.
***
.
7
/
.
+
.
.
.
.
2. W
..
.
.
A
A
...
A. Vse
.
//
XYZ/
W-
mene
SEIENOS
*
.
.
.
*
Y.
.
...
TO
--
WS
.
LP
T.
//
...
.
Z
WA
VAS
.
.
TI
,,
of
When
-
7
"
.
.
.
.
//
.
..
7F..
.
ma
1
-
.
w
.
*
***
tut
*
WYN
the
***
www
V.
w
we
4
.
.
*
**
Sve
WP.
.
.
.
*
.
".
1
.
**
9
.
www.
.
A
N?
the
with mine
***
.
**
..
-
.
***
.
.
***
w::
***
15
DOUNOW
71837
UpUN
ARTES
SCIENTIA
VERITAS
LIBRARY
OF THE
UNIVERSITY OF MICHIGAN
TOITT
MURIDIUS UNISA
B!
Tubin
QUACRIS PENINSULAM'AMOE NAMI:
CIRCUMSPICE
VORW.QWUTNIM WWW.LU.
LITHIBHI
تنتان
PII
MUTUMITIMINE
OIL 1:1
transferio
Huna
1-4-6
PREFACE.
Licla... Ef 12-9537
The following treatise, portions of which have been
delivered in lectures at the South Kensington Museum,
the Royal Academy of Music, and elsewhere, aims at
placing before persons unacquainted with mathematics
an intelligible and succinct account of that part of the
Theory of Sound which constitutes the physical basis
of the Art of Music. No preliminary knowledge, save
of arithmetic and of the musical notation in common
use, is assumed to be possessed by the reader. The
importance of combining theoretical and experimental
modes of treatment has been kept steadily in view
throughout.
The author has incorporated the chief acoustical
discoveries of Professor Helmholtz, but adopted his
own course in explaining them and developing their
connection with the previously established portions of
the subject. The present volume, therefore, even where
4
PREFACE.
its obligations to the great German philosopher are
the deepest, is not a mere epitome of his work, but
the result of independent study.
TRINITY COLLEGE, CAMBRIDGE, June, 1873.
Die Lehre von den Tonempfindungen. Dritte Ausgabe. Braun-
schweig. 1870. Of this profound and exhaustive treatise it is
not too much to say that it does for Acoustics what the “ Prin-
cipia” of Newton did for astronomy.
CONTENTS.
CHAPTER I.-ON SOUND IN GENERAL, AND THE MODE OF ITS TRANSMISSION.
Sensation of sound, and its cause, $ 1~Connection of sound with motion, $ 2- Velocity
of sound, & 3-Stationary media of sound, $ 4Motion of sea-waves, $ In Description of a
wave, $ 6-Length, amplitude, and form of wave, $ ----Length of wave and time of vibra-
tion, $ 8-Periodic vibrations, S$ 9, 10 Form of wave and mode of vibration, SS 11, 12-Ex-
tension of term “wave" to solid bodies, $ 13—Wave on the surface of a field of standing
corn, $ 14-Longitudinal vibrations, SS 15, 16–Condensation and rarefaction, $ 17–Asso-
ciated wave, $ 18Law of pendulum-vibration, $ 19— Pressure and density of air; Mari-
otte's law, $ 20-Transmission of sonorous waves along a tube of uniform bore, $ 21-Un-
constrained motion of sound-waves, $ 22-Musical and non-musical sounds, $ 28.
CHAPTER II.-ON LOUDNESS AND PITch.
Three elements of a musical sound, loudness, pitch, and quality, $ 24-Loudness and ex-
tent of vibration, $ 25—Pitch and rapidity of vibration; the sirene; continuity of pitch,
S$ 26, 27-Measure of pitch; vibration-numbers, $ 28-Limits to the pitch of musical
sounds, $ 29—Relative pitch; intervals, $ 30—Tonic intervals of the major scale; concords
and discords, $ 31-Additional notes required for the minor scale, $ 32Measure of inter-
vals, $ 33— Vibration-fractions, $ 34-Table of vibration-fractions for the tonic intervals of
the major and minor scales, § 35. Calculation of the vibration-numbers of all the notes in a
scale, from the vibration-number of its tonic, $ 36.
CHAPTER III.--ON RESONANCE.
Resonating piano-forte wires and tuning-forks; cause of the phenomenon, SS 37, 38–
Resonance of a column of air; laws of its production, $ 39—Relation between the length
of an air-column and the pitch of its note of maximum resonance, $ 40-Resonance-boxes,
$ 41—Helmholtz's resouators, $ 42.
CHAPTER IV.-ON QUALITY.
Composite nature of musical sounds in general; series of constituent tones, and law
which connects them, $ 43.-Experimental analysis of musical sounds, SS 44, 45—-Nomencla-
ture of the subject, $ 46--Helmholtz's theory of musical quality, as depending on the num-
ber, orders, and relative intensities of the partial-tones present in any giveu clang, $ 47.
CHAPTER V.-ON THE ESSENTIAL MECHANISM OF THE PRINCIPAL MUSICAL
INSTRUMENTS, CONSIDERED IN REFERENCE TO QUALITY,
Sounds of tuning-forks, $ 48—Modes of vibration of an elastic tube, $ 49—Meeting of
equal and opposite pulses; formation of nodes, $ 50-Number of nodes formed, $ 51-Na-
6
CONTENTS.
ture and rate of segmental vibration, SS 52, 53—Motion of a sounding string; quality and
pitch of its note, $$ 54, 55—The piano-forte, $ 50m-Meeting of equal and opposite systems
of longitudinal waves, $ 57-Reflection of sound at a closed and at an open orifice, $ 58-
Modes of segmental vibration in stopped and open pipes, S$ 59, 60—Deepest note obtainable
from a pipe, $ 61–Relation between length of pipe and pitch of note, $ 62—Theory of reso-
Dance-boxes, 63—Flue-pipes and reed-pipes, & 64-Construction of flue-pipes and qual-
ity of their sounds, § 65_Mechanism of a reed; timore of an independent reed, and a reed
associated with a pipe, $ 66—Orchestral wind-instruments, $ 67-Mechanism of tbe human
voice, $ 68mSynthetic confirmation of Helmholtz's theory of quality, $ 69.
CHAPTER VI.-ON TRE CONNECTION BETWEEN QUALITY AND MODE OF
VIBRATION.
Composition of vibrations, $ 70-Superposition of small motions, $ 71-Phase of a vibra-
tion; non-dependence of quality on differences of pbase among partial-tones, 8 72—Simple
and resultant wave-forms; Fourier's theorem, $ 73.
CHAPTER VII.-ON THE INTERFERENCE OF SOUND, AND ON
BEATS."
Composition of vibrations of equal periods, $ 74-Two sounds producing silence, $ 75%.
Beats of simple tones, 8 76/Graphic representation of beats, $ 77— Experimental study of
beats, $ 78.
CHAPTER VIII.-ON CONCORD AND DISCORD.
Helmholtz's discovery of the nature of dissonance; conditions under which it may arise
between two simple tones, $ 79—Mode of determining the whole dissonance produced by
two composite sounds, $ 80—Classification of the tonic intervals of the scale, according to
their freedom from dissonance, SS 81-86--Picture of amount of dissonance for all intervals
not wider than one octave, $ 87-Consonance dependent on quality, $ 88-Apparent objec-
tion to Helmholtz's theory of quality, $ 89—Combinntion-tones, $ 90—Their use in defining
certain consonant intervals for simple tones, $ 91—Divergence from the views of musical
theorists, $ 92--Dissonance due to combination-tones produced between the partial-tones
of clangs forming a given interval, $ 93.
CHAPTER IX.-ON CONSONANT TRIADS.
Rules for the employment of vibration-fractions, SS 94-96–Inversion of intervals, $ 97—
Definition of a consonant triad, $ 99-Determination of all the consonant topic triads within
one octave, $ 99-Major and minor groups, $ 100—Mutual relation between the members
of each group, § 101-Notation of thorough bass, $ 102–Fundamental and inverted posi-
tions of common chords, $ 103-Effects of major and minor chords, $ 104.
CHAPTER X.-ON PURE INTONATION AND TEMPERAMENT.
Successive intervals of the major scale, $ 105—Requisites for pure intonation in keyed
instruments, SS 106-108—Tempering and temperament, $ 109-System of equal tempera-
ment, $ 110-Its defects, $ 111-Its influence on vocal intonation, $ 112-Cumbrousness and
inefficiency of the established pitch-notation for vocal music, § 113-The "Tonic Sol-Fa"
pitch-notation, $ 114-Its simple and effective character, $ 115Relation of the physical
theory of consonance and dissonance to the esthetics of music, $ 116-Importance of ex-
treme discords; conclusion, $ 117.
THE SCIENCE OF MUSIC. .
CHAPTER I.
ON SOUND IN GENERAL, AND THE MODE OF ITS TRANSMISSION.
1. In listening to a sound, all that we are immedi-
ately conscious of is a peculiar sensation. This sensation
obviously depends on the action of our organs of hearing;
for, if we close our ears, the sensation is greatly weakened,
or, if originally but feeble, altogether extinguished. Per-
sons whose auditory apparatus is malformed, or has been
destroyed by disease, may be totally unconscious of any
sound, even during a thunder-storm, or the discharge of
artillery. These simple considerations should prepare us
to expect that what we feel as sound may be represented,
externally to ourselves, by a state of things very different
from the sensation we experience. Indeed, this would only
be in accordance with the modes of action of our other
senses ; for instance, the sensation of warmth, and its
cause, a coal-fire--of fragrance, and its cause, a rose-of
pain, and its cause, a blow, are quite unlike each other.
Analogy, then, indicates that some purely mechanical
phenomena external to the ear will prove to be turned
into the sensation we call Sound by a process carried on
within that organ, and the brain with which it is in direct
communication. This mechanical agency, whatever may
8
SOUND AND ITS TRANSMISSION.
be its nature, is usually set going at a distance from the
ear, and, to reach it, must traverse the intervening space.
In doing so, it can pass through solid and liquid as well
as gaseous bodies. If one end of a felled tree is gently
scratched with the point of a penknife, the sound is dis-
tinctly audible to a listener whose ear is pressed against
the other end of the tree. When a couple of pebbles are
knocked together under water, the sound of the blow
reaches the ear after first passing through the intervening
liquid. That sound travels through the air is a matter
of universal experience, and needs no illustration. In
every case, accessible to common observation, where
sound passes from one point of space to another, it neces-
sarily traverses matter, either in a solid, liquid, or gase-
ous form. We may hence conjecture that the presence
of a material medium of some kind is indispensable to.
the transmission of sound. This important point can be
readily brought to the test of experiment, as follows: Let
a bell, kept ringing by clock-work, be placed under the
receiver of an air-pump, and the air gradually exhausted.
Provided that suitable precautions are taken to prevent
the communication of sound through the base of the
receiver itself, the bell will appear to ring more and more
feebly as the exhaustion proceeds, until, at last, it alto-
gether ceases to be heard. On readmitting the air, the
sound of the bell will gradually recover its original loud-
ness. It results from this experiment that sound cannot
travel in vacuo, but requires for its transmission a mate-
rial medium of some kind. The air of the atmosphere is,
in the vast majority of cases, the medium which conveys
to the ear the meclianical iinpulse which that wonderful
organ translates, as it were, into the language of sound.
2. Having ascertained that a material medium, in
every case, acts as the carrier of sound, we have next to
examine in what manner it performs this function. The
MOTIONS IN THE MEDIUM. .
9
roughest observations suffice to put us on the right track,
in this inquiry, by pointing to a connection between
sound and motion. The passage, through the air, of
sounds of very great intensity is accompanied by effects
which prove the atmosphere to be in a state of violent
commotion. The explosion of a powder-magazine is capa-
ble of shattering the windows of houses at several miles'
distance. Sounds of moderate loudness, such as the rattle
of carriage-wheels, the stamping of feet, the clapping of
hands, are produced by movements of solid bodies which
cannot take place without setting up a very perceptible
agitation of the air. In the case of weaker sounds, the
accompanying air-motion cannot, it is true, be ordinarily
thus recognized; bat, even here, a little attention will
usually detect a certain amount of movement on the part
of the sound-producing apparatus, which is probably
capable of being communicated to the surrounding air.
Thus, a sounding piano-forte-string can be both seen and
felt to be in motion: the moveinents of a finger-glass,
stroked on the rim by a wet finger, can be recognized by
observing the thrills which play on the surface of the water
it contains : sand strewed on an horizontal drum-head is
thrown off when the drum is beaten. These considerations
raise a presumption that sound is invariably associated
with agitation of the conveying medium - that it is im-
possible to produce a sound without at the same time set-
ting the medium in motion. If this should prove to be
the case, there would be ground for the further conjecture
that motion of a material medium constitutes the mechani-
cal impulse which, falling on the ear, excites within it the
sensation we call sound. Let us try to form an idea of the
kind of motion which the conditions of the case require.
3. It will be convenient to begin by determining the
rate at which sound travels. This varies, indeed, with
the nature of the conveying medium. It will suffice,
10.
SOUND AND ITS TRANSMISSION.
however, for our present purpose to ascertain its velocity
in air, the 'medium through which the vast majority of
sounds reach our ears. As long as we confine our atten-
tion to sounds originating at but small distances from us,
their passage through the intervening space appears in-
stantaneous. If, liowever, a gun is fired at a considerable
distance, the flash is seen before the report is heard--a
proof that an appreciable interval of time is occupied by
the transmission of the sound. The occurrence of an
echo, in a position where we can measure the distance
passed over by the sound in traveling from the position
where it is produced to that where it rebounds, gives us
the means of measuring the velocity of sound; since we
can, by direct observation, ascertain how long a time is
spent on the out-and-home journey. The following easy
experiinent gives a near approximation to the actual
velocity of sound-in fact, a much closer one than the
rough nature of the observation would have led one to
expect: In the North cloister of Trinity College, Cam-
bridge, there is an unusually distinct echo from the wall
at its eastern extremity. Standing near the opposite end
of the cloister, I clapped my hands rhythmically, in such
a manner that the strokes and echoes succeeded each other
alternately at equal intervals of time. A friend at my
side, watch in hand, counted the number of strokes and
echoes. The result was that there were 76 in half a min-
ute, i. e., 38 strokes and 38 echoes. A little consideration
will show that the sound traversed the cloister and
returned to the point of its origination regularly once in
each interval between a stroke and its echo. Since each
such interval was exactly equal to that between an echo
and the following stroke, the whole movement of sound
took place in alternate equal intervals, i. e., in half the
observed time, or fifteen seconds. Accordingly, the sound
travelled to and fro in the cloister 38 times in 15 seconds.
VELOCITY OF SOUND.
11
>
The length thus traversed I found to be 419 feet. The
velocity of sound per second thus comes out equal to
38 x 419
or 1,061' feet and a fraction. Sound, then, trav-
15
els through the air at the rate of upward of 1,000 feet in
a second, which is more than 600 miles an hour, or about
15 times the speed of an express-train. In solid and
liquid bodies its velocity is still greater, attaining, in the
case of steel wire, a speed of from 15,000 to 17,000 feet in
a second, or, roughly speaking, about 200 times that of
an express-train.
4. Though the sound-impulse thus advances with a
steady and very high velocity, the medium by which it is
transmitted clearly does not share such a motion. Solid
conductors of sound remain, on the whole, at rest during
its passage, and a slight yielding of their separate parts
is all that their constitution generally admits of. In
fluids, or in the air, a rapid forward motion is equally out
of the question. The movement of the particles compos-
ing the sound-conveying medium will be found to be of a
kind examples of which are constantly presenting them-
selves, but without attracting an amount of attention at
all commensurate with their interest and importance.
5. An observer who looks down upon the sea from a
moderate elevation, on a day when the wind, after blow-
ing strongly, has suddenly dropped, sees long lines of
waves advancing toward the shore at a uniform pace and
at equal distances from each other. The effect, to the
eye, is that of a vast army marching up in column, or of
a ploughed field moving along horizontally in a direction
perpendicular to the lines of its ridges and hollows. The
actual motion of the water is, however, very different
from its apparent motion, as may be ascertained by
1 This is about 50 feet below its real value under the circumstances of
my observation. (Sce Tyndall's "Sound," p. 24.)
Tyndall's “Sound," p. 38.
2
12
SOUND AND ITS TRANSMISSION.
noticing the behavior of a cork, or other body, floating
on the surface of the sea, and therefore sharing its move-
ment. Instead of steadily advancing like the waves, the
cork merely performs a heaving motion as the successive
waves reach it, alternately riding over their crests, and
sinking into their troughs, as if anchored in the position
it happens to occupy. Hence, while the waves travel
steadily forward horizontally, the drops of water compos-
ing them are in a state of swaying to-and-fro motion, each
separate drop rising and falling in a vertical straight line,
but having no horizontal motion whatever.
Thus, when we say that the waves advance horizon-
tally, we mean, not that the masses of water, of which
they at any given instant consist, advance, but that these
masses, by virtue of the separate vertical motions of their
individual drops, successively arrange themselves in the
same relative positions, so that the curved shapes of the
surface, which we call waves, are transmitted without
their materials sharing in the progress. The accompany-
ing figure will show how this happens.
1
Fig. 1.
B
c"
F
А.
с
Z
D)
B
D
Z
Let ABCDEF represent a section of a part of the
sea-surface at any given instant, and suppose that, during,
say, the next ensuing second of time, the separate drops
in ABCDEF move vertically, either upward or down-
ward, as shown by the arrows, so that, at the end of that
second, they all occupy positions along the dotted curved
1 This statement, though not strictly accurate, is sufficiently near the.
truth for our present purpose. (See Weber's "Wellenlehre.")
HOW THE MOTION IS PROPAGATED.
13
line,' A'B'C'D'E'F'. The two portions, ABCDE and
B'C'D'EF', are exactly alike, and, therefore, the effect
is just what it would have been bad we pushed the curve
ABCDE along horizontally until it came to occupy the
position B'C'D'E'F'.
In order furtlier to illustrate this point, let us suppose
that a hundred men are standing in a line, and that the
first ten are ordered to kneel down; a spectator who is
too far off to distinguish individuals will merely see a
broken line like that in the figure below.
(1)
WALAUDIO STAGNACISTICIRATI ARRIUGARDERATANODIGINAL STANDARD
Now, suppose that, after one second, the eleventh man
is ordered to kneel and the first to stand ; after two sec-
onds the twelfth man to kneel and the second to stand,
and so on.
There will theu continue to be a row of ten
kneeling men, but, during each second, it will be shifted
one place along the line. The distant observer will,
therefore, see a depression steadily advancing along the
line. The state of things presented to his eye after two,
six, and nine seconds, respectively, is shown in Fig. 2.
FIG. 2.
TERBANDA LAERIANDESARTARATIBOR MALARIAL E GRANEL
PRODUADRUARIERARIA VARIRAMARIKINARALIN LEITE
There is here no horizontal motion on the part of the
men composing the line, but their vertical motions give
14
SOUND AND ITS TRANSMISSION.
rise, in the way explained, to the horizontal transference
of the depression along the line. The reader should ob-
serve that for no two consecutive seconds does the kneel-
iny row consist of exactly the same men, while in such
positions as those shown in the figure, which are sepa-
rated by more than ten seconds of time, the men who
form it are totally different.
6. Let us now return to the sea-waves, and examine
more closely the elements of which they consist.
Fig. 3 represents a vertical section of one complete
wave.
Fig. 3.
H
N
K
A
B В
G
o
L
I
P
M
The dotted line is that in which the horizontal plane,
forming the surface of the sea when at rest, cuts the plane
of the figure. The distance between the two extreme
points of the wave, measured along this line, is called the
length of the wave. C is the highest point of the crest
DCB; E the lowest point of the trongh AED. CF
and GE are vertical straight lines through C and E;
HCK and LEM are horizontal straight lines through
the same two points. The vertical distance between the
lines HK and LM is called the breadth or amplitude of
the wave. Thus, AB is the length of the wave, and, if
we produce EG and CF to cut the lines HK and LM in
N and P respectively, we have for its amplitude either
of the equal lines EN, PC. Each of these is clearly
equal to FC and GE together, that is to say, the ampli-
tude of the wave is equal to the height of the crest above
the level-line together with the depth of the trough be-
FORMS OF SEA-WAVES.
15
low it. In addition to the length and amplitude of the
wave, we have one more element, its form. The wave
in the figure has its crest shorter than its trough and
higher than its trough is deep. Moreover, the part
DC of the crest is steeper than the part CB, while,
in the trough, the parts AE and EB are equally steep.
Sea-waves have the most varied shapes according to
the direction and force of the wind producing them.
IIence, before we can lay down a wave in a figure, we
inust know the nature of the wave's curve, or, in other
words, its form.
Since the crests of the waves are raised above the
ordinary level of the sea, the troughs must necessarily be
depressed below it, just as, in a ploughed field, the earth
heaped up to form the ridges must be taken out of the
furrows. Each crest being tlus associated with a trough,
it is convenient to regard one crest and one trongh as
forming together one complete wave. Thus, each wave
consists of a part raised above, and a part depressed be-
low, the horizontal plane which would be the surface of
the sea were it not being traversed by waves.
7. The length, amplitude, and form of a wave com-
pletely determine the wave, just as the length, breadth,
and height of an oblong block of wood, i. e., its three
dimensions, fix the size of the block. These three ele-
ments of a wave are mutually independent, that is to say,
we may alter any one of them without altering the other
two. This will be seen by a glance at the accompanying
figures.
FIG. 4 (1).
F
(1) shows variation in length alone; (2) in amplitude
alone; (3) in form alonc.
16
SOUND AND ITS TRANSMISSION. .
(2)
(3)
8. We will next study more closely the motion of an
individual drop of water, in the surface of the sea, while
a wave passes across it. Fig. 5 shows nine positions of the
wave and moving drop at equal intervals of time, each
one-eighth of the period during which the wave traverses
an horizontal distance equal to its own wave-length. In
(1) the front of the ware has just reached the drop pre-
viously at rest in the level-line represented by dots in the
figure. In (2), the drop is a part of the way up the front
of the crest; in (3), at the summit of the crest, and there-
fore at its greatest distance above the level-line. In (4),
it is on the back of the crest, and in (5) occupies its original
position. It then crosses the level-line; is on the front
of the trough in (6), and at it lowest point in (7), where
it attains its greatest distance below the level-line. In
(8), it is on the back of the trough, and in (9) has once
more returned to its starting-point in the level-line.
We have here a vibratory or oscillatory movement
like that of the end or “bob” of a clock-pendulum, but
executed in a vertical straight line. We call the distance
MOTION OF A DROP OF WATER.
17
FIG. 5.
(
1)
(2)
(3)
(4)
(6)
(7)
CL
(8)
(9)
1:8
SOUND AND ITS TRANSMISSION.
between the two extreme positions of the bob the extent
of swing of the pendulum. The extent of the drop's
oscillation will be seen, from (3) and (7), to be equal to
the sum of the height of the wave's crest above the level-
line, and of the depth of the trough below it.
But this, as was shown in § 6, is equal to the ampli-
tude of the wave. Hence, “ extent of drop's vibration "
and "amplitude of corresponding wave” are only differ-
ent ways of expressing the same thing.
Let the line A'04 be that in which the drop under
FIG. 6.
A
A
consideration vibrates, o being in the level-line, A and
A' the limits of oscillation. The whole movement given
in Fig. 5 will then be from 0 to A, from A through O
to A', and from A' back again to O. This is termed one
complete vibration, and since, in the course of it, each por-
tion of the drop's path is passed over twice, one complete
vibration is equal to an upward swing from A to A, to-
gether with a downward swing from A to A'. In the
clock-pendulum we have, during each second, one com-
plete oscillation, consisting of one swing from left to right
and one from right to left.
Reference to Fig. 5 at once shows that, during the
time occupied by the wave in traversing its own wave-
length, the moving drop performs one complete vibra-
tion, or, to express the same fact in the reverse order,
VIBRATION OF WATER-DROPS.
19
that, while the drop makes one complete vibration, the wave
advances through one wave-length. This is a most im-
portant principle, and should be thoroughly mastered and
borne in mind by the student.
9. What has just been proved for a particular drop is,
of course, equally true for any assigned drop in the sur-
face passed over by a wave. All the drops, therefore, go
through exactly the same vibrations in exactly equal times,
but, since each drop can only start at the moment when
the front of the wave reaches it [Fig. 5, (1)], they will in
general occupy different positions in their paths at the
same time. We may illustrate this by supposing a num-
ber of watches, which keep good time, to be set going
successively in such a way that the first shall mark xII at
twelve o'clock, the second at five minutes past twelve,
the third at ten minutes past twelve, and so on. The
hands of each watch will describe the same paths in equal
times, but, at any assigned moment, will occupy different
positions in those paths corresponding to the lateness of
their several starts. The drops in the sea-surface being,
in this manner, thrown successively into the same vibra-
tory motion, give rise, by their consequent varieties of
position at any assigned moment, to the transmission of
the form which we call a wave.
FIG. 7.
When a series of continuous equal waves, such as those
in Fig. 7, is being transmitted, each oscillating drop,
after completing one vibration, will instantly commence
another precisely eqnal vibration, and go on doing so as
long as the series of waves lasts. The kind of motion, in
which the same movement is continuously repeated in
successive equal intervals of time, is called “periodic,"
20
SOUND AND ITS TRANSMISSION.
and the time which any one of the movements occupies is
called its "period.” Thus, to continuous equal waves
correspond continuous periodic drop-vibrations.
10. We will next compare the periods of the drop-
vibrations corresponding to waves of different lengths ad-
vancing with equal velocities.
In Fig. 8, waves of three different lengths are repre-
sented. One wave of (1) is as long as two of (2), and as
three of (3). Therefore, a drop makes one complete vi-
bration in (1) while the long wave passes from A to B,
FIG. 8.
(1)
B
(2) 4
Eivorouoc vu ..... - 4.41094011811AAAA.
B
(3) A
B
two in (2) while the shorter waves there presented pass
over the same distance, and three in the case of the short-
est waves of (3). But the velocities of these waves being,
by our supposition, equal, the times of describing the dis-
tance AB will be the same in (1), (2), and (3). Hence, a
drop in (2) vibrates twice as rapidly, and a drop in (3)
three times as rapidly, as a drop in (1); or, conversely, a
RATES OF DROP-VIBRATION.
21
drop in (1) vibrates half as rapidly as a drop in (2), and
one-third as rapidly as a drop in (3).
The rates of vibration in (1), (2), and (3), (by which
we mean the numbers of vibrations performed in any
given interval of time) are, therefore, proportional to the
numbers 1, 2, and 3, which are themselves inversely pro-
portional to the wave-lengths in the three cases respec-
tively. We may express our result thus : the rate of drop-
vibration is inversely proportional to the corresponding
wave-length. The same reasoning will apply equally well
to any other case; the proposition, therefore, though de-
rived from particular relations of wave-lengths, is true
universally.
11. We have now connected the extent of the drop-
vibration with the amplitude, and its rate with the length,
of the corresponding wave.
It remains to examine what
feature of the oscillatory movement corresponds to the
third element, the form, of the wave.
Fig. 9.
H
B
1
А.
Suppose that two boys start together to run a race
from 0 to A, from A to B, and from B back to 0, and
that they reach the goal at the same moment. They may
obviously do this in many different ways. For instance,
they may keep abreast all through, or one may fall be-
hind over the first half of the course, and recover the lost
ground in the second. Again, one may be in front over
010, and the other over OBO, or each boy may pass,
and be passed by, his competitor, repeatedly during the
We may regard the movement of each boy as con-
stituting one complete vibration, and thus convince our-
selves that an oscillatory motion of given extent and pe-
riod may be performed in an indefinitely numerous
race.
22
SOUND AND ITS TRANSMISSION.
variety of modes. Let us now compare the positions of a
drop at successive equal intervals of time, when coöperat-
ing in the transmission of waves of different forms.
In each of these three cases, in Fig. 10, the front of a
wave-crest is shown in the positions it respectively occu-
pies at the end of ten equal intervals of time (each one-
tenth of that occupied by the wave in traversing a quar-
ter-wave-length), the apex of the wave being successively
at the equidistant points of the level-line 1, 2, 3, 4, etc.
FIG. 10.
А
(1)
0 1 2 3 4 5 6 7 8 9
16
A
(2)
0 1 2 3 3
4 5 6 7 8 9 10
A
(3)
0 1 2
2 3
3 4 4 5 6 7 8
9 10
A drop whose place of rest is O will then assume the
corresponding positions in the vertical line 0 A. Thus
the points where this line cuts the successive wave-frouts
show the positions of the vibrating drop at equal intervals
of time.
VIBRATION-MODE AND WAVE-FORM.
23
By comparing the three cases it will be seen that the
mode of the drop's vibration is distinct in each. In (1),
it moves fastest at 0, and then slackens its pace up to A.
In (2), it starts more slowly than in (1), attains its great-
est speed near the middle of 0 A, and again slackens on
approaching A. In (3), the pace steadily increases from
O to A. The different waves in the figure have been
purposely drawn of the same amplitude and length, in
order that only such variations as were due to differences
of form might come into consideration. The reader should
construct for himself similar figures with other wave-
forms, and so convince himself, more thoroughly, that
every distinct form of wave has its own special mode of
drop-vibration,
12. The converse of this proposition is also true, viz.,
that each distinct inode of drop-vibration gives rise to a
special form of wave. We will show this by actually con-
structing the form of wave which answers to a given mode
of drop-vibration. When a drop vibrates in a given mode,
its position at any assigned moment during its vibration
is of course known. If we also know the amount by
which drops farther on in the level-line are later in their
starts ( 9) than drops less advanced in that line, we can
assign the positions of any number of given drops at any
given instant of time.
Suppose that each drop makes one complete vibration
per second about its position of original rest in the level-
line. The law of its vibration is roughly indicated in Fig.
11. AB is the path described by any drop ; O its position
when in the level-line; 1, 2, 3, 4...16 its positions after
sixteen equal intervals of time, each one-sixteenth of a
second ; 16 coincides with 0-i. e., the drop has returned
to its starting-point.
Next, select a series of drops originally at rest in equi-
distant positions along the level-line, and so situated that
24
SOUND AND ITS TRANSMISSION.
each commences a vibration, identical with that above laid
down in Fig. 11, one-sixteenth of a second after the drop
FIG. 11.
B
5
6
4
3
2
7
1
8
0,16
9
15
10
14
11
13
12
А.
next it has started on an equal oscillation. Fig. 12 shows
tlie rest-positions of the series of drops
do, Cli, (2, Clz ... (169
in the level-line, and their contemporaneous positions
din, ai', az', az ... 16
during a subsequent vibration.
FIG. 12.
12
น
I
a
13
11
a
10
da
14
ang
ܝܚܞ-;
a
15
8
1
|
h
1
1
wania
ict
2
ice
4
a
3
1
5
a
8
NG
ia
7
a
11
20
a
12
13
14
15
gria
an
al
ܘܘܝܘ
az
a
4
The moment selected for the figure is that in which
the first of the series, clo, is on the point of commencing
CONSTRUCTION OF WAVE-FORU:
23
its vibration in a vertical direction. Since the second
drop started one-sixteenth of a second after the first, its
position in the figure will be below the level-line at ai',
making the line ay alı' equal to the line 015 in Fig. 11.
The next drop, which is two-sixteenths behind do in its
path, will be at az', making dla dz' equal to 014 in the
same figure. In this way the positions of all the points,
ai' an' az', etc., in Fig. 12, are determined from Fig. 11.
They give us, at once, a general idea of the form of the
resulting wave. By laying down more points along the
line AB in Fig. 11, we can get as many more points on
the wave as we please, and should thus ultimately arrive
at a continuous curved line. This is the wave-form re-
sulting from the given vibration-mode with which we
started, and, since only one wave-forın can be obtained
from it, we infer that each distinct mode of drop-vibra-
tion gives rise to a special form of wave.
