NOTE:--It is usually taken for granted that the student of

music is familiar with the significance of such terms as

over-tone, equal temperament, etc., and with principles

such as that relating to the relation between vibration rates

and pitches: the writer has in his own experience found,

however, that most students are not at all familiar with such

data, and this appendix is therefore ad
ed in the hope that a

few facts at least regarding the laws of sound may be brought

to the attention of some who would otherwise remain in entire

ignorance of the subject.

1. Acoustics is the science which deals with sound and the laws of its

production and transmission. Since all sound is caused by vibration,

acoustics may be defined as the science which treats of the phenomena

of sound-producing vibration.

2. All sound (as stated above) is produced by vibration of some sort:

strike a tuning-fork against the top of a table and see the vibrations

which cause the tone, or, if the fork is a small one and the vibrations

cannot be seen, hold it against the edge of a sheet of paper and hear

the blows it strikes; or, watch one of the lowest strings of the piano

after striking the key a sharp blow; or, look closely at the heavier

strings of the violin (or better still, the cello) and watch them

oscillate rapidly to and fro as the bow moves across them.

The vibrating body may be a string, a thin piece of wood, a piece of

metal, a membrane (cf. drum), the lips (cf. playing the cornet), the

vocal cords, etc. Often it is a column of air whose vibrations give rise

to the tone, the reed or other medium merely serving to set the air in


3. Sound is transmitted through the air in somewhat this fashion: the

vibrating body (a string for example) strikes the air-particles in its

immediate vicinity, and they, being in contact with other such

air-particles, strike these others, the latter in turn striking yet

others, and so on, both a forward and backward movement being set up

(oscillation). These particles lie so close together that no movement at

all can be detected, and it is only when the disturbance finally reaches

the air-particles that are in contact with the ear-drum that any effect

is evident.

This phenomenon of sound-transmission may perhaps be made more clear by

the old illustration of a series of eight billiard balls in a row on a

table: if the first ball is tapped lightly, striking gently against ball

number 2, the latter (as well as numbers 3, 4, 5, 6, and 7) will not

apparently move at all, but ball number 8 at the other end will roll

away. The air-particles act upon each other in much this same fashion,

the difference being that when they are set in motion by a vibrating

body a complete vibration backward and forward causes a similar

backward and forward movement of the particles (oscillation) instead

of simply a forward jerk as in the case of the billiard balls.

Another way of describing the same process is this: the vibration of

some body produces waves in the air (cf. waves in the ocean, which carry

water forward but do not themselves move on continuously), these waves

spread out spherically (i.e. in all directions) and finally reach the

ear, where they set the ear-drum in vibration, thus sending certain

sound-stimuli to the nerves of hearing in the inner ear, and thus to the


An important thing to be noted in connection with sound-transmission is

that sound will not travel in a vacuum: some kind of a medium is

essential for its transmission. This medium may be air, water, a bar of

iron or steel, the earth, etc.

4. The rate at which sound travels through the air is about 1100 feet

per second, the rapidity varying somewhat with fluctuations in

temperature and humidity. In water the rate is much higher than in air

(about four times as great) while the velocity of sound through other

mediums (as e.g., steel) is sometimes as much as sixteen times as

great as through air.

5. Sound, like light, may be intensified by a suitable reflecting

surface directly back of the vibrating body (cf. sounding board); it may

also be reflected by some surface at a distance from its source in such

a way that at a certain point (the focus) the sound may be very clearly

heard, but at other places, even those nearer the source of sound, it

can scarcely be heard at all. If there is such a surface in an

auditorium (as often occurs) there will be a certain point where

everything can be heard very easily, but in the rest of the room it may

be very difficult to understand what is being said or sung.

Echoes are caused by sound-reflection, the distance of the reflecting

surface from the vibrating body determining the number of syllables that

will be echoed.

