the vaulted roof, and as a more
magnificent form of this, the dome.
The Roman architecture is founded on this new dis-
covery. But the semicircular arch becomes unstable when
of large dimensions; it is found that the pointed arch
answers the purpose better in certain cases. It allows of
and demands greater height in the buildings, and is
accompanied by an admirable development of details
which are perfectly adapted to it, and it is thus that
the Gothic style in all its immense variety was developed.
Thus a simple consideration of stability and strength has
caused different nations to find different solutions, and
from three simple primitive forms, three magnificent styles
of architecture have been developed, which differ from
* Helmholtz, Op. cit.
BIBLICAL MUSIC.
115
each other so much that it would almost be thought that
they had nothing in common.
4. Primitive music is as ancient as history itself. From
the high plains of Asia, where many ancient historical
traces of it are found, it followed man in his wanderings
through China, India, and Egypt. One of the most ancient
books, the Bible, speaks of music often and from its
earliest pages."
David and Solomon were very musical. They composed
psalms full of inspiration, and evidently intended to be
sung. To the latter is due the magnificent organisation of
the singing in the temple at Jerusalem. He founded a
school for singers, and a considerable band, which at last
reached the number of four thousand trumpeters, the
principal instruments being the harp, the cithern, the
trumpet, and the drum.
The history of the development of Greek music has a
more important bearing on the question now under con-
sideration. It is incontestably established that the Greeks
had no true principle of harmony even in their most pros-
perous times. The only thing that they did in this re-
spect was to accompany in octaves when men and boys
executed the same melody.
*
Thus their instrumentation only served to reinforce the
[At Genesis iv. 21, speaking of the generations of Cain, it says:
"And his brother's name was Jubal. He was the father of all such as
handle the harp and organ." These words do not prove in the least
that the organ was in existence at that time, but they are certainly
characteristic.]
116
THE THEORY OF SOUND.
voice part, whether it were played in unison or in octaves,
or whether more or less complicated variations were exe-
cuted between one verse and another, or even between the
parts of a verse. With them music was an auxiliary art,
intended to increase, by idealising it, the effect of words.
The development of their music must be regarded only
from this point of view, and in this respect it must be
admitted that they arrived at a considerable degree of
perfection, notwithstanding the truly primitive form under
which it appears at the present time. It was, in fact, a
sort of lofty declamation, with more variable rhythm and
more frequent and more pronounced modulation than
ordinary declamation. This music was much enjoyed by
the Greeks, and when it is considered that the Greeks
were the most artistic and most creative nation that has
ever existed, it becomes necessary to look with care for
the refinements which their music must, and in fact does
contain.
5. The Greek musical scale was developed by successive
fifths. Raising a note to its fifth signifies, as has been
seen in the fifth chapter, multiplying its number of vibra-
tions per second by 3. This principle was rigorously
maintained by the Greeks; rigorously, because the fourth,
of which they made use from the very beginning, is
only the fifth below the fundamental note raised an
octave. To make the tracing out of these musical ideas
clearer, recourse will be had to our modern nomenclature,
making the supposition that our scale, which will be
1
LYRE OF ORPHEUS.
117
studied later on in its details, is already known to the
reader; calling the fundamental note c, and the successive
notes of our scale d, e, f, g, a, b, c, with the terms sharps
and flats for the intermediate notes, as is done in our
modern music. In this scale the first note, the c, repre-
sents the fundamental note, the others are successively
the second, the third, the fourth, the fifth, the sixth, the
seventh, and the octave, according to the position which
they occupy in the musical scale.
If the c be taken as a point of departure, its fifth is g,
and its fifth below is f. If this last note be raised an
octave, so as to bring it nearer to the other notes, and if
the octave of c be also added, the following four notes are
obtained:-
whose musical ratios are-
c, f, g, c,
1, 1, 1, 2.
3
2
1,,
These four notes, according to an ancient tradition, con-
stituted the celebrated lyre of Orpheus. Musically speaking,
it is certainly very poor, but the observation is interesting,
that it contains the most important musical intervals of
declamation. In fact, when an interrogation is made, the
voice rises a fourth. To emphasise a word, it rises another
tone, and goes to the fifth. In ending a story, it falls a
fifth, &c. Thus it may be understood that Orpheus' lyre,
notwithstanding its poverty, was well suited to a sort of
musical declamation.
Progress by fifths up and down can be further continued.
118
THE THEORY OF SOUND.
1
The fifth of g is d, and if it be lowered an octave, its musical
ratio will be. The fifth below ƒ is bb, whence its musical
ratio when raised an octave is 16. We have then the
following scale:-
whose intervals are-
c, d, f, g, bb, c,
1,
2, 4, 2, 1, 2,
3 16
8, 3,
which is nothing more than a succession of fifths, all
transposed into the same octave, in the following way :-
bb, f, c, g, d.
an
This is the ancient Scotch and Chinese scale, in which
enormous number of popular songs are written,
especially those of Scotland and Ireland, which all have a
peculiar and special colouring.
6. But the scale can be continued further by successive
fifths. Omitting, as the Greeks did, the fifth below bb, and
adding instead three successive fifths upwards, we shall
have a as the fifth of d, and e as the fifth of a; and
finally, b as the fifth of e.
The ratios of these notes, when brought into the same
octave, will be-
243
1289
17, 81, 148,
27
169
whence the scale will be the following:-
c, d, e, f, g, a, b, c,
with the ratios-
1, f, §, †, §, 17, 118, 2.
The first and the second of the last three fifths men-
tioned above, the a and the e, were introduced by Ter-
PYTHAGOREAN SCALE.
119
pandro; the last, the b, by Pythagoras, whence the Greek
scale still bears the name of the Pythagorean scale. It
is formed, as has been seen, by successive fifths-that is
to say, with the fundamental idea of simple ratios.
But it is necessary to observe that the execution of this
idea is not entirely happy. It is true that the law of
formation is very simple, but the individual notes have,
nevertheless, an origin very distant from the fundamental
note. The mode of formation of the scale was well suited
for tuning the strings of the lyre, and this seems to have
been one of the principal motives for adopting this mode
of formation; but the interval between any two notes of
the scale is anything but simple. It may thus be seen
further that some of the notes bear extremely complex
ratios to the fundamental note.
This is especially the case with the three notes last
introduced into the scale-that is to say, those corre-
sponding to our a, e, and b—which no longer bear simple
ratios to the fundamental note, being expressed by the
fractions, 8, 113.
243
128′
The last would not be a matter of much importance.
The b can only be considered as a passing note, which
by its open dissonance leads up to the c, or other
consonant note. Its being more or less dissonant does no
harm, and may in certain cases be pleasing. But that the
third and sixth bear complex ratios is a grave defect, and
this is probably the principal reason why the Greek music
did not develop harmony. The Pythagorean third and
120
THE THEORY OF SOUND.
sixth are decidedly dissonant, and with the fourth and
fifth alone no development of harmony is possible, the
more so that the interval between the fourth and fifth is
rather small, and therefore dissonant.
7. The Pythagorean scale held almost exclusive sway
in Greece. However, in the last centuries before the
Christian era-that is to say, during the period of Greek
decline in politics and art-many attempts at modifying
it are found. Thus, for example, they divided the interval
between the notes corresponding to our c and d into two
parts, introducing a note in the middle. At last they
went so far as to again divide these intervals in two, thus
introducing the quarter tone, which we look upon as dis-
cordant. Others again introduced various intervals, founded
for the most part rather on theoretical speculations than
on artistic sentiment.
All these attempts have left no trace behind them, and
therefore are of no importance. But the Pythagorean
scale passed from Greece to Italy, where it held sovereign
sway up to the sixteenth century, at which epoch began
its slow and successive transformation into our two
musical scales.
It ought to be added that the Greeks, in order to
increase the musical resources of their scale, also formed
from it several different scales, which are distinguished
from the first only by the point of departure.
The law of formation was very simple; in fact, suppose
the scale written as follows:-
AMBROSIAN AND GREGORIAN CHANTS.
c, d, e, f, g, a, b, c.
121
Any note whatever may be taken as a starting-point, and
the scale may be written, for example, thus-
or-
e, f, g, a, b, c, d, e;
a, b, c, d, e, f, g, a, &c.
It is evident that seven scales in all can be formed in
this way, which were not all used by the Greeks at
different epochs, but which were all possible. A musical
piece, founded on one or other of them, must evidently
have had a distinctive character; and it is in this respect,
in the blendings of shades, that Greek melody must be
considered as more rich than ours, which is subject to far
more rigid rules.
8. The different Greek scales underwent much dis-
turbance in Italy. Ambrose, Bishop of Milan, and
later, Pope Gregory the Great, had the merit of re-estab-
lishing the first four; and the second, the rest of the
Greek scales. Thus ecclesiastical music [the Ambrosian
and Gregorian chants] acquired a clearer and more elevated
character. It was a recitative on a long-sustained or
short note, according to the words that accompanied it,
music for a single voice, which is still partially retained,
and which may be said to differ from the Greek music
only by the purpose for which it is intended.
9. In the tenth and eleventh centuries an attempt
was begun, especially in Flanders, at polyphonic music-
that is to say, at music for several voices. It consisted in
1
122
1
THE THEORY OF SOUND.
combining together two different melodies, so as not to
produce discord. This sort of music also advanced rapidly
in Italy. In the time of Guido d'Arezzo, the celebrated
inventor of musical notation, such pieces were composed,
in which frequent use was made of successive fifths—a
thing most displeasing to the ear, and which we now look
upon as a serious mistake in music. By the impulse
of Josquino and Orlando Tasso, the last and perhaps the
most important composer of that school, polyphonic music
was developed in a surprising manner. Three, four, and
more melodies were combined together in a most compli-
cated fashion, in which the art of combination had a much
more considerable part than artistic inspiration-mere
tours de force without any musical worth! Such music
was especially cultivated by church singers, to whom
was thus given a means of displaying their own ability.
The voices were interwoven in a thousand ways, and
the only restraint on the composer was not to produce
unpleasant discords. Luther's great Reformation put
an end to this fictitious and artificial style of music.
Protestantism rising into importance at that time, made
it a necessity that church singing should be executed
by the congregation, and not by a special class of singers.
The music was therefore obliged to be simplified to put it
within the power of all. The ground was already prepared
for this. The Troubadours, Minstrels, and Minnesänger
had developed primitive and simple melody, whence
sprang madrigals and popular songs. And thus for
PALESTRINA.
123
polyphonic music another form was substituted, in which
the different voices sustain each other.
Harmony properly so called arose from these simple
and sustained chords, and from the easy movement of the
different voice parts.