It has now been sufficiently shown that, corresponding
to the three elements of a wave, amplitude, length, and
form, there are three elements of its proper drop-vibra-
tion, ectent, rate, and mode.
13. We have seen that a sea-wave consists of a state
of elevation and depression of the surface above and be-
low the level-plane. The same thing holds of the small
ripples set up by throwing a stone into a pond, and the
non-progressive nature of the motion of individual drops
on the surface can be as easily made out on a small as on
a large sheet of water. Moreover, the characteristic phe-
nomenon on which we have been engaged—viz., a uni-
formly progressive motion arising out of a number of
oscillatory movements—is by no means confined to liquid
bodies. Thus, when a carpet is being shaken, bulging.
forms, exactly like water-waves, are seen running along
it. A flexible string, one end of which is tied to a fixed
point, and the other held in the hand, exhibits the same
2
26
SOUND AND ITS TRANSMISSION.
phenomenon when the loose end is suddenly twitched.
It has accordingly been found convenient to extend the
term “wave” beyond its original meaning, and to desig.
nate as
wave-motion"
any movement which comes
under the definition just laid down. We proceed to an
instance of such motion which is important from its sim-
ilarity to that to which the transmission of sound is due.
14. Any one, who has looked down from a slight ele-
vation on a field of standing corn on a gusty day, must
have frequently observed a kind of thrill running along
its surface. As each ear of corn is capable of only a
slight swaying movement, we have here necessarily an in-
stance of wave-motion, the ear-vibrations corresponding
to the drop-vibrations in water-waves. There is, how-
ever, this important difference between the cases, that the
ears' movements are mainly horizontal-i. e., in the line
of the wave's advance—whereas, the drop-vibrations are
entirely perpendicular to that line. The advancing wave
is therefore.. no longer exclusively a state of elevation or
depression of surface, but of more tightly or less tightly
packed ears. The annexed figure gives a rough idea how
TIG. 13.
с
B
A
WW
this takes place. The wind is supposed to be moving
from left to right and to have just reached the ear A.
Its neighbors to the right are still undisturbed. The
stalk of C has just swung back to its erect position. The
LONGITUDINAL VIBRATIONS.
27
ears about B are closer to, and those about Cfarther
apart from, each other, than is the case with those on
which the wind bas not yet acted.
After this illustration, it will be easy to conceive a
kind of wave-motion in wbich there is no longer (as in
the case of the ears of corn) any movement transverse to
the direction in which the wave is advancing.
15. Let a series of points, originally at rest in equi-
distant positions along a straight line, as in (0), Fig. 14,
be executing equal periodic vibrations in that line, in such
a manner that each point is a certain fixed amount far-
ther back in its path than is its neighbor on one side, and
therefore exactly as much more forward than its neigh-
bor on the other side.
(1) shows the condition of the row of points at the
moment when the extreme point on the left is beginning
its swing from left to right, which, in accordance with the
direction of the arrow in the figure, we may call its for-
ward swing. The equidistant vertical straight lines fix
the extent of vibration for each oscillating point. The
constant amount of retardation between successive points
is, in the instance here selected, one-eighth of the path
traversed by eaclı point during the period of a complete
oscillation. Thus, proceeding from left to right along the
line (1), we have the first point beginning a forward
string, the second, third, 'fourth, and fifth points entering
respectively on the fourth, third, second, and first quar-
ters of a backward swing, and the sixth, seventh, eighth,
and ninth points on the fourth, third, second, and first
quarters of a forward swing.
Since the ninth point is just beginning a forward
swing, its situation is exactly the same as that of the first
point. Beyond this point, therefore, we have only repe-
titions of the state of things between the first and ninth
points. The row (1) is therefore made up of a series of
28
SOUND AND ITS TRANSMISSION.
FIG. 14.
(0)
...
B
C'
a
.GI
ъ
(1)
(2)
(3)
܂1ܫ
(4)
(5)
(6)
(7)
(8)
(9)
7
a!
a
А
groups, or cycles, of the same number of points arranged
in the same manner throughout. Two such cycles, in-
cluded by the large brackets A and B, are shown in (1).
Each cycle is divided by the small brackets a, a' and b, V .
into two parts. In c and the distances between succes-
sive points are less than, and in û and V' greater than,
LONGITUDINAL VIBRATIONS.
29
5
the corresponding distances when the points occupied
their undisturbed positions, as in (0). The cycles cor-
respond to complete waves on the surface of water, the
shortened and elongated portions of each cycle answer-
ing to the crest and trough of which each water-wave
consists.
(2) shows the state of the row of points when an inter-
val of time equal to one-eighth of the period of a com-
plete point-vibration has elapsed from the moment shown
in (1). The wave A has here moved forward into the
po-
sition indicated by the dotted lines.
The following rows (3), (4), (5), etc., show the state of
things when two-eighths, three-eighths, four-eighths, etc.,
of a vibration-period has elapsed since (1). In each, the
wave A moves forward one step farther.
In (9), a whole vibration-period has elapsed since (1).
Accordingly every oscillating point has performed one
complete vibration, and returned to the position it held
in (1). The wave to meanwhile, has traveled constantly
forward so as to be, in (9), where B was in (1), i.e., to have
advanced by one whole wave-length. The proposition
proved for waves due to transverse vibrations in 88 is
thus shown to hold good likewise for waves due to longi-
tudinal vibrations.
16. In the waves shown in Fig. 14, the points in the
bracket a are mutually equidistant, as are also those in
the bracket b. This is due to the fact that, in the case
there represented, the oscillating points move uniformly,
i. e., with equal velocity, throughout their paths. If we
take other modes of vibration, we shall find that this
equidistance no longer exists. Fig. 15 shows three dis-
tinct modes of vibration with the wave resulting from
each, on the planı: of (1), Fig. 14. The extent of vibra-
tion and length of wave are the same in the three
cases.
30
SOUND AND ITS TRANSMISSION.
In (I.), the points move quickest at the middle, and
slowest at the ends, of their paths; in (II.), fastest at the
ends, and slowest in the middle; in (III.), slowest at the
left end, and fastest at the right.
The shortest distance separating any two points con-
tained in a is, in (I.), that between 7 and 8; in (II.), that
between 8 and 9; in (III.), that between 5 and 6. The
corresponding greatest distances are, in (I.), between 2 and
3; in (II.), between 1 and 2; in (III.), between 4 and 5.
The remaining points likewise exhibit differences of rela-
tive distance in the three cases. Thus, the positions of
greatest shortening and greatest lengthening occupy dif-
ferent situations in the wave, and the intermediate
و
Fig. 15.
B
a
(I.)
1
2
3
4
5
6
7
8
9
6
(II.)
1
2
3
4
5
6
7
b
a
(III.)
1
a
3
5
6
8
9
variations between them proceed according to different
laws, when the modes of point-vibration are different.
The more points we lay down in their proper positions
in a and b, the less abrupt will be the changes of distance
between successire points. By indefinitely increasing
the number of vibrating points, we should ultimately
arrive at a state of things in which perfectly continuous
changes of shortening and lengthening intervened be-
tween the positions of maximum shortening and maxi-
mum lengthening in the same wave.
CONDENSATION AND RAREFACTION.
31
17. Let us now replace our row of indefinitely numer-
ous points by the slenderest filament of some material
whose parts (like those of an elastic string) admit of
being compressed or dilated at pleasure. When any por-
tion of the filament is shortened, a larger quantity of
material is forced into the space which was before oc-
cupied by a smaller quantity. The matter within this
space is, therefore, more tightly packed, more dense, than
it was, i. e., a process of condensation has occurred. On
the other hand, when a portion of the filament is length-
ened, a smaller quantity is made to occupy the space
before occupied by a larger quantity. Here the matter
is more loosely packed, more rare, than it was, i. e., a
process of rarefaction has taken place.
Let us now suppose the particles, or smallest conceiv-
able atoms, of the filament, to be thrown into successivo
vibrations in the manner already so fully explained.
Alternate states of condensation and rarefaction will
then travel along the filament. It will be convenient to
call these states “pulses"--of condensation or rarefaction
as the case may be. A pulse of condensation and a pulse
of rarefaction together make up a complete wave.
18. The degree of condensation or rarefaction exist-
ing at any given point of a wave has been shown to
depend on the mode in which the particles of the fila-
inent vibrate. It is therefore desirable to have some
simple method, appealing directly to the eye, of exhibit-
ing the law of any assigned mode of vibration which
takes place in a straight line. We may arrive at such a
method by the following considerations.
When a line of particles vibrates longitudinally, it
gives rise to waves of condensation and rarefaction; when
transversely, to waves of displacement on opposite sides
of the line of particles in their positions of rest. Never-
theless, if the vibrations in the two cases are identical
32
SOUND AND ITS TRANSMISSION.
in all other respects save direction alone, the distance
which, at any moment, separates an assigned particle
from its position of rest will be the same, whether the
vibrations are longitudinal or transverse. It is therefore
only necessary to construct the wave corresponding to
any system of transverse vibrations, in the way shown in
§ 12, to obtain the means of fixing the position of an
assigned particle, at any given moment, for the same
system of vibrations executed longitudinally.
FIG. 16.
a'
A-
--B
a
D
ს
B'
Let AB and CD, Fig. 16, be lines of particles execut-
ing vibrations transverse to AB, and along CD, respec-
tively. Let a and b be corresponding particles in their
positions of rest. Draw the transverse wave for any
given instant of time: the particle originally at a will
now be at a', and that originally at 6, at b', making bb'
equal to aa'.
By performing the same process for different instants,
we can find as many corresponding positions of the longi-
tudinally vibrating particle as we please. It is true that
we learn nothing new by this, since we cannot construct
the wave-curve without knowing beforehand the mode of
the particle's vibration ($ 12]. Still, when we are deal-
ing with longitudinal particle-vibrations, and require to
know the law of the variation of condensation and rare-
faction at different points of a single wave, it is conven-
ient to have a picture of the modle of vibration, by. which,
PENDULUM=VIBRATION.
33
as we know [$ 16], that law is determined. Such a pict-
ure we have in the form of the wave produced by the
same mode of vibration when executed transversely. Let
us call the wave so related to a given wave of condensa-
tion and rarefaction, its associated wave.
19. Before leaving this portion of the subject it will
be advisable to draw the associated wave for that particu-
lar mode of longitudinal vibration in which each particle
moves as if it were the extremity of a pendulum travers-
ing a path which is very short compared to the pendin-
lum's length. The meaning of this limitation will be
easily seen from Fig. 17.
FIG. 17.
B
a
7
d.
D
A
3
a
Let O be the fixed point of suspension ; 0 A the pen-
dulum in its vertical position ; AB a portion of a circle
with centre 0 and radius OA; a, b, c, d, points on this
circle; AD an horizontal straight line through A; aa',
60', cd', dd', verticals through a, b, c, d, respectively. If
the pendulum is placed in the position Oa, and left to it-
self, it will swing through twice the angle a0 A before it
31
SOUND AND ITS TRANSMISSION.
turns back again. Similarly, if started at Ob, it will
swing through twice the angle 60 A; if at Oc, through
twice the angle c0 A, and so on. Now, the extremity of
the pendulum, when at a, is farther from the horizontal
line, AD, than when it is at b, since aa' is greater than
67, and at b farther than at c. If we make the pendu-
lum vibrate through only a small angle, by starting it,
say, in the position Od, its extremity will, throughout its
inotion, be very near to the horizontal straight line AD.
If we make the angle small enough, or, which is the same
thing, take Ad sufficiently small compared with 04, we
may, without any perceptible error, suppose the end of
the pendulum to move in an horizontal straight line, in-
stead of in a circular arc, i. e., along d'A instead of dA.
To take an actual case, let us suppose the pendulum to
be ten. feet long, and that its extent of swing is one inclı
on either side of its vertical position. A very easy geo-
metrical calculation will show that the end of the pendu-
lum will never be so much as oth of an inch out of the
horizontal straight line drawn through it in its lowest
position. This is a vanishing quantity; we may, there-
fore, safely regard the vibration as performed along d'A
instead of dA. Such a vibration, though executed in a
straight line instead of in the arc of a circle, we may
properly call a pendulum-vibration, as expressing the
law according to which it takes place. This law admits
of easy geometrical illustration as follows: Let a ball, or
other sinall object, be attached to some part of an up-
right wheel revolving uniformly about a fixed axis, so
that the ball goes round and round in the same vertical
circle with constant velocity. If the sun is in the zenith,
i. e., in such a position that the shadows of all objects are
thrown vertically, the shadow of the ball on any horizon-
tul plane below it will move exactly as does the bob of a
pendulum.
MOTION IN FILAMENTS.
35
The form of the associated wave for longitudinal pen-
dulum-vibrations is shown in Fig. 17 (a).
Retaining the form of the curve, we may make its
amplitude and wave-length as large or as small as we
please, as in the case of the waves in Fig. 4 (1) and (2),
pages 15, 16.
Fig. 17 (?).
20. We have examined the transmission of waves due
to longitudinal vibrations along a single very slender fila-
ment. Suppose that a great number of such filaments
are placed side by side in contact with each other, so
as to form a uniform material column. If, now, precisely
equal waves are transinitted along all the constituent fila-
ments simultaneously, successive pulses of condensation
and rarefaction will pass along the column. The parts
in any assigned transverse section of the column will.
obviously, at any given moment of time, all have exactly
the same degree of compression or dilatation. When a
pulse of condensation is traversing the section, its parts
· will be more dense, when a pulse of rarefaction is trav-
ersing it, less dense, than they would be were the column
transmitting no waves at all, and its separate particles,
therefore, absolutely at rest. Let the column with which
we have been dealing be the portion of atmospheric air
inclosed within a tube of uniform bore. The pbenomena
just described will then be exactly those which accom-
pany the passage of a sound from one end of the tube
36
SOUND AND ITS TRANSMISSION.
to the other. It reinains to examine the mechanical
cause to which these phenomena are due.
Atinospheric air, in its ordinary condition, exerts a
certain pressure on all objects in contact with it. This
pressure is adequate to support a vertical column of
mercury thirty inches high, as we know by the common
barometer. In Fig. 18 is shown a section of a tube
closed at one end, with a movable piston fitting into
the other. In (1) the air on both sides of the piston is
in the ordinary atmospheric condition, so that the press-
ure on the right face of the piston is counteracted by an
exactly equal and opposite pressure on its left face.
FIG. 19.
(1
)
(
2)
(3)
In (2) the piston has been moved inward, so as to
compress the air on the right of it. That on its left,
being in free communication with the external air, is
not permanently affected by the motion of the piston.
In order to retain the piston in its forward position,
it is necessary to exert a force upon it, in the direc-
tion of the arrow. If this force is relaxed, the piston is
driven back. Since the pressure of the air on the left of
the piston is just what it was before, that on its right
must necessarily have increased. But this increase of
pressure is accompanied by an increase of density, due to
MARIOTTE'S LAW.
37
the compression of the air on the right of the piston.
Hence, increase of pressure accompanies increase of den-
sity. If, as in (3), we reverse the process, by moving the
piston outward, the extraneous force must be exerted in
the opposite direction, as shown by the arrow. The
pressure on the right of the piston is therefore less than
the normal atmospheric pressure on its left, i. e., diminu
tion of pressure accompanies diminution of density. By
experiments such as the above, it was shown, by the
French philosopher Mariotte, that the pressure of the air
varies as its density.
21. Next, let us take a cylindrical tube open at one
end and having a movable piston fitting into the other, as
in Fig. 19.
In (1) the piston is at rest at A, and the air in its
ordinary atmospheric condition of density and pressure.
FIG. 19.
(1
)
A
B
(2)
C
D
(3)
E
F
G
(4)
н H
K
L
M
(5)
E
N
0
р
9
R
In (2) the piston is pushed inward as far as C. While it
is moving up to this position, the air-particles in front of
it are thrown into motion. Suppose that, at the moment
when the piston reaches C, the particles at D are just
beginning to be disturbed. The air which, in (1), occu-
pied AB, is now crowded into CD, and is, therefore,
38
SOUND AND ITS TRANSMISSION.
denser than that farther on in the tabe. Now, let the
piston be drawn back to E, (3), as much to the left of its
original position, A, (1), as, in (2), it was to the right of
it. The air in CD, (2), will, while this is taking place,
expand into EF; for, being denser, it will also be at
greater pressure than the air to the right of it. It will,
therefore, act on the air in advance of it in the same way
as the piston did on the air in contact with it when muv-
ing from A, (1), to C, (2). Hence, the air in FG will be
condensed, G being the point where the air-particles are
just beginning to be disturbed when the piston reaches
the position E. Thus, the air at D advances to F.
Further, in consequence of the backward motion of the
piston, the air in the neighborhood of C, (2), has to move
to E, (3). Thus, the air originally in AB now occupies
EF, which is greater than AB. It is therefore less dense
than in (1), i. e., is in a state of rarefaction. Now, let
the piston again advance to H, (4). The air in FG, be-
ing at a greater pressure than that in its front, and still
more so than that in its rear, will expand in both direc-
tions, causing a new condensation, LM, to be formed
farther on, and itself becoming the rarefaction KL, CO-
operating, at the same time, with the advancing piston to
produce in its own rear the condensation HK. In (5)
the piston is again where it was in (3). HK has expand-
ed into the rarefaction NO, KL contracted into the con-
densation OP, LM expanded into the rarefaction PQ,
and a new condensation, QR, been formed in front.
The figure makes it clear that each forward stroke of
the piston produces a pulse of condensation, and each
backward stroke a pulse of rarefaction ; but that, when
once formed, these pulses travel onward independently of
any external force. They do so, as we have seen, in virtue
of the relation which connects the pressure of the air with
its density, in other words, on the elasticity of the air.
THE EXPANDING SOUND-WAVE.
39
If we suppose our movable piston withdrawn from
the tube, and a vibrating tuning-förk held with the ex-
tremity of one prong close to the orifice of the tube, the
conditions of the problem will not be essentially modified.
Each outward swing of the prong will give rise to a con-
densed, and each inward swing to a rarefied, pulse, and
thus, during every complete vibration of the fork, one
sonorous wave, consisting of a pulse of condensation and
a pulse of rarefaction, will be started on its journey along
the tube.
22. We have examined the transmission of sound
along a column of air contained in a tube of uniform
bore. A more important case is that in which a sound,
originated at an assigned point, spreads out from it freely
in all directions. Here we must conceive a series of
spherical shells, alternately of condensed and of rarefied
air, one inside the other, and all having the point of
origination of the sound as their common centre. All
the shells must be supposed to expand uniformly like an
elastic globular balloon constantly inflated with more and
The great difference between this case and
that last considered lies in this, that, as the spherical
shells of condensation and rarefaction spread, it is neces-
sary, in order to keep up the wave-motion, to throw larger
and larger surfaces of air into vibration; whereas, within
the tube, the transverse section remained the same through-
out. Hence, as the same amount of original disturbing
force has to set a constantly-increasing number of air-
particles into motion, it can only do so by proportionate-
ly shortening the distances through which the individual
particles move, i. e., by diminishing their extent of vibra-
tion. Accordingly, when sound-waves spread out freely
in all directions, the farther any given air-particle is from
the point at which the sound originated, the smaller will
be the extent of the vibration into which it will be thrown
When tlie wares reach it.
more gas.
40
SOUND AND ITS TRANSMISSION.
23. Sounds are either musical or non-musical. The
vast majority of those ordinarily heard--the roaring of
the wind, the din of traffic in a crowded thoroughfare--
belong to the second class. Musical sounds are, for the
most part, to be heard only from instruments constructed
to produce them. The difference between the sensations
caused in our ears by these two classes of sounds is ex-
tremely well marked, and its nature admits of easy analy-
sis. Let a note be struck and held down on the harmo-
nium, or on any instrument capable of producing a sus-
tained tone. However attentively we may listen, we per-
ceive no change or variation in the sound we hear, A
perfectly continuous and uniform sersation is experienced
as long as the note is held down. If, instead of the har-
inonium, we employ the piano-forte, where the sound is
loudest directly after the moment of percussion, and then
gradually dies away, the result of the experiment is that
the diminution of loudness is the only change which oc-
curs; the effect produced is the same as if our harmonium
had, while sounding out its note, been carried gradually
farther and farther away from us.
In the case of non-musical sounds, variations of a dif-
ferent kind can be easily detected. In the howling of
the wind the sound rises to a considerable degree of shrill-
ness, then falls, then rises again, and so on. On parts of
the coast, where a shingly beach of considerable extent
slopes dowų to the sea, a sound is heard in stormy weather
which varies from the deep thundering roar of the great
breakers, to the shrill tearing scream of the shingle dragged
along by the retreating surf. Similar variations may be
noticed in sounds of small intensity, such as the rustling
of leaves, the chirping of insects, and the like. The dif-
ference, then, between musical and non-musical sounds
seems to lie in this, that the former are constant, while
the latter are continually varying. The human voice can
MUSICAL AND NON-MUSICAL SOUNDS.
41
produce sounds of both classes. In singing a sustained
note it remains quite steady, neither rising nor talling.
Its conversational tone, on the other band, is perpetually
varying in height even within a single syllable; directly
it ceases so to vary, its non-musical character disappears,
and it becomes what is commonly called “sing-song."
We may then define a musical sound as a steady sound,
a non-musical sound as an unsteady sound. It is true we
may often be puzzled to say whether a particular sound
is musical or not; this arises, however, from no defect in
our definition, but from the fact that such sounds consist
of two elements, a musical and a non-musical, of which
the latter may be the more powerful, and therefore ab-
sorb our attention, until it is specially directed to the for-
mer. For instance, a beginner on the violin often pro-
duces a sound in which the irregular scratching of the
bow predominates over the regular tone of the string. In
bad flute-playing, an unsteady hissing sound accompanies
the naturally sweet tone of the instrument, and may
easily surpass it in intensity. In the tones of the more
imperfect musical instruments, such as drums and cym-
bals, the non-musical element is very prominent, while in
such sounds as the hammering of metals, or the roar of a
water-fall, we may be able to recognize only a trace of
the musical element, all but extinguished by its boisterous
companion.
We have seen that sound reaches our ears by means
of rapid vibrations of the particles of the atınosphere.
It has also been shown that steadiness is the characteristic
feature of musical, as distinguished from non-musical,
sounds. We may infer hence that the motion of the air
corresponding to a single musical sound will be itself
steady-i. e., that equal numbers of equal vibrations will
be executed in precisely equal times. This conception of
the physical conditions under which musical sounds are
42
SOUND AND ITS TRANSMISSION.
produced will suffice for the present. We proceed to con-
sider in detail the various ways in which such sounds may
differ from each other, and to investigate the mechanical
cause to which each such difference is to be referred. In
what follows, by the word "sound" will always be meant
"musical sound,” unless the contrary be expressly stated.
CHAPTER II.
ON LOUDNESS AND PITCH.
24. A MUSICAL sound may vary in three different re-
spects. Let a note be played, first by a single violin,
then by two, by three, and so on, until we have all the
violins of an orchestra in unison upon it. This is a varia-
tion of loudness only. Next, let a succession of notes be
played on any instrument of uniform power, such as the
harmonium without the expression-stop, or on the princi-
pal manual of an organ, only one combination of stops
being in either case used. Here we have a variation of
pitch alone. Lastly, let one and the same note be suc-
cessively struck on a number of piano-fortes of the same
size, but by different makers. The sounds heard will all
have exactly the same pitch, and about the same degree
of loudness; nevertheless, they will exhibit decided dif-
ferences of character. The tone of one instrument will
be rich and full, of another ringing and metallic, that of
a third will be described as “wiry," of a fourth as “tin-
kling,” and so on.
Sounds thus related to each other are said to vary in
quality only. The instances just considered have the ad-
vantage of simplicity, since they allow of changes in loud-
ness, pitch, and quality, being exhibited separately. They
are, however, less striking than other cases where sounds
vary in two, or in all t!rree, of these respects at the same
44.
LOUDNESS AND PITCH.
time. A practised ear may be requisite to detect the dif-
ference between the tone of two pianofortes, but no one
is in danger of mistaking, for instance, a flute for a
trumpet. There is here, no doubt, considerable difference
of loudness as well as of quality, but let the more power-
ful instrument be placed at such a distance that it sounds
110 louder than the weaker one, and the distinction be-
tween the two kinds of tone will be still quite decisive.
Two assigned musical sounds thus may differ from
each other in loudness or pitch or quality, and agree in
the other two-or they may differ in any two of these,
and agree in the third-or they may differ in all three.
There is, however, no other respect in which they can
differ, and accordingly we know all about a niusical sound
as soon as we know its loudness, its pitch, and its qual-.
ity. These three elements determine the sound, just as
the lengths of the three sides of a triangle determine the
triangle.
25. The loudness of a musical sound depends entirely,
as we shall easily see, on the extent of oscillatory move-
inent performed by the individual particles composing
the medium through which the sound is conveyed to our
A sound-producing instrument can be readily ob-
served to be in a state of rapid vibratory motion. The
vibrations of a tuning-fork are perceptible to the eye
in the fuzzy, half-transparent rim which surrounds its
prongs when it is struck; and to the touch, if, after strik-
ing the fork, we place a finger gently against one of the
prongs. The harder we hit the fork the louder is its
sound, and the larger, estimated by both the above modes
of observation, are its vibrations. The experiment may
Le tried equally well on any piano-forte whose construc-
tion allows the wires to be uncovered. It is natural to
infer that a vibration on the part of a sound-producing
instrument coinmunicates to the particles of the air in
ears.
ON WHAT LOUDNESS DEPENDS.
45
Thus a
contact with it a corresponding movement.
sound of given loudness is conveyed by vibrations of
given extent, and, if the sound increases or diminishes in
intensity, the extent of the vibrations will increase or
diminislı with it.
We conclude, then, that the loudness of a musical
sound depends solely on the extent of excursion of the
particles which coustitute the conveying medium in the
neighborhood of our ears. This last condition is clearly
essential, since a sound grows more and more feeble, the
greater our distance from the point where it is produced.
This diminution of intensity with the increase of distance
from the origin of sound is a direct consequence of the
connection between loudness and the extent of vibration,
We have seen [S 22] that the farther an air-particle is
from the point where a sound is produced, the smaller
will be the extent of the vibration into wủich it is
thrown by the sonorous wave. Hence, as the sound ad-
vances, it will necessarily become feebler, provided al-
ways that the waves are permitted to spread out in all
directions. If they are confined, say, in a tube, the in-
tensity of the sound will not diminish with any thing
like the same rapidity. We have here the theory of mes-
sage-pipes, which are used in large establishments to
enable a conversation to be carried on between distant
parts of a building. A whisper, inaudible to a person
close to the speaker, may, by their means, be perfectly
well heard by a listener at the other end of the tube.
26. We have next to inquire to what mechanical
causes differences in the pitch of musical sounds are to be
referred. Rough observation at once indicates the direc-
tion in which we must look. If we draw the point of a
pencil along a rough surface, first slowly and then more
quickly, the sound heard will be distinctly slıriller the
more rapid the inovement of the pencil. As its point
46
LOUDNESS AND PITCH.
passes over the minute elevations and depressions which
constitute the roughness of the surface, a series of irregu-
lar vibrations is set up in the materials of the surface,
and by them communicated to the air. The more rapid
are these vibrations, the shriller does the sound become.
The instruinent described below, which is called a “si-
rene,” gives us the means of following up with accuracy
the hint just obtained.
FIG. 20.
А A
bo
с
O
B
AB is a thin circular disk of tin or card-board, which,
by means of a multiplying wheel, can be set in rapid
revolution about a fixed axis through its centre, C. A
series of holes (eight in the figure) is punched in the
disk at equal distances along a circle having its centre
at C. A small tube, ab, is held with one end close to one
of the holes. If, while the disk is rotating, we blow
steadily and continuously into the tube at a, a certain
quantity of air will pass through the disk whenever a
hole traverses the orifice 6, of the tube ab. During the
intervals of time which elapse between the passage of ad-
jacent holes across 6, no air can pass through the disk.
Hence, if the disk be revolving uniformly, series of
such discharges will succeed each other at perfectly
regular intervals of time. The air on the other side of
ACTION OF THE SIRENE.
47
the disk will necessarily be agitated by the process.
Every time that air is driven through one of the holes,
an increase of pressure occurs close to it, and accordingly
a pulse of condensation is formed there. The elastic
force of the air will give rise to a pulse of rarefaction
during each interval between successive discharges.
Hence the sirene supplies us with a regular series of
alternate condensations and rarefactions, which, when suf-
ficiently rapid, will, as we have seen, produce a musical
sound.
27. While air is being blown steadily into the tube,
let the disk be made to revolve slowly, and then with
gradually-increasing rapidity. At first nothing will be
audible but a series of faint, intermittent throbs, due to
the impact of the air driven through the tube against the
successive portions of the disk which separate its holes.
This sound may be exactly reproduced by moving the
forefinger to and fro rapidly before the lips, while blow-
ing through thein. It contributes nothing to the proper
musical sound of the instrument, and is only audible in
its immediate neighborhood. Presently, a deep musical
sound begins to be heard, which, as the velocity of rota-
tion increases, constantly rises in pitch. The acuteness
of the sound thus obtainable depends solely on the speed
to which we can urge the instrument, and is therefore
limited only by the driving-power at our command. The
rise of pitch in this experiment is perfectly continuous,
that is to say, the sound of the sirene, in passing from a
graver to a more acute note, goes through every possible
intermediate degree of pitch. It is important that we
should familiarize ourselves with this conception of the
continuity of the scale of pitch, because in the instrument
from which our ideas on this subject are usually obtained
--the piano-forte-the pitch alters discontinuously, i. e.,
by a series of juinps of half a tone each, and we are thus
48
LOUDNESS AND PITCH.
tempted to ignore the intervening degrees of pitch, or
even to suppose them non-existent. The more perfect
musical instruments, such as the human voice or the
violin, are as capable as the sirene of passing through all
degrees of pitch from one note to another in the way
called "portamento" or "slurring."
It is clear, from the nature of the sirene's construction,
that the only change which can take place during the
rise of pitch is the increased number of impulses com-
municated to, and therefore of vibrations set up in, the
external air, during any given interval of time. If, when
a note of given pitch has been attained by the sirene, we
check any further increase of velocity, and cause the disk
to rotate uniformly at the rate which it has just reached,
no further alteration of pitch will occur, and the note
will be steadily held by the instrument, so long as the
uniform rotation of its disk is kept up. Hence the num-
ber of aërial vibrations executed in a given time deter-
mines the pitch of the sound heard.
28. The sirene, besides teaching us this most impor-
tant fact, gives us the means of determining the number
of vibrations corresponding to any given note. If we
know the number of rotations which the disk has per-
formed in a given time, we have only to multiply this
number by the number of holes on the disk in order to
ascertain how many tube-discharges have occurred, and
therefore how many corresponding aërial vibrations have
been performed, in the period in question. The sirene is
provided with a counting-apparatus which registers the
number of times its disk rotates per second.
In order, therefore, to obtain the number of vibra-
tions in a second which correspond to an assigned note,
we have only to proceed as follows: Let the note be
steadily sounded by some instrument of sustained power,
e. g., organ or harmonium, and then cause the sirenc-
LIMITS OF MUSICAL SOUNDS.
49
sound to mount the scale until its pitch coincides with
that of the note under examination. At the instant of
coincidence read the figure indicated by the counting-
apparatus. This, multiplied by the number of holes in
the disk, gives the number of vibrations per second re-
quired. It will be convenient, for the sake of shortness,
to call the number of vibrations per second, to which any
note is due, the vibration-number of the note in question.
It is clear, from what has gone before, that any assigned
degree of pitch can be permanently registered, when
once its vibration-number has been ascertained.
29. The sirene shows that, below a certain rate of vi-
bration, no musical sounds are produced. The position
of the absolute limit thus placed to the gravity of such
sounds cannot be exactly defined, and probably varies
somewhat for different ears. The lowest note on the
largest modern organs has 161 for its vibration-number,
but it is a moot question whether the musical character
of this note can be recognized or not.
