Adolphe Ganot.

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plate B there are equidistant circular holes, and in the disc A are an
equal number of holes of the same size, and the same distance from the
centre as those of the plate. These holes are not perpendicular to the
disc ; they are all inclined to the same extent in the same direction in the
plate, and are inclined to the same extent in the opposite direction in the
disc, so that when they are opposite each other they have the appearance
represented in mn,fig. 171. Consequently, when a current of air from
the bellows reaches the hole ;;?, it strikes obliquely against the sides of
the hole n, and imparts to the disc A a rotatory motion in the direction



For the sake of simplicity, let us first suppose that in the movable
disc A there are eighteen holes, and in the fixed plate B only one, which



-229] Measurement of the Number of Vibrations. 183

faces one of the upper holes. The wind from the bellows striking
against the sides of the latter, the movable disc begins to rotate, and
the space between two of its consecutive holes closes the hole in the
lower plate. But as the disc continues to turn from its acquired velocity,
two holes are again opposite each other, a new impulse is produced,
and so on. During a complete revolution of the disc the lower hole is
eighteen times open and eighteen times closed. A series of effluxes and
stoppages is thus produced, which makes the air vibrate, and ultimately
produces a sound when the successive impulses are sufficiently rapid. If
the fixed plate, like the moving disc, had eighteen holes, each hole would
separately produce the same effect as a separate one, the sound would be
eighteen times as intense, but the number of vibrations would not be in-
creased.

In order to know the number of vibrations corresponding to the sound
produced, it is necessary to know the number of revolutions of the disc
A in a second. For this purpose an endless screw on the rod T transmits
the motion to a wheel, a, with 100 teeth. On this wheel, which moves
by one tooth for every turn of the disc, there is a catch, P, which at each
complete revolution moves one tooth of a second wheel, b (fig. 170).
On the axis of these wheels there are two needles, which move round
dials represented in fig. 169. One of these indices gives the number of
turns of the disc A, the other the number of hundreds of turns. By means
of two screws, D and C, the wheel a can be uncoupled from the endless
screw.

Since the sound rises in proportion to the velocity of the disc A, the
wind is forced until the desired sound is produced. The same current is
kept up for a certain time, two minutes for example, and the number of
turns read off. This number multiplied by 18, and divided by 120, indi-
cates the number of vibrations in a second.

With the same velocity the syren gives the same sound in air as in
water ; the same is the case with all gases ; and it appears, therefore, that
any given sound depends on the number of vibrations, and not on the
nature of the sounding body.

The buzzing and humming noise of certain insects is not vocal, but is
produced by very rapid flapping of the wings against the air or the body.
The syren has been ingeniously applied to count the velocity of the undu-
lations thus produced, which is effected by bringing it into unison with the
sound. It has thus been found that the wings of a gnat flap at the rate of
15,000 times in a second.

229. Bellows. In acoustics a bellows is an apparatus by which wind
instruments, such as the syren and oigan pipes, are worked. Between
the four legs of a table there is a pair of bellows, S (fig. 172), which is
worked by means of a pedal, P. D is a reservoir of flexible leather, in
which is stored the air forced in by the bellows. If this reservoir is
pressed by means of weights on a rod, T, moved by the hand, the air is
driven through a pipe, E, into a chest, C, fixed on the table. In this
chest there are small holes closed by leather valves, which can be opened



1 84



by pressing on keys in front ot the box.
placed in one of these holes.



Acoustics. [229-

The syren or sounding pipe is




Fig. 172.

230. Limit of perceptible sounds. Before Savart's researches,

physicists assumed that the ear could not perceive a sound when the
number of vibrations was below 16 for deep sounds, or above 9,000 for
acute sounds. But he showed that these limits were too close, and
that the faculty of perceiving sounds depends rather on their intensity
than on their height ; so that when extremely acute sounds are not heard,
it arises from the fact that they have not been produced with sufficient
intensity to affect the organ of hearing.

By increasing the diameter of the toothed wheel, and consequently the
amplitude and intensity of the vibrations, Savart pushed the limit of acute
sounds to 24,000 vibrations in a second.

