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225. Harmony and Discord. Two musical sounds are said to produce harmony when, on being sounded together, they produce a result pleasing to the ear. If the result is displeasing, they are said to produce discord. One cause of discord is the presence of beats between the two tones, and the greatest discord, between tones of medium pitch, results when the beats are about 32 per second; if the number of beats is fewer than 10 or greater than 70 per second, they are somewhat unpleasant but do not produce discord.


1. Under what conditions will two wave motions completely neutralize each other? What will be the result if they are water waves? What if they are sound waves?

2. If a tuning fork is put in vibration and then rotated while it is held near the ear, there will be four times per rotation when it can hardly be heard. Explain.

3. If a tuning fork is held over the mouth of a resonator, while vibrating, and slowly rotated, a point can be found at which there is practically no resonance. When a tube is slipped over the upper prong, without touching it, the sound of the resonator will again be heard. Explain.

4. Why cannot one fork be set in motion by another unless its rate of vibration is the same?

5. Why do soldiers break step on crossing a bridge?

6. Why does an auto horn seem to have a higher pitch before you meet it than after it has passed?

7. Why does the chord do, mi, sol, have a richer sound than either of the tones alone?

8. How many keys would it require for an octave on the piano if the sharps and flats were not played on the same key?

9. Does a violin player in a string quartet play the equally tempered or true interval scale? Why?

10. Why does a singer sometimes prefer the piano accompaniment to a song to be played in the key of three flats instead of in the key of four sharps?


1. If a tuning fork gives 384 vibrations per second, what must be the length of a resonator tube for it, when the temperature is 21° C.? (The effect of the diameter of the tube need not be considered in this problem.)

2. A tuning fork gives 256 vibrations. What must be the length of a resonating tube for it at 0° C.?

3. What is the velocity of sound, determined by the apparatus shown in Fig. 191, when the resonator tube is 22 mm. in diameter and 248.5 mm. long, if the fork makes 320 vibrations per second, the temperature being 0° C.?

4. Two tuning forks vibrate 126 and 128 times per second respectively. How many times per second do they reënforce each other?

5. Two tuning forks that give 260 vibrations per second are sounded together, showing by the absence of beats that they are in unison. One of them is now loaded until five beats are heard per second. How many vibrations does it now give?

6. Suppose the inmost row of holes in the siren described in the demonstration in § 216 gives the tone c' 256 vibrations. How many times does the disk rotate per second? What tones will the other rows of holes give?

7. What tones will each row of holes give if the speed of the siren mentioned in problem No. 6 is doubled?

8. A person in an automobile that is being driven at the rate of 25 miles per hour blows a whistle that gives 320 vibrations per second. How many vibrations per second will reach the ear of a man standing by the roadside, first, before, and second, after it has passed, if the temperature is 20° C.?


9. If c' has 256 vibrations, show how to find the number of vibrations in d' and e' as played on the violin; on the piano.

10. Show why a tone must be introduced into the scale of the key of G that cannot be found in the scale of the key of C.

11. In the time of Handel the standard a' fork gave 424 vibrations per second. The present international a' fork gives 435 vibrations. What effect does this have upon the difficulty of singing the high notes of a song written by Handel?


226. The Sonometer. - To investigate the laws of the vibration of strings an instrument called the sonometer is used. This is also called a monochord, since it often has but a single string. The essential parts are a base with a bridge at each end, a pin to which to fasten one end of the string, and some method of stretching the string by attaching a spring balance


FIG. 199.-Sonometer with Two Strings

or weights at the other end. A movable bridge (at E in Fig. 199) is used to change the length of the vibrating string, and a scale is laid off on the base.

227. Laws of the Vibration of Strings.

I. The tension and mass per unit length being the same, the number of vibrations per second varies inversely as the length of the string.

II. The length and tension being the same, the number of vibrations per second varies inversely as the square root of the mass per unit length of the string.

III. The length and mass per unit length being the same, the number of vibrations per second varies directly as the square root of the tension.

The above laws can be expressed by the proportion

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For the first law, under the conditions given, the proportion becomes

N:N' =

N:N' =

If the length of a certain string is taken as unity the parts of the same string that give the other tones of the scale are as follows:

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1 1

Syllable names: do

ti do


Length of string: 1

45 4 3 3 15 1

It will be observed that these ratios which give the relative length of string are the reciprocals of the ratios giving the relative numbers of vibrations.

For the second law the proportion becomes


re mi fa sol la




or, N:N' = l' : l.

228. Nodes and Loops. of the wire spring as in § 189.



, or, N:N' VM': √M;

and for the third law, N: N' =


These laws can be verified on the sonometer.

The strings of the piano illustrate the preceding laws. The lowest tones are made by long, heavy strings without great tension, while the highest tones are made by short, light strings stretched to a high tension.


FIG. 200

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whole by a slight movement of the hand. Quicken the movement, and it can be thrown into vibrations in halves, thirds, quarters, etc., giving a number of complete stationary waves.”

When the spring is vibrating as shown in Fig. 200, the points of no vibration are called nodes, as N, N', while the points of maximum vibration, as L, L', etc., are called loops. The vibrations are caused by waves sent out from A and reflected from B. Whenever the wave starting from A tends to give a certain velocity to any particle, and the reflected wave from B tends to give it an equal velocity in the opposite direction, the two forces neutralize each other, the particle remains at rest, and a node is formed.

When the string of a musical instrument is put into vibration by drawing a bow across it, by striking it a blow, or by plucking it, it vibrates transversely, not only as a whole, giving its fundamental tone, but also in halves, thirds, fourths, etc., each one of which gives its own tone. These different tones, with the fundamental, determine the quality of the tone.

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Demonstration. Place a wire, or heavy bass viol string on the sonometer and stretch it until it gives a suitable tone. Sound the fundamental tone with the bow. Touch the string lightly in the middle with the finger and draw the bow across the string one fourth the length of the string from the end. This will sound the octave. Touch the finger at one third the length of the string from the end. Draw the bow at one sixth and the note sounded will be the second harmonic, etc.

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229. Overtones and Harmonics. It is not necessary to touch the string in order to make it vibrate in parts besides vibrating as a whole. The tones caused by the vibrations in parts can be heard by listening carefully when the string is plucked. These tones are called overtones, and if the numbers of vibrations which produce them are 2, 3, 4, etc., times the number of vibrations of the fundamental, they are called harmonics. Overtones can be very readily pro

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