In any case, we may regard the lower limit of musi-
cal sounds as situated in the immediate neighborhood of
this degree of pitch. For soine distance above the limit
the musical character continues very imperfect, and it is
not until we reach 411 vibrations per second, the lowest
note of the double-bass, that we get a satisfactory musical
sound. There is no corresponding limit absolutely bar-
ring the scale of pitch in the opposite direction, but
sounds abore a certain degree of acuteness become pain-
ful to the ear, and therefore unfit for musical purposes.
The highest note of the piccolo, the shrillest sound heard
in the orchestra, inakes 4,752 vibrations per second. And
this we may regard as constituting a practical superior
limit to the scale of pitch at the disposal of musical art.
The extremest range attainable by exceptional human
voices, from the deepest note of a bass to the highest of a
3
50
LOUDNESS AND PITCH.
soprano, lies, roughly speaking, between 50 and 1,500
vibrations per second. Ordinary chorus - voices range
from 100 to 900 or 1,000 vibrations per second. The
number of sounds within the limits of the musical scale,
which can be recognized as possessing distinct degrees of
pitch, will vary with the acuteness of perception of indi-
vidual observers. Trained violinists are said to be able.
to distinguish about seven hundred sounds in a single
octave, which would give nearly five thousand for the
whole scale. But, since the difficulty of fixing the pitch
with accuracy increases very rapidly with very low or
very high sounds, this estimate would probably much
exceed the limits of what could be achieved by the very
finest ear. We shall, however, be well within the mark if
we assume that an ordinary ear can recognize, on the aver-
age, between one and two hundred sounds in an octave,
or fully one thousand in the whole scale. There is noth-
ing in the continuously shading-off gradations of pitch to
indicate what sounds should be picked out to form agree-
able sequences, or combinations, with each other. Nev-
ertheless, the human mind, working on this seeming
chaos from the earliest dawn of musical art, has reduced
it to order by discovering the following principle:
30. When one sound has been arbitrarily selected as
the starting-point, there are a certain number of other
sounds, having fixed relations of pitch to that previously
chosen, which are capable of forming, with it and with
each other, melodic and harmonic effects especially pleas-
ing to the ear. These are the notes of the ordinary major
and minor scales, the original sound of reference being
the common tonic, or key-note, of those scales. In say-
ing that these sounds have fixed mutual relations of
pitch, we merely state formally an obvious fact. A famil-
iar melody is recognized equally well whether heard in
the deep tones of a man's, or in the shrill notes of a
RELATIONS OF PITCE.
51
as,
child's voice. Whether the singer pitches it on a low or
on a high note of his voice makes no difference in the
melody itself. In fact, the correctness with which an air
is sung no more depends on the exact pitch of the note
on which the singer starts it, than does the faithfulness
of a plan on the precise scale which the draughtsman has
adopted. It is sufficient that the constituent notes of the
melody should have fixed mutual relations of pitch, just
in the plan, the several objects represented need only
be drawn in proportion to their actual dimensions.
The difference in pitch of any two notes is called the
interval between them : it is on accuracy of intervals
that music essentially depends.
31. The most important interval in the scale is the
octave. It is that which separates the highest note of a
peal of eight bells from the lowest. When a bass and a
treble voice sing the same melody together, the notes of
the latter are usually one octave above those of the for-
mer.
The octave has this peculiarity, shared by no
other interval, that, if, starting from any note we choose,
we ascend to that an octave above it, then to that an oc-
tave above the last, and so on, we get a number of notes
which sound perfectly smooth and agreeable wlien heard
all together. The same thing holds good if we descend
by a succession of octaves from the note fixed on as our
starting point. Hence, we may conveniently regard the
whole scale of pitch as divided into a series of octaves,
taken upward and downward from some one sound arbi-
trarily selected. Narrower intervals situated in any one
octave are repeated in all the other octaves, so that, when
we have settled those intervals for a single octave, we have
settled them for all the rest. Within the limits of each oc-
tave, the common major scale presents us with seven notes,
or, if we include that which forms the starting point of
the next octave, with eight. The fact that the eighth
52
LOUDNESS AND PITCH.
note is the octave of the first explains the meaning of the
word "octave," i. e., “ eighth” (Latin,“ octavus”).
The eight notes are those of an ordinary peal of the
same number of bells, or of the white keys of the piano-
forte between two adjacent C's. We may, for convenience
of reference, nunber them 1, 2, 3......8, beginning with
the lowest note, or tonic. The following nomenclature is
used to describe the intervals formed by the several notes
with the tonic:
Notes forming Interval.
Name of Interval.
1
and 2
Second
1 .... 3
Major Third
1
.. 4
Fourth
1
5
Fifth
1
... 6
Major Sixth
1
7 7
Major Seventh
1
8
Eighth or Octave.
When two notes of the same pitch are sounded to-
gether-e. g., by two instruments or by two voices--the
notes are said to be in unison. Though there is here no
difference of pitch whatever, it is convenient to rank the
unison as an interval. With this explanation, we may
add to the above table that 1 and 1 form the interval of
a unison. The reader must carefully avoid giving to
the “thirds,” “ fourths,” etc., which he meets with in
music, the meanings attached to the same words in frac-
tional arithmetic, with which they have absolutely noth-
ing to do. 'A "fiftli," for example, does not stand for a
fifth part of an octave, or indeed for a fifth part of any
CONCORDS AND DISCORDS.
53
thing, but for the difference of pitch between the first
and fifth notes of the scale.
The several pairs of notes forming the intervals laid
down in our table do not all produce smooth and agree-
able effects when sounded together. The following pairs
blend pleasantly :
1-3, 144, 145, 146,1–8;
the remaining two,
1-2 and 1-7,
give rise to decidedly harsh effects. The intervals in the
first line are therefore classed as concords, those in the
second as discords.
32. The minor scale has the notes 1, 2, 4, 5, 8 in com-
mon with the major scale. It substitutes for 3 a sound
lying between that note and 2, which forms with 1 a con-
sonant interval called the minor third. According to cir-
cumstances, it may either retain 6 or replace it by a
sound lying between that note and 5, which makes with
1 a concord called the minor sixth. Similarly, it may
employ 7, or, in the room of that note, a fresh sound sit-
uated below it, but above 6, which with 1 forms a discord
called the minor seventh.
Thus, including the octave, the two scales together
give us a series of eleven notes, which, severally com-
bined with the tonic, furm ten distinct intervals. They
are expressed in musical notation as follows:
Second.
Minor Third. Major Tbird.
Fourth.
Fifth.
12
AE
be
Minor Sixth. Major Sixth. Minor Seventh. Major Seventh.
Octave.
ha
ta
bo
The reader should endeavor to familiarize himself
54
LOUDNESS AND PITCH.
with these intervals, so that, when he hears the pair of
notes which form any one of them successively sounded,
he inay at once be able to name the interval.
33. The sirene enables us to obtain simple numerical
measures of the intervals exhibited on the preceding
page.
Let a second circular row, containing sixteen holes, be
punched in its disk, and the instrument set uniformly ro-
tating. If we now blow alternately against the 8-hole
row and the 16-hole row we shall find that the sound pro-
duced at the latter is precisely one octave higher in pitch
than the sound produced at the former. If we increase
or diminish the velocity of rotation, both sounds will, of
course, rise or fall proportionately, but the interval be-
tween them will remain unaffected, and equal, as before,
to an exact octave. The number of air-discharges corre-
sponding to the more acute sound is, in this case, evident-
ly twice as large, in any given time, as the number, dur-
ing the same time, for the graver sound. Accordingly,
we have the following result:
When two sounds differ by a single octave, the higher
sound makes exactly twice as many vibrations in any as-
signed time as the lower.
Next, let a row of 12 holes be punched in the disk
of the sirene. Taking this row with the 8 hole row, and
proceeding as in the last instance, we find that the more
acute sound forms a fifth with the graver one. The num-
bers of discharges in any given time are here as 12 to 8
-i. e., as 3 to 2. The result, therefore, is as follows:
When two sounds differ by a fifth, the higher sound
makes exactly three vibrations during the time in which
the lower sound makes two.
If we take the 16-hole and 12-hole rows together, the
interval amounts to a fourth ; accordingly, when two
sounds differ by a fourth, the higher sound makes exact-
MEASURE OF INTERVĀLS.
55
ly four vibrations during the time in which the lower
sound makes three.
31. Tlie results just obtained may be somewhat more
concisely stated. During one second of time, the upper
of two sounds differing by an octave makes a number of
vibrations, which is to the number niade by the lower
sound as 2 to 1. For a fifth the ratio is as 3 to 2. For
a fourth it is as 4 to 3. Remembering, then, the defini-
tion of the vibration-number of an assigned sound [$ 28],
we may express our three results as follows:
When two sounds form with each other the intervals
of an octave, a fifth or a fourth, their vibration-numbers
are to each other, in the first case as 2 to 1, in the second
as 3 to 2, in the third as 4 to 3.
A ratio is most easily expressed by a fraction. Thus,
we may regard the fractions as denoting the interval of
a fifth. It may be taken as an abbreviated statement of
the fact that, when two sounds form a fifth with each
other, the more acute makes 3 vibrations while the graver
makes 2.
35. By suitable experiment, similar numerical rela-
tions to those already established may be obtained for all
the intervals already considered. A fraction can thus be
determined for each interval, in the manner exemplified
in the case of the fifth. We will call this fraction the
vibration-fraction of the interval in question. The ac-
companying table gives, in the second colunin, the vibra-
tion-fractions corresponding to the intervals named in the
first; and, in the third, describes the consonant or disso-
nant character of the intervals.
It is noticeable that the dissonant intervals involve
higher numbers in their vibration-fractions than the con-
sonant intervals do; the latter, with the solitary excep-
tion of the minor sixth, having nothing beyond 6, while
the former bring in 9, 15, and 16.
56
LOUDNESS AND PITCH.
Name of Interval.
Vibration-fraction.
Character of Interval.
/
Unison
concord
Second
Oko
discord
Minor Third
ajc
concord
Major Third
5.
concord
Fourth
寺
​concord
to
concord
4 Fifth
6 Minor Sixth
Cybo
concord
Major Sixth
cipo
concord
Minor Seventh
16
discord
Major Seventh
15
discord
Octave
2 옥
​concord
36. By the help of the last table, we can calculate
the vibration-numbers of all the notes within a single
ostave which belong to the major or minor keys as soon
as the vibration-number of the tonic is given. For in-
stance, let middle of the piano-forte (vib.-no. 264) be
the tonic. From the second line of the table, we see
that the vibration-number of D must be to 264 in the
9
ratio of 9 to 8. It must therefore be equal to
8
For Eb, by exactly similar reasoning, we obtain
6.
5
x 264 or 3164 ; for E, x 264, or 330.
5
4
The student should work out the remaining cases for
himself.
The complete results of the major scale are as fol-
lows:
x 264,
or 297.
CALCULATION OF VIBRATION-NUMBERS.
57
6
F
264
297
330
352
396
440
495
528
In order to extend the scale another octave upward,
we have only to multiply each vibration-number by 2.
A second multiplication by 2 will raise it by another
octave, and so on. Conversely, in order to pass to the
octave below, we divide each vibration-number by 2. To
descend a second octave we repeat the operation, and
so on.
Thus the pitch of the tonic absolutely fixes the pitch
of every note, in the scale of which it is the starting-point.
Before we proceed to investigate the mechanical equiv-
alent of the third element [$ 24] of a musical sound, its
quality, it will be convenient briefly to examine a subject
possessing an important bearing on that inquiry. This
we shall do in the next chapter.
CHAPTER III.
.
ON RESONANCE.
66
37. WHEN a sounding body causes another body to
emit sound, we have an instance of a very remarkable
phenomenon called resonance. The German terin for it,
co-vibration" (Mitschwingung), possesses the merit of
at once indicating its essential meaning, namely, the set-
ting up of vibrations in an instrument, not by a blow or
other immediate action upon it, but indirectly as the
result of the vibrations of another instrument. In order
to produce the effect, we have only to press down very
gently one of the keys of a piano-forte, so as to raise the
damper, without making any sound, and then sing loud-
ly, into the instrument, the corresponding note. * When
the voice ceases, the instrument will continue to sustain
the note, which will then gradually fade away. If the
key is allowed to rise again before the sound is extinct, it
will abruptly cease. A similar experiment may be tried,
as follows, on any horizontal piano-forte which allows the
wires to be uncovered: Each note is, it is well known,
produced by two or by three wires. Having, as in the
· previous case, raised one of the dampers without striking
the note, twitch one of the corresponding wires sharply
with the finger nail, and then wait a few seconds. The
vibrations will, in this interval, have communicated
EXPERIMENTS WITE TUNING-FORKS.
59
themselves to the other string, or strings, belonging to
the note pressed down: if, now, the first wire be stopped
by applying the tip of the finger to the point where it
was at first twitched, the same note, produced by these
transmitted vibrations, will continue to be sustained by
the remaining wire or wires.
A more instructive method of studying resonance is
to take two unison tuning-forks, strike one of them, and
hold it near the other, but without touching it. The sec-
ond fork will then commence sounding by resonance, and
will continue to produce its note though the first fork be
brought to silence. It is essential to the snccess of this
experiment that the two forks should be rigorously in
unison. If the pitch of one of them be lowered by caus-
ing a small pellet of wax to adhere to the end of one of
its
prongs,
the effect of resonance will no longer be pro-
duced, even thouglı the alteration of pitch be too small
to be recognized by the ear. Further, the phenomenon
requires a certain appreciable length of time to develop
itself; for, if the silent fork be only momentarily exposed
to the influence of its vocal fellow, no result ensues. The
resonance, when produced, is at first extremely feeble,
and gradually increases in intensity under the continued
action of the originally-excited fork. Some seconds must
elapse before the maximum-resonance is attained. The
conditions of our experiment show, directly, that the
resonance of the second fork was due to the transmission,
by the air, of the vibrations of the first, the successive
air-impulses falling in such a manner on the fork as to
produce a cumulative effect. If we bear in mind the dis-
proportionate mass of the body set in motion compared
to that of the air acting upon it-steel being inore than
six thousand times as heavy as atmospheric air, for equal
bulks-we cannot fail to regard this as a very surpris-
ing fact.
60
ON RESONANCE.
Let us examine the mechanical causes to which it is
due. Suppose a heavy weight to be suspended from a
fixed support by a flexible string, so as to form a pendu-
lum of the simplest kind. In order to cause it to perform
oscillations of considerable extent by the application of a
number of small impulses, we proceed as follows: As
soon as, by the first impulse, the weight has been set
vibrating through a small distance, we take care that
every succeeding impulse is impressed in the direction in
which the weight is moving at the time. Each impulse,
thus applied, will cause the pendulum to oscillate through
a larger angle than before, and, the effects of many im-
pulses being in this way added together, an extensive
swing of the pendulum is the result.
When the distance through which the weight travels
to and fro, though in itself considerable, is small com-
pared to the length of the supporting string, the time of
oscillation is the same for any extent of swing within this
limit, and depends only on the length of the string.
readers will find this important principle illustrated in
any manual of elementary mechanics, and I must ask
them to take it for granted here. For the sake of simpli-
city, let us suppose that we are dealing with a second's
pendulum, i. e., one of such a length as to perform one
complete oscillation in each second, and therefore to
make a single forward or backward swing in each half-
second. It will be clear, from what has been said above,
that the most rapid effect will be produced on the motion
of the pendulum, by applying a forward and a backward
impulse respectively during each alternate half-second,
or, which is the same thing, administering a pair of to-
and-fro impulses during each complete oscillation of the
pendulum. We have a simple instance of such a proceed-
ing in the way in which a couple of boys set a heavily-
laden swing in violent motion. They stand facing each
My
AGENCY OF THE AIR.
61
other, and .each boy, when the swing is moving away
from him, helps it along with a vigorous push.
38. The above considerations enable us to explain how
a sounding-fork excites the vibrations of another fork in
unison with itself, through the medium of the intervening
air. When a continuous musical note is being sounded,
we know that, at any one point we choose to fix upon, the
air is undergoing a series of rapid changes, becoming alter-
nately denser, and less dense, than it would be were the
sound to cease.
The increase of density is accompanied
by an increase of pressure ; its diminution by a diminu-
tion of pressure [S 20]
.
FIG. 21.
e c d
U V
А
В
Let A, Fig. 21, be the sounding-fork, B that whose
vibrations are to be excited by resonance, and let us con-
sider the effect of the alternations of pressure on the air
at с on the prong bc. The increase of pressure will tend
to move the prong into the position bd, its subsequent
diminution will facilitate the elastic recoil of the fork,
supported also by the superior density of the air on the
other side of the prong, and thus tend to bring the prong
into the position be, farther to the left of its original posi-
tion, bc, than bd was to the right it. Thus the alter-
nate condensations and rarefactioñis of the sound-waves
impress on the fork B correspondins impulses in opposite
directions. One pair of such impulses is applied regu-
larly during each completo vibrationtok B, since they are
due to the vibrations of A; which is unison with B.
62
ON RESONANCE.
Further, for the small extent of vibration with which we
have here to deal, the prongs of a tuning-fork move ex-
actly according to the same law as a pendulum.' Ac-
cordingly, these air-impulses are applied under precisely
the conditions which we found to be inost favorable to
the rapid development of vibratory motion. The large
number of such impulses which succeed each other in
few seconds make up for the feebleness of each by itself.
It is in accordance with this that resonance is produced
more slowly between unison-forks of low, than between
those of high, pitch. I find that, with two making 256
vibrations per second, about one second is requisite to
bring out an audible resonance; while, with another pair,
making 1,920 vibrations per second, I am not able to
dainp the first fork sufficiently soon after striking it, to
prevent tle other from making itself heard.
39. A column of air is easily set in resonant vibration
by a note of suitable pitch. The roughest experiment
suffices to establish this fact. We have only to roll up a
piece of paper so as to make a little cylinder six inches
long and an inch or two in diameter, with both ends open,
€ 므
​and to hold a common C tuning-fork
close
to one of the apertures, after striking it briskly. As soon
as the fork reaches the position (1), Fig. 22, its tone will
unmistakably swell out. In order to estimate the increase
of intensity produced, it is a good plan to move the fork
rapidly to and fro a few times between the positions (1)
and (2).
In the first case we have the full effect of resonance,
in the second only the unassisted tone of the fork, and
the contrast is very marked. We may shorten or length-
en our cylinder, within certain limits, and still obtain
1 This will be proved in s 70.
RESONANT AIR-COLUMNS.
63
the phenomena of resonance, but the greatest reënforce-
inent of tone we can obtain with the fork selected will
be produced by an air-column about six inches long.
Fig. 22.
m
(1)
(2)
vit
If we close oue end of the paper cylinder, by placing
it, for instance, on a table, and repeat our experiment at
tlie open end, only a very weak resonance is produced ;
but we obtain a powerful resonance by operating with a
fork
making half as many vibrations per
second as that before employed. In this case, then, a col-
umn of air contained in a cylinder, of which one end was
closed, resounded powerfully to a note one octave below
that which elicited its most vigorous resonance when con-
tained in a cylinder open at both ends.
By operating in this fashion, with, forks of different
pitch, on air-columns of different lengths, we arrive at the
following laws, which are universally true:
1. For every single musical note there is a correspond-
ing air-column of definite length wlich resounds the most
powerfully to that note.
2. The maximum resonance of air in a closed pipe is
produced by a note one octave below that to which an
64
ON RESONANCE.
open pipe of the same length resounds the most power-
fully.
40. In order to ascertain the precise relation between
the pitch of a note and the length of the corresponding
air-column, we will examine the way in which resonance is
produced in a column of air contained in a pipe closed at
one end.
Let A, Fig. 23, be the open, and B the closed, ends of
the pipe, and let us for a moment replace the contained
air by an elastic spiral spring, fastened at B, and of
length equal to AB.
FIG. 28.
8 0 000 000 0 Ꭷ Ꭷ Ꭷ Ꭷ Ꭷ Ꭷ Ꭷ Ꭷ Ꭷ Ꭷ Ꭷ Ꭷ Ꭷ Ꮫ
B
А
Suppose the end of the spring suddenly pushed a little
way from A toward B. The coils of the spring nearest
1 will be squeezed together, and this condensed state of
the spring will travel along it until it reaches B. The
end of the pipe will there cause the condensation to re-
bound, and travel back again to A. If let alone, the
end of the spring would now protrude slightly beyond
tlie open end of the tube, the coils near A would be drawn
somewhat apart, and a rarefaction would in consequence
pass along AB, and, after reflexion at B, return to A,
where it would meet the end of the spring just contract-
ing to its original length. The elasticity of the spring
would thus cause it to lengthen and shorten as a whole,
in consequence of the single push originally given it, and
tliis motion would for a time continue, its successive pe
riods being four times the space of time occupied by a
pulse of condensation or rarefaction in traversing the
length of the tube. The free end of the wire may, how-
ever, be pulled and pushed, alternately, so as to reënforce
RESONANT AIR-COLUMNS.
65
each pulse as it arrives at the open end of the tube, and
in this manner the maximum of motion will be commu-
nicated to the spring. In this case, one outward, and one
inward, impulse of the hand must be communicated to
the free end of the spring, during the time which elapses
while a pulse traverses four times the length of the tube.
Reverting to the actual conditions of our problem, we
have the resonance of the air-column in place of the al-
ternate lengthening and shortening of the spring. For
the to-and-fro motion of the hand at A, we must substi-
tute that of the prong of the vibrating fork. The sound-
pulse traverses four times the length of the tube while
the fork is performing one complete vibration. We know,
however [8S 8 and 15], that, during this latter period, the
sound-pulse due to the fork's action traverses precisely
one wave-length corresponding to the pitch of the note
produced by the fork. Hence, for maximum resonance
in the case of a closed pipe, the wave-length correspond-
ing to the note sounded must be four times as great as the
length of the air-column, or the length of the column one-
quarter of the wave-length.
41. These principles give us the explanation of a use-
ful appliance for intensifying the sound of a tuuing-fork.
Such a fork, when held in the hand after being struck,
communicates but little of its vibration to the surround-
ing air ; when, however, its handle is screwed into one
side of an empty wooden box of suitable dimensions, in
the way shown in Fig. 24, the tone becomes much louder,
The vibrations of the fork pass from its handle to the
wood of the box, and thence to the air-column within,
which is of appropriate length for maximum resonance to
the fork's note. This convenient adjunct to a tuning-fork
goes by the name of a “resonance-box.”
42. When a munber of musical sounds are going on
at once, it is generally difficult, and often impossible, for
66
ON RESONANCE.
the unaided ear to decide whether an individual note is
or is not present in the whole mass of sound heard. If,
however, we had an instrument which intensified the tone
of the note of which we were in search, without similarly
FIG. 24,
reënforcing others which there was any risk of our mis-
taking for it, our power of recognizing the note in ques-
tion would be proportionately increased. Such an instru-
ment has been invented by Helmholtz. It consists of a
hollow ball of brass with two apertures at opposite ends
of a dianieter, as shown in Fig. 25. The larger aperture
FIG. 25.
M
allows the vibrations of the external air to be communi-
cated to that within the ball; the smaller aperture passes
through a nipple of convenient form for insertion in the
ear of the observer. The air contained in the ball re-
ACTION OF RESONATORS.
67
sounds very powerfully to one single note of definite
pitch, whence the instrument has been nained by its in-
ventor a resonator. The best way of using it is, first to
stop one ear closely, and then to insert the nipple of the
instrument in the other : as often as the resonator's own
note is sounded in the external air, the instrument will
sing it into the ear of the observer with extraordinary
emphasis, and thus at once single out that note from
among a crowd of others differing from it in pitch. A
series of such resonators, tuned to particular previously-
selected notes, constitutes an apparatus for analyzing a
composite sound into the simple tones of which it is made
up.
CHAPTER IV.
ON QUALITY.
43. THE laws of resonance enable us to establish a
remarkable, and by most persons utterly unsuspected,
fact, viz., that the notes of nearly every regular musical
instrument with which we are familiar, are not, as they
are ordinarily taken to be, single tones of one determinate
pitch, but composite sounds containing an assemblage of
such tones. These are always members of a regular
series, forming fixed intervals with each other, which
may be thus stated : If we number the separate single
tones, of which any given sound is made up, 1, 2, 3, etc.,
beginning with the lowest, and ascending in pitch, we
have
(1) The deepest or fundamental tone, which is com-
monly treated as determining the pitch of the
whole sound.
(2) A tone one octave above (1).
(3) A tone a Fifth above (2)—i. e., a Twelfth above
(1).
() A tone a Fourth above (3)—i. e., two octaves
above (1)
(5) A tone a Major Third above (4)-i. e., two octaves
and a Major Third above (1).
(6) A tone a Minor Third above (5)—i. e., two octaves
and a Fifth above (1). .
ELEMENTS OF COMPOSITE SOUNDS.
69
These are the most important members of the series.
Their vibration-numbers are connected by a simple law,
which is easily deduced from the above relations. If the
fundamental tone makes 100 vibrations per second, (2)
will make twice as many-i. e., 200; (3), being a Fifth
3
above (2), will have for its vibration-number, x 200, or
2
300. For (4), which is a Fourth above (3), we get simi-
6
larly 300, or 400 ; for (5), Ø x 400, or 500 ; for (6),
5
x 500, or 600. Thus the numbers come out 100, 200,
300, and so on, or, generally, whatever be the vibration-
number of (1), those of (2), (3), (4), etc., are respectively
twice, three times, four times, etc., as great. Subjoined,
in musical notation, is the series of tones complete up
the tenth, taking C in the bass clef as our fundamental
tone, though any other would do equally well.
be 2 2
*
19
1
2
3 4 5 6 y 8 9 10
The asterisk denotes that the pitch of the 7th tone is
not precisely that of the note by which it is here repre-
sented. It is, in fact, slightly less acute.
The reader must not suppose that, because the tones
into which a note of a musical instrument may usually be
decomposed are members of a fixed series, all those which
we have written down are necessarily present in every
such note. All that is meant to be asserted is, that those
which are present, be they few or many, must occupy po-
sitions determined by the law connecting each tone with
its fundamental. The sound may contain, say, (1), (3),
and (5) only, or (1), (4), and (8) only, and so on, the rest
being entirely absent, but in 120 case can a tone interme-
70
ON QUALITY.
diate in pitch between any two consecutive members of the
series make its appearance.
44. Experimental evidence shall now be produced in
support of the extremely iinportant proposition just enun-
ciated.
We will begin with the sounds of the piano-forte. Let
the note 9
be first silently pressed down, and then
gi be vigorously struck, and, after three or four
seconds, allowed to rise again. The lower note is at once
extinguished, but, we now hear its octave sounding with
considerable force from the wires of 9
mit the damper to fall back on these, by releasing the
note hitherto held down, the whole sound is immediately
cut off.
If we per-
Next, retaining the same fundamental note,
9
let
G
be quietly freed from its damper, and the
experiment repeated as before. We shall then hear this
note sounding on after the extinction of 3 Simi-
lar results may be obtained with the three next tones,
2
but they drop off very rapidly in inten-
sity. The tones above
are so weak as to be
practically insensible. Tlie series of tones produced in
this succession of trials can only be due to resonance.
But, as has been already shown, the vibrations of any
instrument are excited, by resonance, only when vibra-
tions of the same period are already present in the sur-
ANALYSIS BY RESONANCE.
71
*
rounding air. Accordingly, the only sound directly
originated in each variation of our experiment, viz., that
of the note 5, must have contained all the tones
successively heard. The reader should apply the method
of proof here adopted to notes in various regions of the
key-board. He will find considerable differences, even
between consecutive notes, in the number and relative
intensities of the separate tones into which he is thus
able to resolve them. The higher the pitch of the funda-
mental tone, the fewer will the recognizable associated
tones become, until, in the region above
the
>
notes are themselves approximately single tones. The
causes of these differences will be explained, in detail, in
a subsequent chapter; it is sufficient here to indicate
their existence. The result arrived at, thus far, is that
the sounds of the piano-forte are, in general, composite, the
number of constituent tones into which they are l'esolv-
able being largest in the lower half of the instrument,
and diminishing in its upper half, until, at last, no analy-
sis is called for.
45. The above resolution lias been effected by means
of the principle of resonance. It can, however, be per-
formed by the ear directly, though only to a sinall ex-
tent, and with less ease. In endeavoring to hear a par-
ticular constituent tone among the assemblage forming
à compound sound, the best plan is, first, to let the upper
tone be heard by itself a few times, so as to prepare the
ear for the precise degree of pitch it is to expect, and
then to develop the compound sound. If, ineanwhile,
the observer has succeeded in keeping his attention un-
swervingly fixed on the tone for which lie is listening, he
will hear it come out clearly from the mass of tones in-
72
ON QUALITY.
cluded in the composite sound. If the piano-forte note,
, be thus examined, the octave,
and
Twelfth,
Twelfth, the
can generally be recognized with con-
siderable ease; the second octave,
with a little
trouble; the next three tones of the series on page 69
with increasing difficulty, and those which succeed them
not at all. The reader approaching this phenomenon for
the first time must not be disappointed if, in trying this
experiment, he fail to hear the tones he is told to expect.
He should vary its conditions by changing the note
struck, in such a way that his attention will not be liable
to be diverted by the presence of distinct tones more
acute than than that of which he is search. Thus, a note
near
may be advantageously chosen to observe
the first octave,
2
; one near J3
to observe
the Twelfth,
one near
9
to observe thie
second octare, 9: He may, however, altogether
fail in performing the analysis with the unassisted ear.
This by no means indicates any aural defect, as he may
at first be inclined to imagine. It rather shows that the
life-long habit of regarding the notes of individual sound-
producing instruments aş single tones cannot be un-
learned all at once. The case is analogous to that of single
vision with two eyes, where two distinct and different
images are so blended together as to appear, to all ordi-
CLANG, OR PARTIAL-TONES.
73
nary observation, as one. The acoustical observer who
is thus situated must rely on the analysis by resovance,
and on the evidence of those who are able to perform the
direct analysis. As he pursues the subject further experi-
mentally, his analytical faculty will no doubt in time ade-
quately develop itself.
46. The composite character of musical sounds, which
we have recognized in the case of the piano-forte, and
shall have ample opportunity of verifying more generally
in the sequel, requires the iutroduction, here, of certain
verbal detinitions and limitations. The phraseology hith-
erto employed, both in the science of acoustics and in the
theory of music, goes on the supposition that the sounds
of individual instruments are single toiles, and therefore,
of course, contains no single term specially denoting com-
pound sounds and their constituents. “Sound,” "note,"
and “tone,” are used as nearly synonymous. It will be
convenient to restrict the meaning of the latter so that
it shall denote a sound which does not admit of resolu-
tion into simple elements. A single sound of determi-
nate pitch, we shall, accordingly, in what follows, call a
tone, or simple tone. For a compound sound the word
clang will be a serviceable term. The series of element-
ary sounds into which a clang can be resolved we shall
call its partial-tones, sometimes distinguishing, among
these, the lowest, or fundamental tone, from the others,
or overtones of the clang. This nomenclature is a direct
adaptation of the German terms employed by Helmholtz.
Its introduction is due to Prof. Tyndall.
47. This long discussion has paved the way for the
complete explanation of musical quality which is con-
tained in the following proposition: The quality of a
clang depends on the number, orders, and relative in-
tensities, of the partial-tones into which it can be re-
solved. We have here three different causes to which
4
74
ON QUALITY.
variations in the quality of composite sounds are as-
signed.
1. A clang may contain only two or three, or it may
coutain half a dozen, or even as many as fifteen or twenty,
well-developed partial-tones.
2. The number of partial-tones present remaining the
same, the quality will vary according to the positions
they occupy in the fixed series on page 69, i. e., on their
orders. Thus, a clang containing three tones may con-
sist of (1), (2), (3), or of (1), (3), (5), or of (1), (7), (10), and
so on, the quality varying in each instance.