For deep sounds, he substituted for the toothed wheel an iron bar
about two feet long, which revolved on a horizontal axis between two
thin wooden plates, about 0-08 of an inch from the bar. As often as the
bar passed, a grave sound was produced, due to the displacement of the
air. As the motion became accelerated, the sound became continuous,
very grave and deafening. By this means Savart found, that with 7 to 8
vibrations in a second, the ear perceived a distinct but very deep sound.

M. Despretz, however, who has investigated the same subject^ disputes
Savart's results as to the limits of deep sounds, and holds that no sound is



-231] Measurement of the Number of Vibrations.



185



audible that is made by less than 16 vibrations per second. Helmholtz
holds that the perception of a sound begins at 30 vibrations, and only has
a definite musical value when the number is more than 40. Below 30
the impression of a number of separate beats is produced. On the other
hand, acute sounds are audible up to those corresponding to 38,000 vibra-
tions in a second.

The discordant results obtained by these and other observers for the
limit of audibility of higher notes, are no doubt due to the circumstance
that different observers have different capacities for the perception of
sounds.

231. Duhamel's graphic method. When the syren or Savart's wheel
is used to determine the exact number of vibrations corresponding to a
given sound, it is necessary to bring the sound which they produce into
unison with the given sound, and this cannot be done exactly unless the
experimenter have a practised ear. M. Duhamel's graphic method is very
simple and exact, and free from this difficulty. It consists in fixing a line
point to the body emitting the sound, and causing it to trace the vibrations
on a properly prepared surface.

The apparatus consists of a wood or metal cylinder, A, fig. 173, fixed




Fig. 173'

to a vertical axis, O, and turned by a handle. The lower part of the axis is
a screw working in a fixed nut, so that, according as the handle is turned
from left to right, or from right to left, the cylinder is raised or depressed.
Round the cylinder is rolled a sheet of paper covered with an inadhesive
film of lampblack. On this film the vibrations register themselves.



1 86 Acoustics. [231-

This is effected as follows : Suppose the body emitting the note to be a
steel rod. It is held firmly at one end, and carries at the other a fine
point which grazes the surfaces of the cylinder. If the rod is made to
vibrate and the cylinder is at rest, the point would describe a short line ;
but if the cylinder is turned, the point produces an undulating trace,
containing as many undulations as the point has made vibrations. Con-
sequently the number of vibrations can be counted. It remains only to
determine the time in which the vibrations were made.

There are several ways of doing this. The simplest is to compare the
curve traced by the vibrating rod with that traced by a tuning-fork
(237), which gives a known number of vibrations per second for example,
500. One prong of the fork is furnished with a point, which is placed
in contact with the lampblack. The fork and the rod are then set
vibrating together, and each produces its own undulating trace. When
the paper is unrolled, it is easy by counting the number of vibrations
each has made in the same distance to determine the number of
vibrations made per second' by the elastic rod. Suppose, for instance,
that the tuning-fork made 150 vibrations, while the rod made 165
vibrations. Now we already know that the tuning-fork makes one
vibration in the ^ part of a second, and therefore 150 vibrations in I, 5 ,;)
of a second. But in the same time the rod makes 165 vibrations;

therefore it makes one vibration in the 1 ^ - of a second, and hence

500 x 165

it makes per second 52? L ..? or 550 vibrations.




CHAPTER III.

THE PHYSICAL THEORY OF MUSIC.

232. Properties of musical tones. A simple musical tone results
from a continuous rapid isochronous vibration, provided the number of
the vibrations falls within the very wide limits mentioned in the last
chapter (230). Musical tones are in most cases compound. The dis-
___^- tinction between a simple and a compound musical tone will be explained
later in the chapter. The tone yielded by a tuning-fork furnished with
a proper resonance box is simple ; that yielded by a wide-stopped organ
pipe, or by a flute, is nearly simple ; that yielded by a musical string is
compound.

