SOUND
(1) SOUND AND WAVE MOTION
=317. What is a Sound?=--This question has two answers, which may be
illustrated as follows: Suppose that an alarm clock is set so that it
will strike in one week and that it is placed upon a barren rock in the
Pacific Ocean by sailors who immediately sail away. If when the tapper
strikes the bell at the end of the week no ear is within a hundred
miles, is any sound produced? The two view-points are now made evident,
for some will answer "no" others "yes." Those answering "no" hold that
sound is a _sensation_ which would not be produced if no ear were at
hand to be affected. Those answering "yes" understand, by the term
sound, _a mode of motion capable of affecting the auditory nerves_, and
that sound exists wherever such motions are present. This latter point
of view is called the _physical_ and is the one we are to use in this
study.
[Illustration: FIG. 314.--The tuning fork is vibrating.]
=318. Source of Sound.=--If we trace any sound to its source, it will be
found to originate in a body in rapid motion usually in what is called a
state of _vibration_. To illustrate, take a tuning fork, strike it to
set it in vibration and place its stem firmly against a thin piece of
wood; the sound will be strengthened materially by the vibration of the
wood. If now the vibrating fork is placed with the tips of the prongs in
water, the vibration is plainly shown by the spattering of the water
(Fig. 314). When one _speaks_, the vibrating body is in the _larynx_ at
the top of the windpipe. Its vibration may be plainly felt by the hand
placed upon the throat while speaking.
=319. Sound Media.=--Usually sounds reach the ear through the air. The
air is then said to be a _medium for sound_. Other substances may serve
as a sound medium, for if the head is under water and two stones, also
under water, are struck together a sharp sound is heard. Also if one end
of a wooden rod is held at the ear and the other end of the rod is
scratched by a pin, the sound is more plainly perceived through the wood
than through the air. Think of some illustration from your own
experience of a solid acting as medium for sound. If an electric bell is
placed in a bell jar attached to an air pump, as in Fig. 315, on
exhausting the air the loudness of the sound is found to diminish,
indicating that in a perfect vacuum no sound would be transmitted. This
effect of a vacuum upon the transmission of sound is very different from
its effect upon radiation of heat and light. Both heat and light are
known to pass through a vacuum since both come to the earth from the sun
through space that so far as we know contains no air or other matter.
Sound differs from this in that it is always transmitted by some
material body and cannot exist in a vacuum.
[Illustration: FIG. 315.--Sound does not travel in a vacuum.]
=320. Speed of Sound.=--Everyone has noticed that it takes time for
sound to travel from one place to another. If we see a gun fired at a
distance, the report is heard a few seconds after the smoke or flash is
seen. The time elapsing between a flash of lightning and the thunder
shows that sound takes time to move from one place to another. Careful
experiments to determine the speed of sound have been made. One method
measures accurately the time required for the sound of a gun to pass
between two stations several miles apart. A gun or cannon is placed at
each station. These are fired alternately, first the one at one station
and then the one at the other so as to avoid an error in computation due
to the motion of wind. This mode of determining the speed of sound is
not accurate. Other methods, more refined than the one just described
have given accurate values for the speed of sound. The results of a
number of experiments show that, at the freezing temperature, 0°C., the
speed of sound in air is 332 meters or 1090 ft. a second. The speed of
sound in air is affected by the temperature, increasing 2 ft. or 0.6
meter per second for each degree that the temperature rises above 0°C.
The speed decreases the same amount for each degree C. that the air is
cooled below the freezing point. The speed of sound in various
substances has been carefully determined. It is greater in most of them
than in air. In water the speed is about 1400 meters a second; in wood,
while its speed varies with different kinds, it averages about 4000
meters a second; in brass the speed is about 3500 meters; while in iron
it is about 5100 meters a second.
=321. The Nature of Sound.=--We have observed that sound originates at a
vibrating body, that it requires a medium in order to be transmitted
from one place to another, and that it travels at a definite speed in a
given substance. Nothing has been said, however, of the _mode_ of
transmission, or of the _nature_ of _sound_. Sounds continue to come
from an alarm clock even though it is placed under a bell jar. It is
certain that nothing material can pass through the glass of the jar.
If, however, we consider that _sound is transmitted by waves through
substances_ the whole matter can be given a simple explanation. In order
to better understand the nature of sound a study of waves and wave
motion will be taken up in the next section.
Important Topics
Sound: two definitions, source, medium, speed, nature.
Exercises
1. Give two illustrations from outside the laboratory of the fact that
sound is transmitted by other materials than air.
2. Name the vibrating part that is the source of the sound in three
different musical instruments.
3. Is sound transmitted more strongly in solids, liquids or gases? How
do you explain this?
4. How far away is a steamboat if the sound of its whistle is heard 10
seconds after the steam is seen, the temperature being 20°C.? Compute in
feet and in meters.
5. How many miles away is lightning if the thunder is heard 12 seconds
after the flash in seen, the temperature being 25°C.?
6. Four seconds after a flash of lightning is seen the thunder clap is
heard. The temperature is 90°F. How far away was the discharge?
7. The report of a gun is heard 3 seconds after the puff of smoke is
seen. How far away is the gun if the temperature is 20°C.?
8. An explosion takes place 10 miles away. How long will it take the
sound to reach you, the temperature being 80°F?. How long at 0°F.?
9. How long after a whistle is sounded will it be heard if the distance
away is 1/4 mile, the temperature being 90°F.?
10. The report of an explosion of dynamite is heard 2 minutes after the
puff of smoke is seen. How far away is the explosion the temperature
being 77°F.?
(2) WAVES[N] AND WAVE MOTION
[N] A wave is a disturbance in a substance or medium that is
transmitted through it.
=322. Visible Waves.=--It is best to begin the study of wave motion by
considering some waves which are familiar to most persons. Take for
example the waves that move over the surface of water (Fig. 316). These
have an onward motion, yet boards or chips upon the surface simply rise
and fall as the waves pass them. They are not carried onward by the
waves. The water surface simply rises and falls as the waves pass by.