3. The number and orders of the partial-tones present
remaining the same, the quality will vary according to
the relative degrees of loudness with which those tones
speak. Thus, in the simplest case of a clang consisting
of (1) and (2), (2) may be twice as loud, or as loud, or
half as loud, as (1), and so on.
It is clear that these three classes of variations are en-
tirely independent of each other, that is to say, any two
clangs may differ in the number, orders, and relative in-
tensities, of their constituent partial-tones. The variety of
quality thus provided for is almost indefinitely great. In
order to form some idea of its extent, let us see how many
clangs of different quality, but of the same pitch, can be
formed with the first six partial-tones, by variations of
number and order only. We will indicate each group of
tones by the corresponding figures inclosed in a bracket;
thus, e. g., (1, 3, 5) represents a clang consisting of the
first, third, and fifth tones.
All the possible groups, each necessarily containing
the same fundamental tone, are given in the following
enumeration :
Two at a time:
(1, 2), (1, 3), (1, 4), (1, 5), (1, 6).
Total 5.
THEORY OF QUALITY.
75
Three at a time:
(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 3, 4),
(1, 3, 5), (1, 3, 6), (1, 4, 5), (1, 4, 6), (1, 5, 6).
Total 10.
Four at a time:
(1, 2, 3, 4), (1, 2, 3, 5), (1, 2, 3, 6),
(1, 2, 4, 5), (1, 2, 4, 6), (1, 2, 5, 6),
(1, 3, 4, 5), (1, 3, 4, 6), (1, 3, 5, 6), (1, 4, 5, 6).
Total 10.
Five at a time :
(1, 2, 3, 4, 5), (1, 2, 3, 4, 6), (1, 2, 3, 5, 6),
(1, 2, 4, 5, 6), (1, 3, 4, 5, 6).
Total 5.
Six at a time: (1, 2, 3, 4, 5, 6).
Total 1.
The whole number of groups is 31, or, if we allow the
fundamental tone (1) to count by itself as a sound of sep-
arate quality, 32. Let us next examine how many clangs
of different quality can be obtained from a single com-
bination of three fixed partial-tones by variations of in-
tensity only, supposing that each tone is capable of but
two degrees of loudness. Representing one of these hyf;
and the other hy p, we indicate, e. g., by (f, P, p) a clang
in which the fundamental tone is sounded forte, and the
two overtones piano. The different cases which present
themselves are the following:
(fif,f), (f, P,f), (P,$,f), (P, P, f), (f, f, p),
(f, P, P), (2,5, p), (P, P, P),
or seven in all, since (P, P, p) has the same quality as
(f. ff). The number of cases increases very rapidly as
we take more partial-tones together. Thus a clang of
four tones will produce 15 sounds of different quality;
one of five tones 31; one of six tones 63, by variations of
76
ON QUALITY.
intensity only. Altogether we could form, with six par-
tial-tones, each susceptible of only two different degrees
of intensity, upward of four hundred clangs of distinct
quality, all having the same fundamental tone. The sup.
position above made utterly understates, however, the
varieties of quality dependent only on changes of relative
intensity. A very slight increase, or diminution, of loud-
ness, on the part of a single constituent tone, is enough
to produce a sensible change of quality in the clang. We
should be still far below the mark if we allowed each
partial tone four different (legrees of intensity, though
even this supposition would bring us more than eight
thousand separate cases.
Since many more variatiors of
intensity are practically efficacious, and also since the
number of disposable partial-tones need by no means be
limited, as has here been done, to the first six, the above
calculation will probably suffice to convince the reader
that the varieties of quality which the theory we are en-
gaged upon is capable of accounting for, are alınost in-
definitely numerous. This is, in fact, no more than we
have a right to expect from the theory, when we reflect
on the fine shades of quality which the ear is able to
distinguish. No two instruments of the same class are
exactly alike in this respect. For instance, grand piano-
fortes by Broadwood and by Erard exhibit unmistakable
differences, which we describe as “Broadwood tone” and
“Erard tone.” Less marked, but still perfectly recogniz-
able, differences exist between individual instruments of
the same class and maker, and even between consecutive
notes of the same instrument. To these we have to add
the variations in quality due to the manner in which the
performer handles his instrument. Even on the piano-
forte the kinds of tone elicited by a dull, slamming touch,
and by a lively, elastic one, are clearly distinguishable.
With other instruments the distinctions are much more
VARIATIONS OF INSTRUMENTS.
77
marked. On the violin we perceive endless gradations
of quality, from the rasping scrape of the beginner up to
the smooth and superb tone of a Joachim (or, as I ought
rather to say, the Joachim). A precisely similar remark
applies to wind-instruments; the differences, for exainple,
between first-rate and inferior playing on the hantboy,
bassoon, horn, or trumpet, being perfectly obvious to
every musical ear.
In the next chapter we will discuss the quality and
essential mechanism of the principal inusical instru-
ments, among which the piano-forte will receive an
amount of attention proportionate to its popularity and
general use. We begin with the elementary tones of
which all composite sounds are made up.
CHAPTER V.
ON THE ESSENTIAL MECHANISM OF THE PRINCIPAL MUSICAL
INSTRUMENTS, CONSIDERED IN REFERENCE TO QUALITY.
1. Sounds of Tuning-forks.
48. WHEN a vibrating tuning-fork is held to the ear,
we perceive, besides the proper note of the fork, a shrill,
ringing, and usually rather discordant, sound. If, how-
ever, the fork is mounted on its resonance-box, as in Fig.
24, p. 66, the fundamental tone is so much strengthened
that the other is by comparison faint, and the sound
heard may be regarded as practically a simple tone. It
is characterized by extreme mildness, without a trace of
any thing which could be called harslı or piercing. As
compared with a piano-forte note of the same pitch, the
fork-tone is wanting in richness and vivacity, and pro-
duces an impression of greater depth, so that one is at
first inclined to think the piano-forte note corresponding
to it must be an octave lower than is actually the case. It
follows immediately, from the general theory of the nature
of quality, that simple tones can differ only in pitch and
intensity. Accordingly, we find that tuning-forks of the
same pitch, mounted on resonance-boxes and set vibrat-
ing by a resined fiddle-bow, exhibit, however various their
forms and sizes, differences of loudness only. When made
to sound with equal intensity by suitable bowing, their
tones are absolutely undistinguishable from each other.
SOUNDS OF VIBRATING STRINGS.
79
2. Sounds of Vibrating Strings.
49. Sounding strings vibrate. so rapidly that their
movements cannot be followed directly by the eye. It
will be well, therefore, that we should examine how the
slower and more easily-controllable vibrations of non-
sounding strings are performed, before treating the
proper subject of this section. Take a flexible caout-
chouc tube, ten or fifteen feet long, and fasten its ends to
two fixed objects, so that the tube is loosely stretched
between them. The tube can be set in regular vibration
by impressing a swaying moveinent upon it with the
fingers near one extremity, in suitable time. According
to the rapidity of the motion thus communicated, the
tube will take up different forms of vibration. The sim-
plest of these is shown in Fig. 26. A and B being its
tixed extremities, the tube vibrates as a whole, between
the two extreme positions AaB and AbB.
FIG. 26.
а.
А.
B
7
The tube may also vibrate in the form shown in Fig.
27, where AabB and AcdB are its extreme positions.
FIG. 27.
a
.d.
А.
B
C
с
ь
In this instance the middle point of the tube, C, re-
mains at rest, the loops on either side of it moving inde-
pendently, as though the tube were fastened at C, as well
80
MUSICAL INSTRUMENTS.
as at A and B. For this reason the point C is called a
node, from the Latin ngdus, a knot.
Fig. 28 shows a form of vibration with two nodes, at
TIG. 28.
B
A
D
C and D, dividing the distance AB into three equal
parts. The portions of the tube AC, CD, DB, vibrate
independently of each other, forming what are called
ventral segments. We may also obtain forms with three,
four, five, etc., nodes, dividing the tube into four, five,
six, etc., equal ventral segments, respectively. The stiff-
ness of very short portions of the tube alone imposes a
limit on the subdividing process. Let us examine the
mechanical causes to which these effects are due.
Fig. 29.
BO
A
50. If we unfasten one end of the tube, and, holding
it in the hand as in Fig. 29, raise a hump upon it, by
moving the hand suddenly through a small distance, the
lump will run along the tube until it reaches its fixed
extremity, B; it will then be reflected and run back to
FIG. 30.
a
А.
B
A
B В
7
A, where it will undergo a second reflection, and so on.
At each reflection the hump will have its convexity re-
OPPOSITE EQUAL PULSES.
81
versed. Thus, if, while traveling from A toward B, its
form was that of a, Fig. 30, on its return it will have the
form 6. After reflection at A, it will resume its first
form c, and so on. Now, instead of a single jerk, let the
hand holding the free end execute a series of equal con-
tinuous vibrations. Each complete vibration will cause
a wave ab, Fig. 31, consisting of crest b and trough a, to
FIG. 31.
7
B
A
a
pass along the tube from A to B, where reflection will
turn crest into trough and trough into crest; su that the
wave will return from B to A stern foremost. Next, let
the tube be again fastened at both ends, as before, and
the vibrations of the hand impressed at some iutermedi-
ate point, as C, Fig. 32.
FIG. 32.
D
D
с
A
B
맙
​We will suppose
Two sets of waves will now start from C in the direc-
tions of the arrows. They will be reflected at A and B,
and then their effects intermingled.
that the tube has been set in steady motion, and, the
hand being removed, continues its vibrations without any
external force acting on it. Two sets of equal waves are
now moving with equal velocities from A toward B and
from B toward A, and we have to determine their joint
effect in fixing the form of vibration in which the tube
swings.
Suppose that a crest a, Fig. 33, moving from A tow-
ard B, meets an equal trough b, moving from B torrard
82
MUSICAL INSTRUMENTS:
A, at the point c. The point c is now solicited by a and
6 in opposite directions and with equal energy, and there-
fore remains at rest. The two opposite pulses then pro-
ceed to cross each other, but, as a moves to the right and
b to the left with equal speed, there is nothing to give
FIG. 33.
a
с
B
A-
Z 2
either of them an influence upon the point c, where they
first met, superior to that exercised in the contrary direc-
tion by the other. Thus, c remains at rest under their
joint influence, and a node is therefore formed at that
point. If a trough had been moving froin A toward B,
and an equal crest from B toward A, the effect would
clearly have been the same.
A node must therefore be formed at every point where
two equal and opposite pulses, a crest, and a trough, meet
each other.
51. The annexed figure represents two series of equal
waves advancing in opposite directions with equal veloci-
ties. The moment chosen is that at which crest coin-
cides with crest, and trough with trough. The joint ef-
fect thus produced does not appear in the figure, our ob-
ject at present being merely to determine the number
and positions of the resulting nodes. For the sake of
clearness, one set of waves is represented slightly below
the other, though, in fact, the two are strictly coincident.
1
Fig. 34.
ъ
Ն
B
А.
re
đ
Za
B
m P 관
​TU
n'
N
Let the waves abdf.... be moving from left to right,
the waves z't's'n'...a' from right to left. The crest klm
NODES AND PULSÉS.
83
meets the trough pn'm at m. After these have crossed
each other, the trough ghk and the crest ro'p will also
meet at m, since km and pm are equal distances. Simi-
larly, the crest efg. and the trouglı ts'r will meet at m.
Accordingly, the point m is a node, and, by exactly the
same reasoning, so are a, c, e, g, k, p, r, t, etc. The dis-
tances between pairs of consecutive nodes are all equal,
each being a single pulse-length-i. e., half a wave-length
of either series.
Two pulse-lengths, as gk and km, give three nodes,
g, k, and m; three pulse-lengthis four nodes, and so on.
There is thus always one node in excess of the number
of pulses. On the other hand, the fixed ends of the tube,
which are the origins of the systems of reflected waves,
occupy two of these nodes. Deducting them, we arrive
at this result:
The number of nodes is one less than the number of
the pulse-lengths (or half wave-lengths), which together
make up the length of the vibrating tube.
52. We will now ascertain low the portions of the
tube between consecutive nodes move under the action of
the two systems of waves passing along it. Let AB,
Fig. 35, be the fixed ends, as before, and let us take five
nodes at the points 1, 2, 3, 4, 5. In (1) the systems of
waves coincide, accordingly each point of the tube is dis-
placed through twice as great a distance as if it liad been
acted on by only one system. The tube thus takes the
form indicated by the strong line in the figure. In (2)
one set of waves has moved half a pulse-length to the
right, and the other the same distance to the left. The
two systems are now in complete opposition at every
point, and the tube is, therefore, momentarily in its un-
disturbed position. In (3) each system has moved through
a pulse-length, and the combined effect is again produced
on the tube, but in the opposite direction to that of (1).
84
MUSICAL INSTRUMENTS.
In (4), where the systems have moved through a pulse-
length and a half, the tube passes again through its un-
disturbed position, and in (5) regains the position it occu-
pied in (1), the systems of waves, meanwhile, having
FIG. 35.
А.
B (1)
2
3
4
5
А
21
B (2)
21
115
А
B (3)
2
5
A
B (4)
11
12
13
14
15
А
13 (5)
each traversed two pulse-lengths, or one wave length.
Thus, the tube exechtes one complete vibration in the
time occupied by a pulse in passing along a length of the
tube equal to twice one of its own ventral segments. In
other words, the tube's rate of vibration varies as the num-
ber of segments into which it is divideil. It noves most
slowly in the form shown in Fig. 26 with but' a single
segment; twice as fast in that of Fig. 27, when divided
into two segments; three tinies as fast with three seg-
ments, and so on. It is easy to confirm this by direct ex-
periment, the swaying movement of the hand on the tube
SEGMENTAL VIBRATIONS.
85
needing to be twice as rapid for a forin of vibration with
two segments as for a form with one, and so on.
53. Instead of comparing the different rates at which
the same tube vibrates, when divided into different num-
bers of ventral segments, we may compare the rates of vi-
bration of tubes of different lengths, divided into the same
number of segments.
Let us take as an example the two tubes, AB, CD,
Fig. 36, each divided by three nodes into four ventral
FIG. 36.
AK
B
1'
2
3
02
UD
1
1
2
3
segments. By wliat has been already shown, the time
of vibration of either tube will be that which a pulse oc-
cupies in traversing two of its ventral segments. There-
fore, the time of vibration of AB will be to that of CD
as A2 is to C2-i. e., as one-half of AB is to one-lialf of
CD, or as AB is to CD. This reasoning is equally ap-
plicable to any other case. Accordingly we have the
general result that, when tubes of different lengtlıs are
divided into the same number of ventral segments, their
times of vibration are proportional to the lengtlıs of the
tubes, or, which comes to the same thing, their rates of
vibration inversely proportional to their lengths. The
reader should observe that it has been throughout this
discussion assumed that the material, thickness, and ten-
„sion of the tube or tubes in question were subject to no
variation whatever. Any changes in these would corre-
spondingly affect the rates of vibration produced.
54. We are now prepared to examine the motion of a
sounding string. Its ends are fastened to fixed points of
attachment and the string is excited at some intermediate
86
MUSICAL INSTRUMENTS.
point, by plucking it witli the finger, as in the barp and
guitar, by striking it with a soft hammer, as in the piano-
forte, or by stroking it with a resined bow, as in the vio-
lin and other instruments of the same class. The impulses
thus set up are reflected at the extremities of the string
(in the violin at the bridge and at the finger of the per-
former), and behave toward each other exactly as in the
case of the vibrating tube considered above. The results
thius obtained are therefore directly applicable to the case
before us. The string may vibrate in a single segment as
in Fig. 26. This is the form of slowest vibration with a
string of given length, material, and tension. According-
ly, when thus vibrating, the string produces the deepest
note of which, all other conditions remaining the same, it
is capable. The string may also vibrate in the forms,
shown in Figs. 27, 28, 35, or in forms with larger numbers
of segments. The rapidity of vibration in any one of
these forms is, as we liave seen [$ 52], proportional to the
number of segments formed, so that, with two segments,
it vibrates twice, with three, thrice, with four, four times,
as fast as in the form with one segment. It follows hence
[S 43] that the notes obtained by causing a string to
vibrate successively in forms of vibration with 1, 2, 3, 4,
5, etc., segments, are all partial-tones of one compourd
sound, the lowest being of course its fundamental-tone.
The modes of eliciting the sounds of stringed, instru-
inents described above are not capable of setting up any
one of the above forms of vibration by itself, but cause
several of them to be executed together. The result is
that each form of vibration called into existence siðgs,
as it were, its own note, without heeding what is being
done by its fellows. Accordingly, a certain number of
tones belonging to one family of partial-tones are simul-
taneously heard.
What precise members of the general series of partial-
MODE OF PERCUSSION.
87
tones [p. €9] are present, and with what relative intensi-
ties, in the sound of a string set vibrating by a blow,
depends on the position of the point at which the blow is
delivered, on the nature of the striking-object, and on the
miaterial of the string. It is clear that a node can never
be formed at the point of percussion. Therefore no par-
tial-tone requiring for its production a node in that place
can exist in the resulting sound. If, for instance, we ex-
cite the string exactly at its iniddle point, the forms of
vibration with an even number of ventral segments, all of
which have a node at the centre of the string, are ex-
cluded, and only the odd partial-tones, i. e., the 1st, 3d,
5th, and so on, are heard. In tliis manner we can always
prevent the formation of any assigned partial-tone, by
choosing a suitable point of percussion. On the other
hand, a vibration-form is in the most favorable position
for development when the middle point of one of its ren-
tral segments coincides with the point of percussion. The
more nearly it occupies this position the louder will be
the corresponding partial-tone, wliile the more it recedes
from this position toward that in which one of its nodes
falls on the point of percussion, the weaker will the par-
tial-tone become.
The form and material of the hammer, or other ob-
ject with which the string is struck, have also a great in-
fluence in modifying the quality of the sound produced.
Sharpness of edge and hardness of substance tend to de-
velop high and powerful over-tones, a rounded form and
soft elastic substance to strengthen the fundamental-tone.
The material of the string itself produces its effect chiefly
by limiting the number of partial-tones. The stiffness of
the string resists division into very short segments, and
this implies, for every string, a fixed limit beyond which
further submission becomes impossible; and where, there-
fore, the series of over-tones is cut short. Hence, very
88
MUSICAL INSTRUVENTS.
thin mobile strings are favorable, thick weighty strings
unfavorable, to the production of a large number of par-
tial-tones.
55. Having examined what determines the quality of
the sound of a vibrating string, we have next to inquire
on what its pitch depends. This term is indeed, strictly
speaking, inappropriate to a composite sound containing
a series of different tones, each having its own vibration-
number and definite position in the musical scale. If,
however, we use the phrase "pitch of a sound” as equiv-
alent to "pitch of the fundamental tone of the sound,"
we shall avoid any confusion arising from this circum-
stance. The pitch of a string-sound depends of course on
the rate at which the string is vibrating. We have seen
that, when the material thickness and tension of a string
remain the same, its rate of vibration varies inversely as
the length of the string. Accordingly, the vibration-
number of a string-sound varies inversely as the length
of the string. It follows hence that the numerical rela-
tions between the vibration-numbers of sounds forming
given intervals with each other, hold equally for the
lengths of the strings by which such sounds are produced.
To verify this by experiment we have only to stretch a
wire between two fixed points A and B, and divide it
into two segments by applying the finger to it at some
intermediate point C. If AC bears to CB any one of the
simple numerical ratios exhibited in the table on page 56,
Vull
+ B
AH
C
we obtain the corresponding interval there given by al-
ternately exciting the vibrations of the two segments at
any pair of points in AC and CB respectively. Thus, if
CB is twice as long as AC, the sound produced by the
THE VIOLIN.
.89
former will be one octave lower than that produced by
the latter. If AC is to CB in the proportion of 2 to 3,
AC's sound will be a Fifth above CB's; and similarly in
utlier cases.
It was by experiments of this kind that the
ancient Greek philosopher, Pythagoras, discovered the
existence of a connection between certain mus.cal inter-
vals and the ratios of certain small integers. He ascer-
tained that an octave was produced by a wire divided
into two parts in the proportion of 2 to 1; that a Fifth
was obtained by division in the proportion of 3 to 2, and
so forth. The relations existing between these lengths
and the vibration-numbers of the notes produced by them
were entirely unknown to Pythagoras and his contem-
poraries ; indeed, it was not until the seventeenth century
that they were discovered by Galileo.
In instruments of the violin class, the pitch of the
notes sounded. varies with the position of the finger on the
vibrating string. The length of string intercepted be-
tween the fixed bridge and the finger admits of being
altered at pleasure, and thus every shade of pitch can be
produced from such instruments. The resined bow
maintains the vibration of the string by alternately drag-
ging it out of its position of rest, letting it Ay back again,
catching it once inore, and so on. The hollow cavity of
the instrument reënforces the string-sound by resonance.
The quality of instruments of the violin class is viva-
cious and piercing. The first eight partial-tones are well
represented in their clang.
The Piano-forte.
56. In this instrument each wire is stretched between
two pegs, which are fixed into a flat plate of wood, called
the sound-board. The string is fastened to one peg, and
coiled round the other, which admits of being turned
about its own axis by means of a key of suitable con-
90
MUSICAL INSTRUMENTS.
struction. In this manner the string can be accurately
tuned, since, by tightening or loosening the wire, we
raise or lower its pitch at pleasure. In small instruments
two, in larger ones three, wires in unison with each other
usually correspond to each note of the key-board. While
the instrument is not in action, a series of small pieces of
wood, covered with list, called “dạmpers," rests upon the
wires. These are connected with the key-board in such
a manner that, when a note is pressed down, the corre-
sponding damper rises from its place, and the wires it pre-
vionsly covered remain free, until the note is allowed to
spring up again, when the damper immediately sinks
back into its original position. Each note is connected
with an elastic hammer, which deals a blow to its own set
of wires, and then springs back from them. The wires
thus set in motion continue to vibrate until either the
sound gradually dies away, or is abruptly extinguished by
the descent of the damper. The action of the two pedals
is as follows: the soft pedal shifts the key-board and as-
sociated hammers in such a way that each hammer only
acts on one of the wires corresponding to it, instead of
on its complete set of two or three wires. The sound
produced by striking a note is therefore proportionally
weakened. The loud pedal lifts all the dampers off the
wires at once. It thus not only allows the notes to con-
tinue sounding after the finger of the player has quitted
them, but places other wires than those actually struck
in a position to sound by resonance. The number of
wires thus brought into play by striking a single note of
the instrument will be easily seen to be considerable.
Suppose, first, that a simple-tone, e. g., that of a tuning-
fork, is sounding near the wires of a piano-forte with the
loud pedal down, its pitch being that of middle C,
the wires of the corresponding note will of course rcso-
EFFECT OF LOUD PEDAL.
91
ñate with it, vibrating in the simplest form with only one
ventral segment. The wires of the note Sone oc-
tave below it, are also capable of producing middle C
when they vibrate in the form with two segments. So are
those of
a Twelfth below it, when vibrating
with three segments, those of
, 9
two octaves below
it, vibrating with four segments, and so on. Proceeding
in this way we determine a series of notes on the key-
board of the piano-fore, the wires of which are able to
produce a simple tone of the pitch of middle C. They
obviously follow the same law as the harmonic over-tones
of a compourd sound with middle C for its fundamental-
tone, except that the successive intervals are reckoned
downward instead of upward. The wires of all these
notes will reinforce the tone of the tuning-fork by reso-
nance. If, now, we remove the fork, and strike middle
C on the piano-forte itself, we obtain, of course, a com-
pound sound consisting of a number of simple-toncs. To
each of these latter there corresponds a descending series
of notes on the key-board, commencing with that whose
fundamental is in unison with the simple-tone in question.
A full chord struck in the middle region of the instrument
will, in this way, command the more or less active ser-
vices of two or three times as many wires as have been
set vibrating by direct percussion. The increase of loud-
ness thus secured is not very considerable, the effect be-
ing rather a heightened richness, like that of a mass of
voices singing pianissimo. The actual intensity of the
sound so heard may be less than could be produced by a
quartet of solo singers, but it possesses a multitudinous
character which the other lacks. The sustaining power
92
MUSICAL INSTRUMENTS.
of the loud pedal renders care in its employment essen-
tial. It should, as a general rule, be held down only as
long as notes belonging to one and the same chord are
struck. Whenever a change of barmony occurs, the pe-
dal should be allowed to rise, in order that the descent
of the dampers may at once extinguish tbe preceding
chord. If this precaution is neglected, perfectly irrecon-
cilable chords become promiscuously jumbled together,
and a series of jarring discords ensues, which are nearly
as distressing to the ear as the striking of actual wrong
notes. The quality of piano-forte notes varies greatly in
different parts of the scale. In the lower and middle re-
gion it is full and rich, the first six partial-tones being
audibly present, though 4, 5, 6, are much weaker than 1,
2, 3. Toward the upper part of the instrument the high-
er partial-tones disappear, until, in the uppermost octave,
the notes are actually simple-tones, which accounts for
their tame and uninteresting character. The piano-forte
shares with all instruments of fixed sounds certain seri-
ous defects, which will be discussed in detail in a subse-
quent chapter.
When a vibrating wire is passing through its undis-
turbed position, its tension is necessarily somewhat less
than at any other moment, since, in order to assume the
curved segmental form, it must be a little elongated,
which involves à corresponding increase of tension.
Hence, the two pegs by which the ends of a wire are at-
tached to the sound-board are submitted to an additional
strain twice during each complete segmental vibration.
The sound-board, being purposely constructed of the most
elastic wood, yields to the rhythmic impulses acting upon
it, and is thrown into segmental vibrations like those of
the wire.
These vibrations are communicated to the air in con-
tact with the sound-board, and then transmitted farther
SOUND-BOARD OF PIANO.
93
in the ordinary way. The amount of surface which a
wire presents to the air is so small that, but for the aid
of the sound-board, its vibrations would hardly excite an
audible sound. The reader will not fail to notice that the
sound-board of the piano-forte plays the same part as the
hollow cavity of the violin, and is, in fact, a solid reso-
nator. In the harp, the framework of the instrument
serves the same purpose. We have, in this combination
of a vibration-exciting apparatus with a resonator, the
type of construction adopted in nearly all musical instru-
ments.
3. Sounds of Organ-pipes.
57. It has been shown [8 51] tliat, when two series of
equal waves, due to transverse vibrations, travel along a
stretched wire, in opposite directions, stationary nodes
are formed at equal distances along it, separated by vi-
brating segments of equal lengths. Let us now suppose
that two series of equal waves due to longitudinal vibra-
tions are traversing, in opposite directions, a column of
air contained in a tube of uniform bore. Each set of such
waves has its own associated wave-form ($ 18]. These
will behave to each other precisely in the same way as
the transverse waves of Fig. 35. We have only, there-
fore, to consider the curves drawn in that figure as con-
stituting the associated waves for the longitudinal air-vi-
brations, in order to make the conclusions of $ 52 at once
applicable to the case before us. The result is a series of
equidistant nodes, or points of permanent rest distributed
along the column of air. The intervening portions of air
vibrate longitudinally at the same rate as the correspond-
ing ventral segments of Fig. 35. We have here, as in
the case of the sounding wire, all the conditions for the
production of a musical note, of pitch corresponding to
the rapidity of vibration obtained. It only remains to
94
MUSICAL INSTRUMENTS.
show that, in the case of every organ-pipe, two sets. of
equal waves traverse in opposite directions the air-column
which it contains.
Organ-pipes are of two kinds, called respectively
“stopped ” and open ”-epithets which, however, ap-
ply only to one end of tlie pipe; the other is in both
kinds open.
To begin with the first variety.
FIG. 37.
c
B
D
58. Let AB, Fig. 37, be the closed end of a stopped
pipe, and let a series of pulses of condensation and rare-
faction be passing along the air within it, in the direction
shown by the arrow. First, let a pulse of condensation,
CABD, have just reached AB. By supposition, the air
in AB is denser, and therefore at a higher pressure than
the air behind it. It will therefore expand. Forward
inotion being barred by AB, the expansion must take
place entirely in the opposite direction. Flence, tlie pulse
of condensation is reflected at the end of the pipe, and
proceeds to traverse its previous course in the reverse
direction. Next, suppose CABD to be a pulse of rare-
faction. The air in it is at a less pressure than that of the
air behind it. Accordingly, it will be condensed between
the pressure from behind and the resistance of the fixed
obstacle in front. The condensed pulse behind it will
expand during the process, and become itself rarefied.
Thus, a pulse of rarefaction, equally with one of conden-
sation, is reflected at the closed end of the pipe. Neither
STOPPED AND OPEN ORGAN-PIPES.
95
pulse suffers any other change except of direction of mo-
tion. Since every pulse is thus regularly reflected at AB,
and made to travel back unchanged along the pipe, it fol-
lows that a system of equal waves advancing in the direc-
tion of the arrow is necessarily met by an exactly equal
system proceeding in the opposite direction. For stopped
pipes, therefore, the point required to be proved is made
out.
Fig. 38,
с
A
B
D
Let AB, Fig. 38, be one end of an open pipe, along
which condensed and rarefied pulses are being alternately
transmitted in the direction of the arrow. First, let
CABD be a pulse of condensation which has just reached
AB. The air in it is at a higher pressure than the outer
air beyond AB, which is in the ordinary atmospheric
condition, neither condensed nor rarefied. Hence, some
of the adynced part of the pulse CABD will escape into
the
open
air. This exit will cease as soon as the air just
in front of AB has been sufficiently condensed by its
means. But, in the mean time, CABD has become rare-
fied by the escape of part of its air. Hence, a rarefaction
will travel back along the tube.
Now, let CABD have been originally a rarefied pulse.
It will be converted by the superior pressure of the air,
both in front and rear, into a condensation, and in this
condition start on its backward route.
By the above reasoning,' which the student should
1 I am indebted for this popular explanation of reflection at the open
end of a pipe to Mr. Coutts Trotter, Fellow and Tutor of Trinity College,
Cambridge.
96
MUSICAL INSTRUMENTS.
carefully compare with that of $ 21, it is clear that reflec-
tion takes place at an open as well as at a closed end of a
pipe; with the difference, however, that, in the former
case, the condensation is turned into rarefaction and rare-
faction into condensation, so that the wave returns hind
part before. We have thus established for an open pipe
what was proved for a stopped one on pages 94, 95.
59. We will now examine what forms of segmental
vibration the air in a stopped pipe can adopt. Every
such form must necessarily have a node coincident with
the closed end of the pipe, since no longitudinal vibra-
tions are possible there. The impulses constituting the
series of direct waves are not, as we shall see presently,
originated, like those of a piano-forte string, at some inter-
mediate point, but enter the pipe at its open end. This
minst, therefore, be a point of maximum vibration. Now,
a glance at Fig. 35 shows that the maxima of vibration
are at the middle points of the ventral segments. Hence
the centre of a segment must coincide with the open end
of the pipe.
The above considerations suffice to solve the problem
before us.
If the closed end of the pipe is placed at A
(Fig. 35), the open end must be midway between A and
1, or between 1 and 2, 2 and 3, 3 and 4, and so on. No
other forms of vibration are possible.
Fig. 39 shows the air in a stopped pipe of given length
vibrating in four such ways. The vertical lines indicate
the positions of the nodes. For the sake of greater clear-
ness, the loops of the associated vibration-forms are in
each case drawn in dotted lines.
In (1) we have half a segment; in (B) a segment and
a half; in (C) two segments and a half; in (D) three seg-
ments and a half. The numbers of segments into which
the length of the air-column is divided, in the four cases,
are, therefore, proportional to
VIBRATION-FORMS FOR STOPPED PIPE.
97
1, 1, 2, and 34,
i. e., to , , , and 1,
or to the whole numbers, 1, 3, 5, and 7.
Now, by $ 53 it appears that the rate of vibration in
any form varies as the number of segments into which it
is divided. The vibration-numbers of the sounds pro-
duced in the present instance are, therefore, proportional
FIG. 39.