Musical tones have three leading qualities, namely, /*'/&, intensity,
and timbre or colour.

i. The pitch of a musical tone is determined by the number of vibra-
tions per second yielded by the body producing the tone.

ii. The intensity of the tone depends on the extent of the vibrations.
It is greater when the extent is greater, and less when it is less. It is, in



-234] Physical Theory of Music. 1 87

fact, nearly or exactly proportional to the square of the extent or amplitude
of the vibrations which produce the tone.

iii. The timbre is that peculiar quality of tone which distinguishes a
note when sounded on one instrument from the same note when sounded
on another. Thus when the C of the treble stave is sounded on a
violin, and on a flute, the two notes will have the same pitch, that is,
are produced by the same number of vibrations per second, and they may
have the same intensity, and yet the two tones will have very distinct
qualities, that is, their timbre is different. The cause of the peculiar
timbre of tones will be considered later in the chapter.

233. Musical intervals. Let us suppose that a musical tone, which
for the sake of future reference we will denote by the letter C, is pro-
duced by m vibrations per second ; and let us further suppose that any
other musical tone, X, is produced by n vibrations per second, n being
greater than m ; then the interval from the note C to the note X is the
ratio n ; ;//, the interval between two notes being obtained by division,
not by subtraction. Although two or more tones may be separately
musical, it by no means follows that when sounded together they produce
a pleasurable sensation. On the contrary, unless they are concordant,
the result is harsh, and usually unpleasing. We have therefore to
enquire what notes are fit to be sounded together. Now when musical
tones are compared, it is found that if they are separated by an interval
of 2 : i, 4 : i, etc., they so closely resemble one another that they may
for most purposes of music be considered as the same tone. Thus, sup-
pose c to stand for a musical note produced by 2m vibrations per second,
then C and c so closely resemble one another as to be called in music by
the same name. The interval from C to c is called an octave, and c is
said to be an octave above C, and conversely C an octave below c. If we
now consider musical sounds that do not differ by an octave, it is found
that if we take three notes, X, Y, and Z, resulting respectively from p, a,
and r vibrations per second, these three notes when sounded together will
be concordant if the ratio of p : q : r equals 4:5:6. Three such notes
form a harmonic triad, and if sounded with a fourth note, which is the
octave of X, constitute what is called in music a major chord. Any of
the notes of a chord may be altered by one or more octaves without
changing its distinctive character; for instance, C, E, G, and c are a chord,
and C, c, e,g, form the same chord.

If, however, the ratio/ : q : r equals 10 : 12 : 15, the three sounds are
slightly dissonant, but not so much so. as to disqualify them from pro-
ducing a pleasing sensation, at least under certain circumstances. When
these three notes and the octave to the lower are sounded together they
constitute what in music is called a minor chord.

234. The musical scale, The series of sounds which connects a
given note C, with its octave, c, is called the diatonic scale or gamut.
The notes composing it are denoted by. the letters C, D, E, F, G, A, B.
The scale is then continued by taking the octaves of these notes, namely,
c, d, e,f, g, a, b, and again the octaves of these last, and so on.

The notes are also denoted by names, viz., do, or ///, re, mi, fa, sol, la, si,



1 88 Acoustics. [234-

do. The relations existing between the notes are these : C, E, G, form
a major triad, G, B, d, form a major triad, and F, A, c, form a major
triad. C, G, and F have, for this reason, special names, being called
respectively the tonic, dominant, and sub-dominant, and the three triads
the tonic, dominant, and sub-dominant triads or chords respectively.
Consequently the numerical relations between the notes of the scale will
be given by the three proportions

C : E : G::4: 5 : 6
G : B : 2D::4 : 5 : 6
F : A : 201:4: 5 :6

Hence if m denotes the number of double vibrations corresponding to
the note C, the number of vibrations corresponding to the remaining
notes will be given by the following table

do re mi fa sol la si do
CDEFGAB^



35,



2111




The intervals between the successive notes being respectively
C to D D to E E to F F to G G to A A to B B to c