The shock of the German movement was felt even in
Italy, where musical reform was initiated in a truly genial
way by Palestrina, partly, indeed, to follow the deliberations
of the Council of Trent. Palestrina abandoned the artificial
method in use up to that time, and laid the most stress on
simplicity and deeply-artistic inspiration. His composi-
tions ("Crux fidelis," "Improperia," "Missa papæ Mar-
celli," &c.) are, and always will be, a model of that style.
10. But so radical a transformation could not be brought
about by one individual, nor in a short time. The Pytha-
gorean scale, which was in general use at the time, was
opposed to a true development of harmony, and the more
so when the execution of the music was intrusted to
human voices in which every discord becomes doubly
perceptible. True harmony could only be developed by
means of the successive transformations of the musical
scale into another, in which the ratios of the notes to the
fundamental note, and to each other, were as simple as
possible. It is thus that the different Greek scales have
been transformed by degrees into our two modern scales-
that is, into the major scale and the minor scale. The first
was more easily to be found, but the second, with its two
variations for the ascending and descending movement, is
'124
THE THEORY OF SOUND.
•
not found completely developed until the seventeenth
century, when music had attained an admirable degree of
development, and when there were magnificent schools of
music and singing in the principal cities of Italy.
Yet another idea characterises our modern music: the
idea of the fundamental note and chord. This idea did
not exist in Greek music, although certain passages of
Aristotle point to something similar. It did not exist in
the Ambrosian chant, but began to be developed with
polyphonic music. The interlaced singing of the Middle
Ages demanded, as a practical condition, that the different
singers should frequently return to one note, as to a firm
resting-place, in order to keep together. The more com-
plicated the harmony was, the more necessary such a
resting-place became. It is thus that the idea of the
fundamental note or tonic was developed, and later, the
idea of the fundamental chord and of key. This precept
has become more and more rigid, as music has become
more complicated. It is now required that a piece of
music should begin and end with the fundamental chord,
which can only be a perfect major or minor chord, and
that in the following out of the musical idea, and in the
development of the great masses of chorus and orchestra
the fundamental note should often recur, as a necessary
resting-place for our comprehension.
This idea was only slowly developed. In the music of
Palestrina and of his successors it had not reached that
clearness which is required at the present day. And this
}
MODERN SCALES-MAJOR AND MINOR.
125
is perhaps the principal reason why this music, notwith-
standing its simplicity and great beauty, appears some-
what confused, and rather quaint than beautiful.
11. These facts being premised, it is now time to
examine in detail the formation and the properties of our
musical scale, as they have been thus necessarily developed
by the requirements of polyphonic music and of harmony.
The scales which are adopted now are two in number: the
major scale and the minor scale. The last having yet
another modification, which may be considered as the
point where the one passes into the other.
In order to hear our scale in its most perfect possible
form, use may be made of an instrument (fig. 29) invented
by Seebeck and perfected by König. A strong clockwork
movement, enclosed in a box P, enables a regular move-
ment of rotation to be given to a metallic disc placed at
O. The disc carries eight series of holes, placed con-
centrically to the same circle at equal distances from each
other. Tubes of indiarubber, A, B, C, are so arranged
that any circle of holes can be blown against at will, so as
to produce different notes. It is, in fact, a siren in which
the rotatory movement is produced independently of the
current of air. The eight circles, beginning with the inner
one, have the following numbers of holes :-
24, 27, 30, 32, 36, 40, 45, 48.
But as the number of vibrations per second of the notes
produced is proportional to the number of impulses, and
this to the number of holes, it follows that at equal

126
THE THEORY OF SOUND.
velocities the above numbers will express the relative
number of the vibrations. Dividing by 24, the following
ratios are obtained:-
1, 2, 4, 3, 2, 5, 15, 2.
9 5 4 3
8
When the disc has acquired a uniform velocity, if the
'P
Fig. 29.
different circles of holes be successively blown against, it
will be heard that the major scale is obtained. And it
may be added that this scale is perfect, because it satisfies
even in theory, the law of exact ratios.
!
MODERN AND PÝTHAGOREAN SCALES COMPARED. 127
The major scale is therefore constituted by the following
ratios:-
which are called-
·
1, 1, 5, 1, 3, 5, 15, 2,
c, d, e, f, g, a, b, c.
In it, as may be seen, the major third, , the fourth,,
the fifth, 3, the sixth, 5, form consonant harmonies with
the fundamental note, which harmonies have already
been deduced from theory. They are the most simple
ratios which can be mentioned. It may further be seen
that these ratios have been substituted with advantage
for some of the more complicated ones of the Pythagorean
scale; especially the third and the sixth, which in it were
dissonant, and which are now consonant. The second,
is the same as in the Pythagorean scale; the seventh,
15, is considerably simplified. It may be said, then, that
this scale is nothing more than a modification of the
Pythagorean scale, made with the idea of keeping the
same notes in such a form, however, that they should
satisfy the law of simple ratio. It is clear and evident
that this scale must therefore lend itself to harmony,
whilst the Pythagorean scale was decidedly opposed to it.
8,
But it is not enough that the ratio of the note to the
fundamental note should be simple, it is necessary that
the ratios of the notes to each other should also be simple.
It will not be possible in this treatise to examine the
question with all the necessary details; the most remark-
able properties of our major scale may, however, be made
128
THE THEORY OF SOUND.
known. The ratio between one note of the scale and its
antecedent note is called a musical interval, a ratio which
is found by dividing one by the other. In this respect
the comparison with the Pythagorean scale is very in-
structive. The Pythagorean scale is expressed by the
following ratios:-
4 3 27 243 2.
1, 8, 81, 3, 2, 16, 128,
The intervals are therefore as follows:-
243,
9 9 256 9 9 9 256
8 8 243 8' 8' 8' 243′
On this scale, therefore, the intervals are of two classes:
the one a somewhat large interval, g, which is called a
whole tone; the other smaller and more complicated,
25, which is called a semitone. The Pythagorean scale
is composed of five whole tones all equal to each other,
and of two complicated semitones, one placed between
the third and fourth, the other between the seventh and
octave.
Our major scale has the following ratios:-
Its intervals are-
3
1, 2, 5, 4, 5, 15, 2.
8, 2,
9 10 16 9 10 9 16
81 9, 15, 8, 9' 8' 1 5°
In it the intervals are therefore of three classes: the largest,
g, which occur three times, are the same as the Pythagorean
intervals; others, a little smaller, 10, are met with twice.
They are both called whole tones, but to distinguish one
from the other, is called a major tone, and 10 a minor
tone. The third interval, considerably smaller than the
9
8
MODERN AND PYTHAGOREAN SCALES COMPARED. 129
16
243.
two first, 19, is met with twice, and in the same place as
the Pythagorean interval 258. It is called a major semi-
tone, to distinguish it from another smaller semitone, 25,
which will be spoken of later on, and which is called a
minor semitone. If no distinction be made between the
major and minor tone, it will be seen that the major scale
presents, like the Pythagorean, five tones, and two semi-
tones, distributed in the same order. The difference
between the two scales lies in this, that we make a
distinction between the major and minor tone, and that
our semitone is much more simple than the Pythagorean.
We have thus three different intervals instead of two,
and therefore a greater complication, which however is
amply compensated for by the simpler ratios and intervals.
To this may be added that the greater complication which
has been pointed out constitutes even by itself a greater
variety, and hence a greater richness, in our music.
Certain musical niceties-as, for example, the somewhat
different character possessed by the different keys—find
their most natural explanation in this greater variety
of musical intervals; and in truth, since, for example, the
interval between c and d is not equal to that between
d and e, to play c d is not the same thing as to play
d e.
The same reasoning applied to a whole piece of
music, shows that the selection of the first fundamental
note and of the key modifies somewhat the order of the
intervals, and thence also the musical character of the
piece.
130
THE THEORY OF SOUND.
12. The second of our scales is the minor scale, in which
the minor third is substituted for the major third, and
the sixth and seventh are modified. It is composed of the
following ratios:-
9 6 4 3
1, ,,2.
Its intervals are the following:-
10.
9 16 10 9 16 9 10
8' 15
%, 18, 1, 1, 18, 2, .
In it the same intervals are found as in the major scale;
10
the intervalg three times, twice, and 1 twice. The
16
15
minor scale, then, only differs from the major in that the
same intervals are differently distributed.
The minor scale is also used in the following form :-
9
6
3
1, 2, 4, 5, 15, 2.
81 51 31 ZI
In which the intervals are-
9 16
9
9
8, 18, 10, 8, 10, 2, 18.
9
This scale is composed, for the first half, of the minor
scale, and for the second half, of the major scale. The
intervals are again the same, only differently distributed.
The second form is adopted by preference for the
ascending scale-that is to say, for the movement from
bass towards treble-whilst the first form is adopted for
the descending scale, from treble to bass.
The minor scale, then, has as its characteristic note the
minor third, whilst the major scale has the major third.
These two scales may therefore be considered as the
further development of, and emanation from, the perfect
CHARACTERS OF MAJOR AND MINOR SCALES. 131
major and minor chords, with which they are closely
allied. A piece of music, written in the major scale, has
the major chord as its basis; the minor chord is reserved
for those pieces which move within the minor scale.
16
1
25
The characteristic difference between the two scales lies
in the thirds. Now the interval between the major third,
5, and the minor third, g, is found by dividing one by the
other. It is, then, 24, a smaller interval than the major
semitone, ; and on this account is called the minor
semitone. It may therefore be concluded that the whole
of the great difference which exists between the major
and minor scale may be reduced to the presence or
absence of one minor semitone in the interval of the third.
However small this difference may be mathematically
speaking, it is very great from an artistic point of view.
Those points discussed with respect to the two perfect
chords [see chapter vi.] also hold good for the two
scales. The major scale and all pieces founded on it have
an open, cheerful, bold character; pieces that are based on
the minor scale are sad, melancholy, and, above all,
undecided. The reason for this is to be found in the
different resultant notes, which are produced by the two
perfect chords; they are also again met with in the
scales. In conclusion, it may be said that variations so
small as to appear unimportant considerably modify the
character of a piece of music.
In this respect the scale is like the skeleton of organised
beings, who show different characters, tendencies, and
$132
THE THEORY OF SOUND.
developments as soon as characteristic differences are set
up in its construction.
13. One last step remained to be made in music, and it
is not probable that it is as yet entirely made. The scale
must now be considered from a fresh point of view, when
it will be seen that it is infinitely richer in resource than
would be at first sight imagined.
This step consists of transposition, and of modulating
from one key to another in the most artistic manner.
Consider an instrument with fixed notes, such as the
pianoforte, and suppose that the scale begins on c, and
that some piece of music-for example, a melody-is
founded on this scale. Its fundamental note or tonic is
then c.