Consider also the waves that may be seen to move across a field of tall
grass or grain. Such waves are produced by the bending and rising of the
stalks as the wind passes over them. Again, waves may be produced in a
rope fastened at one end, by suddenly moving the other end up and down.
These waves move to the end of the rope where they are _reflected_ and
return. The three types of waves just mentioned are illustrations of
_transverse_ waves, the ideal case being that in which the particles
move at _right angles_ to the path or course of the wave. Such waves are
therefore called _transverse_ waves.
[Illustration: FIG. 316.--Water waves.]
=323. Longitudinal waves.=--Another kind of wave is found in bodies that
are elastic and compressible and have inertia, such as gases and coiled
wire springs. Such waves may be studied by considering a wire spring as
the medium through which the waves pass. (See Fig. 318.)
[Illustration: FIG. 317.--The compression wave travels through the
spring.]
If the end of the wire spring shown in Fig. 317 is struck the first few
turns of the spring will be compressed. Since the spring possesses
elasticity, the turns will move forward a little and compress those
ahead, these will press the next in turn and so on. Thus a _compression_
wave will move to the end of the spring, where it will be reflected and
return. Consider the turns of the spring as they move toward the end.
On account of their _inertia_ they will continue moving until they have
separated from each other _more_ than at first, before returning to
their usual position. This condition of a greater separation of the
turns of the spring than usual is called a _rarefaction_. It moves along
the spring following the wave of compression. The condensation and
rarefaction are considered as together forming a complete wave. Since
the turns of wire move back and forth in a direction parallel to that in
which the wave is traveling, these waves are called _longitudinal_.
[Illustration: FIG. 318.--Longitudinal waves (1) in a spring, (2) in
air, and (3) graphic representation showing wave length, condensations,
and rarefactions.]
=324. The transmission of a sound by the air= may be understood by
comparing it with the process by which a _wave is transmitted by a wire
spring_. Consider a light spring (Fig. 318, 1) attached at the end of a
vibrating tuning fork, _K_, and also to a diaphragm, _D_. Each vibration
of the fork will first compress and then separate the coils of the
spring. These impulses will be transmitted by the spring as described in
Art. 315, and cause the diaphragm to vibrate _at the same rate_ as the
tuning fork. The diaphragm will then give out a sound similar to that of
the tuning fork. Suppose that the spring is replaced by air, and the
diaphragm, by the ear of a person, _E_, (Fig. 318, 2.) when the prong
of the fork moves toward the ear it starts a compression and when it
moves back a rarefaction. The fork continues vibrating and these
impulses move onward like those in the spring at a speed of about 1120
ft. in a second. They strike the diaphragm of the ear causing it to move
back and forth or to vibrate at the same rate as the tuning fork, just
as in the case of the diaphragm attached to the spring.
=325. Graphic Representation of Sound waves.=--It is frequently
desirable to represent sound waves graphically. The usual method is to
use a curve like that in (Fig. 318, 3). This curve is considered as
representing a train of waves moving in the same direction as those in
Fig. 318 1 and 2, and also having the same length. The part of the wave
_A-B_ represents a condensation of the sound wave and the part _B-C_
represents a rarefaction. A complete wave consisting of a condensation
and a rarefaction is represented by that portion of the curve _A-C_. The
portion of the curve _B-D_ also represents a _full wave length_ as the
latter is defined as _the distance between two corresponding parts of
the adjacent waves_. The curve, Fig. (318, 3) represents not only the
wave length, but also the height of the wave or the amount of movement
of the particles along the wave. This is called the _amplitude_ and is
indicated by the distance _A-b_. Since the _loudness_ or intensity of a
sound is found to depend upon the amount of movement of the particles
along the wave, the _amplitude_ of the curve is used to indicate the
loudness of the sound represented. All of the characteristics of a sound
wave may be graphically represented by curves. Such curves will be used
frequently as an aid in explaining the phenomena of wave motion both in
sound and in light.
=326. Reflections of Sound.=--It is found that a wave moving along a
wire spring is reflected when it reaches the end and returns along the
spring. Similarly a sound wave in air is reflected upon striking the
surface of a body. If the wave strikes perpendicularly it returns along
the line from which it comes, if, however, it strikes at some other
angle it does not return along the same line, but as in other cases of
reflected motion, the _direction_ of the _reflected_ wave is described
by the _Law of Reflected Motion_ as follows: _The angle of reflection is
always equal to the angle of incidence_. This law is illustrated in Fig.
319. Suppose that a series of waves coming from a source of sound move
from _H_ to _O_. After striking the surface _IJ_ the waves are reflected
and move toward _L_ along the line _OL_. Let _PO_ be perpendicular to
the surface _IJ_ at _O_. Then _HOP_ is _the angle_ of incidence and
_LOP_ is the _angle of reflection_. By the law of reflected motion these
angles are equal. In an ordinary room when a person speaks the sound
waves reflected from the smooth walls reinforce the sound waves moving
directly to the hearers. It is for this reason that it is usually easier
to speak in at room than in the open air. Other illustrations of the
reinforcement of sound by reflection are often seen. Thus an _ear
trumpet_ (Fig. 320), uses the principle of reflection and concentration
of sound. So-called _sounding boards_ are sometimes placed back of
speakers in large halls to reflect sound waves to the audience.
[Illustration: FIG. 319.--Law of reflection.]
[Illustration: FIG. 320.--An ear trumpet.]
=327. Echoes.=--_An echo is the repetition of a sound caused by its
reflection from some distant surface_ such as that of a building, cliff,
clouds, trees, etc. The interval of time between the production of a
sound and the perception of its echo is the time that the sound takes to
travel from its source to the reflecting body and back to the listener.
Experiments have shown that the sensation of a sound persists about
one-tenth of a second. Since the velocity of sound at 20°C. is about
1130 ft. per second, during one-tenth of a second the sound wave will
travel some 113 ft. If the reflecting surface is about 56 ft. distant a
_short_ sound will be followed immediately by its echo as it is heard
one-tenth of a second after the original sound. The reflected sound
tends to strengthen the original one if the reflecting surface is less
than 56 ft. away. If the distance of the reflecting surface is much more
than 56 ft. however, the reflected sound does not blend with the
original one but forms a distinct echo. The echoes in large halls
especially those with large smooth walls may very seriously affect the
clear perception of the sound. Such rooms are said to have poor
_acoustic_ properties. Furniture, drapery, and carpets help to deaden
the echo because of diffused reflection. The Mormon Tabernacle at Salt
Lake City, Utah, is a fine example of a building in which the reflecting
surfaces of the walls and ceiling are of such shape and material that
its acoustic properties are remarkable, a pin dropped at one end being
plainly heard at the other end about 200 ft. away.