(4)
(B)
(C)
(D)
to 1, 3, 5 and 7, i. e., we get the first four odd partial-tones
of a sound of which (A) gives us the fundamental-tone
[S 43]. The reasoning here adopted evidently applies
equally well to cases in which the air-column is subdivided
to any assigned extent. It follows, therefore, that the
notes obtainable from a stopped pipe are all odd partial-
tones belonging to one and the same clang.
60. The case of the open pipe shall next be investi-
gated. Here, as in the previous case, the end at which
the direct pulses enter must be at the centre of a ventral
segment. The considerations alleged on 1, 95 indicate
5
98
MUSICAL INSTRUMENTS.
that the same thing must also hold good at the opposite
orifice.
Referring once more to Fig. 35, we obtain all the pos-
sible modes of vibration which satisfy both the above con-
ditions by placing one end of the pipe midway between A
and 1, and the other successively half-way between 1 and
2, 2 and 3, 3 and 4, and so on. The first four of the cases
thus obtained, for a tube of constant length, are shown in
the next figure, which is drawn on precisely the same plan
as Fig. 39.
In each case, the two half segments at the ends of the
pipe make up one whole segment. The numbers of seg-
ments into which the air-column is divided are, therefore,
in (A), 1; in (B), 2; in (C), 3; in (D), 4. The same law
BIG. 40.
(A)
(B)
(C)
(D)
would obviously hold for higher subdivisions. Hence, in
the case of an open pipe, the rates of all the possible
modes of segmental vibration are as the numbers 1, 2, 3,
4, 5, etc. The notes obtainable from such a pipe are,
STOPPED AND OPEN PIPES.
99
therefore, the complete series of partial-tones belonging to
one and the same clang.
61. If the slowest forms of vibration, shown at (A) in
Figs. 39 and 40, are compared with each other, it will be
at once seen that the vibrating segment of Fig. 39 is ex-
actly twice as long as that of Fig. 40. Hence, the deepest
tone obtainable from a stopped pipe is always precisely
one octave lower than the gravest tore producible from an
open pipe of the same length. It has been shown in $ 89
that this result of theory is borne out by experiment.
62. In order to complete this investigation, it is neces-
sary to determine the pitch of the lowest note which a
pipe of given length is capable of uttering. By § 52 we
know that a complete segmental vibration is perforined
during the time occupied by a pulse in traversing twice
the length of a single segment. In (A) Fig. 39 this is
equal to four times the length of the tube. The velocity
of the pulse is here the velocity of sound in the air,
which, under ordinary conditions of temperature, etc., we
may put at 1,125 feet per second.' The vibration-num-
ber of a stopped pipe's lowest tone is therefore found by
dividing 1,125 by four times the length of the pipe ex-
pressed in feet. Conversely, the length of a stopped pipe,
which is to have as its deepest tone a note of given pitch,
is found by dividing 1,125 by four times the vibration-
number of the note to be produced. The quotient gives
the required length in feet. For example, iniddle C of
the piano-forte makes 264 vibrations per second. The
1,125
required length in this case would be expressed by
1,056
which is rather more than 1 ft. 1 in., i. e., roughly speak-
ing, one foot. An open pipe, to produce the same note,
would therefore liave to be two feet in length.
It has just been shown that the vibration-number of
1 Tyndall's “ Sound,” r. 24.
100
MUSICAL INSTRUMENTS.
SO on.
the lowest tone producible, either from a stopped or an
open pipe, varies inversely as the length of the pipe. The
length of the pipe, therefore, varies inversely as the vibra-
tion-number. Hence, the relations established in $ 55 for
strings, hold also for columns of air contained in pipes.
The case of the pipe-sounds is, however, somewhat siin-
pler than that of the string-sounds, since the pitch of the
latter depends on the tension of the strings as well as on
their lengths, whereas, in the former, pitch depends, under
given atmospheric conditions, on length alone. Hence,
we may define a note of assigned pitch by merely stating
the length of the stopped or open pipe, whose fundament-
al tone it is. . The open pipe is commonly preferred for
this purpose, and, accordingly, organ-builders call middle
O “2-foot tone; » the octave below it,“ 4-foot tone,” and
The lowest C on modern piano-fortes is “ 16-foot
tone; " that one octave lower, which is found on the very
largest organs,
"32-foot tone." The highest note of the
piano-forte, usually A, would be about“ 2-inch tone.”
63. The reader should observe that, in the course of
this discussion, we have incidentally obtained a more
complete theory of resonance than could be given in
Chapter III. When a tuning-fork is held at the orifice
of a tube, the strongest resonance will be produced if the
note of the fork coincides with the fundamental tone of
tlie tube. A decided, though less powerful, resonance
ought also to ensue if the fork-note coincides with one of
the higher tones of the tube, which, as we know, are all
over-tones of its fundamental. A resonance-box is only
a stopped pipe under another name. We may therefore
employ it to test the truth of our result, that the only
tones obtainable from a stopped pipe are the odd partial-
tones of a clang, of which the first is the fundamental
tone. I possess a series of forks giving the first seven
partial-tones of a clang. When I strike 1, 3, 5, or 7, and
CONSTRUCTION OF FLUE-PIPES.
101
hold them before the open end of the resonance-box cor-
responding to 1, a decided reënforcement of their toues is
heard. If I do the same with 2, 4, or 6, hardly any reso-
nance is produced. Thus, our theoretical result is ex-
perimentally verified.
64. Organ-pipes are divided into two classes, accord-
ing as the sounds, which they are to strengthen by reso-
pance, are originated-
(1) by blowing against a sharp edge,
(2) by blowing against an elastic tongue.
Those of the first class are called flue-pipes; those of
the second class, reed-pipes. We will consider each class
by itself.
TIG, 41,
с
3
al
65. Flue-pipes.—Here the wind is driven through a
narrow slit against a sharp edge placed exactly opposite
to it, in the manner shown in Fig. 41, which represents a
vertical section of a portion of the pipe near the end at
which the sound originates.
The air is forced by the bellows through the tube ab,
into the chamber c, and escapes through the slit d, thus
102
MUSICAL INSTRUMENTS.
impinging against the edge e, where it produces a sharp,
bissing sound which may be imitated by blowing with
the mouth against a knife-edge held in front of it. This
sound, as we shall see in the sequel, may be regarded as
consisting of a great variety of notes of different pitch.
Of these the pipe is able to reënforce, by the resonance
of its air column, such notes as coincide with its own es-
sential tones. The quality of tbe sound thus resulting
will, of course, depend on the number, orders, and rela-
tive intensities of the partial-tones present in the clang
heard. The original hissing sound contribntes nothing
directly to the whole effect, being, with well-constructed
mechanism, inaudible except close to the pipe.
Stopped wooden flue-pipes of large aperture, blown
by only a light pressure of wind, produce sounds which
are nearly simple tones; only a trace of partial-tone No.
3 being perceptible. Such tones, like the fork tones with
which they are in fact almost identical, sound sweet and
mild, but also tame and spiritless. A greater pressure of
wind develops 3 distinctly, in addition to 1, and, if it be-
comes excessive, may spoil the quality by giving the over-
tone too great an intensity compared to that of the fun-
damental, or may even extinguish the latter altogether,
and so cause the whole sound to jump up an octave and
a Fifth. This result may easily be obtained by blowing
with the mouth into a small 6-inch stopped pipe, which
can easily be obtained at any organ-factory.
Stopped pipes of narrow aperture develop 5 audibly,
as well as 1 and 3.
In the case of an open pipe the fundamental-tone is
never produced by itself. According to the dimensions
of the pipe, and the pressure of wind, it is accompanied
by from two to five over-tones. Open flue-pipes present,
therefore, various degrees of timbre which are exhibited
in the different "stops” of a large organ.
SOUNDS OF REED-PIPES.
103
66. Reed-pipes.—The apparatus by which the sounds
of pipes of this class are originated is the following: One
end of a thin narrow strip of elastic metal, called a
tongue,” is fastened to a brass plate, while the other
end is free. A rectangular aperture, very slightly larger
than the tongue, is cut through the plate, so as to allow
the tongue to oscillate into and out of the aperture, like
a door with double hinges, without touching the edges
of the aperture as it passes them. The accompanying
figure shows this piece of mechanism, which is called a
"reed," in its position of rest.
TIG. 42.
It is set in motion by a current of air being driven
against the free end of the tongue, which is thus made
to swing between limiting positions as shown in the
annexed sections.
FIG. 43.
(A)
(B)
When the tongue occupies a position intermediate
· between that of (A) and its position of equilibrium, the
air passes through the aperture in the direction indicated
by the arrows in (A). At the moment that the tongue
passes through its equilibrium position toward that shown
in (B), the current of air is barred by the accuracy with
which the tongue fits into the aperture beneath it. Only
when the tongue again emerges can the air resume its
passage. The reed thus produces a series of equal dis-
continuous impulses of air at equal intervals of time.
The principle of the instrument is identical with that of
the sirene, and it therefore gives rise to a regular inusical
104
MUSICAL INSTRUMENTS.
sound. Its note is a highly-composite clang, containing
distinctiy-recognizable partial-tones up to the 16th or
20th of the series. Thus a reed does not require to be
associated with a resonating column in order to produce
a musical sound; in fact, the instrument called the har-
monium consists of reeds without such adjuncts. The
timbre of an independent reed is, however, characterized
by too great intensity on the part of the higher partial-
tones. It is desirable to correct this defect by strength-
ening the fundamental-tone of the clang. This is done
by placing the reed in the mouth of a pipe whose deepest
tone coincides with the fundamental-tone of the reed-
clang. This tone will then be most powerfully reënforced
by resonance. The other partial-tones of the clang (the
odd ones only in the case of a stopped pipe) will also be
strengthened by resonance, but to a smaller and smaller
extent as their order rises. The force required to throw a
column of air into rapid vibration is greater than suffices
to set up a slow vibration. Hence, if two partial-tones in
the reed-clang were exactly equally intense, the lower of
them would cause a more powerful resonance than the
higher. Since the force necessary to produce segmental
vibration increases very rapidly, as the subdivisions of the
air-column become more numerous, the very high partial-
tones of the reed-clang are practically unsupported by
the resonance of the associated pipe. It will be seen
hereafter how the quality of the resulting sound is ini-
proved by this circunstance.
It is clear that sounds differing widely in quality may
be obtained by associating a reed with pipes of different
lengths and forms. If the pipe's fundamental-tone coin-
cides with that of the reed-clang, in the case of a stopped
pipe, only odd, in that of an open pipe, both odd and
even, partial-tones are strengthened by resonance. If the
fundamental-tone of the pipe coincides with one of the
ORCHESTRAL WIND-INSTRUMENTS.
105
over-tones of the reed-clang, the quality of the resulting
sound is correspondingly affected. The form of the pipe
may also be modified, so as to be conical, or of any other
slape, which will bring in other changes in its resonating
properties. In these ways we have provision for the
great variety of quality among reed-pipes, which we find
represented in organ-stops of that class.
4. Sounds of Orchestral Wind-Instruments and of the
Human Voice.
67. The flute is in principle identical with an open
flue-pipe. The lips, and a hole near the end of the tube,
play the parts of the narrow slit and opposing edge. The
quality of the instrument is sweet, but too nearly simple
to be heard during a long solo without becoming weari-
some. Its most lovely effects are produced by contrast
with the more brilliant timbre of its orchestral colleagues.
The clarionet, hautboy, and bassoon, have wooden
reeds. The clarionet has a stopped cylindrical tube, pro-
ducing only odd partial-tones, whence its characteristic
quality. The hautboy and bassoon have conical tubes.
In the horn and trumpet the lips of the performers
supply the place of the reed.
68. The apparatus of the human voice is essentially a
reed (the vocal chords), associated with a resonance-cav-
ity (the hollow of the moutlı).
The vocal chords are elastic bands, situated at the top
of the windpipe, and separated by a narrow slit, which
opens and closes again with great exactness, as air is
forced through it from the lungs. The form and width
of the slit allow of being quickly and extensively modi-
fied by the changing tension of the vocal chords, and
thus sounds widely differing from each other in pitch
may be successively produced with surprising rapidity.
106
MUSICAL INSTRUMENTS.
In this respect, the human " reed” far exceeds any that
we can artificially construct.
The size and shape of the cavity of the mouth may be
altered by opening or closing the jaws, raising or drop-
ping the tongue, and tightening or loosening the lips.
We should expect that these movements would not be
without effect on the resonance of the contained air, and
such proves, on experiment, to be the fact. If we hold a
vibrating tuning-fork close to the lips, and then modify,
successively, the resonating cavity, in the ways above de-
scribed, we shall fivd that it resounds most powerfully to
the fork selected when the parts of the mouth are in one
definite position. If we try a fork of different pitch, the
attitude of the mouth, for the strongest resonance, is no
longer the same.
Hence, when the vocal chords have originated a reed-
clang containing numerous well-developed partial-tones,
the mouth-cavity, by successively throwing itself into dif-
ferent postures, can favor by its resonance, first one par-
tial-tone, then another; at one moment this group of par-
tial-tones, at another that. In this manner endless vari-
eties of quality are rendered possible. The art of vocal-
izing consists in so placing the resonating apparatus of
the voice as to modify the clang due to the vocal chords
in the way most attractive to the ear.
The complete analysis of the sounds of the human
voice into their separate partial-tones presents peculiar
clifficulties to the unassisted ear, and can hardly be effect-
ed without the help of resonators such as those described
in § 42. By their aid we can detect in the lower notes
of a bass voiee, when vigorously sung, shrill over-tones
reaching as far as No. 16, which is four octaves above its
fundamental-tone. Under certain conditions these high
over-tones can be readily heard without recourse to reso-
nators. When a body of voices are singing fortissimo
COALESCENCE OF PARTIAL-TONES.
107
without any instrumental accompaniment, a peculiar
shrill, tremulous sound is heard, which is obviously far.
above the pitch of any note actually being sung. This
sound is, to my ear, so intensely shrill and piercing as to
be often quite painful. I have also observed it when lis-
tening to the lower notes of an unusually fine contralto
voice. The reason why these acute sounds are tremulous
will be given later.
69. We close this discussion by describing a mode of
submitting Helmholtz's general theory of musical quality
to a further and very severe test.
The sounds of tuning-forks, when mounted on their
appropriate resonance-boxes, are, as we know, very ap-
proximately simple tones. If, therefore, we allow a num-
ber of such sounds, coincident in pitch with the funda-
mental-tone, and with individual over-tones, of one and
the same clang, to be simultaneously produced, the effect
on the ear ought, if Helmholtz's theory is true, to be that
of a single musical sound, not that of a series of indepen-
dent tones. To try the experiment in the simplest form,
take two mounted forks forming the interval of an oc-
tave, and cause them to utter their respective tones to-
gether. For a short time we are able to distinguish the
two notes as coming from separate instruments, but soon
they blend into one sound, to wlich we assign the pitch
of the lower fork, and a quality more brilliant than that
of either. So strong is the illusion, that we can hardly
believe the higher fork to be really still contributing to
its note, until we ascertain that placing a finger on its
prongs at once changes the timbre, by reducing it to the
dull, uninteresting quality of a simple tone. The charac-
ter of a clang consisting of only one over-tone and the
fundamental, may be shown to admit of many different
shades of quality, by suitably varying the relative inten-
sities of the two fork-tones in this experiinent. If we add
. 108
MUSICAL INSTRUMENTS.
a fork a Fifth above the higher of the first two, and there-
·fore yielding the third partial-tone of the clang of which
they form the first and second, the three tones blend as
perfectly as the two did before ; the only difference per-
ceptible being an additional increase of brilliancy. The
experiment admits of being carried further with the saine
result.
If we were able to produce by means of tuning-forks
as many simple tones of the series on page 69 as we
pleased, and also to control at will their relative intensi-
ties, it would be possible to imitate, in this manner, the
varying timbre of every musical instrument. The un-
manageable character of very high forks has as yet pre-
vented this being done for sounds containing a very large
number of powerful over-tones, but an apparatus on this
principle has been devised by Prof. Helinholtz, which im-
itates, very successfully, sounds not involving more than
the first six or eight partial-tones. His theory of quality
is thus experimentally demonstrated, both analytically
and synthetically. We will examine in the next chapter
some important theoretical considerations by which this
theory is further elucidated and confirmed.
CHAPTER VI.
OY THE CONNECTION BETWEEN QUALITY AND MODE OF VI-
BRATION.
70. It was stated on page 60 that, when a pendulum
performs oscillations whose extent is small compared to
the length of the pendulum itself, the period of a vibra-
tion is the same for any extent of swing within this limit.
We will apply this fact to prove that the prongs of a tun-
ing-fork vibrate in the same mode (S 11] as does a pendu-
lun.
When a sustained simple tone is being transmitted by
the air, we may regard it as originated by a tuning fork
of appropriate pitch and size. But we know experiment-
ally that, by suitable bowing, we may elicit from such a
fork tones of various degrees of intensity, though having
all the same pitch. Here, therefore, the extent of vibra-
tion varies, while the period remains constant, which is
the pendulum-law. Accordingly, the vibrations of a tun-
ing-fork are identical in mode with those of a pendulum.
The same thing will hold good of the aërial vibrations to
which those of a fork give rise. Hence, in general, a sim-
ple tone is due to vibrations executed according to the
pendulum-law.
Such vibrations, when performed longitudinally, will,
therefore, give rise to waves of condensation and rarefac-
tion whose associated wave-forın is that drawn in Fig. 17
110
QUALITY AND MODE OF VIBRATION.
(a), page 35. It will be convenient to call the vibrations
to which a simple tone is due simple vibrations; and
the associated waves simple waves. We proceed to ex-
amine the modes of vibration corresponding to composite
sounds.
Let us, first, take the case of a sustained clang con-
sisting of but two simple tones, the fundamental and its
first over-tone. A particle of air engaged in transmit-
ting this sound is simultaneously acted upon by two sets
of vibratory movements, and we have to investigate what
its motion will be under their joint influence. In fact,
the problem before us is the composition of two simple vi-
brations. In order to solve it, we must employ a principle
of mechanics, called the “superposition of small motions,"
the nature of which can be illustrated experimentally as
follows:
71. Suppose a cork to be floating on the undisturbed
surface of a sheet of water into which two stones are
thrown at different points. From each origin of agita-
tion concentric circular waves will spread out, and pres-
ently the cork will be influenced by both sets of disturb-
ances at once. Either series of waves, if it acted sepa-
rately on the cork, would cause it to execute a vibratory
movement in a vertical straight line. The mechanical
principle which we are explaining asserts that the joint
effect of the two sets on the cork will be exactly equal to
the sum or difference of their separate effects, according
as these are produced in the same, or in opposite direc-
tions. The accompanying figures show the four different
cases which may arise. In each, a and b are the points
which the cork, originally at rest at 0 in the level-line,
would occupy at the moment indicated in the figure, were
it acted on by either set of waves alone; c is its contem-
poraneous position under their joint action.
Iu (1), crest falls on crest; in (2), trough on trough,
SUPERPOSITION OF SMALL MOTIONS.
111
FIG. 44.
(1)
(2)
..
a
h
(3)
(4)
10
C
and the displacement, Oc, of the cork from its position
of rest, 0, is equal to the sum of the displacements due
to the two crests separately-mviz., Oa and 0b. In (3) and
(4), where the crest of one wave meets the trough of an-
other, Oc is equal to the difference between Oa and Ob;
c being above or below the level-line, according as Oa is
greater than Ob, as in (3), or less than Ob, as in (4).
Thus, each wave produces its own full effect on the cork
in its own direction, or, in other words, the motion due to
one wave is “superposed "on that due to the other.
In order, then, to determine the form of the joint
wave which results from the combination of two constitu-
ent waves, we have only to apply the above principle suc-
cessively to points in the level-line which both sets of
disturbances simultaneously affect. We thus obtain an
assemblage of points constituting the joint wave required.
In the instance now before us we proceed as follows:
Let each simple tone be represented by its associated
wave-system. Ascertain, by the process just described,
to what joint form the combination of the two associated
wave-systems leads. The result will be the associated sys-
tein correspondir:g to the mode of particle-vibration to
which the compound sound is due.
112
QUALITY AND MODE OF VIBRATION.
72. Before, Lowever, we can lay down the two tribu-
tary systeins of waves, an important point remains to
be settled. We will, for a moment, suppose that the
two simple tones on which we are engaged are originated
and sustained by two tuning-forks, situated as in the an-
nexed figure, and that we are examining the transmission
of their resulting clang along the dotted line with respect
to which they are symmetrically situated.
FIG. 45,
c/
Let the forks have been set sounding by precisely si-
multaneous blows. They will then commence swinging
out of their positions of rest in the same direction at the
same instant. The points in the associated wave-forms,
where a vibrating particle is momentarily in its position
of rest, are those in which it cuts the level-line. Hence,
in laying down the two tributary wave-systems along the
same level line, we must make them both cut that line in
FIG. 46.
0
some one point, taking care that their convexities at that
point are both turned the same way, as at O, Fig. 46.
In this case the two vibrations are said to start in the
same phase.
If the two forks are set in vibration at different mo-
SIMPLE AND RESULTANT WAVE-FORMS.
113
ments, they may not swing out of their equilibrium posi-
tions in the same direction together. Hence, we no
longer necessarily have a point where both sets of waves
cut the level-line. The result is of the kind shown in
Fig. 47, where three different cases are represented.
Here we have vibrations starting in different phases.
It is clear from the figure that all phase-differences
can be properly represented by merely causing the wave-
systems engaged to assume different positions upon the
level line, with reference to each other. The second and
third cases are obtainable from the first by sliding the
system of shorter waves bodily along the level-line, while
the other system of waves retains its position.
FIG. 47.
By the help of the instrument mentioned ou page
108, Prof. IIelmholtz has demonstrated that, when a
number of partial-tones are independently produced, the
clang into which they coalesce has the same quality, what-
ever differences of phase may exist among the systems of
simple vibrations to which the constituent partial tones are
due. Accordingly, we may expect to find that not one
single wave-form, but many such forms, correspond to a
sound of given quality and pitch.
114.
QUALITY AND MODE OF VIBRATION.
1
1
1
1
1
1
FIG. 48.
FIG. 49.
FIQ. 60.
1
.
1
1
f
1
1
1
VARIETY OF RESULTANT WAVE-FORMS.
115
In Figs. 48, 49, 50, the associated wave-form corre-
sponding to our clang of two partial-tones (page 110] is
constructed for three degrees of phase-difference. The
simple constituent waves are shown in thin, the result of
their composition in full lines. In each case two complete
wave-lengths of the latter are exhibited.
Figs. 51 and 52 present two wave-forms drawn in the
same way, for a clang of constant pitch and quality con-
tiining the partial-tones 1, 2, and 3.
FIG, 51.
FIG. 52.
The dissimilarity of form, and therefore of corre-
sponding particle-vibration, is, in both sets of figures,
inost marked.
73. It has been shown that, by mere alteration of
phase, a very great variety of resultant wave-forms can
116
QUALITY AND MODE OF VIBRATION.
be obtained from two sets of simple waves of given
lengths and amplitudes. Each one of these forms will
give rise to a cycle of others, if we allow the relative am-
plitudes of the constituent systems to be changed, while
keeping the difference of phase constant. If, therefore,
we have at our disposal the systems of simple waves cor-
responding to an unlimited number of partial-tones, and
can assign to them any degrees of intensity and phase-
difference that we choose, it is manifest that we may pro-
duce by their coinbination an endless series of different
resulting wave forms. On the other hand, it is not evi-
dent that, even out of this rich abundance of materials,
we could build up every form of wave which could possi-
bly be assigned.
The French mathematician Fourier
has, however, demonstrated that there is no form of wave
which (unless itself simple) cannot be compounded out of
a number of simple waves, whose lengths are inversely
as the numbers 1, 2, 3, 4, etc. He has further shown that
each individual wave-form admits of being thus com-
pounded in only one way, and has provided the means of
calculating, in any given case, how many, and what,
members of the partial series will appear, their relative
amplitudes and their differences of phase.
When translated from the language of Mechanics into
that of Acoustics, the theorem of Fourier asserts that
every regular musical sound is resolvable into a definite
number of simple tones whose relative pitch follows the
law of the partial-tone series. It thus supplies a theoret-
ical basis for the analysis and synthesis of composite
sounds which have been experimentally effected in Chap-
ters IV. and V.
When we are listening to a sustained clang, the air, at
any one point witliin the orifice of the ear, can have only
one definite mode of particle-vibration at any one mo-
ment. Ilow does the ear behave toward any such given
ANALYZING POWER OF THE EAR.
117
vibration? It proceeds as follows: If the vibration is
simple, it leaves it alone. If composite, it analyzes it into
a series of simple vibrations whose rates are once, twice,
three times, etc., that of the given vibration, in accordance
with Fourier's theorein. In the former event, the ear
perceives only a simple tone. In the latter, it is able to
recognize, by suitably directed and assisted efforts, partial-
tones corresponding to the rate of each constituent into
which it has analyzed the composite vibration originally
presented to it. The ear being deaf to differences of
phase in partial-tones (p. 113), perceives no distinction
between such modes of vibration as those exhibited on page
114, but merely resolves them into the same single pair
of partial-tones. Since, however, only one such resolution
of a given vibration-mode is possible, the ear can never
vary in the series of partial-tones to which it reduces an
assigned clang.
The power possessed by the ear of thus singling out
the constituent tones of a clang, and assigning to them
their relative intensities, is unlike any corresponding
capacity of the eye. Take, for instance, the two curves
shown in Figs. 51 and 52, and try to determine, by the
eye alone, what simple waves, present with what ampli-
tudes, must he superposed in order to reproduce those
forins. The eye will be found absolutely to break down
in the attempt.
We have seen that the loudness of a composite sound
depends on extent of vibration, and its pitch on rate of
vibration. There remains only one variable element, viz.,
mode of vibration, to account for the quality of the sound.
From this consideration it follows that some connection
must exist between the quality of a sound and the mode
of aërial vibration to which the sound is due. Up to the
time of Helmholtz no advance had been made in clearing
up the nature of this connection. It was reserved for him
118
QUALITY AND MODE OF VIBRATION.
to show that, while no two sounds of different quality can
correspond to the same mode of vibration, many different
modes of vibration may yet give rise to a sound of only
one degree of quality. In other words, mode of vibration
determines quality, but quality does not determine mode
of vibration.
CHAPTER VII.
ON THE INTERFERENCES OF SOUND, AND ON
BEATS."
74. IN $71 we examined the principle on which the
problem of the composition of vibrations is generally
solved. We now approach certain very important par-
ticular cases of that problem, which it will be worth
while to solve both independently, and also as instances
of the method repeatedly applied in $72.
Suppose that a particle of air is vibrating between the
extreme positions A and B, under a sustained simple tone
FIG. 58.
A
B
produced by a tuning-fork, or stopped flue-pipe. Now
let a second instrument of the same kind be caused to
emit a tone exactly in unison with the first. We will as-
sume that, when the vibrations constituting the second
tone fall on the particle, it is just on the point of starting
from A toward B, under the influence of those of the first.
Two extreme cases are now possible, depending on the
movement which the particle would have executed, had
it been affected by the later-impressed vibration alone.
First, suppose that to be from A along the line AB,
either through a greater or less distance than AB, back
again to A, and so on. Here the separate effects of the
two sets of vibrations will be added together, the particle
120
INTERFERENCE AND BEATS.
will, therefore, perform vibrations of larger extent than it :
would under either component separately. Next, suppose
that, under the second set of vibrations alone, the particle
would move from A in the opposite direction to its former
course, i. e., along BA produced, shown by a dotted line
in the figure. In this case the separate effects are abso-
lutely antagonistic; accordingly, the joint result is that
due to the difference of its components. The particle will,
therefore, execute less extensive vibrations than it would
have done under the more powerful of the two compo-
nents acting alone.
The most striking result presents itself when the two
systems of vibrations, besides being in complete opposition
to each other, are also exactly equal in extent. In this
case, the air-particle, being solicited with equal intensity
in two diametrically opposite directions, remains at rest,
the two systems of vibrations completely neutralizing each
other's effect. 'In general, however, these systems, even
when equal in extent of vibration, are neither in complete
opposition nor in complete accordance, but in an inter-
mediate attitude, so as only partially to counteract, or
support, each other
These conclusions admit of being
exhibited in a more complete manner by means of asso-
ciated waves. We have on y to. lay down the simple
wave-forms corresponding to the constituent vibrations,
and superpose them as in $ 72. The reader will have
noticed that the differences of relative motion described
above are merely phase-differences.
In Fig. 54, (1), (2), (3), we have two waves of unequal
amplitudes in complete accordance, complete antagonism,
and an intermediate condition respectively. In Fig. 55,
a case of equal and opposite waves is shown. In (1) Fig.
54, the resultant wave is the sum, and in (2) the difference
of the component waves. In (3), we get a wave of inter-
mediate amplitude. These three resulting waves are ne-
INTERFERENCE OF SOUNDS.
121
FIG. 5.
(
1)
(2)
(3)
.
cessarily simple, as otherwise two simple tones in unison
would give rise to a composite sound, which would be
absurd. In Fig. 55 the wave-form degenerates into the
level line, i. e., no effect whatever occurs.
Fig. 55,
73. Thus, when one simple tone is being heard, we by
no means necessarily obtain an increase of loudness by
exciting a second simple tone of the same pitch. On the
6
122
INTERFERENCE AND BEATS.
contrary, we may thus weaken the original sound, or
even extinguish it entirely. When this occurs we have
an instance of a phenomenon which goes by the name of
Interference. That two sounds should produce absolute
silence seems, at first sight, as absurd as that two loaves
should be equivalent to no bread. This is, however, only
because we are accustomed to think of sound as some-
thing with an external objective existence; not as con-
sisting merely in a state of motion of certain air-particles,
and therefore liable, on the application of an opposite
system of equal forces, to be absolutely annihilated.
A siugle tuning-fork presents an example of this very
important phenomenon. Each prong sets up vibrations
corresponding to a simple tone, and the two notes so pro-
duced are of the same pitch and intensity. If the fork,
after being struck, is held between the finger and thumb,
and made to revolve slowly about its own axis, four posi-
tions of the fork with reference to the ear will be found
where the tone completely goes ont. These positions are
midway between the four in which the faces of the prongs
are held flat before the ear.. As the fork revolves from
one of these prositions of loud tone to that at right angles
to it, the sound gradually wanes, is extinguished in pass-
ing the Interference-position, reappears very feebly im-
mediately afterward, and then continues to gain strength
until its quarter of a revolution has been completed.
76. The case of coexistent unisons has now been ade-
quately examined: we proceed to inquire what happens
when two simple tones, differing slightly in pitch, ako
simultaneously produced. The problem is, in fact, to
compound two sets of pendulum-vibrations whose periods
are no longer exactly equal. Let us fix on a moment of
time at which the two component vibrations, simultane-
ously soliciting an assigned particle of air, are in complete
accordance, and suppose that the particle, under their joint
BEATS OF SIMPLE TONES.
123
influence, is just commencing a vibration from left to
right. It will be convenient to call this an outward, and
its opposite an inward swing. Since the periods of the
two component vibratious are unequal, one of them will
at once. begin to gain on the other, and therefore, directly
after the start, they will cease to be in complete accord-
ance.
It is easy to ascertain what their subsequent bear-
ing toward each other will be, by considering two ordi-
nary pendulums of unequal periods, both beginning an
outward swing at the same instant. Let A be the slower,
B the quicker pendulun. When A has just finished its
outward swing, В will have already turned back and per-
formed a portion of its next inward swing. Thus, during
each successive swing of A, B will gain a certain distance
upon it. When B has, in this manner, gained one whole
swing, i. e., half a complete oscillation, upon A, it will
begin an inward swing at the moment when A is com-
mencing an outward swing. The two vibrations are bere,
for the moment, in complete opposition. After another
interval of equal length, B, having gained another whole
swing, will be one complete oscillation ahead of A, and
they will therefore start on the next outward swing to-
gether, i. e., the vibrations will be momentarily in com-
plete accordance. Thus, during the time requisite to
enable B to perform one entire oscillation more than A,
there occur the following changes : Complete accordance
of vibrations, lasting only for a single swing of the more
rapid pendulum, followed by partial accordance, in its
turn gradually giving way to discordance, which culmi-
nates in complete opposition at the middle of the period,
and then, during its latter half, gradually yields to re-
turning accordance, which regains completeness just as
the period closes.