But it may happen that a singer who is to
execute this melody finds it too low or too high for his
voice, and he prefers to transpose it, so that the tonic may
be, for example, g.
This is permissible, since it has been seen as a general
· proposition that any note, whatever may be its number of
vibrations per second, may be used as a point of departure,
since all the successive notes remain in determinate ratios
to the first or fundamental note; which signifies, with
respect to the scale, that the intervals remain the same
when the tonic is moved from c to g.
Now, if the major scale be examined, it will be found
that if the fundamental note be c, there is the interval of
a semitone between e and f-that is to say, between the
third and fourth; the same interval is found between b and
1
TRANSPOSITION.
133
c, or between the seventh and octave. If the piece, then,
be transposed from c to g, in order not to alter the disposi-
tion of the semitones, it must be referred to a scale which
begins on g, and has the interval of a semitone between
the third and fourth, and seventh and octave. But if this
scale be written simply in the following way, as the
Greeks wrote theirs :-
g, a, b, c, d, e, f, g,
it will be seen that between the third and fourth-that is,
between b and c-there is the required interval of a semi-
tone. This is not, however, the case between the seventh
and octave—that is, between ƒ and g. The semitone is, on
the other hand, found between the sixth and seventh-
that is to say, between e and f. The scale, then, does not
preserve the original order of the intervals. The desired
order of intervals can, however, easily be obtained by
raising ƒ a semitone, since then the interval between the
sixth and seventh becomes a tone, whilst that between the
seventh and octave becomes a semitone.
14. Raising a note a semitone signifies raising that note
to its sharp, as lowering it a semitone signifies lowering
it to its flat. In theory, raising a note to its sharp means
multiplying the number of vibrations per second of that
note by the ratio 25, which is the ratio of the number of
vibrations per second of two notes at an interval of a
minor semitone. To lower it to its flat, on the other
hand, means to multiply its number of vibrations per
second by the inverse ratio 2.
24
7
134
THE THEORY OF SOUND.
Every transposition from one tonic to another carries
with it the necessity of raising certain of the notes to
their sharps, or lowering some others to their flats. That
which has been explained for one example holds good for all
cases. The tonic can be changed from c to g, to e, to ƒ, &c.—
that is to say, to each note of the scale. By this operation
a quantity of new notes is obtained, and experiment shows
that every note of the scale ought to be able to be raised
to its sharp. To return to the example of the pianoforte,
it is seen that it is necessary to add to the seven white
keys of an octave seven black keys, and not five, as is
done in practice; as raising to a sharp means rising through
the interval 25, and this interval does not exist in the
simple musical scale.
15.
This idea may be better explained by an example.
Between e and ƒ there is an interval of a major semitone,
16 If e then be raised to its sharp, a note is found by
rising through the smaller interval 25, which is near ƒf,
but lower. e sharp, then, does not coincide with f, as
pianoforte-players are prone to believe.
But there is no need for music to stop at this first
transposition. When it is once admitted that the tonic
can be changed, for example, from c to g, it can be then
changed from c to c sharp. The same considerations,
therefore, must be made for the superadded seven black
keys that have already been made for the white keys. If,
then, the tonic be changed successively to each of the
sharps, in order to keep the intervals the same it is
TRANSPOSITION.
135
necessary to raise some of the sharps a semitone, or
through another interval of 25. Double sharps are thus
24
obtained, which being composed of two minor semitones,
are not exactly equivalent to a whole tone, whether major
or minor. In fact, to take a single example, the double
sharp of c is obtained by multiplying its number of vibra-
tions per second by 25 twice over-that is, by 25. This
fraction, then, represents the interval between c and c
double sharp, whilst the interval between c and d is
represented by the considerably greater ratio. These
two notes, then, are not identical with each other.
24
576.
The conclusion is thus arrived at, that to take into ac-
count all possible transpositions of this kind, it would be
necessary to have the seven original notes-seven notes
for the sharps, and seven notes for the double sharps.
15. Another class of transpositions carries with it the
necessity of lowering by a semitone one or other of the
notes of the scale by multiplying by 24, which is called
lowering a note to its flat. The same considerations which
have been made for the sharps hold good also for the
flats.
Reasoning in the same way, the conclusion is arrived at,
that seven new notes occur for the flats, and seven others
for the double flats. It may be added as a general propo-
sition, that these new notes do not coincide with those
already mentioned for the sharps. In executive music,
and especially for the pianoforte, it is generally admitted
that the sharp of a note is equivalent to the flat of the
136
THE THEORY OF SOUND.
successive note, that, for example, c sharp is equivalent to
d flat; but this is not strictly accurate. In fact, taking c
at 1, c sharp is expressed by . But d is, whence d flat
8
25
24
27
would be multiplied by 2, which gives 21, a value
different from and greater than the first. c sharp is
therefore somewhat lower than d flat. Similar considera-
tions can be made for all the other notes at whole intervals.
As to the semitones of the scale, the considerations remain
the same. If the interval.e to ƒ be taken, for example,
which is equivalent to a major semitone, and therefore is
represented by 1%, as sharps and flats correspond respec-
tively to the intervals 24 and 2, it follows that e sharp
does not coincide either with ƒ flat or ƒ, and that therefore
four distinct notes must be distinguished-e, f flat, e sharp,
and f. Indeed, in composition a distinction is made
between these four notes, and a composer may not con-
found them one with another.
16
25
•
It follows that the pianoforte, in order to take into strict
account all these exigencies, ought for each octave to have
one set of seven keys for the original notes of the scale,
and also four other sets, each of seven keys, for the sharps,
double sharps, flats, and double flats-that is to say,
thirty-five keys in all to each octave. It is true that
some of these many notes sensibly coincide with each
other, so that even a somewhat smaller number of notes
would satisfy scientific exigencies; but it is none the less
true that music, especially instrumental music, cannot
follow out such great complications.
TEMPERATE SCALE
137
The
16. As music, then, has developed in respect to all
possible transpositions, many attempts have arisen to
diminish the excessive number of notes, and to render
practical execution more easy. These attempts are all
founded on the principle of abandoning strictly mathe-
matical ratios, and of being content, on the other hand,
with approximately correct notes, so long as the error is
not too perceptible to the ear, and of considering as the
same notes differing but little from each other.
fruit of these manifold attempts, which arose particularly
towards the end of the seventeenth and beginning of the
eighteenth centuries, is the temperate scale, which reached
its full development at the middle of the last century,
especially by means of the works of Sebastian Bach, who
dedicated to it forty-eight of his most inspired preludes
and fugues. It starts with the principle of making no
distinction between the major and minor tone, of con-
founding the major semitone with the minor semitone,
and of considering the sharp of a note as equal to the flat
of the succeeding note, so that all the notes of an octave
are reduced to twelve only, which are considered equi-
distant from each other; and it is on this principle that
our pianofortes, organs, &c., are constructed, where all
the notes comprised in an octave are provided by seven
white keys and five black ones.
The temperate scale has become generally accepted;
has so come into daily use that, for the most part, our
modern executant musicians no longer know that it is an
}
1
138
THE THEORY OF SOUND.
incorrect scale, born of transition in order to avoid the
practical difficulties of musical execution. The great pro-
gress made in instrumental music is due to this scale,
and, above all, the ever-increasing importance of the
pianoforte in social life is to be attributed to it.
But, no doubt, it does not represent all that can be done
in this respect. It would certainly be very desirable to
return to the exact scale, with a few difficulties smoothed
over to meet the requirements of practice; for it cannot
be denied that the temperate scale has destroyed many
delicacies, and has given to music, founded on simple and
exact laws, a character of almost coarse approximation.
But before continuing this argument, I will give, as an
example, the values of the different scales, so as to
make the measure of their differences more striking.
Supposing the fundamental note makes 240 vibrations
in a second of time, and calling it c, our major scale is
represented by the following figures:-
с
d
e
f g α b c
240, 270, 300, 320, 360, 400, 450, 480.
The minor scale by-
240, 270, 288, 320, 360, 384, 432, 480.
The Pythagorean scale, comparable with our major scale—
240, 270, 303, 320, 360, 405, 455, 480.
Lastly, the temperate major scale has-
240, 269, 302, 320, 359%, 403%, 453, 480.
If the temperate scale be compared with the mathe-
matical scale, it is seen that with the exception of the
EXACT AND TEMPERATE SCALES.
139
fundamental note and the octave, none of the other notes
coincide exactly. In the temperate scale all the notes are
somewhat modified to a greater or less degree.
Taking, for example, the third, this gives in our example
300 vibrations per second for the exact scale, and 302%
for the temperate scale. Therefore there is a difference
in this case of 23 vibrations per second. It may be said
that this difference is small; however, when it is con-
sidered that it is equal to about of that furnished by the
28
Pythagorean scale, in which the third is represented by
3033, it must be admitted that it cannot be neglected.
It has been seen that in Greek music there was no
development of harmony, especially because the third and
sixth were dissonant. It must be concluded that our
harmony founded on the temperate scale is also very
defective.
Another argument must be put forward. The differ-
ence in our example between the major and minor third
is 12 vibrations per second-the first being 300, and the
second 288 vibrations per second. But if 12 vibrations
per second in this most delicate and sensitive note are
enough to change the character of the fundamental chord
of the scale, and of the whole piece of music founded on
it, it must be allowed that 22 vibrations per second cannot
be quite immaterial, and must therefore produce a sensible
discord.
Lastly, it is known that whenever a chord is not in
perfect tune, it gives rise to beats. On the pianoforte
140
THE
THE THEORY OF SOUND.
$
·
these are not very perceptible, but on instruments with
full, loud, and sustained notes, such as the organ, they
may be recognised in a most unpleasant degree, especially
when they are frequent. In the example chosen, they
would only be 2% in a second, which becomes sufficiently
unpleasant, although there have been here produced low
notes of 200 and 300 vibrations per second. For the high
notes the beats are much more frequent. Applying the
example to the next higher octave, the number of vibra-
tions per second of the two notes, and with this the
number of beats per second, are doubled, which must
produce a most unpleasant effect.
Even the resultant notes are considerably altered, and
instead of reinforcing existing notes, and entering into the
general harmony of the chord, they appear discordant—
not indeed loud enough to produce true dissonance, but
sufficiently so to disturb the serene quiet and the trans-
parency of a chord.
It follows that music founded on the temperate scale
must be considered as imperfect music, and far below our
musical sensibility and aspirations. That it is endured,
and even thought beautiful, only shows that our ears have
been systematically falsified from infancy.*
* Cornue has lately made some most ingenious experiments, in which
he measured directly the number of vibrations per second of the notes
produced by good singers and violin-players, whilst they executed a pure
melody with the greatest possible care. He found that they made use
neither of the exact nor of the temperate scale, but of a scale differing
but little from the Pythagorean; from which he concludes that the
EXACT AND TEMPERATE SCALES.