The acoustics of an auditorium (i.e., its hearing properties) depend

upon the position and nature of the reflecting surfaces and also upon

the length of time a sound persists after the vibrating body has

stopped. If it persists longer than 2-1/4 or 2-1/3 seconds the room will

not be suitable for musical performances because of the mixture of

persisting tones with following ones, this causing a blurred effect

somewhat like that obtained by playing a series of unrelated chords on

the piano while the damper-pedal is held down. The duration of the

reverberation depends upon the size and height of the room, material of

floor and walls, furniture, size of audience, etc.

6. Sound may be classified roughly into tones and noises although

the line of cleavage is not always sharply drawn. If I throw stones at

the side of a barn, sounds are produced, but they are caused by

irregular vibrations of an irregularly constructed surface and are

referred to as noise. But if I tap the head of a kettle-drum, a

regular series of vibrations is set up and the resulting sound is

referred to as tone. In general the material of music consists of

tones, but for special effects certain noises are also utilized (cf.

castanets, etc.).

7. Musical tones have three properties, viz.:

1. Pitch.

2. Intensity.

3. Quality (timbre).

By pitch is meant the highness or lowness of tone. It depends upon

rate of vibration. If a body vibrates only 8 or 10 times per second no

tone is heard at all: but if it vibrates regularly at the rate of 16 or

18 per second a tone of very low pitch is heard. If it vibrates at the

rate of 24 the pitch is higher, at 30 higher still, at 200 yet higher,

and when a rate of about 38,000 per second has been reached the pitch is

so high that most ears cannot perceive it at all. The highest tone that

can ordinarily be heard is the E[flat] four octaves higher than the

highest E[flat] of the piano. The entire range of sound humanly audible

is therefore about eleven octaves (rates 16-38,000), but only about

eight of these octaves are utilized for musical purposes. The tones of

the piano (with a range of 7-1/3 octaves) are produced by vibration

rates approximately between 27 and 4224. In the orchestra the range is

slightly more extended, the rates being from 33 to 4752.

Certain interesting facts regarding the relation between vibration-rates

and pitches have been worked out: it has been discovered for instance

that if the number of vibrations is doubled, the pitch of the resulting

tone is an octave higher; i.e., if a string vibrating at the rate of

261 per second gives rise to the pitch c', then a string one-half as

long and vibrating twice as rapidly (522) will give rise to the pitch

c'', i.e., an octave higher than c'. In the same way it has been found

that if the rate is multiplied by 5/4 the pitch of the tone will be a

major third higher; if multiplied by 3/2, a perfect fifth higher,

etc. These laws are often stated thus: the ratio of the octave to the

fundamental is as two is to one; that of the major third as five is to

four; that of the perfect fifth as three is to two, and so on through

the entire series of pitches embraced within the octave, the ratio

being of course the same for all octaves.

9. The intensity (loudness or softness) of tones depends upon the

amplitude (width) of the vibrations, a louder tone being the result of

vibrations of greater amplitude, and vice versa. This may be verified by

plucking a long string (on cello or double-bass) and noting that when

plucked gently vibrations of small amplitude are set up, while a

vigorous pluck results in much wider vibrations, and, consequently, in a

louder tone. It should be noted that the pitch of the tone is not

affected by the change in amplitude of vibration.

The intensity of tones varies with the medium conveying them, being

usually louder at night because the air is then more elastic. Tone

intensity is also affected by sympathetic vibrations set up in other

bodies. If two strings of the same length are stretched side by side and

one set in vibration so as to produce tone the other will soon begin to

vibrate also and the combined tone will be louder than if only one

string produced it. This phenomenon is the basis of what is known as

resonance (cf. body of violin, resonance cavities of nose and mouth,

sounding board of piano, etc.).