It follows from this, combined with what is said on
pp. 119, 120, that in the case of two simple tones, we must
121
INTERFERENCE AND BEATS.
hear a sound going through regularly-recurring alter-
nations of loudness in equal successive intervals of time,
its greatest intensity exceeding, and its least intensity
falling short of, that of the louder of the two tones.
Each recurrence of the maximun intensity is called a
beat, and it is clear that exactly one such beat will be
heard in each interval of time during which the acuter
of the two simple tones performs one more vibration than
the graver tone. Accordingly, the number of beats heard
in any assigned time will be equal to the number of com-
plete vibrations which the one tone gains on the other in
that time. We may express this result more briefly as
follows: the number of beats per second due to two simple
tones is equal to the difference of their respective vibra-
tion-numbers.
77. By means of the associated wave-forms we can
obtain a graphic representation of beats, which will prob-
ably be more directly intelligible than any verbal descrip-
tion. In Fig. 56, the constituent simple waves are laid
down, and their resultant constructed, for the interval of
a semi-tone. The vibration-fraction for this interval is
16
i.
e.,
16 vibrations of the higher tone are performed
in the same time as 15 of the lower. The figure repre-
sents completely the state of things from a maximum of
intensity to the adjacent minimum. The time during
which this change occurs is one-half of that above men-
tioned: accordingly, the figure shows 8 and 72 wave-
lengths of the respective systems. Thus half a beat isa
here pictorially represented, the amplitude of the result-
ant waves steadily diminishing during this period. We
have only to turn the figure upside down, to get a picture
of what occurs in the next following equal period. The
amplitudes here again increase, until they regain their
former proportions. One whole beat is this accounted for
15;
NUMBER OF BEATS PER SECOND. .
125
FIG. 56.
In addition to the alternations of
intensity which characterize beats,
they also contain variations of pitch.
The existence of such variations is
both theoretically demonstrable and
experimentally recognizable, but
they are too minute to require ex-
amination here."
78. The most direct way of study-
ing the beats of simple tones experi-
mentally is to take two unison tun-
ing-forks and attach a small pellet
of wax to the extremity of a prong
of one of them. The fork so operated
on becomes slightly heavier than be-
fure; its vibrations are therefore pro-
portionately retarded, and its pitch
lowered. When both forks are struck
and held to the ear, beats are heard.
These will be most distinct when the
fork's tones are exactly equally loud,
for in this case the minima of inten-
sity will be equal to zero, and the
beats will therefore be separated by
intervals of absolute, though but
momentary, silence. The increase in
rapidity, of the beats, as the interval
between the beating-tones widens,
may be shown by gradually loading
one of the forks more and more
1 The reader may, if he wishes to pursue this
subject, refer to a paper, by the author, on “Va-
riations of Pitch in Beats,” in the Philosophical
Magazine for July, 1872, from which Fig. 56 has
been copied.
126
INTERFERENCE AND BEATS.
heavily with wax pellets, or by a sınall coin pressed upon
them. If it is desired to exhibit these phenomena to a
large audience, the forks should first be mounted on their
resonance-boxes, and, after the pellets have been attached,
stroked with the resined bow, care being taken to produce
tones as nearly as possible equal in intensity. Slow beats
may also be obtained from any instrument capable of pro-
ducing tones whose vibration-numbers differ by a suffi-
ciently small amount. Thus, if the strings corresponding
to a single note of the piano-forte are not strictly in uni-
son, such beats are heard on striking the note. If the
tuning is perfect, a wax pellet attached to one of the
wires will lower its pitch sufficiently to produce the
desired effect. Beats not too fast to be readily counted
arise between adjacent notes in the lower octaves of the
harmonium, or, still more conspicuously, in those of large
organs. They are also frequently to be heard in the
sounds of church-bells, or in those emitted by the tele-
graph-wires when vibrating powerfully in a strong wind.
In order to observe them in the last instance, it is best to
press one ear against a telegraph-post and close the other:
the beats then come out with remarkable distinctness. It
should be noticed that, when we are dealing with two
composite sounds, several sets of beats may be heard at
the same time, if pairs of partial-tones are in relative
positions suited to produce them. Thus, suppose that
two clangs coexist, each of which possesses the first six
partial-tones of the series audibly developed. Since the
second, third, etc., partial-tones of each clang make twice,
three times, etc., as many vibrations per second as their
respective fundamentals (page 69), it follows that the
difference between the vibration-numbers of the two sec-
ond-tones will be twice, that between those of the two
third-tones three times, etc., as great as the difference be-
tween the vibration-numbers of the two fundamentals.
A
BEATS OF COMPOSITE SOUNDS.
127
Accordingly, if the fundamental-tones give rise to beats,
we may hear, in addition to the series so accounted for,
five other sets of beats, respectively twice, three, four, five,
and six times as rapid as they. In order to determine
the number of beats per second, for any such set, we need
only, multiply the number of the fundamental beats by
the order of the partial-tones concerned. The beats of
two simple tones necessarily become more rapid if the
liigher tone be sharpened, or the lower flattened; i. e., if
the inverval they form with each other be widened. The
beats may, however, also be accelerated, without altering
the interval, by merely placing it in a higher part of the
scale. In this way greater vibration-numbers are ob-
tained, and the difference of these is proportionally large,
though their ratio to each other remains what it was
before. Thus the rapidity of the beats due to an assigned
interval depends jointly on two circumstances, the width
of the interval, and its position in the musical scale; in
other words, on both the relative and absolute pitch of
the tones forming the given interval.
CHAPTER VIII.
ON CONCORD AND DISCORD.
79. A QUESTION of fundamental importance now pre-
serts itself, viz., What becomes of beats, when they are so
rapid that they can no longer be separately perceived by
the ear? In order to answer it, the best plan is to take
two unison-forks, of medium pitch, mounted on their
resonance-boxes, attach a small pellet of wax to a prong
of one of them, and then gradually increase the quantity
of wax. At first very slow beats are heard, and, as long
as their number does not exceed four or five in a second,
the ear can follow and count them without difficulty.
As they become more rapid, the difficulty of counting
them augments, until at last they cannot be recognized
as distinct strokes of sound. Even so, however, the ear
retains the conviction that the sound it hears is a series
of rapid alternations, and not a continuous tone. Its in-
termittent character is not lost, although the intermit-
tances tbemselves pass by too rapidly for individual
recognition. Exactly the same thing may be observed in
the roll of a side drum, which no one is in danger of mis-
taking for a continuous sound.
Rapid beats produce a decidedly harsh and grating
effect on the ear; and this is quite what the analogy of
our other senses would lead us to expect. The disagree-
able impressions excited in the organs of sight by. a flick-
CAUSE OF DISCORD.
129
ering, unsteady light, and in those of touch by tickling
or scratching, are familiar to every one. The effect of
rapid beats is, in fact, identical with the sensation to
which we commonly attach the name of discord. Let us
exainine, in somewhat greater detail, the conditions ne-
cessary for its production between two simple-tones. If
we take a pair of middle C unison-forks, and gradually
throw them more and more out of tune with each other
in the way already described, the roughness due to their
beats reaches its maximum when the interval between
them is about a half-tone: for a whole tone, it is decided-
ly less marked, and, when the interval amounts to a
Minor Third, scarcely a trace of it remains. Hence, in
order that dissonance may arise between two simple-
tones, they must form with each other a narrower inter-
val than a Minor Third. If we call this interval the
beating-distance for two such tones, we may express the
above condition thus: Dissonance can arise directly be-
tween two simple tones, only when they are within beat-
ing-distance of one another. It follows at once from this
that the amount of discord heard by no means exclusive-
ly depends on the rapidity of the beats produced, since
the same interval will give rise to a very different number
of beats per second according as it occupies a high or a
low position in the scale. Nevertheless this circumstance
exerts a considerable influence in modifying the intensity
of the dissonance of given intervals, according to the ab-
solute pitch of the tones which form them. Thus the
beats of a whole tone, which, in low positions, are power-
ful and distinct, become less marked as we ascend in the
scale, and in its highest portions practically inaudible.
Accordingly, the beating-distance, which, for tones of
medium pitch, we have roughly fixed at a Minor Third,
must be supposed to contract somewhat for very high
tones, while, for very low ones, it correspondingly ex-
130
ON CONCORD AND DISCORD.
pands. In consequence of this, a difference of relative
pitch, whiclı, in the lower part of the scale produces beats
so slow as not perceptibly to interfere with smoothness
of unison, may, in its higher region, cause a harsh dis-
sonance. We have here the reason why the ear is more
sensitive to slight variations of pitch in liigh than in low
notes, and why, therefore, greater accuracy in tuning is
essential to obtain a good unison effect from the former,
than from the latter.
The general partial-tone series consists of simple tones
which, up to the seventh, are mutually out of beating-
distance: above the seventh they close in rapidly upon
each other. In the neighborhood of 10, the interval be-
tween adjacent partial-tones is about a whole tone; near
16, a semi-tone; higher in the series they come to still
closer quarters. These high partial-tones are therefore so
situated as to produce harsh dissonances with each other.
Where they are strongly developed in a clang, there will
therefore be a certain inevitable roughness in its timbre.
This is the cause of the harsh quality of trumpet or trom-
bone notes, and also of the shrill tremulous sounds some-
times observed in the human voice (page 107). In fact,
we may regard all the portion of the partial-tone series
above the eighth tone as contributing mere noise to the
clang. Thus a noise may, conversely, be regarded as due to
many sinple tones within beating-distance of each other.
80. It has been shown that, when two simple-tones
are simultaneously sustained, beats can arise directly
between then only under one condition, viz., that the *
tunes shall differ in pitch by less than a Minor Third, or
thereabouts. When, however, the two coexisting sounds
are no longer simple tones, but composite clangs, each
consisting of a series of well-developed partial-tones, the
case becomes altogether different. Let us examine the
state of things which then presents itself.
DISSONANCE OF TWO CLANGS.
131
The sounds of most musical iustruments do not con-
tain more than the first six partial-tones; we will, there-
fore, assume this to be the case with the clangs before us.
No beats can then arise between partial-tones of the same
clang, for the reason assigned on page 130. Dissonance
due to beats will, however, be produced if a partial-tone
belonging to one clang is within the specified distance of
a partial-tone belonging to the other clang. Several pairs
of tones may be thus situated, and, if so, each will con-
tribute its share of roughness to the general effect. The
intensity of the rougliness due to any such pair will
depend chiefly on the respective orders to which the beat-
ing partial-tones belong, and on the interval between
them. The lowest partial-tones being the loudest, pro-
duce the most powerful beats, and half-tone beats are, in
general, harsher than those of a whole tone. In deter-
mining the general effect of a combination of two clangs,
we have to ascertain what pairs of partial-tones come
within beating-distance, and to estimate the amount of
rougliness due to each pair. The sum of all these rough-
nesses, if there are several such pairs, or the roughness
of a single pair if there be but one, constitutes the dis-
sonance of the combination. If there be no dissonance,
the combination is described as a perfect concord. When
dissonance is present, it will depend on its amount
whether the combination is called an imperfect concord
or a discord. The line separating the two must, of course,
be somewhat arbitrarily drawn.
81. Let us examine the principal consonant intervals,
in the manner above described, beginning with the octave.
3.
3-
27
81
1 12
132
ON CONCORD AND DISCORD.
The minims here represent the fundamental-tones; the
crotchets above them corresponding over-tones. Those
belonging to the higher clang are only written down as
far as the third, since the fourth, fifth, and sixth, have no
corresponding tones of the lower clang disposable with
which to forin beating pairs. As long as the tuning is
perfect, each partial-tone of the higher clang coincides
exactly with one belonging to the lower: No dissonance
can consequently occur, and the combination is a perfect
concord. But, suppose the higher C to be out of tune:
each of its partial tones will be correspondingly too
sharp or too flat, and three sets of beats will be leard be-
tween the partial tones 2-1, 4-2, and 6–3. When the
higlier C is as much as a semi-tone wrong, the result is
3
6
4.
3-
2.
21 1
1 1?
若
​9:
12
The pair 2–1 is of the most importance, and gives in
each case sixteen beats per second. The two others give
respectively 32 and 48 beats per second. A semi-tone
corresponds to about maximum roughness in the middle
region of the scale, so that we have before us an exceed-
ingly harsh discord. As the pitch of the higher note is
gradually corrected, the rapidity of the beats diminishes,
but the tuning must be extremely accurate to make them
entirely vanish. If the note makes but one vibration per
second too many, or too few, which corresponds to a dif-
ference in pitch of only about a thirtieth part of a whole
tone, the defect of tuning makes itself felt by three sets
of beats, of 1, 2, and 3 per second respectively. The
tunist must keep slightly altering the pitch until he at
CONSONANCE AND DISSONANCE.
133
length hits on that which completely extinguishes the
beats. We saw in an earlier part of this inquiry (p. 53)
that, when two sounds form with each other the interval
of an octave, their vibration-numbers must be in the ratio
of 2:1. Long after it had been experimentally ascer-
tained that rigorous compliance with this arithmetical
condition was essential for securing a perfectly smooth
octave, the reason for this necessity remained entirely un-
known, and nothing but the raguest and most fanciful
suggestions were offered to account for it-such as, for
instance, that “the human mind delights in simple
numerical relations." This attempt at explanation over-
looked the obvious fact that many people, who knew
nothing either about vibrations or the delights of simple
numerical relations, could tell a perfect octave from an
imperfect one a great deal better than the majority of
men of science. The true explanation, which it was left
for Helmholtz to discover, lies in the fact that only by
cxactly satisfying the assigned numerical relation, can
the partial-tones of the higher clang be brought into exact
coincidence with partial-tones of the lower, and thus all
beats and consequent dissonance prevented.
82. No narrower interval than an octave can be found
which gives an absolutely perfect concord. The nearest
approach to such a concord is the Fifth.
4.
3
3-
itim
1-2-
21
99
12
Here we get two pairs of coincidences 3—2 and 6-4,
but a certain roughness is caused by 3 of the higher clang
being within beating-distance of both 4 and 5 of the lower
134
.
ON CONCORD AND DISCORD.
clang. The tuning must be perfectly accurate, this inter-
val being closely bounded by harsh discords. The result
of an error of a semi-tone is as follows:
-3
م م |ح
-2-
1
9
1
l'or every single vibration per second by which the
higher clang is out, there will be two beats per second
from the pair 3-2 with others of greater rapidity, but
less intensity, from the higher pairs. The result, for the
Fifth, is, therefore, that, liowever accurately tuned, it in-
volves an aſ preciable roughness. It is true that since 4
and 5 are generally weak, and the beating-intervals are
whole tones, the roughness will be very slight: still the
trace of dissonance due to it prevents our classing the
Fifth as an interval quite equally smooth with the octave.
83. For the Fourth we have
-4
13
4
3
2
2
93
-1
12
The amount of dissonance is greater than in the case
of the Fifth, since 3 and 2 are usually both powerful tones,
and produce, therefore, londer beats than those of 43
and 5—3. There are, in addition, the beating pairs 6–4
and 6–5. Moreover, the first pair of coincident or partial
tones, 4—3, are, in general, weaker than the beating pair
below thein, 3—2. The Fourth is bounded only on one
MAJOR THIRD AND MAJOR SIXTH.
135
side by a harsh discord. If its upper clang is half a note
too sharp, we have the interval C-F#, which is treated
in the last figure but one. Slight flattening of the F
will set the pair 4—3 beating slowly; the disappearance
of their beats thus secures the accurate túning of the in-
terval. On the other hand, lowering F weakens the
beats of 3—2, by widening the distance between those
tones, and, therefore, tends to lessen the whole amount
of roughness. These considerations go far to explain the
fact that a long dispute runs through the history of music,
as to whether the Fourth ought to be treated as a concord
or as a discord. The decision ultimately arrived at in
favor of the first of these alternatives was perhaps, as
Helmholtz suggests, due more to the fact that the Fourth
is the inversion of the Fifth, than to the inherent smooth-
ness of the former interval.
84. Next come the intervals of the Major Third and
Major Sixth, which shall be taken together, as they are
very nearly equally consonant.
6
5
4
3-6
4
3
+4
13
2
-2
24
1
2:
1
17
The dissonance, due to the pair 3-2, separated by a
tone in the Sixth, is perhaps, about equal to that of the
weaker pair 4—3, which are only a semi-tone apart, in the
Third. The definition of these intervals depending, as it
does (in both) on the fifth tone of the lower clang, will in
general be but feebly marked.
85. The last remaining intervals, less than an octave,
which rank as concords, are those of the Minor Third and
Minor Sixth.
136
ON CONCORD AND DISCORD. .
Each contains strong elements of dissonance; in fact,
we are here near the boundary-line between concord and
discord. As regards sharpness of definition, the tones
5
-0-3
- 2
-3-
2
2.
ban
by1
12.
C12
6 and 5, on which it depends in the first of the two inter-
vals, are, in the sounds of most instruments, weak or even
entirely absent, while for the second interval the series of
partial-tones must be extended as far as the 8th of the
lower clang in order to reach the first coincident pair.
Accordingly, the Minor Sixth can hardly be said to be de-
fined at all, for claugs of ordinary quality, by coincidence
of partial-tones. Its powerful beating pair 3—2, separated
by the interval of greatest dissonance, a semi-tone, makes
it the roughest of all the concords. On the piano-forte,
and other instruments with fixed tones, the same notes
(CAD) which represent the Minor Sixth have also to do
duty as one of the harshest discords, the sharp Fifth
(CG#). The extremely defective consonance of the Minor
Sixth could hardly be more conclusively shown than by
the fact just mentioned.
86. As regards the dissonant intervals of the scale, we
have, in addition to those incidentally examined above,
the seni-tone, tone, and Minor Seventh. The first two
need not be examined, since obviously each pair of corre-
sponding over-tones is brought within the same beating
intervals as the two fundamentals. The dissonance re-
sulting is, of course, harsher for the half than for the
whole tone. The Minor Seventh is constituted thus :
DISSONANT INTERVALS.
137
-3
po-2
3
2
271
ระ
16
It is the mildest of the discords, in fact, in actual smoothie
ness it decidedly surpasses the Minor Sixth.
87. In order that the reader may see at a glance the
whole result of this somewhat laborious discussion, we
subjoin a graphical representation of the amount of disso-
nance contained in the several intervals of the scale. The
figure is taken, with some slight alterations, from that
given at page 519 of Helmholtz's work. The intervals,
reckoned from C, are denoted by distances measured along
the horizontal straight line. The dissonance for each in-
terval is represented by the vertical distance of the curved
line from the corresponding point on the horizontal line.
The calculations on which the curve is based. were made
by Helmholtz for two constituent clangs of the quality
of the violin. For piano-forte sounds the form of the
curve would be slightly different; for those of stopped
organ-pipes, etc., very different indeed.
FIG. 57.
Dammed
If I
The figure indicates the sharpness of definition of an
interval by the steepness with which the curve ascends
in its vicinity. If we regarded the outline as that of a
138
ON CONCORD AND DISCORD.
mountain-chain, the discords would be represented by
peaks, and the concords by passes. The lowness and nar-
rowness of a particular pass would measure the smooth-
ness and definition of the corresponding musical interval.
88. The theory of musical consonance and dissonance,
our examination of which is now concluded, necessarily
leads us to regard the distinctions between different con-
cords laid down by theoretical musicians as not in them-
selves absolute, but dependent on the quality of the sounds
experimented upon. The results we have arrived at are
generally true for sounds containing the first six partial-
tones, but they will not apply, without modification, to
clangs differently constituted. To take a case or two in
point: Suppose, for instance, we are dealing with sounds
such as those of stopped organ-pipes which contain only
odd partial-tones (p. 97). It is at once clear from p. 134
that the interval of the Sharp Fourth C-F#, which owes
its dissonant character to the beating pairs 3—2, 43,
and 6–4, will become something quite different when
the dissonance due to all these pairs disappears, as it must
do, since each of them contains at least one partial-tono
of an even order. The Minor Sixth would also gain in
smoothness in such a timbre, by the removal of the loudly-
discordant pair 3-2.
Helmholtz has examined the case of a hautboy taking
one note of an interval and a clarionet the other, and
shown that some concords sound best when the former
instrument plays the upper note and the latter the lower,
while with others the reverse is the case. The bautboy
produces the uninterrupted series of partial-tones, the
clarionet only its odd members. Reference to page 135
slows that, in the case of the Major Third, we can only
get rid of the dissonant pair 4–3 by assigning the higher
note to the hautboy. The Fourth, on the contrary, will
be seen, by page 134, to sound smoothest when the clari-
CASE OF SIMPLE TONES.
139
onet is above the hautboy, since by this arrangement we
divorce the quarrelsome couple 3—2, whose bickerings
will, in the opposite position, continue to be heard.
These conclusions, which experiment confirms, are, I be-
lieve, in advance of any obtained empirically by musical
theorists. Corresponding rules might easily be elicited
for other instruments.
89. It is possible to draw from the general theory of
consonance and dissonance an inference which.
1.seeins, at
first sight, fatal to the truth of the theory itself. “It," it
may be said, “the difference between a consonant and a
dissonant interval depends entirely on the behavior tow-
ard each other of certain pairs of over-tones, then, in
the case of sounds like those of large stopped Alue-pipes,
where there are no over-tones at all, the distinction between
concords and discords ought entirely to disappear, and
the interval of a Seventh, for instance, to sound just as
smooth as that of an octave. As this is notoriously not
the fact, the theory cannot be true.”
In order to meet this objection, it will be necessary
first to acquaint the reader with certain known experi-
mental facts which Helmholtz has dragged out of the
obscurity in which they had lain for fully a century, and
forced to deliver their testimony in favor of his theory.
90. Let two tuning-forks of different pitch mounted
on their respective resonance-boxes, and therefore pro-
ducing simple tones, be thrown into powerful vibration
by the use of a resined fiddle-bow. With adequate atten-
tion, it is possible to recognize, in addition to the tones
of the forks themselves, certain new sounds, which usually
differ in pitch from both the former ones.
These tones,
called, from the manner of their production, combination-
tones, fall into two classes, with only one of which, dis-
covered in 1740 by a German organist named Sorge, we
need here concern ourselves. It consists of a series of
140
ON CONCORD AND DISCORD.
tones called combination-tones of the first second, third,
etc., orders, of which the first is of the most importance,
as it can be heard without difficulty. Its pitch is deter-
mined by the following law: The combination-tone of the
first order of two simple primary-tones has for its vibra-
tion-number the difference between the respective vibra-
tion-numbers of the primaries. Thus, e. g., if the two
primaries make 200 and 300 vibrations per second, and
therefore form a Fifth with each other, the first combina-
tion-tone will make 100 vibrations per second, and, ac-
cordingly, lie exactly one octave below the graver of the
two primary tones. In this manner we can determine
the combination-tones of the first order for pairs of simple
primaries forming any given interval with each other.
The following table, copied from Helmholtz's work,
shows the results for all the consonant intervals not ex-
ceeding one octave:
Interval.
Vibration-ratio.
Difference.
Depth of the Combination-tone
below the graver primary.
Octave
Fifth
Fourth
Major Third
Minor Third
Major Sixth
Minor Sixth
1:2
2: 3
3:4
4:5
5 : 6
3 : 5
5 : 8
1
1
1
1
1
2
3
Unison
Octave
Twelfth
Two Octaves
Two Octaves and Major Third
Fifth
Major Sixth
In inusical notation the same thing stands thus, the
primaries being denoted by minims, and the cornbination-
tones by crotchets :
12
略
​9 :
Ebe
be
.
COMBINATION-TONES.
141
Combination-tones are produced with remarkable dis-
tinctness by the harmonium. If the primaries shown in
the treble stave are played on that instrument while the
pressure of air in the bellows is vigorously sustained, the
corresponding combination-tones of the first order, writ-
ten in the bass, come out with unmistakable clearness.
They are in fact much better heard thus than from tun-
ing-forks.
Combination-tones of the second order may be treated
as if they were first-order tones produced between one or
other of the primaries and the combination-tone of the
first order. Similarly we may regard each combination-
tone of the third order as due to a second-order tone,
paired either with one of the primaries, with the first-
order tone or with its own fellow of the second order.
Successive subtraction will therefore enable us to deter-
mine the vibration number of a combination-tone of any
order from the vibration-numbers of the two primaries.
Combination-tones grow rapidly feebler as their order
becomes higher. Those of the first order are usually
distinct enough, and those of the second to be heard with
a little trouble. The third order is only recognizable
when entire stillness is secured, and the greatest, atten-
tion paid. It is a moot point whether the fourth-order
tones can be heard at all.
91. We can now show that the existence of combina-
tion-tones prevents intervals formed by two simple tones
from altogether lacking the characteristic differences of
consonance and dissonance, though those differences are
far less marked than in the case of composite sounds. To
begin with the octave: Let us suppose that we have two
simple tones forming nearly this interval, but that the
higher of them is a little sharp, so that the octave is not
strictly in tune, is in fact slightly impure. Let the lower
tone make 100, the higher 201, vibrations per second.
They will give rise to a combination-tone making 101
142
ON CONCORD AND DISCORD.
vibrations per second (page 140), and this with the lower
primary will produce one beat per second. If the higher
primary had been flat, instead of sharp, making, say, 199
vibrations per second, we should have bad 99, as com-
bination-tone, giving rise, with 100, to beats of the same
rapidity as before. These beats cannot be got rid of ex-
cept by making the vibration-ratio exactly 1:2, i. e., the
octave perfectly pure. The roughness must increase both
when the interval widens and when it contracts, so that
the octave, in simple tones, is a well-defined concord
bounded on either side by decided discords. This result
may be easily verified experimentally by taking two
tuning-forks forming an octave with each other, and
throwing the interval slightly out of tune by causing a
pellet of wax to adhere to a prong of one of them. On
exciting the forks, the beats will be distinctly heard.
The octave is the only interval which is defined by
the beats of a combination-tone of the first order with
one of the primary-tones. For the next smoothest con-
cord, that of the Fifth, we are obliged to have recourse
to the second order. Thus, proceeding as in the case of
the octave, we havem
Primaries
200
301
C'T. of 1st order
C. Ts. of 2d order
99
200
No. of beats per sec.
2
The Fifth is, thus, a fairly well-defined consonance,
thongh decidedly less sharply bounded than the octave,
owing to the feebleness of the C. T. of 2d order. For
the Fourth we have
/ 101
300
401
101
Primaries
C. T. of 1st order
C. Ts. of 2d order
C. Ts. of 3d order
No. of beats per sec.
199
101
300
98
202
3
CONSONANCE AND QUALITY.
143
The third-order toue being excessively weak, the inter-
val of a Fourth can scarcely be said to be defined at all.
Still less can the remaining consonant intervals of the
scale, by the evanescent beats of still higher orders of
combination-tones.
92. With simple tones, then, the case stands thus:
The intervals of a Second and a Major Seventh are pal-
pably dissonant, owing to the beats of the primaries, in
the former, and of a first-order combination-tone with a
primary, in the latter. There is a certain amount of dis-
sonance in intervals slightly narrower or slightly wider
than a Fifth, but of a feebler kind than in the case of the
octave, inasmuch as it is due to only a second-order com-
bination-tone. Whatever dissonance may exist near the
Fourth is practically imperceptible. All other intervals
are free from dissonance. Accordingly all intervals from
the Minor Third nearly up to the Fifth, and from a little
above the Fifth up to the Major Seventh, ought to sound
equally smooth. This conclusion is probably very incon- .
sistent with the views of musical theorists, who regard
concord and discord as entirely independent of quality,
but it is strictly borne out by experiment. With the
tones of tuning-forks the intervals lying between the
Minor and Major Thirds, and between the Minor and
Major Sixths, though sounding somewhat strange, are
entirely free from rougliness, and therefore cannot be
described as dissonant. As a further verification of this
fact, Helmholtz advises such of his readers as have access
to an organ to try the effect of playing alternately the
smoothest concords and the most extreme discords which
the musical scale contains, on stops yielding only simple
tones, such, e. g., as the flute, or stopped diapason. The
vivid contrasts which such a proceeding calls out on in-
struments of bright timbre, like the piano-forte and har-
monium, or the more brilliant stops of the organ, such as
144
ON CONCORD AND DISCORD.
principal, hautboy, trumpet, etc., are here blurred and
effaced, and every thing sounds dull and inanimatę, in
consequence. Nothing can slow more decisively than
such an experiment that the presence of over-tones con-
fers on music its most characteristic charms.
Thus the remark put into the mouth of a supposed
objector in $ 89 turns out to be no objection whatever to
Helmholtz's theory of consonance and dissonance, but, so
far as it represents actual facts, to be valid against the
prevalent views of musical theorists.
93. It may be well to advert briefly, in this place, to
a point connected with combination-tones which may
otherwise occur as
a difficulty to the reader's mind.
When two clangs coexist, combination-tones are produced
between every pair which can be formed of a tone from
one clang with a tone from the other. These intrusive
tones will usually be very numerous, and, for aught that
appears, may interfere with those originally present, to
such an extent as to render useless a theory based on the
presence of partial-tones only. Helmholtz bas removed
any such apprehension, by showing that, in general, dis-
sonance due to combination-tones produced between over-
tones, never exists except where it is already present by
virtue of direct action among the over-tones themselves.
Thus the only effect attributable to this source is a some-
what increased roughness in all intervals except absolute-
ly perfect concords. No modifications, therefore, have to
be introduced, on this score, in the conclusions of SS
81-86.
CHAPTER IX.
ON CONSONANT TRIADS.
94. In the ensuing portion of this inquiry we shall
have to make more frequent use than hitherto of vibra-
tion-fractions. It may, therefore, be well to explain the
rules for their employment, in order that the student may
acquire some facility in handling them. The vibration-
fraction of an assigned interval expresses the ratio of the
numbers of vibrations performed in the same time by the
two notes which forin the interval. The particular length
of time chosen is a matter of absolute in difference. The
upper note of an octave, for instance, vibrates twice as
often as the lower does in any time we choose to select, be
it an hour, a minute, a second, or a part of a second. It is
often convenient to determine the vibration-fraction of an
interval from the vibration-numbers of its constituent
notes: in such a case we choose one second as our time
of comparison, and in this way vibration fractions were
defined in § 34: any other standard is, however, equally
legitimate, though, in general, less convenient. To illus-
trate these remarks on a particular case, the vibration-
fraction indicates that, while the lower of two notes
forming a Major Third makes four vibrations, the higher
of them makes five. Therefore, while the lower makes
one vibration, the higher makes aths of a vibration, or 11
vibration. Conversely, while the higher note makes one
116
ON CONSONANT TRIADS.
vibration, the lower makes this of a vibration. The same
reasoning being equally applicable to other cases, it fol-
lows that any fraction greater than unity denotes the
number of vibrations, and fractions of a vibration, made
by the higher of two notes forming a certain interval,
while the lower note is making a single vibration. Simi-
larly, any fraction less than unity indicates the proportion
of a whole vibration performed by the lower note, while
the upper is making one complete vibration.
The rules for adding and subtracting intervals shall
next be laid down.
95. Suppose that, starting from a given note, a second
note, a Fifth above it, is sounded, and then a third note,
a Major Third above the second. What will be the vibra-
tion-fraction of the interval formed by the first and third
notes, i. e., of the sum of a Fifth and a Major Third ?
We will, for shortness, call the three notes (1), (2), (3), in
order of ascending pitch. The vibration-fractions being,
for (1)—(2), ), and, for 2–3, , tre proceed thus :
While (2) makes 4 vibrations, (3) makes 5 vibrations.
Therefore, while (2) makes 1 vibration, (3) makes & vibrations.