Important Topics
1. Waves: transverse, longitudinal; wave length, condensation,
rarefaction.
2. Wave motion: in coiled spring, in air, on water.
3. Reflection of waves: law, echoes.
Exercises
1. A hunter hears an echo in 8 seconds after firing his gun. How far is
the reflecting surface if the temperature is 20°C.?
2. How far is the reflecting surface of a building if the echo of one's
footsteps returns in 1 second at 10°C.?
3. Why is it easier to speak or sing in a room than out of doors?
4. Draw a curve that represents wave motion. Make it exactly three full
wave lengths, and state why your curve shows this length. Indicate the
parts of the curve that correspond to a condensation and to a
rarefaction.
5. How long does it take the sound of the "pin drop" to reach a person
at the farther end of the building mentioned at the end of Art. 327?
6. An echo is heard after 6 seconds. How far away is the reflecting
surface, the temperature being 70°F.?
7. Why are outdoor band-stands generally made with the back curving over
the band?
8. A man near a forest calls to a friend. In 4 seconds the echo comes
back. How far away is he from the forest?
9. Would it be possible for us ever to hear a great explosion upon the
moon? Explain.
10. If a sunset gun was fired exactly at 6:00 P.M. at a fort, at what
time was the report heard by a man 25 miles away, if the temperature was
10°C.?
(3) INTENSITY AND PITCH OF SOUNDS
[Illustration: FIG. 321.--Graphic representations of (_a_) a noise,
(_b_) a musical sound.]
=328. Musical Sounds and Noises Distinguished.=--The question is
sometimes raised, what is the difference between a _noise_ and a
_musical sound_? The latter has been found to be produced by an even and
regular vibration such as that of a tuning fork or of a piano string. A
noise on the other hand is characterized by sudden or irregular
vibrations such as those produced by a wagon bumping over a stony
street. These differences may be represented graphically as in Fig. 321,
(a) represents a noise, (b) a musical tone.
[Illustration: FIG. 322.--Curve _b_ represents a tone of greater
intensity.]
=329. Characteristics of Musical Sounds.=--Musical tones differ from one
another in three ways or are said to have _three characteristics_, viz.,
_intensity_, _pitch_, and _quality_. Thus two sounds may differ only in
intensity or _loudness_, that is, be alike in all other respects except
this one, as when a string of a piano is struck at first gently, and
again harder. The second sound is recognized as being louder. The
difference is due to the greater _amplitude_ of vibration caused by more
energy being used. Fig. 322 shows these differences graphically. Curve
_b_ represents the tone of greater intensity or loudness, since its
amplitude of vibration is represented as being greater.
=330. Conditions Affecting the Intensity of Sound.=--The intensity of
sounds is also affected by the _area_ of the vibrating body. This is
shown by setting a tuning fork in vibration. The area of the vibrating
part being small, the sound is heard but a short distance from the fork.
If, however, the stem of the vibrating fork is pressed against the panel
of a door or the top of a box, the sound may be heard throughout a room.
The stem of the fork has communicated its vibrations to the wood. The
vibrating area, being greater, the sound is thereby much increased in
intensity, producing a wave of greater amplitude. The same principle is
employed in the sounding boards of musical instruments as in the piano,
violin, etc. It is a common observation that sounds decrease in
loudness as the distance from the source increases. This is due to the
increase of the surface of the spherical sound waves spreading in all
directions from the source. Careful experiments have shown that in a
uniform medium _the intensity of a sound is inversely proportional to
the square of the distance from its source_. If a sound is confined so
that it cannot spread, such as the sound moving through a speaking tube,
it maintains its intensity for a considerable distance. An _ear trumpet_
(see Fig. 320) also applies this principle. It is constructed so that
sound from a given area is _concentrated_ by reflection to a much
smaller area with a corresponding increase in intensity. The _megaphone_
(Fig. 323), and the _speaking trumpet_ start the sound waves of the
voice in one direction so that they are kept from spreading widely,
consequently by its use the voice may be heard several times the usual
distance. The intensity of a sound is also affected by the _density_ of
the transmitting medium. Thus a sound produced on a mountain top is
fainter and thinner than one produced in a valley. The sound of a bell
in the receiver of an air pump becomes weaker as the air is exhausted
from the latter. _Four_ factors thus influence the intensity of a sound,
the _area_ of the vibrating body, its _amplitude_ of vibration, the
_distance_ of the source and the _density_ of the transmitting medium.
It is well to fix in mind the precise effect of each of these factors.
[Illustration: FIG. 323.--The megaphone.]
=331. Pitch.=--The most characteristic difference between musical sounds
is that of _pitch_. Some sounds have a high pitch, such as those
produced by many insects and birds. Others have a low pitch as the notes
of a bass drum or the sound of thunder. How notes of different pitch
are produced may be shown by the siren (Fig. 324). This is a disc
mounted so as to be rotated on an axis. Several rows of holes are
drilled in it in concentric circles. The number of holes in successive
rows increases from within outward. If when the siren is rapidly rotated
air is blown through a tube against a row of holes a clear musical tone
is heard. The tone is due to the succession of pulses in the air
produced by the row of holes in the rotating disc alternately cutting
off and permitting the air blast to pass through at very short
intervals. If the blast is directed against a row of holes nearer the
circumference the pitch is higher, if against a row nearer the center
the pitch is lower. Or if the blast is sent against the same row of
holes the pitch rises when the speed increases and lowers when the speed
lessens. These facts indicate that the pitch of a tone is due to the
number of pulses or vibrations that strike the ear each second; also
that _the greater the rate of vibration, the higher the pitch_.
