=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. [Illustration: FIG. 326.--One tuning fork will vibrate in sympathy with the other, if they have exactly equal rates of vibration.]