Therefore, while (2) makes 3 vibrations, (3) makes 3 x vibrations.
But while (2) makes 3 vibrations, (1) makes 2 vibrations.
Therefore, while (1) makes 2 vibrations, (3) makes 3 x vibrations.
Therefore, while (1) makes 1 vibration, (3) makes { x vibrations.
Our result, then, is the two vibration-numbers multi-
plied together. The reasoning is perfectly general, and
gives us the following rule:
To find the vibration-fraction for the sum of two in-
tervals, multiply their separate vibration fractions together.
96. Next take the opposite case. Let (2) be a Major
Third above (1), and (3) a Fifth above (1), and let the
vibration-fraction for the interval (2)—(3) be required.
ADDITION OF INTERVALS.
147
While (1) makes 4 vibrations, (2) makes 5 vibrations.
Therefore, while (1) makes 1 vibration, (2) makes vibrations.
But, while (1) makes 2 vibrations, (3) makes 3 vibrations.
Therefore, while (1) makes 1 vibration, (3) makes į vibrations.
Hence, while (2) makes vibrations, (3) makes vibrations.
Therefore, while (2) makes of a vibration, (3) makes x of a vibration,
Therefore, while (2) makes 1 vibration, (3) makes x vibrations.
1
The result here is the quotient resulting from the di-
vision of the larger vibration-fraction by the smaller :
hence we have this general rule :
To find the vibration fraction for the difference of two
intervals, divide the vibration fraction of the wider by
that of the narrower interval.
Thus multiplication and division of vibration-frac-
tions correspond to addition and subtraction of intervals.
97. One of the simplest cases of our second rule occurs
when an interval has to be inverted. The "inversion"
of any assigned interval narrower than an octave is the
difference between it and an octave, i. e., the interval
which remains after the first has been subtracted from an
octave. Thus to find the vibration-number for the inver-
sion of the Minor Third we merely have to divide 2 by
, or in other words invert the vibration fraction of the
interval and multiply by 2. This applies to all cases. In
the particular example selected, the result is ; the inver-
sion of the Minor Third is therefore the Major Sixth. The
relation between an interval and its inversion is obviously
mutual, so that each may be described as the inversion of
the other. Accordingly, the inversion of the Major Sixth
is the Minor Third.
The following table shows the three pairs of con-
By simply reducing the numerical results, obtained in SS 95, 96, the
student will establish the following propositions :
“ A Major Third added to a Fifth produces a Major Seventh.”
“A Major Third subtracted from a fifth Icaves a Minor Third.”
59
1
148
ON CONSONANT TRIADS.
sonant intervals narrower than an octave, which stand to
each other in the mutual relation of inversions :
Minor Third ()-Major Sixth (1)
Major Third (1)
-Minor Sixth ()
Fourth (1)-Fifth (1)
The student is advised to verify, by the method of
page 147, the fact that each of these intervals is the in-
version of that placed by its side.
98. A combination of musical sounds of different
pitch is called a “chord.” Hitherto we have considered
only chords of two notes, or “binary” chords. We now
go on to chords of three notes, or, as they are usually
called, "triads." A binary chord is, of course, consonant
if its two notes form a consonant interval. A triad con-
tains three intervals, one between its extreme notes, and
one between the middle note and each of the other two.
In order that the chord may be free from dissonance,
those intervals must all three be concords.
99. We may, then, search for consonant triads in the
following manner. Having selected the lowest of the
three notes at pleasure, choose two others, each of which
forms with the bottom note a consonant interval. Next,
examine whether the interval fornied by the last chosen
notes with each other is also a concord. If so, the triad
itself is consonant. In order to determine all the con-
sonant triads within an octave above the fixed bottom
note, we must assign to the middle and top notes every
possible consonant position with respect to the bottom
note, and reject all such relative positions as give rise to
dissonant intervals between those notes themselves. The
remaining positions will constitute all the .consonant
triads which have for their lowest note that originally
selected. The intervals at our disposal are, for the middle
note, from the Minor Third to the Minor Sixth, and, for
the upper note, from the Major Third to the Major Sixth.
INVERSION OF INTERVALS.
149
In the annexed table the possible positions of the
middle note with respect to the bottom note are shown
in the left-hand vertical column, the name of each inter-
val being accompanied by its vibration-fraction. Thie
possible positions of the top note are similarly shown in
the highest horizontal column. Each space common to
an horizontal and a vertical column contains the vibration-
fraction of the interval formed between the simultaneous
positions of the middle and upper notes named at their
extremities. Where these intervals are dissonant, their
vibration-fractions are inclosed in square brackets. When
they are concords the name of the interval is, in each
case, appended :
Major Third Fourth Fifth Minor Sixth Major Sixth
sha
23 을
​喜
​5.
croo
Minor Third
5.
4
25.
24
25
송
​Fourth
10
Major Third
Major Third
eros
용
​Minor Third
32
25
5
Fourth
5
Fourth
o 各
​8
he
Minor Third Major Third
Fifth
关 ​3 을
​16.
15
IL
9
Y
Minor Sixth
23
24
sfat
100. The following, then, are all the cases :
Middle Note.
Upper Note.
Minor Third
Fifth, or Minor Sixth
Major Third
Fifth, or Major Sixth
Fourth
Minor Sixth, or Major Sixth
1 This table is copied, with slight modifications, from Helmholtz's work.
150
ON CONSONANT TRIADS.
or, in musical notation-
123
Minor Sixth
be
22
多
​g
多
​-Z
They give us two groups of three major, and three
minor, triads, which may be arranged tlus :
Fifth.
Minor Sixth.
(a)
(6)
C)
Minor Third. Fourth.
Fifth.
Major Sixth.
Minor Sixth.
a
)
Minor Third.
Major Third.
Fonrth.
101. Instead of defining our six consonant triads, as
we have done, by the intervals formed by their middle
and top notes with the bottom note, we may define them
by the intervals separating the middle from the bottom
note, and the top from the middle note. In order to
make this change we have, in eacli case, a process of sub-
traction of intervals to perform. Thus the difference be-
tween a Fifth and a Major Third is * x , i. e., , or a
Minor Third.
Proceeding in this way, we find that the top and
middle notes are separated by the following intervals:
(a)
(0)
(c)
(a)
(B)
(y)
Minor Third Fourth Major Third Major Third Fourth Minor Third
(a) {
Hence we may write our two groups as follows:
Minor Third. Fourth.
Major Third.
Minor Third. Fourth,
Major Third. Fourth.
Minor Third.
Minor Third.
Fourth.
(6) { Mine Third
(69)
(a) {
Major Third: (67)
(8) }
Major Third. (a)}
(){Fourth
.
It will now be easy to show that the triads of each
group are very closely connected together. Take (a'),
and let us form another triad from it, by causing its bot-
MAJOR AND MINOR GROUPS.
151
{
tom note to ascend one octave, the other two remaining
where they were. The middle will then become the bot-
tom note, the top the middle note, and the octave of the
former bottom note the top note. Hence the lower inter-
val of the new triad will be the upper interval of the old
one, i. ,
e., a Minor Third. The upper interval of the new
triad will necessarily be the inversion of the interval
which separated the extreme notes of the old triad. This
interval is a Fifth (see (a), p. 150], and its inversion,
by the table on p. 148, is a Fourth. Hence the new
Fourth,
triad is
which is identical with (8').
Minor Third,
If we modify (@') in the same way, the new interval is
the inversion of the Minor Sixth, i. e., tlie Major Third,
, ,
is identi-
cal with (c'). This triad, when similarly treated, brings
is back to (a'), and the cycle of changes is complete. By
an extension of the word "inversion," it is usual to call
the triads (6') and (c) the first and second inversions of
the triad a').
Exactly similar relations hold between the members
of the second group of triads: (B) and (y) are, accord-
ingly, called the first and second inversions of the triad
(a). The proof is exactly like that just given, and will be
easily supplied by the reader.
102. If we choose C as the bottom note of (a) and (a'),
the major and minor groups will be expressed in musical
notation by
and the resulting triad, viz., ( Major, Third,
参 ​be
and
DO
-3 3
(0)
boo!
be
治
​(u')
8 3
(33)
(a)
()
(y')
They may also be defined in the language of Thorough
Bass, which refers erery chord to its lowest note, in ac-
152
ON CONSONANT TRIADS.
cordance with the mode adopted in (a), (6), (C); (a), (B),
(y). Thus the triads (a'), (6'), (c) would be indicated by
the figures , §, respectively, and so would the triads,
(a'), (B'), (y); the differences between Minor and Major
Thirds and Sixths being left to be indicated by the key.
signature.
The positions (a) and (a') are regarded as the funda-
mental ones of each group, (b) (c) and (B) (7) being
treated as derived from them respectively by inversion.
103. The fundamental triads bear the name of their
lowest notes, thus (a) and (a) are called respectively the
major and minor common-chords of C.
The remaining members of each group are not named
after their own lowest note, but after that of tbeir funda-
mental inversion; thus (8) (c) and (B') (y) are respec-
tively major and minor common chords of C in their first
and second inversions.
The reason of this, as far as the major group
cerned, follows, directly from Helmholtz's theory of con-
sonance and dissonance. The notes of the triads (a'), (6'),
(c) are all coincident with individual over-tones of a clang
whose fundamental-tone is the low C,
93
is con-
for (a)
and (6), and the octave above that note for (c): hence
they may be regarded as forming a part of the clang of a
C'sound, and therefore each triad may be appropriately
called by its name.
With the Minor triads this is not so
completely true, because the Eb in (a'), (8'), (y) is not co-
incident with an over-tone of C. The other two notes,
however, are in each case leading partial-tones of the clang
of C, and therefore these triads belong at any rate more
to C than to any other note.
Common-chords of more than three constituent sounds
can only be formed by adding to the consonant triads notes
which are exact octaves above or below those of the triads.
MAJOR AND MINOR EFFECTS.
153
104. The marked distinction existing, for every musi-
cal ear, between the bright open character of major, and
the gloomy veiled effect of minor chords, is attributed by
Helmholtz to the different way in which combination-
tones enter in the two cases. The positions of the first-
"order coinbination-tones, for each of the six consonant
triads, are shown in crotchets in the appended stave, the
primaries being indicated by minims. Each interval gives
rise to its own combination-tone, but, in the cases of the
fundamental position and second in version of the C-
Major triad, two combination-tones happen to coincide.
The reader will at once notice that in the major group
no note extraneous to the harmony is brought in by the
combination-tones. In the minor group this is no longe
the caże.
The fundamental position, and the first inver-
sion, of the triad, both bring in an Ab, which is foreign
to the harmony, and the second inversion involves an
additional extraneous note, Bb. The position of these
adventitious sounds is not such as to produce dissonance,
for which they are too far apart from each other and from
the notes of the triad ; but they cloud the transparency
of the harmony, and so give rise to the effects character-
istic of the ininor inode.
DO
3
The unsatisfying character of Minor, compared with
Major, triads, comes out with peculiar distinctness on the
harmonium; as indeed, from the powerful combination-
tones of that instrument, we should naturally have an-
ticipated.
CHAPTER X.
ON PURE INTONATION AND TEMPERAMENT.
9
89
8)
103. THE vibration-fractions of the intervals formed
by the successive notes of the Major scale with the tonic,
are, including the octave of the tonic, these :
吾​,言​, 是​, 8 $ 10 2.
The intervals between successive notes of the scale are
determined by dividing each of these fractions by that
which precedes it. Thus the consecutive intervals of the
Major scale come out as follows:
:!
d
D
E
F
G
А В
a
coles
10
9
1 부
​16
15
రా120
10
g
ocko
16
15
Only three different intervals are obtained. A is
slightly wider than 10; 1. decidedly narrower than the
other two. f and 10 are called whole tones, 19 a half-
tone or semi-tone, though, strictly speaking, two intervals
of this width added together somewhat exceed the greater
of the two whole tones; since 1 x1f or is to f in the
ratio of 2048 to 2025.
Suppose we had a keyed instrument containing a
number of octaves, each divided into seven notes, forming
the ordinary scale as above: any music could be played
on it which did not introduce notes foreign to the key of
PURE INTONATION.
155
C Major. But, now, suppose we wanted to be able to
play in another Major key as well as in that of C, for in-
stance G. It would be necessary for this purpose to in-
troduce two new notes in every octave of the key-board.
If G is the new tonic, A will not serve us as the second
of its scale, because the step between tonic and second is,
not 1, but f. Hence we must have a fresh note lying
between A and B. Further, F will not do for the ser-
enth of the scale of G, as it is separated from G by 8, in-
stead of 14. This necessitates a second additional note
lying between F and G. If we take, as our original oc-
tave, that from middle-C upward, we bave the following
vibration-numbers:
C D E F
А B C
264 297 330 352 396 410 495 528
The new notes, being respectively & above, and is be-
low G, have for their vibration numbers & * 396, and 18 x
396, i. e., 4451 and 3711. The other notes of the scale
of G Major can be supplied froin that of C Major. Hence
these two scales are closely connected with each other.
Another key nearly related to the key of C is that of F.
Its Fourth is $ x 352, or 4691, which falls between A and
B. Its Major Sixth is x 352, or 5863, which is clearly
not an exact octave of any note between C and C". The
corresponding note in our octave, found by division by 2,
is 293}, which comes between C and D. Thus, two more
new notes in the octare must be introduced, to make the
key of F inajor attainable.
106. In order that the reader may see, at a glance,
the variety of sounds which are requisite to supply com-
plete Major scales for the seven keys of C, D, E, F, G,
A and B, the vibration-numbers for all the notes of these
scales are calculated out and cxbibited in the following
table:
156
INTONATION AND TEMPERAMENT.
Tonic
Second Major Third Fourth Fifth Major Sixth Major Seventh
C, 264
297
330
352
396
440
495
D, 297
334
3712
396
4451
4.95
5567
E, 330
3714
4121
440
495
550
6184
F, 352
396
440
4694
528
586%
660
G, 396
4451
495
528
594
660
7421
A, 440
495
550
5863
660
733}
825
B, 495
5563
6184
660
7427
825
9287
Reducing those notes which lie beyond the octave, by
dividing them by 2, and arranging the results in order of
magnitude, we have twelve notes foreign to the scale of C
Major, the positions of which, with reference to the notes
of that scale, are as follows:
C, 275, 278, 293}, D, 3099, E, 3343, F, 3663, 3714,
G, 4121, 4171, A, 4453, 4641, 4697, B.
107. If it is desired to be able to play in the Minor
mode of each of the seven keys, as well as in the Major,
additional notes will be called for. Each scale must con-
tain three Minor intervals, viz., Minor Third, Minor
Sixth, and Minor Seventh. The following subsidiary
table exhibits the vibration-numbers of the sounds form-
ing these intervals, with the successive key-notes:
Reducing these to one octave, as before, we find seven
notes not included in the previous list, occupying the fol-
lowing positions :
C, 281}, D, 312, 3164, E, F, 356, 3911, G, 422, A,
4757, B.
108. Hence, to play perfectly in tune in both Major
and Minor modes of the seven keys C, D, E, F, G, A,
REQUISITES FOR PURE INTONATIONS.
157
Tonic
Minor Third
Minor Sixth
Minor Seventh
C, 264
316
422
4697
D, 297
3563
4755
528
E, 330
396
528
586
}', 352
422
563}
6253
G, 396
475}
633
704
A, 440
528
704
7823
B, 495
594
792
880
B, it is necessary to have a key-board with twenty-six
notes in every octave. This number, large as it is, by no
means includes all necessary notes. Modern music is
written in sharp and flat keys, i, e., in such whose tonics
are not coincident with any one of the notes CDE...B.
Moreover, the sharp and flat key-notes are different from
each other. Thus G#, being a Major Third above E, is,
as the first table shows, 4121; while Ab is seen, by the
second table, to be 422}, which is a somewhat sharper
note. As the seven keys which have been already ex-
amined require 26 notes in the octave, we may anticipate
that the ten additional sharp and flat keys will bring in a
still larger number. It is needless to institute a detailed
inquiry into these scales, but, after what he has already
seen, the student will feel no surprise when he learns that
a competent authority,' who has examined the subject
most minutely in reference both to melodic and harınonic
requisites, fixes 72 notes in the octave as the number es-
sential to theoretically complete command over all the
keys used in modern music.
109. Without, however, assuming this result, the facts
1 Mr. A. J. Ellis, “Proceedings of the Royal Society," vol. xiii , p. 98.
158
:
INTONATION AND TEMPERAMENT.
coul'se.
1
we have already ourselves established are amply sufficient
to show how serious are the imperfections of tune which
inevitably beset instruments with fixed tones, such as the
piano-forte, harmonium, and organ, containing only twelve
notes in each octave. Pure intonation in the natural »
keys alone, i. e., those whose tonics are white notes on the
board, demands, as has been seen, more than twice this
number of available sounds; and many more still, if the
keys with tonics on the black notes are to be included.
Perfect tuning in all the keys being entirely out of the
question, a compromise of some kind is the only possible
Thus we may tune a single key, say C, perfect-
ly; in which case most of the other keys will be so out
of tune as to be unbearable. Or, again, we may dis-
tribute the errors over certain often-used keys, and ac-
cumulate them in others which are of less frequent occur-
rence.
Expedients of this kind are described as modes of
tempering,” and the system adopted in tuning any par-
ticular instrument is called its “temperament.
temperament.” A vast
number of different methods of tempering have been pro-
posed and tried during the history of the organ and
piano-forte.
110. That which has at last been almost universally
adopted is the system of equal temperament. It consists
in dividing each octave into twelve precisely equal inter-
vals. Each of these intervals is called a semi-tone, and
any two of them together a.whole tone.
The octave of which C, 264, is the lowest note, will
contain, on the equal-temperament system, the followiifg
sounds. The vibration-numbers are given true to the
nearest integer. When a note is slightly sharper than
that so indicated, this is shown by the sign + attached to
the vibration-number in question; when slightly flatter,
by the sign -. For the sake of comparison, the perfect
" TEMPERING" AND " TEMPERAMENT."
159
intervals of the same scale are written below the tem-
pered ones :
O, C#, D, D#, E, F, F# G
G, G# A, A#, B,
264, 290_, 296+, 314-, 333, 352+, 373+, 395+, 419+, 444–, 470+, 498+.
C,
D, Ib, E, F,
G, Ab A., Гь, В,
264,
297, 317—, 330, 352,
396,
440,
495.
424+,
469 +,
It is clear that the regions of the tempered scale
where the tuning is the most imperfect, are in the neigh-
borhood of the Thirds and Sixths. E and A are nearly
three vibrations per second too sharp. The Fourth and
Fifth are less out of tune-in fact, only wrong by a frac-
tion of a vibration per second.
111. The intervals of the tempered scale are so near-
ly equal to those of the perfect scale, that, when the
notes of the former are sounded successively, it requires
a delicate ear to recognize the defective character of the
tuning. When, however, more than one note is heard at
a time, the case becomes quite different. We saw in
Chapter VIII. how rigorously accurate the tuning of a con-
sonant interval must be, to secure the greatest smooth-
ness of which it is capable. It was also shown that such
intervals are generally very closely bounded by harsh dis-
cords. Now, since, in the system of equal temperament,
no interval except that of the octave is accurately in
tune, it follows that every representative of a concord, in
its scale, must be less smooth than it would be were the
tuning perfect. One of the greatest charms of music, and
especially of modern music, lies in the vivid contrast
presented by consonant and dissonant chords in close
juxtaposition. Temperament, by impairing, even though
but slightly, the perfection of the concords, necessarily
somewhat weakens this contrast, and takes the edge off
the musical pleasure which, in the hands of a great com-
poser, it is capable of giving us. A fact already once ad-
verted to (page 126) may be again adduced here, as
160
INTONATION AND TEMPERAMENT.
illustrating the effect of temperament in blurring distinc-
tions of consonance and dissonance, viz., that on the key-
board of the piano-forte, the same two notes which repre-
sent C Ab, which is a concord, though not a very smooth
one, also appear in C-G#, which is a decided discord. A
reference to $ 108 will show that, with perfect tuning,
G# and Ab are different notes having vibration-numbers
in the proportion of 4124 to 422.
One of the readiest ways of recognizing the defective
character of equal temperament tuning is, first, allow a
few accurate voices to sing a series of sustained chords in
three or four parts, without accompaniment, and then,
after noticing the effect, to let them repeat the phrase
while the parts are at the same time played on the piano-
forte. The sour character of the concords of the accom-
panying instrument will be at once decisively manifested.
Voices are able to sing perfect intervals, and their clear
transparent concords contrast with the duller substitutes
provided by the piano-forte in a way obvious to every
moderately acute ear.
112. Since the voice is endowed with the power of
producing all possible shades of pitch within its compass,
and thus of singing absolutely pure intervals, it is clear
that we ought to make the most of this great gift, and,
especially in the case of those persons who are to be pub-
lic singers, allow, during the years of preparation, no
contact save with the purest examples of intonation. Un-
fortunately, the practice of most singing-masters is the
very reverse of this. The pupil is systematically accom-
panied, during vocal practice, on the piano-forte, and thus
accustomed to habitual familiarity with intervals which
are never strictly in tune. No one can doubt the tenden-
cy of such constant association to impair the sensitive-
ness to minute differences of pitch on which delicacy of
musical perception depends. Evil communications are
VOCAL INTONATION.
161
not less corrupting to good ears than to good manners. I
feel convinced that we have here the reason why so com-
paratively few of our trained vocalists, whether amateurs
or professionals, are able to sing perfectly in tune. The
untutored voice of a child who has never undergone the
ear-spoiling process, often gives more pleasure by the
natural purity of its intonation, than the vocalization of
an opera-singer who cannot keep in tune. The remedy
is to practise without accompaniment, or with that of an
instrument like the violin,' which is not tied down to a
few fixed sounds. Even with the piano-forte something
might be done, by having it, when intended to be used
only in assisting vocal practice, put into perfect tune in
one single key, and using that key only.
The services of such an instrument would, no doubt,
be comparatively very restricted, but this might not be
without a corresponding advantage, if the vocalist were
thereby compelled to rely a little more on his own un-
aided ear, lay aside his corks, and swim out boldly into
the ocean of sound.
113. The musical notation in ordinary use evidently
takes for granted a scale consisting of a limited number
of fixed sounds. Moreover, it indicates, directly, absolute
pitch, and, only indirectly, relative pitch. In order to
ascertain the interval between any two notes on the stave,
we must go through a little calculation, involving the
clef, the key-signature, and, perhaps, in addition, “acci-
dental” sharps or flats. Now, these are complications,
1 That a violinist can play pure intervals has been established by Prof.
Helmboltz by the following decisive experiment, performed with the aid of
Herr Joachim: A harmonium was employed which had been tuned so as
to give pure intervals with certain stops and keys; and tempered intervals
with others. A string having been tuned in unison with a common tonic
of both systems, it was found that the intervals played by the eminent
violinist agreed with those of the natural, not with those of the tempered
scale.
102
INTONATION AND TEMPERAMENT.
which, if necessary for piano-forte inusic, are perfectly
gratuitous in the case of: vocal music. The voice wants
only to be told on what note to begin, and what inter-
vals to sing afterward, i. e., it is concerned with absolute
pitch only at its start, and needs to be troubled with it
no further. Hence, to place the ordinary notation before
a child who is to be taught to sing, is like presenting him
with a manual for learning to dance, compiled on the
theory that human feet can only move in twelve different
ways. Not only does the established notation encumber
the vocalist with information which he does not want; it
fails to communicate the one special piece of information
which he does want. It is essential to really good music
that every note heard should stand in a definite relation-
ship to its tonic or key-note. Now, there is nothing in
the established notation to mark clearly and directly
what this relation ought, in each case, to be. Unless the
vocalist, besides his own “part,” is provided with that of
the accompaniment, and possesses some knowledge of
harinony, he cannot ascertain how the notes set down for
him are related to the key-note and to each other. The
extreme inconvenience of this must have become pain-
fully evident to any one who has frequently sung con-
certed music from a single part.
A bass, we will suppose, after leaving off on F#, is
directed to rest thirteen bars, and then come in fortissimo
on liis high Eb. It is impossible for him to keep the ab-
solute pitch of F# in his head during this long interval,
which is perhaps occupied by the other voices in modu-
lating into soine remote key; and his part vouchsafes no
indication in what relation the Eb stands to the notes, or
chords, immediately preceding it. There remains, then,
nothing for him to do but to sing, at a venture, some
note at the top of his voice, in the hope that it may prove
to be Eb, though with considerable dread, in the oppo-
ESTABLISHED MUSICAL NOTATION.
163
site event, of the conspicuous ignominy of a fortissimo
blunder,
The essential requisite for a system of vocal notation,
therefore, is that, whenever it specifies any sound, it shall
indicate, in a direct and simple manner, the relation in
which that sound stands to its tonic for the time being.
A method by which this criterion is very completely
satisfied shall now be briefly described.
114. The old Italian singing-masters denoted the sev-
en notes of the Major scale, reckoned from the key-note
upward, by the syllables
do, re, mi, fa, sol, la, si,
pronounced, of course, in the Continental fashion. As
long as a melody moves only in the Major mode, without
modulation, it clearly admits of being written down, as
far as relations of pitch only are concerned, by the use of
these syllables. The opening phrase of “Rule Britan-
nia,” for instance, would stand thus :
do, do, do, re, mi, fa, sol, do, re, re, mi, fa, mi.
In order to abridge the notation, we may indicate
each syllable by its initial consonant. The ambiguity
which would thus arise between sol and si is got rid of
by altering the latter syllable into ti. In order to distin-
guish a note from those of the same name in the adjacent
octaves above and below it, an accent is added, either
above or below the corresponding initial. Thus d' is an
octave above d; d, an octave below d.
Where a modulation, i. e., a change of tonic, occurs, it
is shown in the following manner.: A note necessarily
stands in a twofold relation to the outgoing and the in-
coming tonic. The interval it forms with the new tonic
is different froin that which it formed with the old one.
Each of these intervals can be denoted by a suitable sylla-
ble-initial, and the displacement of one of these initials by
the other represents in the aptest manner the superses-
*164
INTONATION AND TEMPERAMENT.
sion of the old by the new tonic. The old initial is writ-
ten above and to the left of the new one. Thus f indi-
cates that the note re is to be sung, but its name changed
to fa. As this is a somewhat difficult point, a few modu-
lations are appended, expressed both in the established
notation, and in that now under consideration. The in-
stances selected are, from C to G; from C to F; from
E to C; from G to F#.
E
A
ਸbe
bo
a
f
การ
a
d
ds
f
m
d
#
RD
ta
#
8
St
d
m
mf
m2
a
Immediately after a modulation, the ordinary syllable-
initials come into use again, and are employed until a
fresh modulation occurs. It will be seen at once that
the difficulty of “remote keys,” which is so serious in the
established notation, thus altogether disappears. For in-
stance, a vocal phrase from Spohr's “ Last Judgment,"
which, in the established notation, is as follows:
be
e-bebe-bebe
96
boke
takes, in the notation before us, the perfectly siinple form,
sit amfs's fillslf m.
As another example, take the following from the same
work:
180
220
冲
​fт 7 8 t a
a r r f t a
TONIC SOL-FA NOTATION.
165
115. The system of notation, of which a cursory sketch
has just been given, originated, it is said, with two Nor-
wich ladies named Glover, but has received its present
form at the hands of Mr. J. Curwen, to whom it also owes
the name of " Tonic Sol-Fa," by which it is now so widely
known. As it is no part of the plan of the present work
to go into technical details, only so much has been said
about Mr. Curwen's system as was necessary to enable
the reader to grasp its essential principle. No mention
has been made of the notation for Minor and Chromatic
intervals, nor of that for denoting the relations of time by
measures appealing directly to the eye, instead of by
mere symbols. On these and all other points connected
with his system, Mr. Curwen's published works on Tonic
Sol-Fa give full and thoroughly lucid and intelligible ex-
planations. Mr. Curwen has also created a very extensive
literature of the best vocal music, printed in his own nota-
tion; given a most remarkable impulse to choral singing;
and established a system of graded certificates examina-
tions, guaranteeing the attainment, by their holders, of
corresponding stages of musical cultivation.
I have enjoyed some opportunities of watching the
progress of beginners taught on the old system, and on
that of the Tonic Sol-Fa, and assert, without the slightest
hesitation, that, as an instrument of vocal training, the
new system is enormously, overwhelmingly, superior to
the old. In fact, I am prepared to maintain that the com-
plicated repulsiveness of the pitch-notation, in the old
system, must be held responsible for the bumiliating fact
that, of the large number of musically well-endowed per-
sons of the opulent classes who have undergone at school
an elaborate instrumental and vocal training, compara
tively few are able to play, and still fewer to sing, even
the very simplest music at sight. Set an average young
lady to accompany a ballad, or to sing a psalm-tune she
166
INTONATION AND TEMPERAMENT.
has never seen before, and we all know what the result is
likely to be. Now, there is no more inherent difficulty in
teaching a child with a fairly good ear to sing at sight,
than there is in making him read ordinary print at sight.
A vocalist who can only sing a few elaborately prepared
songs ought to be regarded as on a level with a school-
boy who should be unable to read except out of his own
book. If evidence be wanted to make good this asser-
tion, it is at once to hand in the fact that the youngest
children, when well trained on the Tonic Sol-Fa system,
soon obtain a power of steady and accurate sight-singing,
and will even tell you whether a new tune pleases them
or not, after merely glancing through it, without uttering
a note.
The reader will please to observe that the above re-
marks are strictly limited to the achievements of the
Tonic Sol-Fa system in teaching singing. I express no
opinion as to the applicability of its notation to instru-
mental music, nor do I wish to maintain that even in the
vocal branch it has arrived at absolute perfection. On
the contrary, I am doubtful whether its time-notation,
when applied to very complicated rhythmic divisions,
does not become more difficult than the system in ordinary
use, and I consider the notation adopted for the Minor
inode to be capable of decided improvement. On the
main point, however, viz., the decisive superiority of its
pitch-notation over that of the established system, and the
vitally important consequences as to purity of intonation
which necessarily follow from this superiority, I desire to
express the most confident and uncompromising opinion.
116. In closing the inquiry which occupies the pre-
ceding chapters, it will be advisable to examine, very
concisely, the bearing of our principal result, the theory
of consonance and dissonance, on the æsthetics of music.
Dissonance was shown to arise from rapid beats, and the
TONIC SOL-FA NOTATION.
16'7
concords were classed in order, according to their more
or less complete freedom from dissonance; the octave
coming first, followed by the Fifth, Fourth, Major Third
and Sixth, and Minor Third and Sixth. This classifica-
tion was strictly physical, depending exclusively on
smoothness of combined effect. On its own ground,
therefore, it is absolutely unassailable, and whoever says,
for instance, that a Major Third is a smoother concord
than a Fifth or octave, asserts what is as demonstrably
false as that the moon goes round the earth in an exact
circle. Nevertheless, it by no means necessarily follows
that the smoothest concords must be the most gratifying
to the ear. There may be some other property of an in-
terval which gives us greater satisfaction than mere con-
sonance. Assuming, for the moment, that such a prop-
erty does in fact exist, the ear, if called on to arrange the
consonant intervals in the order of their pleasantness,
might very well bring out a different arrangement from
that adopted by physical science on grounds of smooth-
ness alone. Æsthetic considerations come in here, with
the same right to be heard as mechanical consideratious
have within their own domain.
117. Now, unquestionably the ear's order of merit is
not the same as the mechanical order. It places Thirds
and Sixths first, then the Fourth and Fifth, and the octave
last of all. The constant appearance of Thirds and Sixtlis
in two-part music, compared with the infrequent employ-
ient of the remaining concords, leaves no doubt on this
point. In fact, these intervals have a peculiar richness
and permanent charm about them, not possessed by the
Fourth or Fifth to any thing like the same extent, and
by the octave not at all.
