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Music Notation and Terminology

APPENDIX C

Acoustics

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 added 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 vibration.

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 brain.

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 four octaves higher than the highest E 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 above.)

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.

 

[Listen]

 

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 Clavichord."

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 instruments.

References

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.



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