141
17. Professor Helmholtz has had an harmonicon con-
structed, on which he can play at will in the exact or tem-
perate scale, on purpose to see if there really is an appreci-
able difference between them. As soon as the ear becomes
a little practised, the difference becomes most striking.
In the exact scale the consonant chords become much
sweeter, clearer, and more transparent, the dissonant
chords stronger and rougher; whilst in the temperate scale
all these things are mixed together in one uniform tint
without any distinct character. The resultant notes
have greater importance in the former, and in general the
music acquires a more decided, open, robust, and sweet
character. Fact, therefore, demonstrates that the results.
of theory are not mere speculations or pedantic exagge-
rations, but that they have a true and real value, such as
ought to be accepted even in practice.
The wish may then be expressed that there may be a
new and fruitful era at hand for music, in which we shall
abandon the temperate scale and return to the exact scale,
and in which a more satisfactory solution of the great
Pythagorean scale ought still to be looked upon as the scale of melody,
whilst our modern scale must be considered as the scale of harmony.
In truth, this distinction, even if it exist, is of no great practical im-
portance, since pure melody without harmony does not exist at the present
day, and would be no longer enjoyed. And it is enough for a song to be
accompanied by the most simple harmony, to oblige the singer to adopt
the scale of harmony, if he would avoid discord. The fact, however, is
very interesting in itself, and merits careful examination. It demonstrates
a certain tendency in us to select the Pythagorean scale in melody, and
gives a very natural basis to the melodious music of the Greeks.
142
THE THEORY OF SOUND.
:
difficulties of musical execution will be found than that
furnished by the temperate scale, which, simple though it
may be, is too rude.
But all the stringed instruments, which are the very
soul of the orchestra, and the human voice, which will al-
ways be the most satisfactory and most mellow musical
sound, have their notes perfectly free, and can therefore be
shifted at the will of the artist. The return to the exact
scale does not present any serious difficulty to them. The
same may be said with respect to the wind instruments,
which are still very imperfect, notwithstanding the great
progress already made, but in which the player can,
by changing his lip, somewhat raise or lower the note.
A flute or trumpet player can therefore play in the exact
scale, just as he already plays in the temperate scale;
and the same considerations hold good for the greater part
of the wind instruments. It does not therefore appear
impossible, or even really difficult, for the full orchestra
and chorus to perform a piece of music in the exact
scale.
Greater difficulties are met with in the case of instru-
ments with fixed notes, such as the pianoforte and the
organ. The pianoforte is, indeed, so imperfect an instru-
ment, that notwithstanding the great reputation which
it enjoys, it cannot be given any important place in
executive music properly so called. In fact there is no
pianoforte in the orchestra, and it is thus confined to the
drawing-room and the concert-room. The great defect
i
THE TEMPERATE SCALE AND THE PIANOFORTE. 143
of the pianoforte is that the notes die away rapidly, what-
ever may be the skill of the player. Beats and resultant
notes are therefore perceived with difficulty. As a con-
sequence, even discord becomes less perceptible, and this
is the reason why the sound of an instrument that does
not keep in tune from one day to another is tolerated.
The pianoforte is, truly, the instrument of the temperate
scale; it has been developed, has lived, and probably will
decay with it. Its defects have had an observable
influence on the music written for it. Sustained melody
has been more and more obscured; for it have been sub-
stituted infinite and complicated musical figures, scales,
cadences, shakes, &c., calculated rather to call up the
pride of a brilliant executant than the musical sentiment
of the hearers. For the few simple lines of great musical
works, were substituted infinite arabesques of a new order
of the grotesque.
Reforms of this nature may appear superfluous in the
case of an instrument which does not keep in tune. But
this is not the case with the organ, which has almost lost
its reputation, together with the church in which it is princi-
pally used; but, musically speaking, it will always be an in-
strument of great value. In it the beats and resultant notes
are strongly pronounced, and ought therefore to induce us
to undertake the important reform of the musical scale.
But it is evident that if it be desired to provide the organ
with all the notes necessary, it would lead to very great
complication; but it appears, according to a proposition of
144
THE THEORY OF SOUND.
Helmholtz's, that by starting from somewhat different
considerations, it is possible to provide for everything in
a very satisfactory manner, with twenty-four notes per
octave. This is double the number of keys which we
have at present; but when the great ability of our piano-
forte-players with twelve keys per octave is taken into
consideration, it may well be believed that even twenty-
four keys well arranged would not offer insurmountable
difficulties of execution; and even if musical compli-
cations and arabesques had to be abandoned, true and
serious music would only be the gainer.
I will therefore end this chapter by expressing the hope
that the temperate scale will eventually be abandoned.
It has had its day, and has no longer any real raison d'être.
Man is capable of a much finer class of music than that
performed at the present day. I say this without con-
sidering the different schools which at present divide
musical Europe, since these considerations hold good for
all. But those who consider that the musical province of
Italy is to cultivate and develop melody, ought to be the
first to try for and to favour such a reform. Singing
would gain enormously by it, and a melodious form of
music accompanied by simple quiet chords-much more,
music formed by several voices-would be enormously in-
creased in value by means of such a reform. The ancient
Italian music, the choral music of Palestrina and Basili,
acquires a totally different colouring, and becomes much
more intelligible, when it is performed in the exact scale.
RETURN TO THE EXACT SCALE.
145
This would be a great reform, which could not be
effected in a short time; but once made, it would of itself
constitute a great progress. If some musical instruments
would in part require more or less considerable modifica-
tion in order to adapt them to these new exigencies, the
quartet and choral music can be at once taught and per-
formed in the exact scale.
CHAPTER VIII.
1. QUALITY OR TIMBRE OF MUSICAL SOUNDS—2. FORMS ASSUMED BY THE
VIBRATIONS, OPTICAL METHOD-3..ANOTHER OPTICAL METHOD-4. PHON-
AUTOGRAPHIC METHOD · 5. LAWS OF HARMONICS -6. QUALITY OR
TIMBRE OF STRINGS AND OF INSTRUMENTS 7. GENERAL LAWS OF
CHORDS-8. NOISES ACCOMPANYING MUSICAL SOUNDS-9. QUALITY OR
} TIMBRE OF VOCAL MUSICAL SOUNDS.
1. THE third characteristic difference of musical sounds
is their quality or timbre. Suppose that the same note is
sung by different human voices, and played on the piano-
forte, violin, flute, &c., it does not require a delicate musical
ear to recognise that these notes, although of the same
loudness and pitch, are nevertheless different from each
other. Our ear goes even farther in this direction, and
not only distinguishes between violin and flute, but even
between one violin and another by a different maker. The
difference is very marked, and makes itself felt in a most
remarkable manner in the price of the instrument. Thus,
for example, whilst an ordinary violin costs a few pounds,
many hundreds are paid for a good Stradivarius or N.
Amati. The same may be said of all musical instru-
ments, although the difference of price is not so great for
most of them, as the modern manufacturers are in a posi-
TIMBRE.
147
tion to furnish them in any desired number; whilst violins
increase in excellence and value with their age.
The difference of timbre is therefore very important,
and very characteristic. In the human voice, which con-
stitutes the most agreeable and richest monotone musical
instrument, the variety is immense. There are scarcely
any two individuals who have exactly the same timbre
of voice. Timbre and inflection are the safest means we
have of recognising a person.
But the loudness of a note depends on the width,
height, and length of the oscillations producing it. It
may then be asked, in what two oscillations, of the same
width and length, can differ so as to produce so marked a
difference as that of timbre.
There are two methods of procedure possible in the
study of different forms of oscillation, and of the causes
that influence timbre. The curve of the oscillations may
be traced graphically, and the differences between them
may be examined thus, or the sounds produced by
different instruments may be analysed in order to see if,
besides the principal note that is heard, there are not
other concomitant sounds or noises which alter the timbre
of the simple note, and impress a special character on it.
The question will be studied in this treatise by both
methods, and they will be illustrated by the most impor-
tant examples in each. As to the form of vibrations, it
will be shown that account must be taken not only of
the width and length of the oscillations, but also of the
148
THE THEORY OF SOUND.
}
special form of the curve which represents them. Thus, for
example, the curves 1, 2, 3, in fig. 30, have all three the
same width ab, and the same length AB; but the form is
different for each one of them, and it is precisely on this
special form that that which is called timbre depends. It
will now be explained how such curves may be made
visible.
a
A
a
b
A
b
a
a
b
a
В 1
B2
B 3
Fig. 30
a
2. Of all sonorous bodies the tuning - fork gives the
simplest vibrations-vibrations comparable with the os-
cillations of the pendulum, and which have been called
simple vibrations in the first chapter. They have the
form of the curve 1 (fig. 30), if they be traced on paper
by the graphic method, described in the first chapter.

LISSAJOUS' METHOD.
149
But they may also be made visible in the following
way: A somewhat large tuning-fork F has a small
mirror attached near the extremity of one of its branches,
and a counterpoise to the other [fig. 31]. A ray of solar
m
Fig. 31.
light introduced into the room is made to fall on the
mirror of the tuning-fork, and is reflected thence on to
a concave mirror, and thence again is reflected on to a
diaphragm of translucent paper. An image of the hole in
the window shutter is thus obtained on the diaphragm
under the form of a luminous spot. This spot remains
at rest as long as the tuning-fork is at rest, but when the
latter vibrates, its mirror takes part in the vibration, and
there appears a vertical luminous line instead of the spot
on the diaphragm. This line is caused because our eyes
are not able to follow the rapid movements of the luminous
spot, which occupies a new position at every instant;
n
150
THE THEORY OF SOUND.
but if the concave mirror be rapidly moved by hand, so
that it turns round a vertical axis, the different luminous
spots, which correspond to different points of time, strike
different parts of the diaphragm, and a beautiful curve mn
is thus formed, which represents the form of the vibra-
tions. In it the width evidently depends on the greater
or less degree of energy with which the tuning - fork
vibrates; the length depends on the velocity of rotation
of the concave mirror, and therefore can be varied at will.
But if the mirror be turned with a suitable velocity,
neither too fast nor too slow, a most beautiful sinuous
curve is obtained, which exactly represents the form of
the simple vibrations.
This beautiful optical method of showing the vibrations
of tuning-forks, invented by Lissajous, is very fertile, and
allows a great number of experiments to be made. It is
interesting to study what happens when two vibrations are
combined together in different ways, when somewhat com-
plex compound movements are the result. One example
out of many may be given here, especially as it affords an
admirable illustration of the theory of beats (Chapter v.)