[Illustration: FIG. 324.--A siren.]
=332. The Major Scale.=--If a siren is made with eight rows of holes, it
may indicate the relation between the notes of a _major scale_. To
accomplish this, the number of holes in the successive rows should be
24, 27, 30, 32, 36, 40, 45, 48. If a disc so constructed is rapidly
rotated at a uniform rate, a blast of air sent against all of the rows
in succession produces the tones of the scale. These facts indicate that
the relative vibration numbers of the notes of any _major scale_ have
the same relation as the numbers 24, 27, 30, 32, 36, 40, 45, 48.
The note called middle C is considered by physicists as having 256
vibrations a second. This would give the following _actual vibration_
numbers to the remaining notes of the major scale that begins with
"Middle C" D.-288, E.-320, F.-341.3, G.-384, A.-426.6, B.-480, C'.-512.
Musicians, however, usually make use of a scale of slightly higher
pitch. The _international_ standard of pitch in this country and in
Europe is that in which "A" has 435 vibrations per second. This
corresponds to 261 vibrations for middle C.
=333. The Relation between Speed, Wave Length, and Number of Vibrations
per Second.=--Since the notes from the various musical instruments of an
orchestra are noticed to harmonize as well at a distance as at the place
produced, it is evident that notes of all pitches travel at the same
rate, or have the _same speed_. Notes of high pitch, having a high
vibration rate produce more waves in a second than notes of low pitch,
consequently the former are shorter than the latter. The following
formula gives the relation between the speed (_v_), wave length (_l_),
and number of vibrations per sec. (_n_):
_v_ = _l_ × _n_, or _l_ = _v/n_
that is, _the speed of a sound wave is equal to the number of vibrations
per second times the wave length, or the wave length is equal to the
speed divided by the number of vibrations per second_. This formula may
also be employed to find the _number_ of vibrations when the wave length
and speed are given.
Important Topics
1. Difference between noise and music.
2. Factors affecting intensity: area, amplitude, density, distance.
3. Pitch, major scale, relative vibration numbers.
4. Relation between speed, wave length and vibration rate.
Exercises
1. Give an illustration from your own experience of each of the factors
affecting intensity.
2. Write the relative vibration numbers of a major scale in which _do_
has 120 vibrations.
3. What is the wave length of the "A" of international concert pitch at
25°C.? Compute in feet and centimeters.
4. At what temperature will sound waves in air in unison with "Middle C"
be exactly 4 ft. long?
5. Explain the use of a megaphone.
6. What tone has waves 3 ft. long at 25°C.?
7. What is the purpose of the "sounding board" of a piano?
8. Two men are distant 1000 and 3000 ft. respectively from a fog horn.
What is the relative intensity of the sounds heard by the two men?
9. The speaking tone of the average man's voice has 160 vibrations per
second. How long are the waves produced by him at 20°C.?
(4) MUSICAL SCALES AND RESONANCE
=334. A musical interval= _refers to the ratio between the pitches[O] of
two notes_ as indicated by the results of the siren experiment. The
simplest interval, or ratio between two notes is the _octave_, C':C, or
2:1 (48:24). Other important intervals with the corresponding ratios are
the _fifth_, G:C, or 3:2 (36:24); the _sixth_, A:C, or 5:3 (40:24); the
_fourth_, F:C, 4:3 (32:24); the _major third_, E:C, or 5:4 (30:24); and
the _minor third_, G:E, 6:5. The interval between any two notes may be
determined by finding the ratio between the vibration numbers of the two
notes. Thus, if one note is produced by 600 vibrations a second and
another by 400, the interval is 3:2, or a _fifth_, and this would be
recognized by a musician who heard the notes sounded together or one
after the other. Below is a table of musical nomenclatures, showing
various relations between the notes of the major scale.
[O] Pitch as used here, means _vibration rate_.
TABLE OF MUSICAL NOMENCLATURES
---------------------+---+------+------+------+------+------+-------+----
Name of note | C | D | E | F | G | A | B | C´
---------------------+---+----- +----- +------+------+------+-------+----
Frequency in terms of|_n_|9/8_n_|5/4_n_|4/3_n_|3/2_n_|5/3_n_|15/8_n_|2_n_
"do" | | | | | | | |
---------------------+---+--+---+--+---+--+---+--+---+--+---+---+---+----
Intervals | 9/8 | 10/9 | 16/15| 9/8 | 10/9 | 9/8 | 16/15
---------------------+---+--+---+--+---+--+---+--+---+--+---+---+---+----
Name of note in vocal|do | re | mi | fa | sol | la | ti | do
music | | | | | | | |
---------------------+---+------+------+------+------+------+-------+----
Treble clef.
[Music]
Bass clef.
[Music]
---------------------+---+------+------+------+------+------+-------+----
International pitch | | | | | | | |
of treble clef |261| 293.6| 326.3| 348. | 391.5| 435 | 489.4 | 522
---------------------+---+------+------+------+------+------+-------+----
Scientific scale |256| 288 | 320 | 341.3| 384 | 426.6| 480 | 512
---------------------+---+------+------+------+------+------+-------+----
Relative vibration | 24| 27 | 30 | 32 | 36 | 40 | 45 | 48
numbers | | | | | | | |
---------------------+---+------+------+------+------+------+-------+----
=335. Major and Minor Triads.=--The notes C, E, G (_do_, _mi_, _sol_)
form what is called a _major triad_. The _relative vibration numbers_
corresponding are 24, 30, 36. These in simplest terms have ratios of
4:5:6. Any three other tones with vibration ratios of 4:5:6 will also
form a major triad. If the octave of the lower tone is added, the four
make a major chord. Thus: F, A, C´ (_fa_, _la_, _do_), 32:40:48, or
4:5:6, also form a major triad as do G, B, D´ (_sol_, _ti_, _re_),
36:45:54, or 4:5:6. Inspection will show that these three major triads
comprise all of the tones of the major scale D´ being the octave of D.
It is, therefore, said that the major scale is based, or built, upon
these three major triads. The examples just given indicate the
mathematical basis for harmony in music. Three notes having vibration
ratios of 10:12:15 are called _minor triads_. These produce a less
pleasing effect than those having ratios of 4:5:6.
