Page:Elementary Text-book of Physics (Anthony, 1897).djvu/183

§ 142] taken as the tonic. But if we consider that the whole tones are not all the same, and propose to preserve exactly all the intervals of the transposed scale, the problem becomes much more difficult, and can only be solved at the expense of too great complication in the instrument. Instead of attempting it, a system of tuning, called temperament, is used by which the twelve tones referred to above are made to serve for the several scales, so that while none are perfect, the imperfections are nowhere marked. The system of temperament usually employed, or at least aimed at, called the even temperament, divides the octave into twelve equal semitones, and each interval is therefore the twelfth root of 2. With instruments in which the tones are not fixed, like the violin for instance, the skilful performer may give them their exact value.

For convenience in the practice of music and in the construction of musical instruments, a standard pitch must be adopted. This pitch is usually determined by assigning a fixed vibration number to the tone above the middle C of the piano, represented by the letter $$A'$$. This number is about 440, but varies somewhat in different countries and at different times. In the instruments made by König for scientific purposes the vibration number 256 is assigned to the middle C. This has the advantage that the vibration numbers of the successive octaves of this tone are powers of 2.