The thin effect of the octave undoubtedly depends on
the fact that every partial-tone of the higher of two clangs
forming that interval coincides exactly with a partial-
168
INTONATION AND TEMPERAMENT.
tone of the lower clang. Thus no new sound is intro-
duced by the higher clang; the quality of that previously..
heard is merely modified by the alteration of relative
intensity among the constituent partial-tones. Major and
Minor Thirds bring in a greater variety of pitch in the
resulting mass of sound than does the Fifth ; but this can
hardly be said of the Major and Minor Sixths compared
with the Fourth. On the whole, I am inclined to attribute
the predilection of the ear for Thirds and Sixths, over
the other concords, to circumstances connected with its
perception of key-relations, though I am not able to give
a satisfactory account of them. The ear enjoys, in alter-
nation with consonant chords, dissonances of so harsh a
description as to be barely endurable when sustained by
themselves. This constitutes a marked distinction be-
tween it and the other organs of sense. A stench is not
improved by alternating with the most fragrant odors,
nor nauseous food rendered palatable when administered
at intervals between the most delicious plats. A kick
remains a kick, even though it be preceded and followed
by caresses; and repulsive hideousness forms no welcome
element in pictorial or plastic art. As instances of the
kind of discords in which the ear can find delight, take
the following. The chord '
marked * should in each case
be played first by itself, and then in the place assigned to
it by the composer. The effect of the isolated discord is
so intensely harsh, that it is at first difficult to understand
how any preceding and succeeding concords can make it
at all tolerable; yet the sequence, in both the phrases
cited, is of the rarest beauty.
IMPORTANCE OF DISCORDS.
169
*
Last Chorus, BACH'S
* Passion” (St. Matthew).
3
4
923
3 끔
​Þ=64
24
IT
bo
Dik3
-5
64
a
Considerations such as those just alleged tend to show
that, while physical science is absolutely authoritative in
all that relates to the constitution of musical sounds, and
the smoothness of their combinations, the composer's
direct perception of what is musically beautiful must
mainly direct him in the employment of his materials. It
would be a serious error to force upon him a number of
rules planned, on scientific principles, to secure the max-
imum smoothness of effect; since mere smoothness is
often a matter of extremely secondary importance, com-
pared with grandeur of harmony, and masterly move-
ment of parts. The nature of the subject may sometimes
call for a mode of treatment needing exceptional smooth-
In such a case the rules may become of considera-
ble importance. It is well, therefore, that a composer
should know and be able to handle them, but he should
never allow them to fetter his freedom in wielding the
higher and more spiritual weapons of his warfare.
ness.
THE END.
Opinions of the Press on the “ International Scientific Series."
I.
Tyndall's Forms of Water.
I vol., 12mo. Cloth. Illustrated.
Price, $1.50.
“In the volume now published, Professor Tyndall has presented a noble illustration
of the acuteness and subtlety of his intellectual powers, the scope and insight of his
scientific vision, his singular command of the appropriate language of exposition, and
the peculiar vivacity and grace with which he unfolds the results of intricate scientific
research."-N. Y. Tribune.
“ The 'Forms of Water,' by Professor Tyndall, is an interesting and instructive
little volume, admirably printed and illustrated. Prepared expressly for this series, it
is in some measure a guarantee of the excellence of the volumes that will follow, and an
indication that the publishers will spare no pains to include in the series the freshest in-
vestigations of the best scientific minds."-Boston Journal.
“This series is admirably commenced by this little volume from the pen of Prof.
Tyndall. A perfect master of his subject, he presents in a style easy and attractive his
methods of investigation, and the results obtained, and gives to the reader a clear con-
ception of all the wondrous transformations to which water is subjected.”- Churchman.
II.
Bagehot's Physics and Politics.
I vol., 12mo.
Price, $1.50.
t
"! If the 'International Scientific Series ' proceeds as it has begun, it will more than
fulfil the promisc given to the reading public in its prospectus. The first volume, by
Professor Tyndall, was a model of lucid and attractive scientific exposition; and now
we have a second, by Mr. Walter Bagehot, which is not only very lucid and charming,
but also original and suggestive in the highest degree. Nowhere since the publication
of Sir Henry Maine's 'Ancient Law,' have we seen so many fruitful thoughts sug-
gested in the course of a couple of hundred pages. To do justice to Mr. Bage-
hot's fertile book, would require a long article. With the best of intentions, we are
conscious of having given but a sorry account of it in these brief paragraphs. But we
hope we have said enough to commend it to the attention of the thoughtful reader.”-
Prof. JOHN FISKE, in the Atlantic Monthly.
"Mr. Bagehot's style is clear and vigorous. We refrain from giving a fuller ac-
count of these suggestive essays, only because we are sure that our readers will find it
worth their while to peruse the book for themselves; and we sincerely hope that the
forthcoming parts of the International Scientific Series' will be as interesting."
Atheneum,
“Mr. Bagehot discusses an immense varicty of topics connected with the progress
of societies and nations, and the development of their distinctive peculiarities; and his
book shows an abundance of ingenious and original thought."-ALFRED PUSSELI
WALLACE, in Nature.
D. APPLETON & CO., Publishers, 549 & 551 Broadway, N. V.
Opinions of the Press on the “Internationai Scientific Series.”
III.
Foods.
By Dr. EDWARD SMITH.
I vol., 12mo. Cloth. Illustrated.
Price, $1.75.
In making up THE INTERNATIONAL SCIENTIFIC Series, Dr. Edward Smith was se-
lected as the ablest man in England to treat the important subject of Foods. His services
were secured for the undertaking, and the little treatise he has produced shows that the
choice of a writer on this subject was most fortunate, as the book is unquestionably the
clearest and best-digested compend of the Science of Foods that has appeared in our
language.
“The book contains a series of diagrams, displaying the effects of sleep and meals
on pulsation and respiration, and of various kinds of food on respiration, which, as the
results of Dr. Smith's own experiments, possess a very high value. We have not far
to go in this work for occasions of favorable criticism; they occur throughout, but åre
perhaps most apparent in those parts of the subject with which Dr. Smith's name is es-
pecially linked."--London Examiner.
“The union of scientific and popular treatment in the composition of this work will
afford an attraction to many readers who would have been indifferent to purely theoreti-
cal details... Still his work abounds in information, much of which is of great value,
and a part of which could not easily be obtained from other sources. Its interest is de-
cidedly enhanced for students who demand both clearness and exactness of statement,
by the profusion of well-executed woodcuts, diagrams, and tables, which accompany the
volume. The suggestions of the author on the use of tea and coffee, and of the va,
rious forms of alcohol, although perhaps not strictly of a novel character, are highly in.
structive, and form an interesting portion of the volume.”-N. Y. Tribune.
IV.
Body and Mind.
THE THEORIES OF THEIR RELATION.
By ALEXANDER BAIN, LL. D.
I vol., 12mo. Cloth.
Price, $1.50.
PROFESSOR BAIN is the author of two well-known standard works upon the Science
of Mind The Senses and the Intellect,” and “The Emotions and the Will.” He is
onc of the highest living authorities in the school which holds that there can be no sound
or valid psychology unless the mind and the body are studied, as they exist, together.
" It contains a forcible statement of the connection between mind and body, study-
ing their subtile interworkings by the light of the most recent physiological investiga-
tions. The summary in Chapter V., of the investigations of Dr. Lionel Beale of the
embodiment of the intellectual functions in the cerebral system, will be found the
freshest and most interesting part of his book. Prof. Bain's own theory of the connec-
tion between the mental and the bodily part in man is stated by himself to be as follows:
There is 'one substance, with two sets of properties, two sides, the physical and the
mentala double-faced inity'. While, in the strongest manner, asserting the union
of mind with brain, he yet denies the asscciation of union in place, but asserts them
urion of close succession in time,' holding that 'the same being is, by alternate fits, un-
der extended and under unextcnded consciousness."'-Christian Register.
D. APPLETON & CO., Publishers, 549 & 551 Broadway, N. Y.
Opinions of the Press on the “ International Scientific Series."
V.
The Study of Sociology.
By HERBERT SPENCER.
I vol., 12mo. Cloth.
Price, $1.50.
“The philosopher whose distinguished name gives weight and influence to this vol-
ume, has given in its pages some of the finest specimens of reasoning in all its forms
and departments. There is a fascination in his array of facts, incidents, and opinions,
which draws on the reader to ascertain his conclusions. The coolness and calmness of
his treatment of acknowledged difficulties and grave objections to his theories win for
him a close attention and sustained effort, on the part of the reader, to comprehend, fol-
low, grasp, and appropriate his principles. This book, independently of its bearing
upon sociology, is valuable as lucidly showing what those essential characteristics are
which entitle any arrangement and connection of facts and deductions to be called a
science,"_Episcopalian.
This work compels admiration by the evidence which it gives of immense re-
search, study, and observation, and is, withal, written in a popular and very pleasing
style. It is a fascinating work, as well as one of deep practical thought."-Bost. Post.
“Herbert Spencer is unquestionably the foremost living thinker in the psychological
and sociological fields, and this volume is an important contribution to the science of
which it treats... It will prove more popular than any of its author's other creations,
for it is more plainly addressed to the people and has a more practical and less specu-
lative cast. It will require thought, but it is well worth thinking about."-Albany
Evening Hournal.
VI.
The New Chemistry .
By JOSIAH P. COOKE, JR.,
Erving Professor of Chemistry and Mineralogy in Harvard University.
I vol., 12mo. Cloth.
Price, $2.00.
“The book of Prof. Cooke is a model of the modern popular science work. It has
just the due proportion of fact, philosophy, and true romance, to make it a fascinating
companion, either for the voyage or the study."--Daily Graphic.
“This admirable monograph, by the distinguished Erving Professor of Chemistry
in Harvard University, is the first American contribution to The International Scien-
tific Series,' and a more attractive piece of work in the way of popular exposition upon
a difficult subject has not appeared in a long time. It not only well sustains the char-
acter of the volumes with which it is associated, but its reproduction in European coun:
tries will be an honor to American science."--New York Tribune.
"All the chemists in the country will enjoy its perusal, and many will seize upon it
as a thing longed for. For, to those advanced students who have kept well abreast of
the chemical tide, it offers a calm philosophy. To those others, youngest of the class,
who have emerged from the schools since new methods have prevailed, it presents a
generalization, drawing to its use all the data, the relations of which the newly-fledged
fact-seeker may but dimly perceive without its aid. ... To the old chemists, Prof.
Cooke's treatise is like a message from beyond the mountain. They have hcard of
changes in the science; the clash of the battle of old and new theories has stirred them
from afar. The tidings, too, had come that the old had given way; and little more than
this they knew. Prof. Cooke's 'New Chemistry'must do wide service in bringing
to close sight the little known and the longed for. As a philosophy it is elemen-
tary, but, as a book of science, ordinary readers will find it sufficiently advanced.”_
Utica Morning Herald.
D. APPLETON & CO., Publishers, 549 & 551 Broadway, N. Y.
Opinions of the Press on the “International Scientific Series."
VII.
The Conservation of Energy.
By BALFOUR STEWART, LL. D., F. R. S.
With an Appendix treating of the Vital and Mental Applications of the Doctrine.
I vol., 12mo.
Cloth. Price, $1.50.
“The author has succeeded in presenting the facts in a clear and satisfactory manner,
using simple language and copious illustration in the presentation of facts and prin-
ciples, confining himself, however, to the physical aspect of the subject. In the Ap-
pendix the operation of the principles in the spheres of life and mind is supplied by
the essays of Professors Le Conte and Bain."-Ohio Farmcr.
“ Prof. Stewart is one of the best known teachers in Owens College in Manchester.
“The volume of THE INTERNATIONAL SCIENTIFIC SERIES now before us is an ex-
cellent illustration of the true method of teaching, and will well compare with Prof.
Tyndall's charming little book in the same series on 'Forms of Water," with illustra-
tions enough to make clear, but not to conceal his thoughts, in a style simple and
brief."-Christian Register, Boston.
“The writer has wonderful ability to compress much information into a few words.
It is a rich treat to read such a book'as this, when there is so much beauty and force
combined with such simplicity.-Eastern Press.
VIII.
Animal Locomotion;
Or, WALKING, SWIMMING, AND FLYING.
With a Dissertation on Aëronautics.
By J. BELL PETTIGREW, M. D., F.R.S., F. R. S. E.,
F. R.C.P.E.
I vol., 12mo.
Price, $1.75.
“ This work is more than a contribution to the stock of entertaining knowledge,
though, if it only pleased, that would be sufficient excuse for i:s publication. But Dr.
Pettigrew has given his time to these investigations with the ultimate purpose of solv-
ing the difficult problem of Aëronautics. To this he devotes the last fifty pages of his
book. Dr. Pettigrew is confident that man will yet conquer the domain of the air.”—
N. Y. Journal of Commerce.
“Most persons claim to know how to walk, but few could explain the mechanical
principles involved in this most ordinary transaction, and will be surprised that the
movements of bipeds and quadrupeds, the darting and rushing motion of fish, and the
erratic flight of the denizens of the air, are not only anologous, but can be reduced to
similar formula. The work is profusely illustrated, and, without reference to the theory
it is designed to expound, will be regarded as a valuable addition to natural history.
-Omaha Republic.
D. APPLETON & CO., PUBLISHERS, 549 & 551 Broadway, N. Y.
Opinions of the Press on the “ International Scientific Series."
IX.
Responsibility in Mental Disease.
By HENRY MAUDSLEY, M. D.,
Fellow of the Royal College of Physicians; Professor of Medical Jurisprudence
in University College, London,
I vol., 12mo. Cloth,
Price, $1.50.
“Having lectured in a medical college on Mental Disease, this book has been a
feast to us.
It handles a great subject in a masterly manner, and, in our judgment, the
positions taken by the author are correct and well sustained.”-Pastor and People.
“The author is at home in his subject, and presents his views in an almost singu-
larly clear and satisfactory manner.
The volume is a valuable contribution to one
of the most difficult, and at the same time one of the most important subjects of inves-
tigation at the present day."-N. Y. Observer.
“It is a work profound and searching, and abounds in wisdom.”-Pittsburg Com-
mercial.
“ Handles the important topic with masterly power, and its suggestions are prac-
tical and of great value.”-Providence Press.
X.
The Science of Law.
By SHELDON AMOS, M. A.,
Professor of Jurisprudence in University College, London; author of " A Systematic
View of the Science of Jurisprudence," " An English Code, its Difficulties
and the Modes of overcoming them," etc., etc.
I vol., Izmo. Cloth.
Price, $1.75.
“The valuable series of International Scientific' works, prepared by eminent spe-
cialists, with the intention of popularizing information in their several branches of
knowledge, has received a good accession in this compact and thoughtful volume. It
is a difficult task to give the outlines of a complete theory of law in a portable volume,
which he who runs may read, and probably. Professor Amos himself would be the last
to claim that he has perfectly succeeded in doing this. But he has certainly done much
to clear the science of law from the technical obscurities which darken it to minds which
have had no legal training, and to make clear to his 'lay' readers in how true and high a
sense it can assert its right to be considered a science, and not a mere practice."-The
Christian Register.
“The works of Bentham and Austin are abstruse and philosophical, and Maine's
rcquire hard study and a certain amount of special training. The writers also pursue
different lines of investigation, and can only be regarded as comprehensive in the de-
partments they confined themselves to. It was left to Amos to gather up the result
and present the science in its fullness. The unquestionable merits of this, his last book,
are, that it contains a complete treatment of a subject which has hitherto heen handled
by specialists, and it opens up that subject to every inquiring mind. . . . To do justicc
to 'The Science of Law' would require a longer review than we have space for. We
have read no more interesting and instructive book for some time. Its themes concern
every one who renders obedience to laws, and who would have those laws the best
possible. The tide of legal reform which set in fifty years ago has to sweep yet higher
if the flaws in our jurisprudence are to be removed. The process of change cannot be
better guided than by a well-informed public mind, and Prof. Amos has done great
service in materially helping to promote this end."-Buffalo Courier.
D. APPLETON & CO., PUBLISHERS, 54.9 & 551 Broadway, N. V.
Opinions of the Press on the “ International Scientific Series.”
XI.
Animal Mechanism,
A Treatise on Terrestrial and Aërial Locomotion.
By E. J. MAREY,
Professor at the College of France, and Member of the Academy of Medicine.
With
I17
Illustrations, drawn and engraved under the direction of the author,
I vol., 12mo. Cloth.
Price, $1.75
We hope that, in the short glance which we have taken of some of the most im.
portant points discussed in the work before us, we have succeeded in interesting our
readers sufficiently in its contents to make them curious to learn more of its subject-
matter. We cordially recommend it to their attention.
“The author of the present work, it is well known, stands at the head of those
physiologists who have investigated the mechanism of animal dynamics-indecd, we
inay almost say that he has made the subject his own. By the originality of his con-
ceptions, the ingenuity of his constructions, the skill of his analysis, and the persever-
ance of his investigations, he has surpassed all others in the power of unveiling the
complex and intricate movements of animated beings."-Popular Science Monthly.
XII.
History of
of the
the Conflict between
Religion and Science.
By JOHN WILLIAM DRAPER, M. D., LL.D.,
Author of “The Intellectual Development of Europe."
I vol., 12mo.
Price, $1.75.
"This little 'History' would have been a valuable contribution to literature at any
ime, and is, in fact, an admirable text-book upon a subject that is at present engross-
ing thc attention of a large number of the most serious-minded people, and it is no
small compliment to the sagacity of its distinguished author that he has so well gauged
the requirements of the times, and so adequately met them by the preparation of this
volume. It remains to be added that, while the writer has Alinched from no responsi-
bility in his statements, and has written with entire fidelity to the demands of truth
and justice, there is nol a word in his book that can give offense to candid and fair-
mindeci rcaders."--N. V. Evening Post.
“The key-notc to this volume is found in the antagonism between the progressive
tendencies of the human mind and the pretensions of ecclesiastical authority, as devel-
oped in the history of modern science. No previous writer has treated the subject
from this point of view, and the present monograph will be found to possess no less
originality of conception than vigor of reasoning and wealth of erudition. . The
method of Dr. Draper, in his treatment of the various questions that come up for dis-
cussion, is marked by singular impartiality as well as consummate ability. Through-
out his work he maintains the position of an historian, not of an advocate. His tone is
tranquil and serene, as becomes the scarch aſter truth, with no trace of the impassioned
ardor of controversy. He endeavors so far to identify himself with the contending
parties as to gain a clear comprehension of their motives, but, at the same time, he
submits their actions to the tests of a cool and impartial examination." -N. Y. Tribune.
D. APPLETON & CO., PUBLISHERS, 549 & 551 Broadlway, N. v.
THE GREVILLE MEMOIRS.
COMPLETE IN TWO VOLS.
A JOURNAL OF THE REIGNS OF
King George IV. & King William IV.
By the Late CHAS, C. F. GREVILLE, Esq.,
Clerk of the Council to those Sovereigns.
Edited by HENRY REEVE, Registrar of the Privy Council.
12mo. PRICE, $4.00.
This edition contains the complete text as published in the three volumes
of the English edition.
“The sensation created by these Memoirs, on their first appearance, was not out of
proportion to their real interest. They relate to a period of our history second only in
importance to the Revolution of 1688; they portray manners which have now disap-
peared from society, yet have disappeared so recently that middle-aged men can recol-
lect them; and they concern the conduct of very eminent persons, of whom some are
still living, while of others the memory is so fresh that they still seem almost to be con-
temporaneous."'--The Academy.
“Such Memoirs as these are the most interesting contributions to history that can
be made, and the most valuable as well. The man deserves gratitude from his pos-
terity who, being placed in the midst of events that have any importance, and of people
who bear any considerable part in them, sits down day by day and makes a record of
his observations."-Buffalo Courier.
" The Greville Memoirs, already in a third edition in London, in little more than
two months, have been republished by D. Appleton & Co., New York. The three
loosely-printed English volumes are here given in two, without the slightest abridg-
ment, and the price, which is nine dollars across the water, here is only four. It
is not too much to say that this work, though not so ambitious in its style as Horace
Walpole's well-known Correspondence,' is much more interesting. In a word, these
Greville Memoirs supply valuable materials not alone for political, but also for social
history during the time they cover. They are additionally attractive from the large
quantity of racy anecdotes which they contain.”—Philadelphia Press.
“These are a few among many illustrations of the pleasant, gossipy information con-
veyed in these Memoirs, whose great charm is the free and straightforward manner in
which the writer chronicles his impressions of men and events."-Boston Daily Globe.
“As will be seen, these volumes are of remarkable interest, and fully justify the en-
comiums that heralded their appearance in this country. They will attract a large cir-
cle of readers here, who will find in their gossipy pages an almost inexhaustible ſund of
instruction and amusement."-Boston Saturday Evening Gazette.
“Since the publication of Horace Walpole's Letters, no book of greater historical
interest has seen the light than the Greville Memoirs. It throws a curious, and, we
may almost say, a terrible light on the conduct and character of the public men in Eng-
land under the reigns of George IV. and William IV. Its descriptions of those kings
and their kinsfolk are never likely to be forgotten.”-N. Y. Times.
D. APPLETON & CO., PUBLISHERS, 549 & 551 Broadway, N. Y.
RECENT PUBLICATIONS.
THE NATIVE RACES OF THE PACIFIC STATES.
By HERBERT H. BANCROFT. To be completed in 5 vols. Vol. I. now
ready. Containing Wild Tribes: their Manners and Customs.
I vol., 8vo. Cloth, $6; sheep, $7.
“We can only say that if the remaining volumes are executed in the same spirit of
candid and careful investigation, the same untiring industry, and intelligent good sense,
which mark the volume before us, Mr. Bancroft's 'Native Races of the Pacific States
will form, as regards aboriginal America, an encyclopaedia of knowledge not only un
equaled but unapproached. A literary enterprise more deserving of a generous sym-
pathy and support has never been undertaken on this side of the Atlantic.”-FRANCIS
PARKMAN, in the North American Review.
The industry, sound judgment, and the excellent literary style displayed in this
work, cannot be too highly praised.”_Boston Post.
A BRIEF HISTORY OF CULTURE.
By JOHN S. HITTILL. I vol., 12mo. Price, $1.50.
“He writes in a popular style for popular use. He takes ground which has never
been fully occupied before, although the general subject has been trcated more or less
distinctly by several writers. Mr. Hittell's method is compact, embracing a wide
field in a few words, often presenting a mere hint, when a fuller treatment is craved by
the reader; but, although his book cannot be commended as a model of literary art, it
may be consulted to great advantage by every lover of free thought and novel sugges-
tions."-N. Y. Tribune,
THE HISTORY OF THE CONFLICT BETWEEN RE-
LIGION AND SCIENCE.
By John W. DRAPER, M. D., author of « The Intellectual Develop-
ment of Europe.” I vol., 12mo. Cloth. Price, $1.75.
“The conflict of which he treats has been a mighty tragedy of humanity that has
dragged nations into its vortex and involved the fate of empires. The work, though
small, is full of instruction regarding the rise of the great idcas of science and philos-
ophy; and he describes in an impressive manner and with dramatic effect the way re-
ligious authority has employed the sccular power to obstruct the progress of knowledge
and crush out the spirit of investigation. While there is not in his book a word of dis-
respect for things sacred, he writes with a directness of specch, and a vividness of char-
acterization and an unflinching fidelity to the facts, which show him to be in thorough
carnest with his work. The History of the Conflict between Religion and Science
is a fitting sequel to the 'History of the Intellectual Development of Europe,' and will
add to its author's already high reputation as a philosophic historian.”-N. Y. Tribune.
THEOLOGY IN THE ENGLISH POETS.
COWPER, COLERIDGE, WORDSWORTH, and BURNS. By
Rev. STOPFORD BROOKE. I vol., 12mo. Price, $2.
"Apart from its literary merits, the book may be said to possess an independent
value, as tending to familiarize a certain section of the English public with more en-
lightened views of theology."--London Athenæum.
BLOOMER'S COMMERCIAL CRYPTOGRAPH.
A Telegraph Code and Double Index-Holocryptic Cipher. By J. G.
BLOOMER.
I vol., 8vo. Price, $5.
By the use of this work, business communications of whatever nature may be tele.
graphed with secrecy and economy.
D. APPLETON & CO., Publishers, New York,
RECENT PUBLICATIONS.-SCIENTIFIC.
THE PRINCIPLES OF MENTAL PHYSIOLOGY. With their Ap-
plications to the Training and Discipline of the Mind, and the Study of its
Morbid Conditions. By W. B. CARPENTER, F.R. S., etc. Illustrated. 12mo.
737 pages. Price, $3.00.
“The work is probably the ablest exposition of the subject which has been given to the world, and goes
far to establish a new system of Mental Philosophy, upon a much broader and more substantial basis than
it has heretofore stood."-St. Louis Dernocrat.
“Let us add that nothing we have said, or in any limited space could say, would give an adequate con-
ception of the valuable and curious collection of facts bearing on morbid mental conditions, the learned
pliysiological exposition, and the treasure-house of useful hints for mental training, which make this Inrge:
and yet very amusing, as well as instructive book, an encyclopædia of well-classified and often vury
startling psychological experiences."- London Specta:or.
THE EXPANSE OF HEAVEN. A Series of Essays on the Wonders of
the Firmament. By R. A. PROCTOR, B. A.
"A very charming, work; cannot fail to liſt the reader's mind up through Nature's work to Nature's
God.'»--London Standard,
“Prof. R. A. Proctor is one of the very few rhetorical scientists who have the art of making science
popular without making it or themselves contemptible. It will be hard to find anywhere else so much
skill in effective expression, combined with so much genuine astronomical Icarning, as is to be seen in his
new voluine.”—Christian Union.
PHYSIOLOGY FOR PRACTICAL USE. By various Writers. Edited
by James HINTON. With 50 Illustrations. I vol., 12mo. Price, $2.25.
“This book is one of rare value, and will prove useful to a large class in the community. Its chief
recommendation is in its applying the laws of the scienco of physiology to cases of the deranged or diseased
operations of the orgaus or processes of the human system. It is as thoroughly practical as is a book of
fórmulas of medicine, and the style in which the information is given is so entirely devoid of the mystification
of technical or scientific terms that the most simple can easily comprehend it." Boston Gazelle.
Of all the works upon health of a popular character which we have met with for some time, and we
are glad to think that this most important branch of knowledge is becoming more enlarged every day,
the work before us appears to be the simplest, the soundest, and the best."--Chicago Inter-Ocean.
THE GREAT ICE AGE, and its Relations to the Antiquity oť
Man. By James GEIKIE, F. R. S. E. With Maps, Charts, and numerous
Illus-
trations. i vol., thick 12mo. Price, $2.50.
"The Great Ice Age' is a work of extraordinary interest and value. The subject is peculiarly
attractive in the immensity of its scope, and exercises a fascination over the imagination so absorbing that
it can scarcely find espression in words. It has all the charms of wonder-tales, and excites scientific and
unscientific minds alike."-Borloni Gazette.
"Every step in the process is traced with admirable perspicuity and fullness by Mr. Geikie."-Lon-
clon Suurday Review.
66. The Great Ice Age,' by Jaines Geikie, is a book that unites the popular and abstruse elements of
scientific research to a remarkable degree. The author recounts a story that is more romantic than nine
novels out of ten, and we have read the book from first to last with unllagging interest."-Baslon Commer-
cial Bulletin
ADDRESS DELIVERED BEFORE THE BRITISH ASSOCIA-
TION, assembled at Belfast. By JOHN TYNDALL, F. R. S., President. Re-
vised, with additions, by the author, since the delivery. I2mo.
120 pages.
Paper. Price, 50 cents.
This edition of this now famous address is the only one authorized by the author, and contains addi-
tions and corrections not in the newspaper reports.
THE PHYSIOLOGY OF MAN. Designed to represent the Existing State
of Physiological Science as applicd to the Functions of the Human Body. By
Austin FLINT, Jr., M. D. Complete in Five Volumes, octavo, of about 500
pages each, with 105 Illustrations. Cloth, $22.00; sheep, $27.00. Each vol-
ume sold separately. Price, cloth, $4.50; sheep, $5.50. The fifth and last
volume has just been issued.
The above is by far the most complete work on human physiology in the English language. It trents
of the functions of the human body from a practical point of view, and is enriched by many original ex-
periments and observations by the author. Considerable space is given to physiological anatomy, par-
ticularly the structure of glandular organs, the digestive system, norvous system, blood vessels, organs of
special sense, and organs of generation. It not only considers the various functions of the body, from an
experimental stand-point, but is peculiarly rich in citations of the literature of physiology. It is therefore
invaluable as a work of reference for those who wish to study the subject of physiology exhaustively. As
a coinplete treatise on a subject of such interest, it should be in the libraries of literary and scientific men,
as well as in the hands of practitioners and students of medicine. Illustrations are introduced wherever
they are necessary for the elucidation of the text.
D. APPLETON & CO., PUBLISHERS, 549 & 551 Broadway, N. Y.
A New Magazine for Students and Cultivated Readers.
Τ Η Ε
POPULAR SCIENCE MONTHLY,
CONDUCTED BY
Professor E. L. YOUMANS.
THE growing importance of scientific knowledge to all classes of the
community calls for more efficient means of diffusing it. THE POPULAR
SCIENCE MONTHLY has been started to promote this object, and supplies a
want met by no other periodical in the United States.
It contains instructive and attractive articles, and abstracts of articles,
original, selected, and illustrated, from the leading scientific men of differ-
ent countries, giving the latest interpretations of natural phenomena, ex-
plaining the applications of science to the practical arts, and to the opera-
tions of domestic life.
It is designed to give especial prominence to those branches of science
which help to a better understanding of the nature of man; to present the
claims of scientific education; and the bearings of science upon questions
of society and government. How the various subjects of current opinion
are affected by the advance of scientific inquiry will also be considered.
In its literary character, this periodical aims to be popular, without be-
ing superficial, and appeals to the intelligent reading-classes of the commu-
nity. It seeks to procure authentic statements from men who know their
subjects, and who will address the non-scientific public for purposes of ex-
position and explanation.
It will have contributions from HERBERT SPENCER, Professor HUXLEY,
Professor TYNDALL, Mr. DARWIN, and other writers identified with
lative thought and scientific investigation.
THE POPULAR SCIENCE MONTHLY is published in a large
octavo, handsomely printed on clear type. Terms, Five Dollars per annum,
or Fifty Cents per copy.
specu-
OPINIONS OF THE PRESS,
"Just the publication needed at the present day.”-Montreal Gazette.
' It is, beyond comparison, thc best attempt at journalism of the kind ever made in this
country.”-Home Fournal.
- The initial number is admirably constituted."-Evening Mail.
“In our opinion, the right idea has been happily hit in the plan of this new monthly."
--Buffalo Courier.
A journal which promises to be of eminent value to the cause of popular education in
this country.”—N. ¥. Tribune.
IMPORTANT TO CLUBS.
THE POPULAR SCIENCE MONTHLY will be supplied at reduced rates with any periodi-
cal published in this country.
Any person remitting Twenty Dollars for four yearly subscriptions will receive an ex-
tra copy gratis, or five yearly subscriptions for $20.
THE POPULAR SCIENCE MONTHLY and APPLETONS' JOURNAL (weekly), per annum, $8.00
Payment, in all cases, must be in advance.
Remittances should be made by postal money-order or check to the Publishers,
D. APPLETON & CO., 549 & 551 Broadway, New York.
UNIVERSITY OF MICHIGAN
3 9015 00944 7189
.
..
othes
S.
*
Le
ELV
..
.
.
MAX
TY
r.
:
w
:
WW
V.
V.
A.
A1M
.
*
4*.
WY
V
.
M内
​neur
_
.
.
.
.
.
.
A.
"aa'
***
.
-
SO,
PS
Y
US
""
.
24,
.
.
-
ser
..
NA
.
Y 11
..
wit."
W.
EVENA
.
49
.
.
berita tentan
WA
w..',
T.
M.
T.
.
***
M
ANA
,
.
Pin*
w
**
***
*
M
.
UITS
***
***
>
.
w
ZT
.
A.
os
.
u
.
A