=336. The Need for Sharps and Flats.=--We have considered the key of C.
This is represented upon the piano or organ by white keys only (Fig.
325). Now in order (a) to give variety to instrumental selections, and
(b) to accommodate instruments to the range of the human voice, it has
been necessary to introduce other notes in musical instruments. These
are represented by the _black keys_ upon the piano and organ and are
known as _sharps_ and _flats_. To illustrate the necessity for these
additional notes take the major scale starting with B. This will give
vibration frequencies of 240, 270, 300, 320, 360, 400, 450, and 480. The
only white keys that may be used with this scale are E 320 and B 480
vibrations. Since the second note on this scale requires 270 vibrations
about halfway between C and D the black key C sharp is inserted. Other
notes must be inserted between D and E (D sharp), between F and G (F
sharp), also G and A sharps.
[Illustration: FIG. 325.--Section of a piano keyboard.]
=337. Tempered Scales.=--In musical instruments with fixed notes, such
as the harp, organ, or piano, complications were early recognized when
an attempt was made to adapt these instruments so that they could be
played in all keys. For the vibration numbers that would give a perfect
major scale starting at C are not the same as will give a perfect major
scale beginning with any other key. In using the various notes as the
keynote for a major scale, 72 different notes in the octave would be
required. This would make it more difficult for such instruments as the
piano to be played. To avoid these complications as much as possible, it
has been found necessary to abandon the simple ratios between successive
notes and to substitute another ratio in order that the vibration ratio
between any two successive notes will be equal in every case. The
differences between semitones are abolished so that, for example, C
sharp and D flat become the same tone instead of two different tones.
Such a scale is called a _tempered scale_. The tempered scale has 13
notes to the octave, with 12 equal intervals, the ratio between two
successive notes being the ¹²√2 or 1.059. That is, any vibration rate on
the tempered scale may be computed by multiplying the vibration rate of
the preceding note by 1.059. While this is a necessary arrangement,
there is some loss in perfect harmony. It is for this reason that a
quartette or chorus of voices singing without accompaniment is often
more harmonious and satisfactory than when accompanied with an
instrument of fixed notes as the piano, since the simple harmonious
ratios may be employed when the voices are alone. The imperfection
introduced by _equal temperament tuning_ is illustrated by the following
table:
C D E F G A B C
Perfect Scale of C 256.0 288.0 320.0 341.3 384.0 426.6 480.0 512.0
Tempered Scale 256.0 287.3 322.5 341.7 383.6 430.5 483.3 512.0
=338. Resonance.=--If two tuning forks of the same pitch are placed near
each other, and one is set vibrating, the other will soon be found to be
in vibration. This result is said to be due to _sympathetic vibration_,
and is an example of _resonance_ (Fig. 326). If water is poured into a
glass tube while a vibrating tuning fork is held over its top, when the
air column has a certain length it will start vibrating, reinforcing
strongly the sound of the tuning fork. (See Fig. 327.) This is also an
example of resonance. These and other similar facts indicate that _sound
waves started by a vibrating body will cause another body near it to
start vibrating if the two have the same rate of vibration_. Most
persons will recall illustrations of this effect from their own
experience.
[Illustration: FIG. 326.--One tuning fork will vibrate in sympathy with
the other, if they have exactly equal rates of vibration.]
[Illustration: FIG. 327.--An air column of the proper length reinforces
the sound of the tuning fork.]
=339. Sympathetic vibration= is explained as follows: Sound waves
produce very slight motions in objects affected by them; if the
vibration of a given body is exactly in time with the vibrations of a
given sound each impulse of the sound wave will strike the body so as to
increase the vibratory motion of the latter. This action continuing, the
body soon acquires a motion sufficient to produce audible waves. A good
illustration of sympathetic vibration is furnished by the bell ringer,
who times his pulls upon the bell rope with the vibration rate of the
swing of the bell. In the case of the resonant air column over which is
held a vibrating tuning fork (see Fig. 328), when the prong of the fork
starts downward from 1 to 2, a condensation wave moves down to the water
surface and back just in time to join the condensation wave _above_ the
fork as the prong begins to move from 2 to 1; also when the prong
starts upward from 2 to 1, the rarefaction produced under it moves to
the bottom of the air column and back so as to join the rarefaction
_above_ the fork as the prong returns. While the prong is making a
_single_ movement, up or down, it is plain that the air wave moves twice
the length of the open tube. During a _complete_ vibration of the fork,
therefore, the sound wave moves four times the length of the air column.
In free air, the sound progresses a wave length during a complete
vibration, hence the resonant air column is one-fourth the length of the
sound wave to which it responds. Experiments with tubes cf different
lengths show that the diameter of the air column has some effect upon
the length giving best resonance. About 25 per cent. of the diameter of
the tube must be added to the length of the air column to make it just
one-fourth the wave length. The sound heard in seashells and in other
hollow bodies is due to resonance. Vibrations in the air too feeble to
affect the ear are intensified by sympathetic vibration until they can
be heard. A tuning fork is often mounted upon a box called a
_resonator_, which contains an air column of such dimensions that it
reinforces the sound of the fork's sympathetic vibration.
[Illustration: FIG. 328.--Explanation of resonance.]
Important Topics
1. Musical intervals: octave, sixth, fifth, fourth, third.
2. Major chord, 4:5:6.
3. Use of sharps and flats. Tempered scale.
4. Resonance, sympathetic vibration, explanation, examples.
Exercises
1. What is a major scale? Why is a major scale said to be built upon
three triads?
2. Why are sharps and flats necessary in music?
3. What is the tempered scale and why is it used? What instruments need
not use it? Why?
4. Mention two examples of resonance or sympathetic vibration from your
own experience out of school.
5. An air column 2 ft. long closed at one end is resonant to what wave
length? What number of vibrations will this sound have per second at
25°C.?
6. At 24°C. What length of air column closed at one end will be resonant
to a sound having 27 vibrations a second?
7. A given note has 300 vibrations a second. What will be the number of
vibrations of its (a) octave, (b) fifth, (c) sixth, (d) major third?