10. Quality depends upon the shape (or form) of the vibrations which

give rise to the tone. A series of simple vibrations will cause a simple

(or colorless) tone, while complex vibrations (giving rise to overtones

of various kinds and in a variety of proportions) cause more

individualistic peculiarities of quality. Quality is affected also by

the shape and size of the resonance body. (Cf. last part of sec. 9


11. Practically every musical tone really consists of a combination of

several tones sounding simultaneously, the combined effect upon the ear

giving the impression of a single tone. The most important tone of the

series is the fundamental, which dominates the combination and gives

the pitch, but this fundamental is practically always combined with a

greater or less number of faint and elusive attending tones called

overtones or harmonics. The first of these overtones is the octave

above the fundamental; the second is the fifth above this octave; the

third, two octaves above the fundamental, and so on through the series

as shown in the figure below. The presence of these overtones is

accounted for by the fact that the string (or other vibrating body) does

not merely vibrate in its entirety but has in addition to the principal

oscillation a number of sectional movements also. Thus it is easily

proved that a string vibrates in halves, thirds, etc., in addition to

the principal vibration of the entire string, and it is the vibration of

these halves, thirds, etc., which gives rise to the harmonics, or

upper partials as they are often called. The figure shows Great C

and its first eight overtones. A similar series might be worked out from

any other fundamental.

It will be recalled that in the section (10) dealing with quality the

statement was made that quality depends upon the shape of the

vibrations; it should now be noted that it is the form of these

vibrations that determines the nature and proportion of the overtones

and hence the quality. Thus e.g., a tone that has too large a

proportion of the fourth upper partial (i.e., the third of the

chord) will be reedy and somewhat unpleasant. This is the case with

many voices that are referred to as nasal. Too great a proportion of

overtones is what causes certain pianos to sound tin-panny. The tone

produced by a good tuning-fork is almost entirely free from overtones:

it has therefore no distinctive quality and is said to be a simple

tone. The characteristic tone of the oboe on the other hand has many

overtones and is therefore highly individualistic: this enables us to

recognize the tone of the instrument even though we cannot see the

player. Such a tone is said to be complex.

12. The mathematical ratio referred to on page 134, if strictly carried

out in tuning a keyboard instrument would cause the half-steps to vary

slightly in size, and playing in certain keys (especially those having a

number of sharps or flats in the signature) would therefore sound out of

tune. There would be many other disadvantages in such a system, notably

the inability to modulate freely to other keys, and since modulation is

one of the predominant and most striking characteristics of modern

music, this would constitute a serious barrier to advances in

composition. To obviate these disadvantages a system of equal

temperament was invented and has been in universal use since the time

of Bach (1685-1750) who was the first prominent composer to use it

extensively. Equal temperament means simply dividing the octave into

twelve equal parts, thus causing all scales (as played on keyboard

instruments at least) to sound exactly alike.

To show the practicability of equal temperament Bach wrote a

series of 48 preludes and fugues, two in each major and two

in each minor key. He called the collection The Well-tempered


13. Various standards of pitch have existed at different times in the

last two centuries, and even now there is no absolute uniformity

although conditions are much better than they were even twenty-five

years ago. Scientists use what is known as the scientific standard

(sometimes called the philosophic standard), viz., 256 double

vibrations for middle C. This pitch is not in actual use for musical

purposes, but is retained for theoretical purposes because of its

convenience of computation (being a power of 2). In 1885 a conference of

musicians at Vienna ratified the pitch giving Middle C 261 vibrations,

this having been adopted by the French as their official pitch some 26

years before. In 1891 a convention of piano manufacturers at

Philadelphia adopted this same pitch for the United States, and it has

been in practically universal use ever since. This pitch (giving Middle

C 261 vibrations) is known as International Pitch.

Concert pitch is slightly higher than International, the difference

between the two varying somewhat, but being almost always less than

one-half step. This higher pitch is still often used by bands and

sometimes by orchestras to give greater brilliancy to the wind



Lavignac--Music and Musicians, pp. 1-66.

Broadhouse--The Student's Helmholz.

Helmholtz--Sensations of Tone.

Hamilton--Sound and its Relation to Music.

NOTE:--For a simple and illuminating treatment of the subject

from the standpoint of the music student, the books by

Lavignac and Hamilton are especially recommended.