Fig. 32 shows the arrangement best suited for the present
purpose. From the window, or from a closed box L, a
pencil of rays of solar or artificial light is caused to enter
the room, and having been suitably concentrated by the
lens I, falls upon the mirror of a tuning-fork T', thênce
by reflection on to that of a second tuning-fork T, and
thence, finally, on a diaphragm of paper, where the

LISSAJOUS' METHOD.
151
image of a luminous spot is formed. The tuning-forks
are both equal, and give the same note. If the tuning-
fork T be now rubbed with a violin bow, it vibrates and
changes the luminous spot on the diaphragm into a long
vertical line; but if the other tuning-fork T' be also
rubbed, the vertical line becomes longer or shorter,
according as the vibratory movements of the two tuning-
I
T
T
S
Fig. 32.
forks are made in the same or in reverse directions at the
same instant of time, and therefore reinforce or enfeeble
each other's effect.
This being so, if a small coin be attached by means of a
little wax to the tuning-fork T, it will be retarded in its
very
vibrations, and will thus give with the other T marked
beats. The vertical line will then be of variable length
-sometimes long and sometimes short, and one of these
152
THE THEORY OF SOUND.
changes to each beat. The reason is simple. To each beat
there corresponds one reinforcement and one enfeebling of
the sound, and therefore there is one moment at which the
movements of the two tuning-forks are added together,
and another at which they enfeeble each other's effect.
This vertical line can easily be converted into an undulat-
ing line by rapidly moving the tuning-fork T. Fig. 33
Fig. 33.
is thus obtained, where each reinforcement and enfeeble-
ment are clearly indicated, as also is the whole beautiful
form of the phenomenon.
3. It may also be shown that the vibrations of a string
are much more complicated than those of a simple tuning-
fork. Fig. 34 shows the arrangement necessary for this
purpose. A mirror S is so fixed to the window F as to
be capable of being turned in all directions; this receives
a pencil of solar rays, and sends it into the room through
the slit S'. This illuminates the string (or better, a part
of the string) of a sonometer A. The image of the string
is magnified and projected on a distant screen (not shown
in the figure) by means of the lens L.
But if the string be caused to vibrate, the image becomes
larger and fainter, because the eye is not able to follow
the string in its rapid movement.
In order to see the image of the string clearly at any
MOMENTARY ILLUMINATION OF A STRING.
153
particular instant, it is necessary to illuminate it only
momentarily, whilst the observers remain in darkness.
This can be done, for example, by the electric spark, which
has an extremely short duration. But the illumination
thus obtained is feeble, and the phenomenon cannot be
observed at a little distance off. The slit S', through
L
A
Fig. 34.
B
F
which the pencil of solar rays comes which illuminates
the string, can be closed and opened again for only a very
short time. By this means a very brilliant illumination
of the vibrating string is obtained for an instant, and the
state of the string at that given instant can thus be made
visible. But even this method has the fault that the
•154
THE THEORY OF SOUND.
1
phenomenon lasts for an exceedingly short time, so that
it is difficult to observe it in all its minute details.
This method, however, may be improved by being
arranged so as to give an illumination of the string at
very short intervals each time that it returns to the same
position. An intermittent series of illuminations is thus
obtained, and they succeed each other so rapidly that the
illumination appears constant, and causes the string to
appear as if fixed in the given position. The point, then,
is to discover some means of illuminating the string each
time that it reaches one and the same position.
For this purpose use may be made of a disc of card-
board D (fig. 34), which carries a certain number of narrow
slits (eight in all) disposed at equal distances round the
circumference of a circle. It can be rapidly turned by
means of an excellent rotatory apparatus B, set in motion
by the weight P, and regulated by the fly V—an apparatus
which allows a uniform motion to be given to the disc,
and also allows its velocity to be varied at will. This
latter result is easily obtained by increasing more or less.
the weight P, and modifying the form of the fly; after a
few trials it is easy to find the required velocity.
The apparatus then operates as follows: The disc being
placed before the slit through which the solar rays enter,
intercepts them completely; but every time that one of
the slits in the disc comes before the slit in the shutter,
the rays pass and illuminate the string. If, then, the
velocity of the disc be so chosen that in the time that it
FORM OF A VIBRATING STRING.
155
takes for the second, third, &c., slit in the disc to come
before the slit in the shutter, the string makes a complete
movement, and returns to its first position, the string
will be illuminated intermittently at equal intervals of
time, always when it is in the same position. The string
ought to appear fixed, and present the form which it has
at the given moment and in the given position. The
figures obtained are very clear and complicated, and can
easily be drawn, because the phenomenon remains as it
were fixed for as long a time as may be wished. But it
changes its form according to the way in which the string
is made to vibrate. If it be plucked at a third, at a fourth
of its length, or at a seventh or eighth, if it be rubbed
with a bow at one point or another, sensibly different
figures are obtained.
Fig. 35 shows, as an example, the form observed when
a
b
ď
Ր
Fig. 35.
the string is plucked at one-seventh of its length. The
form is complicated, which is the interesting circumstance.
The image is not equally clear and sharp at all its points.
At the points a, b, c, d, there is a thin black line; at the
intermediate points a', b', c', d', the image is enlarged and
fainter. The different portions of the string at these
156
THE THEORY OF SOUND.
'
latter points undergo very rapid vibrations, which the
disc, on account of its too slow rate of rotation, does not
succeed in analysing. These portions, then, behave as in
the ordinary case of a string illuminated by continuous
light, which does not show its form, but only the limits
between which it vibrates.
4. An apparatus of much use in many acoustic re-
searches, and especially in the present question, is Scott's
Phonautograph (fig. 36). A large sounding-board in the
form of a paraboloid, open at its large end, and closed at its
narrowest part by a very thin animal membrane M, which
is stretched more or less by means of three screws v (of
which only one is shown in the figure). A piece of elder
pith a cut into a right angle is attached to the membrane,
and carries at its extremity a very light and flexible
point p.
A note produced opposite the paraboloid A is reinforced
by it, the membrane M vibrates, the piece a takes part in
these vibrations leverwise, and the point p vibrates strongly
in a direction perpendicular to its own length. The vibra-
tions are further made visible by means of the graphic
method. The point p touches the surface of the revolving
cylinder C, and the screw V serves to regulate this con-
tact. The cylinder is covered by a sheet of very smooth
paper smoked in a petroleum flame, and a movement of
rotation is imparted to the cylinder by means of the
handle M, or better still, by means of a clockwork move-
ment arranged to produce uniform rotation, driven by the

THE PHONAUTOGRAPH.
157
weight P. The axis of the cylinder C further has a screw
cut on it, so that the traces taken by different turns are
not superposed.
The phonautograph is a very useful instrument, which
P
Fig. 35.
may be used for many and various researches. The vibra-
tions of a note are traced by it with great regularity. If
two organ-pipes differing slightly from each other be taken
8
158
THE THEORY OF SOUND.
1
and beats be produced, they will be traced with great
regularity on the revolving cylinder. Very beautiful
curves are thus obtained, similar to those obtained by the
optical method [fig. 33].
But this instrument serves not only to demonstrate the
vibrations, but may also serve most usefully to make
known the form of the special corresponding curve.
In this respect, however, the instrument is not altogether
to be relied upon; it may be that it does not register all
the partial notes which enter into the formation of the
compound note, and it may also be the case that it adds
something on its own account.*
But notwithstanding these small defects, it shows
admirably that different and very characteristic curves
correspond to sounds of different timbre. The modus
operandi is the same in every case.
All that is necessary
is to produce the sound to be examined before the mouth
of the instrument with a sufficient degree of energy,
and the instrument then of itself traces the required
curve.
Fig. 37 represents some of the most important cases in
this respect. Each horizontal line contains two somewhat
different curves, which however belong to the same sound,
and which are obtained according to the greater or less
* The effect is more certain, and the curves obtained are more beautiful
and clearer, when the pith lever is reversed and only attached by one end
to the middle of the membrane, and the rectangular lever provided with a
joint so that it can move freely. The curves lately obtained thus are
much more beautiful and characteristic than those shown in fig. 37.
CURVES TRACED BY THE PHONAUTOGRAPH.
159
degree of energy and clearness with which the sound is
produced. Generally distinct musical sounds give more
complicated and more characteristic curves, than faint
sounds.
The first five lines contain the curves obtained by the
A
E
Ι
U
CI
יפ
St
1
2
3
4
5
6
7
8
Fig 37.
vibrations produced by the strong voice of a baritone
singing the note g, and pronouncing more or less clearly
the vowels A, E, I, O, U. The sixth and seventh lines
contain the curves obtained from the same note sounded
on the clarionet and trumpet. The last contains by way of
160
THE THEORY OF SOUND.
complement the curve obtained by Quincke, by another
method, for a vibrating string-the first half when the
string was rubbed at one-third, the second when it was
rubbed at one-twentieth of its length.
5. All these curves, in their great variety, are certainly
interesting, and give rise to another question: In what
relation do these complex vibrations stand to the simple
vibrations found in the case of the tuning-fork? It may
be easily demonstrated that complex vibrations may be
considered as the sum of 2, 3, or more simple vibrations.
The proposition thus stated is too general for the purposes
of this treatise, and must be restricted to the simplest case
before us.
As an example, suppose there to be given three vibra-
tions, so chosen that their lengths are inversely as 1 : 2:3
[fig. 38]. If it be admitted that these three vibrations are
executed simultaneously by the same body, it is fairly easy
to determine the curve representing the resultant motion.
For this purpose all that is needed is to draw the
vertical lines ab, a'b', a"b", and to set off on these the
values for each curve and add them together. The curve
4 is thus drawn, in which each vertical line is equal to the
algebraic sum of the corresponding vertical lines of the
first three curves. This curve thus produced is the curve
required.
In this way the curves of fig. 30 have a clearly deter-
minate meaning. Curve 1 is simple; the second is the
result of two sets of vibrations whose lengths are in the
COMPOUND VIBRATIONS.
161
proportion of 2 : 1; the third is formed by vibrations whose
lengths are as 8: 1.
It is seen by these examples that a complex curve may
αν
ara"
a
2
4
いじじ
​Fig. 38.
be considered as the resultant of certain simple curves, and
that the compound curve differs according to the form and
number of the simple curves composing.it. As a general
162
THE THEORY OF SOUND.
proposition, it is the more complex, as the number of the
curves that compose it is greater and their form more
complex.
As simple curves combined together give a complex
curve, it may be asked whether every complex curve can
be decomposed into simple curves. Proposed in these
terms the problem is very general, and can only be solved
by mathematical calculation. It will not therefore be
possible to enter on it here; but the great importance of
the subject makes it necessary to give, at all events, the
result arrived at. Calculation shows that such problems.
can always be solved.
