=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._

[Ill.u.s.tration: 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_ ?v(_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_.

[Ill.u.s.tration: 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 E and G, with C. This experiment shows that the sound of the C string contains these tones as overtones. It also ill.u.s.trates sympathetic vibration.

Important Topics

1. Interference, beats, production, effects.

2. Vibration of strings, three laws.

3. Three cla.s.ses 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.

[Ill.u.s.tration: FIG. 337.--Chladni's plate.]

[Ill.u.s.tration: FIG. 338.--Chladni's figures.]

=346. Chladni's Plate.=--The fact that vibrating bodies are capable of many modes of vibration is well ill.u.s.trated by what is known as Chladni's plate. This consists of a circular or square sheet of bra.s.s 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 acc.u.mulating 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.)

[Ill.u.s.tration: 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 part.i.tion or diaphragm. Two outlets are provided on one side of the part.i.tion, one, _C_, leads to a gas pipe, the other is a gla.s.s tube, _D_. On the other side of the part.i.tion 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 princ.i.p.al 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 a.n.a.lysis 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.

[Ill.u.s.tration: 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_.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: FIG. 342.--Graphic representation of sound waves.]

[Ill.u.s.tration: 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 sh.e.l.l-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.

[Ill.u.s.tration: 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 compa.s.s 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