8. In the violin or guitar what takes the place of the sounding board of
the piano?
9. Can you explain why the pitch of the bell on a locomotive rises as
you rapidly approach it and falls as you recede from it?
10. Do notes of high or low pitch travel faster? Explain.
11. An "A" tuning fork on the "international" scale makes 435 vibrations
per second. What is the length of the sound waves produced?
(5) WAVE INTERFERENCE, BEATS, VIBRATION OF STRINGS
=340. Interference of waves.=--The possibility of two trains of waves
combining so as to produce a reduced motion or a _complete destruction_
of motion may be shown graphically. Suppose two trains of waves of equal
wave length and amplitude as in Fig. 329 meet in _opposite phases_. That
is, the parts corresponding to the _crests_ of _A_ coincide with the
_troughs_ of _B_, also the troughs of _A_ with the crests of _B_; when
this condition obtains, the result is that shown at _C_, the union of
the two waves resulting in complete destruction of motion. _The more or
less complete destruction of one train of waves by another similar train
is an illustration of_ =interference=. If two sets of water waves so
unite as to entirely destroy each other the result is a level water
surface. If two trains of sound waves combine they may so interfere that
silence results. The conditions for securing interference of sound waves
may readily be secured by using a tuning fork and a resonating air
column. If the tuning fork is set vibrating and placed over the open end
of the resonating air column (see Fig. 328), an increase in the sound
through resonance may be heard. If the fork is rotated about its axis,
in some positions no sound is heard while in other positions the sound
is strongly reinforced. Similar effects may be perceived by holding a
vibrating fork near the ear and slowly rotating as before. In some
positions interference results while in other positions the sound is
plainly heard. The explanation of interference may be made clear by the
use of a diagram. (See Fig. 330.) Let us imagine that we are looking at
the two square ends of a tuning fork. When the fork is vibrating the two
prongs approach each other and then recede. As they approach, a
condensation is produced at 2 and rarefactions at 1 and 3. As they
separate, a rarefaction is produced at 2 and condensations at 1 and 3.
Now along the lines at which the simultaneously produced rarefactions
and condensations meet there is more or less complete interference. (See
Fig. 331.) These positions have been indicated by dotted lines extending
through the ends of the prongs. As indicated above, these positions may
be easily found by rotating a vibrating fork over a resonant air column,
or near the ear.
[Illustration: FIG. 329.--Interference of sound waves.]
[Illustration: FIG. 330.--At 2 is a condensation; at 1 and 3 are
rarefactions.]
[Illustration: FIG. 331.--The condensations and rarefactions meet along
the dotted lines producing silence.]
[Illustration: FIG. 332.--Diagram illustrating the formations of beats.]
=341. Beats.=--If two tuning forks of slightly different pitch are set
vibrating and placed over resonating air columns or with the stem of
each fork upon a sounding board, so that the sounds may be intensified,
a peculiar pulsation of the sound may be noticed. This phenomenon is
known as _beats_. Its production may be easily understood by considering
a diagram (Fig. 332). Let the curve _A_ represent the sound wave sent
out by one tuning fork and _B_, that sent out by the other. _C_
represents the effect produced by the combination of these waves. At _R_
the two sound waves meet in the _same phase_ and reinforce each other.
This results in a louder sound than either produces alone. Now since the
sounds are of slightly different pitch, one fork sends out a few more
vibrations per second than the other. The waves from the first fork are
therefore a little shorter than those from the other. Consequently,
although the two waves are at one time in the _same_ phase, they must
soon be in opposite phases as at _I_. Here interference occurs, and
silence results. Immediately the waves reinforce, producing a louder
sound and so on alternately. The resulting rise and fall of the sound
are known as _beats_. The number of beats per second must, of course, be
the same as the difference between the numbers of vibrations per second
of the two sounds. One effect of beats is _discord_. This is especially
noticeable when the number of beats per second is between 30 and 120.
Strike the two lowest notes on a piano at the same time. The beats are
very noticeable.
[Illustration: FIG. 333.--Turkish cymbals.]
[Illustration: FIG. 334.--The cornet.]
=342. Three Classes of Musical Instruments.=--There are three classes or
groups of musical instruments, if we consider the vibrating body that
produces the sound in each: (A) Those in which the sound is produced by
a vibrating _plate_ or _membrane_, as the drum, cymbals (Fig. 333),
etc.; (B) those with vibrating _air columns_, as the flute, pipe organ,
and cornet (Fig. 334), and (C) with vibrating _wires_ or _strings_, as
the piano, violin, and guitar. It is worth while to consider some of
these carefully. We will begin with a consideration of vibrating wires
and strings, these often producing tones of rich quality.
Let us consider the strings of a piano. (If possible, look at the
strings in some instrument.) The range of the piano is 7-1/3 octaves.
Its lowest note, A_{4}, has about 27 vibrations per second. Its highest,
C⁴, about 4176. This great range in vibration rate is secured by varying
the length, the tension, and the diameter of the strings.
=343. The Laws of Vibrating Strings.=--The relations between the
vibration rate, the length, the tension and the diameter, of vibrating
strings have been carefully studied with an instrument called a
_sonometer_ (Fig. 335). By this device it is found that the pitch of a
vibrating string is raised one octave when its vibrating length is
reduced to one-half. By determining the vibration rate of many lengths,
the following law has been derived: (Law I) _The rate of vibration of a
string is inversely proportional to its length._
[Illustration: FIG. 335.--A sonometer.]
Careful tests upon the change of vibration rate produced by a change of
_tension_ or pull upon the strings show that if the pull is increased
four times its vibrations rate is _doubled_, and if it is increased nine
times its rate is tripled, that is: (Law II) _The vibration rates of
strings are directly proportional to the square roots of their
tensions._
Tests of the effects of diameter are made by taking wires of equal
length and tension and of the same material but of different diameter.
Suppose one is twice as thick as the other. This string has a tone an
octave lower or vibrates one-half as fast as the first. Therefore: (Law
III) _The vibration rates of strings are inversely proportional to the
diameters._ These laws may be expressed by a formula _n_ ∝√(_t_)/_dl_.