However complicated the form of a periodic curve may
be, it can always be decomposed into a greater or less num-
ber of simple curves, if they be so chosen that the relative
numbers of the vibrations they represent are in the simple.
proportion of the progressive numbers 1, 2, 3, 4, 5,
&c.
This signifies that a complex curve, representing, for
example, the vibrations of a string, can always be decom-
posed into a curve representing a simple vibration of the
same number per second; and, further, into one represent-
ing a vibration of double that number, and others repre-
senting vibrations of a number triple, quadruple, quintuple,
&c., per second. But as vibrations whose numbers per
second are as 1:2:3:4:5, &c., form the notes in that
which has been called the harmonic series, the result of
mathematical investigation may be acoustically expressed
in the following manner: Every musical sound whose
HARMONICS.
163
vibrations have a complex form, as those of a string, can
be decomposed into a series of simple notes all belonging
to the harmonic series. With this theorem in hand, we
ought to be able to discover in compound musical
sounds all the individual simple notes which compose
them, and we have thus initiated a second experi-
mental method of studying the timbre of musical
sounds.
6. Experience shows that the notes of a string are not
always simple, but are accompanied by notes of the har-
monic series; if a sonometer be taken, for example, and
the string be caused to sound so that it gives the funda-
mental note. The reader already knows that the har-
monics are obtained by dividing the string into 2, 3, 4, &c.,
parts (see Chapter i.), and that the fundamental note being
equal to 1, the harmonics are represented by the pro-
gressive numbers 2, 3, 4, &c.
Seebeck's siren (fig. 29) also serves admirably for pro-
ducing the harmonics. If there be placed on the axis of
rotation a disc which contains the following concentric
series of holes :-
8, 16, 24, 32, 40, 48, 56, 64,
which are in the ratios of—
1:2:34:56:7:8,
the harmonics of the fundamental note are produced in
absolutely correct proportions. These notes must not be
confounded with those of the musical scale. They are,
especially the first, very distant from each other.
164
THE THEORY OF SOUND.
In fact, from the fundamental note 1 we pass to
its octave 2, thence to the fifth of its octave 3, and
thence to the second octave above 4; whilst in the
musical scale the successive notes are considerably closer
together.
This being understood, it is not difficult to demonstrate
that in the case of a vibrating string the fundamental note
is accompanied by its harmonics. With a little attention,
and without recourse to special experiments, the presence
of the third harmonic may be perceived by the ear alone.
The latter becomes especially perceptible when the sound
of the string has grown faint, because the fundamental
note dies away more rapidly than the third harmonic, and
therefore makes it more prominent; also the fifth harmonic
is easily perceptible. The second and fourth harmonics
are less quickly perceived, because they represent the first
and second octaves of the fundamental note, and are easily
confounded with it.
The same fact is met with in the strings of a piano-
forte. Sounding a somewhat low note-a c, for example-
the octave of the next, g, is easily perceived, which is its
third harmonic, and the e of the next octave, which is its
fifth harmonic. The other harmonics can also be heard,
lut with greater difficulty. The seventh harmonic is
wanting, because pianofortes are generally so constructed
that the hammer strikes the string almost exactly at the
point which corresponds to a node of the seventh harmonic
-that is to say, strikes at a seventh of the length of the
DETECTION OF HARMONICS BY RESONATORS.
165
string, which prevents the formation of a node at that
point.
Harmonics are met with also in other instruments; but
if they are difficult to perceive, as often happens, either
because they are too much mixed up with the fundamental
note, as is the case with the human voice, or because they
are too feeble, recourse may be had with great success to
Helmholtz's resonators (see fig. 22).
It has been seen [Chapter iii.] that these are nothing more
than sounding-boards of spherical or cylindro-conical form
which are fitted to the ear, and which reinforce, each one,
according to its volume, some one particular note. It is
enough to be provided with a series of these reso-
nators for the different notes that may be met with, to
have a very efficacious means of proving the presence
of any particular note, however feeble it may be, in the
midst of many others. The researches made on this point
demonstrate that musical sounds produced by different
instruments have different harmonics, or at least in a
different measure.
A note not accompanied by its harmonics may some-
times be sweet, but it is always thin and poor, and there-
fore but little musical. This is the case with tuning-forks.
Even the stopped organ-pipe is almost without harmonics;
the result is, that it gives a hollow and by no means
agreeable sound, somewhat like the vowel u. The har-
monics become, therefore, an almost indispensable condition
of musical sounds properly so called. When the funda-
166
THE THEORY OF SOUND.
mental note is accompanied by the lower harmonics 2, 3,
&c., it acquires a broad, open, soft character. If, on the
other hand, it is the higher harmonics that prevail, the
sound acquires a shrill or clanging character-as, for ex-
ample, in the trumpet, &c.
The richest in harmonics are the sounds of the human
voice and of strings, and it is for this reason that instru-
ments of this class are, and always will be, the most musical.
7. The timbre of musical sounds is produced, then, by
the presence, in greater or less number and degree, of the
harmonics which accompany the fundamental note. A
musical sound is always a compound sound, its vibrations
are more or less complicated, and it by itself alone consti-
tutes a true harmony, especially if the seventh harmonic
be wanting, which does not form a part of our musical
system. It follows that in combining two, three, or more
musical sounds in order to form a chord, it is not enough
that the fundamental notes should bear simple ratios to
each other, but it is also necessary that the harmonics
should obey this law. Also the resultant notes, which
may be formed by all these notes, must enter into the
same system of harmony. We can thus formulate a law
for harmony not only vaster and more general, but also, as
will be hereafter seen, even more simple.
In fact, suppose that there be given the fundamental
note 1; the harmonics will be represented by 2, 3, 4, 5, &c.,
whence the complication of notes will be expressed by-
1, 2, 3, 4, 5, &c.
HARMONICS OF THE TONIC AND FIFTH.
167
Suppose that the octave be also given, which is represented
by 2, and with it the harmonics of the doubled num-
bers-
It is seen that,
2, 4, 6, 8, &c.
in fact, the octave does not possess a
single new note. It only repeats certain of the harmonics
of the fundamental note; therefore by combining together
the fundamental note and the octave, the only effect
produced is the reinforcement of some of the harmonics
already existing in the fundamental note. It follows that
the fundamental note and octave cannot strictly be consi-
dered as two separate musical sounds; their harmony consti-
tutes one single musical sound of somewhat modified timbre.
And this is what really takes place in instruments that
are rich in harmonics-as, for example, the violin.
Adding the fifth, which is represented by, to this
chord, the harmonics comprised in the fifth itself will be-
2, 3, 2, 6, &c.
Some of these notes, as 3, 6, &c., are comprised in the
fundamental note; but 2, 2, &c., are new notes. The
chord with the fifth is less perfect than that with the
octave; but we can make it more perfect by adding as a
reinforcement the octave below the fundamental note,
which is expressed by, and has for harmonics-
3
1, 1, 2, 2, 5, 3, &c.
ܪ,1,1
Referring the fifth to this low note, we see that all its
harmonics are already comprised in those of the latter; in
fact, 2, 2, &c., are the third, ninth, &c., harmonics of the
168
THE THEORY OF SOUND.
fundamental note. In this way the harmony becomes
more perfect. This is the reason why the chord of the
fundamental note with the fifth and octave sounds rather
hollow and poor, and is considerably improved when the
lower octave of the fundamental note is added.
A somewhat similar conclusion is arrived at by adding
to this same chord the major third, which with its
harmonics is expressed by-
5, 5, 15, &c.
These notes do not exist in the fundamental note, but
if the second octave below the fundamental note, which
is expressed by, be added, they all become harmonics,
though distant ones, of this note.
The conclusion to be drawn from this is that it is
necessary to add to the perfect chord of fundamental note,
major third and octave, the two octaves below, in order to
render it really agreeable. This conclusion agrees with
that which experience has taught for a great length of
time. In fact, in practical music this chord is always
written as is shown below at 1, and never as at 2 or 3.
1.
2.
3.
9:
170710
}
It may appear strange that in the perfect chord the
<
NOISES ACCOMPANYING MUSICAL SOUNDS.
169
fundamental note and its octaves should occupy four
places, whilst the third and fifth occur only once, and
it might appear that in this arrangement there was a
want of balance between the different notes.
Certainly this preponderance of the fundamental note
would be inexplicable by the laws developed in the
preceding chapters, when account was taken only of
the simple ratios. The reason becomes evident by the
considerations just now made; it is, that in the perfect
chord all the notes must be considered as a simple rein-
forcement of the fundamental note. The perfect chord
becomes, in fact, a single musical sound, of very musical
quality, and with a very rich and varied timbre.
The conclusion is thus arrived at, that chords are so
much the more agreeable to the ear as fewer new notes
are added to the fundamental note; in the most agreeable
chord of all, the perfect chord, not a single new note is
added. This conclusion becomes the more evident, the
richer the musical sounds at our disposal are in harmonics.
It is also easily explained by the conformation of the ear,
which finds a chord the more agreeable, the less effort is
necessary to understand it.
8. The timbre of musical sounds is not only caused by
the harmonics which accompany in different degrees the
fundamental note, but also by the more or less distinct
noises which are caused by the special method by which
the sounds are produced. A string rubbed by a bow
always allows the sound of something scraping to be
170
THE THEORY OF SOUND.
heard; at the embouchure of an organ-pipe the blowing
of the air is heard; in the pianoforte the hammer is
distinctly heard as it strikes the string, and so on. Gene-
rally speaking, as we are accustomed to hear these noises
from our earliest infancy, it is these above all that teach
us to distinguish one instrument from another, whilst the
harmonics are unnoticed, although they may be much
louder than the noises spoken of.
In fact, if the harmonics that accompany a note on the
violin are always the same, as, with the exception of very
slight differences, is really the case, there is no reason in
practical life to induce us to go farther into the matter, or
to analyse and examine as to the extent to which they
may be present. It is for this reason that the harmonics
remained so long unobserved, and that even now many
practical musicians are unacquainted with them, or look
upon them as subjective phenomena.
Our ear does not advance in analysis farther than it
ought; it obeys, in this respect, also that which appears
to be a fundamental law of nature-that is, it obtains its
purpose with the least exertion and the least labour pos-
sible. It may be easily demonstrated that the ear does
not separate notes to the concomitance of which it has
been long accustomed. By taking a series of tuning-
forks giving the harmonic series, and setting them in
action, in a short time their notes become so mixed
together as to appear only one note. If, then, another
tuning-fork, which by itself gives a good harmony with
KÖNIG'S HARMONIC DETECTOR.