9 10 1_6 9 10 9 16

8 ' 15 8 9 8 15

It will be observed here that there are three kinds of intervals, , ^
and \~ ; of these the two former are called a tone, the last a semitone,
because it is about half as great as the interval of a tone. The two tones
however are not identical, but differ by an interval of , which is called a
comma. Two notes which differ by a comma can be readily distinguished
by an educated ear. The interval between the tonic and any note is de-
nominated by the position of the latter note in the scale ; thus the interval
from C to G is a fifth. The scale we have now considered is called the
major scale, as being formed of major triads. If the minor triad were
substituted for the major, a scale would be formed that could be strictly
called a minor scale. As scales are usually written, however, the
ascending scale is so formed that the tonic bears a minor triad, the
dominant and self-dominant bear major triads, while in the descending
scale they all bear minor triads. Practically, in musical composition, the
dominant triad is always major. If the ratios given above are examined,
it will be found that in the major scale the interval irom C to E equals ,
while in the minor scale it equals, \. The former interval is called a
major third, the latter a minor third. Hence the major third exceeds the
minor third by an interval of If. This interval is called a semitone, though
very different from the interval above called by that name.

A complete discussion of the number of notes, and the intervals between
them, will be found in an article by Mr. Ellis, in vol. xiii. of the Proceed-
ings of the Royal Society (p. 93), ' On a perfect Musical Scale.'

235. On semitones and on scales with different key-notes. It will
be seen from the last article that the term * semitone' does not denote
a constant interval, being in one case equivalent to ?| and in another to
||. It is found convenient for the purposes of music to introduce notes



-236] Musical Temperament. 189

intermediate to the seven notes of the gamut ; this is done by increasing
or diminishing those notes by an interval of ff. When a note (say C) is
increased by this interval, it is said to be sharpened, and is denoted by the
symbol Cff , called ' C sharp ' ; that is, CjJ -*- C = If. When it is decreased
by the same interval, it is said to be flattened, and is represented thus
B b , called ' B flat ' ; that is, B -5- B b = ff . If the effect of this be examined,
it will be found that the number of notes in the scale from C up to c
has been increased from seven to twenty-one notes, all of which can be
easily distinguished by the ear. Thus reckoning C to equal i, we have

C Ctt Db D DS Eb E etc.

T 25 27 9 75 6 5 ,,,,._

24 23 8 64 5 4 CtC '

Hitherto we have made the note C the tonic or key note. Any other
of the twenty-one distinct notes above mentioned, e.g. G, or F, or Ctf , etc.,
may be made the key note, and a scale of notes constructed with refer-
ence to it. This will be found to give rise in each case to a series of notes,
some of which are identical with those contained in the series of which
C is the key note, but most of them different. And of course the same
would be true for the minor scale as well as for the major scale, and
indeed for other scales which may be constructed by means of the funda-
mental triads.

236. On musical temperament. The number of notes that arise from
the construction of the scales described in the last article is so great as
to prove quite unmanageable in the practice of music : and particularly
for music designed for instruments with fixed notes, such as the piano-
forte. Accordingly, it becomes practically important to reduce the
number of notes, which is done by slightly altering their just proportions.
This process is called temperament. By tempering the notes, however,
more or less dissonance is introduced, and accordingly several different
systems of temperament have been devised for rendering this dissonance
as slight as possible. The system usually adopted is called the system
of equal temperament. It consists in the substitution between C and
c of eleven notes at equal intervals, each interval being, of course, the
twelfth root of 2, or 1-05946. By this means the distinction be-
tween the semitones is abolished, so that, for example, C# and D 1 ?
become the same note. The scale of twelve notes thus formed is
called the chromatic scale. It of course follows that major triads become
slightly dissonant. Thus, in the diatonic scale, if we reckon C to be I, E
is denoted by 1*25000, and G by 1-50000. On the system of equal tem-
perament, if C is denoted by i, E is denoted by 1-25992 and G by
1-49831.

If individual intervals are made pure while the errors are distributed
over the others, such a system is called that of unequal temperament.
Such a one is Kirnberger's in which nine of the tones are pure.

Although the system of equal temperament has the advantage of
affording with as small a number of notes as possible, the greatest
variety of tones, yet it has the disadvantage that no chord of equally-
tempered instruments, such, as the piano, is quite pure. And as musical



190



Acoustics.



[236-



education mostly has its basis on the piano, even singers and instrumen-
talists usually give equally-tempered intervals. Only in the case of
string quartet players, who have freed themselves from school rules, and
in that of vocal quartet singers, who sing much without accompaniment,
does the natural pure temperament assert itself, and thus produce the
highest effect.