The vibration of a string is rarely a simple matter. It usually vibrates
in parts at the same time that it is vibrating as a whole. The tone
produced by a string vibrating as a whole is called its _fundamental_.
The vibrating parts of a string are called _loops_ or _segments_ (see
Fig. 336), while the points of least or no vibration are _nodes_.
Segments are often called _antinodes_.
[Illustration: FIG. 336.--A string yielding its fundamental and its
first overtone.]
=344. Overtones.=--The _quality_ of the tone produced by a vibrating
string is affected by its vibration in parts when it is also vibrating
as a whole. (See Fig. 336.) The tones produced by the vibration in parts
are called _overtones_ or _partial_ tones. The presence of these
overtones may often be detected by the sympathetic vibration of other
wires near-by. What is called the _first_ overtone is produced by a
string vibrating in _two_ parts, the _second_ overtone by a string
vibrating in _three parts_, the _third_ overtone by its vibration in
_four_ parts and so on. In any overtone, the number of the parts or
vibrating segments of the string is one more than the number of the
overtone. For example, gently press down the key of middle C of a piano.
This will leave the string free to vibrate. Now strongly strike the C an
octave lower and then remove the finger from this key. The middle C
string will be heard giving its tone. In like manner try E¹ and G¹,
with C. This experiment shows that the sound of the C string contains
these tones as overtones. It also illustrates sympathetic vibration.
Important Topics
1. Interference, beats, production, effects.
2. Vibration of strings, three laws.
3. Three classes of musical instruments.
4. Fundamental and overtones, nodes, segments, how produced? Results.
Exercises
1. What different means are employed to produce variation of the pitch
of piano strings? For violin strings?
2. How many beats per second will be produced by two tuning forks having
512 and 509 vibrations per second respectively?
3. A wire 180 cm. long produces middle C. Show by a diagram, using
numbers, where a bridge would have to be placed to cause the string to
emit each tone of the major scale.
4. How can a violinist play a tune on a single string?
5. What are the frequencies of the first 5 overtones of a string whose
fundamental gives 256 vibrations per second?
6. One person takes 112 steps a minute and another 116. How many times a
minute will the two walkers be in step? How many times a minute will one
be advancing the left foot just when the other advances the right?
7. Why is it necessary to have a standard pitch?
8. How can the pitch of the sounds given by a phonograph be lowered?
9. How many beats per second will occur when two tuning forks having
frequencies of 512 and 515 respectively, are sounded together?
10. Which wires of a piano give the highest pitch? Why?
(6) TONE QUALITY, VIBRATING AIR COLUMNS, PLATES
=345. Quality.=--The reason for the _differences in tone quality_
between notes of the same pitch and intensity as produced, _e.g._, by a
violin and a piano, was long a matter of conjecture. Helmholtz, a German
physicist (see p. 397) first definitely proved that tone quality is due
to the _various overtones_ present along with the fundamental and _their
relative intensities_. If a tuning fork is first set vibrating by
drawing a bow across it and then by striking it with a hard object, a
difference in the _quality_ of the tones produced is noticeable. It is
thus evident that the manner of setting a body in vibration affects the
overtones produced and thus the quality. Piano strings are struck by
felt hammers at a point about one-seventh of the length of the string
from one end. This point has been selected by experiment, it having been
found to yield the best combination of overtones as shown by the quality
of the tone resulting.
[Illustration: FIG. 337.--Chladni's plate.]
[Illustration: FIG. 338.--Chladni's figures.]
=346. Chladni's Plate.=--The fact that vibrating bodies are capable of
many modes of vibration is well illustrated by what is known as
Chladni's plate. This consists of a circular or square sheet of brass
attached to a stand at its center so as to be held horizontally. (See
Fig. 337.) Fine sand is sprinkled over its surface and the disc is set
vibrating by drawing a violin bow across its edge. The mode of
vibration of the disc is indicated by the sand accumulating along the
lines of least vibration, called _nodal lines_. A variety of nodal lines
each accompanied by its characteristic tone may be obtained by changing
the position of the bow and by touching the fingers at different points
at the edge of the disc. They are known as Chladni's figures. (See Fig.
338.)
[Illustration: FIG. 339.--Manometric flame apparatus.]
=347. Manometric Flames.=--The actual presence of overtones along with
the fundamental may be made _visible_ by the _manometric flame
apparatus_. This consists of a wooden box, _C_, mounted upon a stand.
(See Fig. 339.) The box is divided vertically by a flexible partition or
diaphragm. Two outlets are provided on one side of the partition, one,
_C_, leads to a gas pipe, the other is a glass tube, _D_. On the other
side of the partition a tube, _E_, leads to a mouthpiece. A mirror, _M_,
is mounted so as to be rotated upon a vertical axis in front of _F_ and
near it. Gas is now turned on and lighted at _F_. The sound of the voice
produced at the mouthpiece sends sound waves through the tube and
against the diaphragm which vibrates back and forth as the sound waves
strike it. This action affects the flame which rises and falls. If now
the mirror is rotated, the image of the flame seen in the mirror rises
and falls, showing not only the fundamental or principal vibrations but
also the overtones. If the different vowel sounds are uttered in
succession in the mouthpiece, each is found to be accompanied by its
characteristic wave form (Fig. 340). In some, the fundamental is
strongly prominent, while in others, the overtones produce marked
modifications. Other devices have been invented which make possible the
accurate analysis of sounds into their component vibrations, while still
others unite simple tones to produce any complex tone desired.
=348. The Phonograph.=--The _graphophone_ or _phonograph_ provides a
mechanism for cutting upon a disc or cylinder a groove that reproduces,
in the varying form or depth of the tracing, every peculiarity of the
sound waves affecting it. The reproducer consists of a sensitive
diaphragm to which is attached a needle. The disc or cylinder is rotated
under the reproducing needle. The irregularities of the bottom of the
tracing cause corresponding movements of the needle and the attached
diaphragm, which start waves that reproduce the sounds that previously
affected the recorder. The construction of the phonograph has reached
such perfection that very accurate reproduction of a great variety of
sounds is secured.