5
2
171
the fundamental note-as, for example, but which is not
a harmonic of the fundamental note 1, be set vibrating,
the presence of this new note almost immediately disturbs
the equilibrium of the harmonics; the ear is then induced
to analyse what it perceives, and all the notes are dis-
tinctly heard. For the same reason it is very difficult
to perceive the harmonics of the human voice, however
numerous and strongly pronounced they may be. It is
necessary for this to have recourse to more minute
methods of analysis, and in this respect Helmholtz's
resonators are of the greatest use. But at the same
time I will describe an instrument founded on the use
of resonators, and constructed by the mechanician König
for the purpose of making the results visible even to a
numerous audience.
9. The apparatus (fig. 39) is composed of eight resona-
tors adapted to the harmonic series of the fundamental
note c.
At the back of each one an indiarubber tube
puts the orifice in communication with a capsule, closed
by an elastic membrane. In front of this, gas enters
and burns under the form of a small, very mobile flame.
Eight flames, therefore, correspond to the eight resonators.
When the air vibrates in one of the resonators, the vibra-
tion is communicated to the flame, and its vibrations are
observed, as is described in the first chapter, by means
of a revolving mirror, which is turned by a handle.
In order to know if the sound of a given instrument or
of the human voice. contains harmonies, and what they

172
THE THEORY OF SOUND.
may be, all that is needful is to produce close to the
apparatus a note corresponding in pitch to the largest
resonator-that is to say, c-which represents the funda-
mental note. Then, if there be harmonics, they will set
93
c8
c2
Fig. 39.
the resonators in action, and thence the corresponding
flames, and a glance at the revolving mirror is all that is
required in order to recognise them immediately.
HARMONICS OF THE VOWELS.
173
By means of this most ingenious instrument it may be
demonstrated that all musical instruments have harmonics,
and it is immediately seen which these harmonics are.
The same may be said of the human voice, which is
very rich in these notes. If a c be sung near the instru-
ment, many flames will be seen to vibrate. But, following
the example of Helmholtz, it may be demonstrated that the
different vowels pronounced whilst singing a note give
different harmonics. In order to show this, sing the
fundamental note, and at the same time pronounce suc-
cessively the different vowels. The harmonics can thus
be easily observed, which accompany the fundamental
note in different degrees. The following results are
obtained in the case we are considering :-
The vowel U is composed of the fundamental note
very strong, and the third harmonic sufficiently pro-
nounced.
O contains the fundamental note, the second harmonic
very strong, and the third and fourth harmonics slightly.
The vowel A contains, besides the fundamental note, the
second harmonic feeble, the third strong, and the fourth
feeble.
E has the fundamental note feeble, the second harmonic
rather strong, the third feeble; on the other hand, the
fourth is very strong and the fifth feeble.
I has very high harmonics, especially the fifth, strongly
marked.
These differences, which are easily observed, arise from
174
THE THEORY OF SOUND.
3
the forms assumed by the mouth, tongue, and lips in the
pronunciation of the different vowels. They are not
exactly the same, as they depend on the timbre of the
voice of the person pronouncing them, on the special
• character of the language in which they are pronounced,
and also on the pitch of the note selected as the funda-
mental note. This last fact may be understood when it
is considered that the mouth itself acts as a resonator of
variable form and size. The results, therefore, are compli-
cated, and I will only draw attention to them.
But if it be true that the timbre of the vowels is con-
stituted in the way briefly indicated above, we ought to
be able to reproduce the different vowels by synthesis, by
combining the fundamental note with its harmonics in
the proper proportions. This work has been actually
accomplished by Helmholtz, who made use for this purpose
of stopped organ-pipes, which gave sensibly simple notes.
He arranged them according to the harmonic series, and
combining them together in the way indicated above,
succeeded in making the fundamental pipe speak the
different vowels in a clearly pronounced manner. The
demonstration is therefore complete, and certainly con-
stitutes one of the great triumphs of science.
CHAPTER IX.
1. DIFFERENCE BETWEEN SCIENCE AND ART-2. ITALIAN AND GERMAN
MUSIC-3 AND 4. SEPARATION OF THE TWO SCHOOLS-5. INFLUENCE
OF PARIS-6. CONCLUSION.
1. THE laws of timbre are the basis of the theory of
instrumentation, and also embrace the whole of harmony.
Thanks to them, all that has been explained heretofore is
reducible to one single principle—that is, That musical
notes must satisfy the laws of harmony, and that this is the
more perfect, the more the notes of a chord reinforce the
fundamental note. Thus the idea of the tonic and the
fundamental chord loses its character of only practical
utility; it becomes a necessary consequence of this
law.
Science has succeeded in taking in from one single
point of view this grand and admirable assemblage of
facts, which are required in the history and development
of music. It is in a position to deduce rigorously the
rules of the art of music, and could easily create it anew
if it happened to be lost.
But I should not wish these words of mine to raise in
the minds of my readers the idea that science can be or
176
THE THEORY OF SOUND.
:
desires to be substituted for art. In art there is one thing
that escapes all calculation, which science indeed can
explain up to a certain point when it has taken a palpable
form, but which it can neither predict or modify this is
Poetic Inspiration. As the most profound knowledge of
grammar, of syntax and prosody, is not sufficient to make
even a mediocre poet, so the most accurate study of the
laws of harmony and of instrumentation would not be
enough to create a composer. Composition and criticism
are two diametrically opposite operations of the human
mind: they ought to go hand in hand, and as far as pos-
sible with a common agreement, and be complementary
one to the other; but the critic will never be a great com-
poser, nor the composer a true critic.
I have endeavoured to pass before my readers in rapid
review the most important facts of the history of music, and
have especially endeavoured to show how even the most
fantastic creations of man may be linked to certain simple
laws which science has revealed. These laws were cer-
tainly unknown to those great men of genius who have left
an imperishable record in their works. They were only
guided in their paths by feeling, fancy, and inspiration.
Science came afterwards and only explained. And it
will always be thus even in the future. The thought,
therefore, will never enter into our mind to prophesy
what music will be fifty or a hundred years hence, and,
artistically speaking, it will be on the ascending or
descending branch of the parabola; the more so as the
{
ITALIAN AND GERMAN MUSIC.
177
æsthetic principles to which the art successively conforms
have no absolute value. But it may be said with cer-
tainty that no single thing will ever be accepted which is
clearly contrary to the broad principles already established
by science. And with this my task would be finished;
but I do not wish to leave this interesting subject, or to
close this treatise, without touching on some few other
questions which have latterly been much agitated,
and which belong to the artistic patrimony of modern
Europe.
2. Much has been said about the great substantial
difference separating the Italian and German music. The
former is called simple, intelligible, and melodious; the
latter complicated, studied, obscure, and transcendental.
And it is sought to find in this one of the characteristic
features of difference between the two nations. It is true
that in the last and present centuries Italian musicians
have cultivated by choice melody and song, and it is also
true that in German music the study of harmony and
of choral and instrumental scoring has been carried to an
admirable degree of perfection; but it is not true that
this has always been the case, and it would be a great
error to endeavour to find in these facts a distinctive
characteristic of the two nations.
In the Middle Ages matters were exactly reversed; the
first centuries of polyphonic music were marked in Italy
by immense complication. Voices linked together in a
most artificial manner, different movements fused to-
178
THE THEORY OF SOUND.
gether according to complicated and obscure rules, were
the characteristics of Italian polyphonic music up to
the time of Palestrina. The Protestant Reformation pro-
duced in Germany simple harmony and simple song, a
form of music at once clear, easy, and transparent. No
comparison is possible as to simplicity between the first
Protestant chants and the Italian music of this same
Palestrina, who was, however, the great reformer and
simplifier of Italian polyphonic music.
After this epoch the two nations continued, in re-
spect to style, almost on the same road. Italy decidedly
took the lead, thanks to her enormous musical activity,
and to the considerable number of men of real creative
power.
From this moment progress was rapid and continuous.
Viadana wrote the first melodies, and added as their
accompaniment the continuous bass; Carissimi and Scar-
latti may be considered as the inventors of expressive
recitative. To this last true musical genius we owe the in-
vention of the aria, which with its first and second part and
repetition represents in music almost that which the column
represents in architecture. In his operatic essays he intro-
duced the recitativo obligato, and initiated by this the change
from the first Italian style to the second, a change which
his great disciples and emulators Durante, Leo, and Greco
thoroughly effected. Thanks to their efforts, music threw
off its character of great severity, and its rigid rules of har-
mony and counterpoint. In their hands, and in those of
DEVELOPMENT OF OPERA.
179
the bold innovator Claudio Monteverde, it, on the contrary,
underwent considerable instrumental development, with
more broadly and freely distributed melody, followed by
accompaniments of greater simplicity and of freer scope.
For the austere movement was substituted clear, simple,
ingenuous sentiments; plastic beauty, exact time, main-
tained with grace and fine discrimination in the midst of
most beautiful melodies, was the character assumed by music
in the seventeenth century, a character which is especially
to be recognised in the ecclesiastical music; whilst in opera,
notwithstanding all efforts, the form still remained very
primitive. This movement continued up to the eighteenth
century. Following close on church music, opera deve-
loped more and more, and with the history of this move-
ment the names of Pergolese, Piccini, Sacchini, Jomelli,
Cimarosa, and Paesiello rcmain connected. This creative
activity was communicated to Germany, where it took
a new form and a surprising development. Men like
Händel, Haydn, Bach, Gluck, Mozart, gave a wonderful
breadth of idea to music; but, with the exception of
Gluck, they must be considered as the fruitful and genial
continuers of the Italian movement, which they carried
out in much the same spirit as the Italian composers
themselves. To show how little the two differed, we need
only compare the "Matrimonio segreto" of Cimarosa with
Mozart's "Nozze di Figaro." One would say that they were
two works of the same school, and composed by two
brothers in art, of which the first was the most facile, most
180
THE THEORY OF SOUND.
cheerful, and most elegant; the second the broader, richer,
and more profound.
3. The distance between the German and Italian music
is most evident in the case of the works of Gluck and
Beethoven on the one hand, and the works of Rossini on
the other. Whilst in the middle of the last century the
two schools did not differ much from each other, and
the Italian music then closely resembled the German, the
executive part of the art took a different direction in Italy.
The last century is the century of grand Italian song.
Italy surprised the world by the number of distinguished
singers she produced, and by the serious and solid method
on which her schools of singing were organised. These
singers overran Europe, passing from triumph to triumph,
fêted by all, and idolised in an almost incredible manner.
But it was precisely the great importance to which the
Italian school of singing rose which became the cause of
its downfall in real value. The singers began to consider
themselves as the thing of greatest importance, and as the
basis on which the grandeur of Italian music reposed.
For them the composition was the pretext; their principal
aim, to shine as much as possible. It then came about
that the music being too simple to afford them the means
of shining, they substituted for simple melody a more
complicated form, interpolating turns and shakes, cadences
and ornamentations of every kind, to the manifest detri-
ment of the composer and of musical good taste. The
great masters of that time submitted to this state of
}
ROSSINI'S TWO STYLES.