237. The number of vibrations producing: each note. The tuning-
fork. Hitherto we have denoted the number of vibrations corresponding
to the note C by M, and have not assigned any numerical value to that
symbol. In the theory of music it is frequently assumed that the middle C
corresponds to 256 double vibrations in a second. This is the note
which, on a pianoforte of seven octaves, is produced by the white key on
the left of the two black keys close to the centre of the keyboard. This
number is convenient as being continuously divisible by 2. It is, however,
arbitrary. An instrument is in tune provided the intervals between the
notes are correct, when c is yielded by any number of vibrations per
second not differing much from 256. Moreover, two instruments are in
tune with one another if, being separately in tune, they have any one
note, for instance, C, yielded by the same number of vibrations. Con-
sequently, if two instruments have one note in common, they can then be
brought into tune jointly by having their remaining notes separately ad-
justecjwith reference to the fundamental note. A tuning-fork or diapason

is an instrument yielding a constant sound,
and is used as a standard for tuning musical
instruments. It consists of an elastic steel
rod, bent as represented in fig. 174. It is
made to vibrate either by drawing a bow .
across the ends, or by striking one of the
legs against a hard body, or by rapidly sep-
arating the two legs by means of a steel
rod, as shown in the figure. The vibration
produces a note which is always the same
for the same tuning-fork. The note is
strengthened by fixing the tuning-fork on a
box open at one end, called a resonance box.
The standard tuning-fork in any country
represents its accepted concert pitch.

It has been remarked for some years that
not only has the pitch of the tuning fork
been getting higher in the large theatres of
Europe, but also that it is not the same in
London, Paris, Berlin, Vienna, Milan, etc.
Fig. 174- This is a source of great inconvenience both

to composers and singers, and a commission was appointed in 1859 to es-
tablish in France a tuning-fork of uniform pitch, and to prepare a
standard which would serve as an invariable type, In accordance with
the recommendations of that body, a normal tuning-fork has been estab-
lished, which is compulsory on all musical establishments in France, and




-238] Musical Notation. 191

a standard has been deposited in the Conservatory of Music in Paris. It
performs 437*5 double vibrations per second, and gives the standard note
a or la, or the a in the treble stave (238). Consequently, with reference to
this standard, the middle c or do would result from 261 double vibrations
per second.

In England a committee appointed by the Society of Arts, recom-
mended that a standard tuning-fork should be one constructed to yield
528 double vibrations in a second and that this should represent d in the
treble stave. This" number has the advantage of being divisible by 2
down to 33, and is in fact the same as the normal tuning-fork adopted in
Stuttgardt in 1834, which makes 440 vibrations in the second, and like
the French one corresponds to a in the same stave.

238. Musical notation. Musical range. It is convenient to have
some means of at once naming any particular note in the whole range of
musical sounds other than by stating its number of vibrations. Perhaps a
convenient practice is to call the octave, of which the C is produced
by an eight foot organ pipe, by the capital letters C, D, E, F, G, A, B ;
the next higher octave by the corresponding small letters, c,d, e, f, g, a, b,
and to designate the octaves higher than this by the index placed over
the letter thus, </, #, e',f',g f , a', b', and the higher series in a similar
manner. The same principle may be applied to the notes below C thus
the octave below C is C /5 and the next lower one C,,.

Thus we have the series

C /y C,Ccc> d" c'" c*.

In musical writing the notes are expressed by signs which indicate the
length of time during which the note is to be played or sung, and are
written on a series of lines called a stave. Thus







stands for the octave in the treble clef ; of which the top note is the
standard c f and the bottom is the middle c. Where the five lines are
insufficient they are continued both above and below the stave by what are
called ledger lines. In order to avoid confusion, a bass clef is used for the

lower notes ; and it may be remarked that g- , - and HEEEEE

stand for the same note (237) which is the middle c.

The deepest note of orchestral instruments is the E / of the double
bass which makes 41 1 vibrations, taking the keynote as making 440
vibrations in a minute. Some organs and grand pianofortes go as low as
C y with 33 vibrations in a minute, some grand pianos even as low as A ,
with 27^ vibrations. But the musical character of all these notes below
E, is imperfect, for we are near the limit at which the ear can combine



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