[Illustration: FIG. 340.--Characteristic forms of manometric flames.]
=349. Wind Instruments.=--In many musical instruments as the _cornet_,
_pipe-organ_, _flute_, etc., and also in _whistles_, the vibrating body
that serves as a source of sound is _a column of air_, usually enclosed
in a tube. Unlike vibrating strings, this vibrating source of sound
changes but little in tension or density, hence changes in the pitch of
air columns is secured by changing their length. The law being similar
to that with strings, _the vibration rates of air columns are inversely
proportional to their lengths_.
[Illustration: FIG. 341.--(_R_) Cross-section of an organ pipe showing
action of tongue at _C_. (_a_) The fundamental tone in a closed pipe has
a wave length four times the length of the pipe; (_b_) and (_c_) how the
first and second overtones are formed in a closed pipe; (_d_) the
fundamental tone of an open pipe has a wave length equal to twice the
length of the pipe; (_e_) and (_f_) first and second overtones of open
pipe.]
If an _open_ organ pipe be sounded by blowing gently through it, a tone
of definite pitch is heard. Now if one end is closed, on being sounded
again the pitch is found to be an octave lower. Therefore, _the pitch of
a closed pipe is an octave lower than that of an open one of the same
length_.
=350. Nodes in Organ Pipes.=--Fig. 341, _R_ represents a cross-section
of a wooden organ pipe. Air is blown through _A_, and strikes against a
thin tongue of wood _C_. This starts the jet of air vibrating thus
setting the column of air in vibration so that the sound is kept up as
long as air is blown through _A_. To understand the mode of vibration of
the air column a study of the curve that represents wave motion (Fig.
342) is helpful Let _AB_ represent such a curve, in this 2, 4 and 6
represent nodes or points of least vibration, while 1, 3 and 5 are
antinodes or places of greatest motion. A full wave length extends from
1-5, or 2-6. Now in the open organ pipe (Fig. 341_d_), the end of the
air column _d_ is a place of great vibration or is an antinode. At the
other end also occurs another place of great vibration or an antinode;
between these two antinodes must be a place of least vibration or a
node. The open air column therefore extends from antinode to antinode
(or from 1-3) or is _one-half_ a wave length. _The closed air column_
(Fig. 341_a_) extends from a place of _great_ vibration at _a_ to a
place of _no_ vibration at the closed end. The distance from an antinode
to a node is that from 1-2 on the curve and is _one-fourth_ a wave
length.
[Illustration: FIG. 342.--Graphic representation of sound waves.]
[Illustration: FIG. 343.--A clarinet.]
When a pipe is blown strongly it yields overtones. The _bugle_ is a
musical instrument in which notes of different pitch are produced by
differences in blowing. (See Fig. 341.) (_d_), (_e_), (_f_). In playing
the _cornet_ different pitches are produced by differences in blowing,
and by valves which change the length of the vibrating air column. (See
Fig. 334.) The _clarinet_ has a mouthpiece containing a reed similar to
that made by cutting a tongue on a straw or quill. The length of the
vibrating air column in the clarinet is changed by opening holes in the
sides of the tube. (See Fig. 343.)
=351. How we Hear.=--Our hearing apparatus is arranged in three parts.
(See Fig. 344.) _The external ear_ leads to the _tympanum_. _The middle
ear_ contains three bones that convey the vibrations of the tympanum to
the _internal ear_. The latter is filled with a liquid which conveys the
vibrations to a part having a coiled shell-like structure called the
_Cochlea_. Stretched across within the cochlea are some 3000 fibers or
strings. It is believed that each is sensitive to a particular vibration
rate and that each is also attached to a nerve fiber. The sound waves of
the air transmitted by the tympanum, the ear bones and the liquid of the
internal ear start sympathetic vibrations in the strings of the cochlea
which affect the auditory nerve and we hear. The highest tones
perceptible by the human ear are produced by from 24,000 to 40,000
vibrations per second. The average person cannot hear sounds produced by
more than about 28,000 vibrations. The usual range of hearing is about
11 octaves. The tones produced by higher vibrations than about 4100 per
second are shrill and displeasing. In music the range is 7-1/3 octaves,
the lowest tone being produced by 27.5 vibrations, the highest by about
4100 per second.
[Illustration: FIG. 344.--The human ear.]
The tones produced by men are lower than those of women and boys. In men
the vocal cords are about 18 mm. long; in women they are 12 mm. long.
The compass of the human voice is about two octaves, although some
noted singers have a range of two and one-half octaves. In ordinary
conversation the wave length of sounds produced by a man's voice is from
8 to 12 ft. and that of a woman's voice is from 2 to 4 ft.
Important Topics
1. Tone quality. Fundamental and overtones. Chladni's plate.
2. Manometric flame apparatus.
3. Phonograph recorder and reproducer.
4. Air columns and wind instruments.
5. How we hear.
Exercises
1. What determines the pitch of the note of a toy whistle?
2. The lowest note of the organ has a wave length of about 64 ft. What
is the length of a closed pipe giving this note? Of an open pipe?
3. What is the first overtone of C? What are the second and third
overtones? Give vibration numbers and pitch names or letters.
4. Why is the music of a band just as harmonious at a distance of 400
ft. as at 100 ft.?
5. A resonant air column 60 cm. long closed at one end will respond to
what rate of vibration at 10°C.?
6. Can you find out how the valves on a cornet operate to change the
pitch of the tone?
7. How is the trombone operated to produce tones of different pitch?
8. The lowest note on an organ has a wave length of about 64 ft. What
must be the length of a closed pipe giving this note?
9. What is the approximate length of an open organ pipe which sends out
waves 4 ft. long?
Review Outline: Sound
Sound--definition, source, medium, speed, nature.
Waves--longitudinal, transverse, illustrations.
Characteristics of { intensity--area, amplitude, density, distance.
Musical Sounds: { pitch--scales; major, tempered, triads, _N_ =
{ _V_/_L_ quality--fundamental and overtones.
Sympathetic Vibrations--resonance, interference, beats, discord.
Musical Instruments--string, air column, membrane or plate.
Laws of; (a) vibrating strings (3), (b) vibrating air columns (2).