181
things, being powerless to remedy it. Then came Rossini,
who thought it better to write complicated melodies with
scales, cadences, and difficulties of every sort himself,
since thus alone could good taste be even partially saved.
He acted like certain politicians, who put themselves at
the head of a movement in the hope of being thus better
able to control it.
The richness and variety of his forms are truly admir-
able; but it is evident that true musical conception must
suffer under these continued shakes and turns. There is
only one purpose for which this light and varied style
seems fitted, and that is for comic opera; and in this
respect Rossini has left an undying model of grace and
freshness in the "Barbiere di Siviglia." In his later works,
in the style of grand opera, Rossini abandoned this style
almost completely. His last opera, "Guglielmo Tell," is
entirely without ornamentation, and in some parts―as,
for example, in the trio and in the conspiracy in the
second act-rises to an incomparable elevation.
But this more chaste and correct style of Rossini's
was formed away from Italy, by tendencies and ideas
different from those that there held sway. For Italy,
the alienation was effected, and could not easily be again
undone. Under his more important successors, as Bellini
and Donizetti, music acquired the character of simple
song-often deep and feeling, often light, superficial, and
cloying.
The impression which the composer of "Norma" has
9
182
THE THEORY OF SOUND.
produced, and still produces by his beautiful and deeply-
felt melodies, the interest that Donizetti inspires by the
elegance of style displayed in his best works, ought not
to make us lose sight of the fact that the superabundance
of melody is not suited to the requirements of the modern
stage. Save for many very beautiful exceptions, senti-
mentality took the place of real sentiment; and dramatic
expression was to a great extent obscured, and often never
sought after. Then came Verdi, who felt that this
continual melody would at last corrupt all minds. For
beautiful melody he substituted movement, which was
not yet dramatic feeling, but contained strength and
vigour, even though it were sometimes in a rather rugged
form. This style of writing was pleasingly in accord-
ance with the national aspirations. Italy at that time
was awaking to a new life; she felt the want of move-
ment and strong emotions. Patriotism therefore laid
hold of the music of Verdi, made it entirely popular,
and used and abused it freely. But musical good taste
and the school of singing suffered immensely. Latterly,
Verdi has considerably modified his style of writing,
and has openly drawn nearer to the German school, or
has at all events diminished the great distance which
formerly separated our school of music from the Ger-
man school.
From "Nabucco" and "Ernani" to "Rigoletto" and
"Il Ballo in Maschera," and from these to "Aida," the
progress in this direction is continuous. These examples,
46
MUSIC AS A MEANS OF DRAMATIC EXPRESSION.
183
besides being well known, are discussed everywhere with
interest and eagerness.
4. Although the rupture came from the Italian side, it
was first brought about on the side of Germany. Gluck
introduced and developed wonderfully the conception of
dramatic music, which proposes as its object to adapt the
music better to the words, and to create a musical work
of art capable of producing on the hearers the same
sensations that the text which accompanies it produces.
In this respect music is an inexhaustible mine of truly
artistic effects. It is superior in many points to poetry-
sometimes in the expression of the terrible, and sometimes
in the expression of the really gentle emotions. To prove
this, it is enough to recall in modern works the love scene
between Faust and Marguerite described musically by
Gounod, to conclude that the grand poetry of Goethe has
not only not suffered here, but that the effect is somewhat
modified and idealised rather than diminished. We need
only recall again the duet between Raoul and Valentina, in
"Gli Ugonotti," in which all emotions from patriotism to love,
and from love to terror, are described with vividness and
incomparable feeling, which, in spite of some exaggerations,
strike us profoundly; lastly, the terrible scene in Weber's
"Der Freischütz," where terror is carried to the highest
degree of musical expression. Music, which in many
respects remains inferior to poetry, shows itself, on the
other hand, superior to it in others where dramatic effect
and feeling are strongly marked,
184
THE THEORY OF SOUND.
•
A greater separation took place, caused by Beethoven, the
great and true creator of modern instrumental music.
From this time the German school separated more and
more from the common road it had trodden in company
with the Italian school. Mendelssohn, Schuman, and
lastly Wagner, are only a continuous gradation on this
path. Music acquired a more and more instrumental
character, and free singing became obscured. To use
a celebrated phrase, which is perhaps exaggerated, but
which sharply depicts the actual state of things, it may
be said that in Italian music the orchestra had become
a big guitar, only intended to accompany the singing;
but it may also be said that in the German music the
singers had become walking orchestral instruments. It
must, however, be admitted that whilst Italian music has
undergone in this century a sensible decline, in Germany
it has remained elevated. The study of harmony and
grand orchestral movements and deep dramatic feeling and
expression, notwithstanding some too realistic exaggera-
tions and some trivialities, have been brought to a high
degree of perfection by the intellectual influence of Richard
Wagner. We owe it to him, that for libretti-almost
always insipid, and which served as bare excuses for the
musical part of the work-has been substituted a more
masculine and independent form of poetry. The closer
union of poetry and music, in which both arts advance
with regular step without one being smothered by the
other, constitutes perhaps the most salient and most
MUSIC OF THE FUTURE.
185
beautiful character of his music, which is almost always
lofty, most rich, and which transports us to an ideal
sphere.
I say this in spite of all the clamour which has been
raised on both sides of the Alps against the "music of the
future." It is reproached with being too studied and
too refined, and with leaning not towards feeling, but
calculation and combination.
But we need only listen to it with attention and
without prejudice to be convinced of the greatness and
number of the beauties which it contains. The prelude
to "Lohengrin," the song to the swan, many passages in
Tannhauser," and other things prove the contrary. This
music has had the great and sad privilege of exciting,
almost incredibly, passions in its favour and against it;
but when these passions have grown calm, I believe that
it will be impossible to deny to it the character of a great
musical poem whose limits will extend far beyond the
national circle for which it was written.
5. Finally, we must take into account a third impor-
tant agent in musical history-that is, the influence
exercised by Paris on the conduct of musical ideas. If
we except opera comique (which must not be confounded
with the Italian opera buffa), in which Grétry, Boieldieu,
Hérold, Auber, and others excelled, it may be said that the
French have never been true creators in music. Not-
withstanding this, the influence of Paris on the history of
the art is great and incontestable. Placed, as it were,
186
THE THEORY OF SOUND.
1
midway between the two musical nations, thanks to the
splendours of Parisian life and its mania for amusement,
Paris became one of the important centres where many
grave musical problems were worked out. It was there
that arose the struggle between Gluck's severe music and
the melodious music of Piccini. It was there that the
Italian Cherubini found a highly honoured post, with his
tendency towards German music. It was there that Meyer-
beer, abandoning his first style, created "Roberto il Diavolo,”
"Gli Ugonotti," and the "Prophète," which by their gran-
deur of conception will make his fame immortal. Finally,
it is there that the best Italian masters have gone in search
of competent criticism, and have modified their style.
Rossini's "Guglielmo Tell," Donizetti's "Favorita" and
"Don Sebastian," finally, most of Verdi's works, have arisen
in this manner. The influence of Paris may be thus de-
fined: that she insists on the creation of a type of music
which should contain the good points of the German and
Italian schools without their exaggeration. This school is
therefore eminently eclectic, and has found the solution
of its problems by clinging closely to dramatic music.
It thus maintained the Italian melody and song, but
limited it to those cases in which it is compatible with
dramatic expression. It has adopted the grand choral
and orchestral movements of Germany, giving them a
suitable importance. Lastly, it tried to obtain an inti-
mate relation between words and music, with the desire,
1
INFLUENCE OF PARIS.
187
rather expressed than carried out, of subordinating neither
to the other.
The character of this school is best recognised in the
French composers who have written dramatic operas.
Halévy, Gounod, Auber, himself in his "Masaniello,” have
trodden this path. Whatever may usually be thought of
eclectic things, the eclecticism of the Paris school has
been of real importance. It must be considered as an
earnest and partly successful attempt to unite from one
common point of view two schools whose tendencies
were very different. And from this attempt have arisen
noble ideas and grand works of art which will exert a true
and great influence even on posterity.
6. As for the future itself, it is not for a musical or
scientific critic to wish to foresee its steps. I shall there-
fore take care not to give an opinion on the subject.
That which it was important for us to demonstrate was
that music has been developed according to certain rules
which depended on unknown laws of nature since dis-
covered, that it cannot be separated from these laws, and
that within them there is a field large enough for all
the efforts of human fancy.
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General History of Greece, from the Earliest Period to the Death of Alex-
ander the Great. By the Rev. GEORGE W. Cox. I vol., 12mo. Cloth. $2.50.
"We envy those schoolboys and undergraduates who will make their first ac-
quaintance with Greek history through Mr. Cox's admirable volume. It ought to
supersede all the popular Histories of Greece which have gone before it."-The Hour.
"The book is worthy, in every way, of the author's reputation.
It is alto-
gether a most interesting and valuable book."-Educational Times.
•
•
•
A Short History of Natural Science and of the Progress of Discovery
from the Time of the Greeks to the Present Day. By ARABELLA B. BUCKLEY.
With Illustrations. I vol., 12mo. $2.00.
"Miss Buckley, the friend of Sir Charles Lyell, and for many years the secretary of
the great geologist, in this volume has given a continuous, methodical, and complete
sketch of the main discoveries of science from the time of Thales, one of the seven
wise men, B. C. 700, down to the present day. The work is unique in its way, being
the first attempt ever made to produce a brief and simple history of science. The au-
thor has entirely succeeded in her labors, evincing judgment, learning, and literary
skill."-Episcopal Register.
A Hand-Book of Architectural Styles.
Translated from the Ger-
man by W. COTLETT-SANDers. I vol., 8vo. With 639 Illustrations. $6.00.
"There is a great amount of information in the book, in a small compass. For one
who simply wishes to gain a full knowledge of the various styles of architecture,
written in a clear and interesting manner, the volume has not its equal nor rival in the
English language. This knowledge will be facilitated by the profuse illustrations, of
which there are not less than six hundred and thirty-nine, nearly all handsome spe-
cimens of engraving, among which figure a large number of famous buildings, ancient
and modern."-Evening Mail.
D. APPLETON & CO., 549 & 551 Broadway, N. Y.
1
NAV 18 "76
FER
2 1981
DATE DUE
JAN 25 RECD
MAR -151985
TAN 25 R
MAY 19871
APR 15 RECO
NOV 01 1988
NOV 221988
RECO
410021000
FEB 2
REC'DOCT-11993
}
}
:
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་ ་
ML
3805
B643 BOUND
i
NOV 18 1
UNIV. OF
LIBRARY
UNIVERSITY OF MICHIGAN
3 9015 00944 5936
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