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.
Fig. 326.—One tuning fork will vibrate in sympathy with the other, if they have exactly equal rates of vibration.
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.
Fig. 328.—Explanation of resonance.