Page:Elementary Text-book of Physics (Anthony, 1897).djvu/182

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The derivation of the names of the intervals will now be apparent. For example, an interval of a third is the interval between any tone of the scale and the third one from it, counting the first as 1. If we consider the intervals from tone to tone, it is seen that the pitch does not rise by equal steps, but that there are three different intervals, $$\tfrac{9}{8}, \tfrac{10}{9}$$, and $$\tfrac{16}{15}$$. The first two are usually considered the same, and are called whole tones. The third is a half-tone or semitone.

It is desirable to be able to use any tone of a musical instrument as the first tone or tonic of a musical scale. To permit this, when the tones of the instrument are fixed, it is plain that extra tones, other than those of the simple scale, must be provided in order that the proper sequence of intervals may be maintained. Suppose the tonic to be transposed from C to D. The semitones should now come, in the major scale, between F and G, and C' and D', instead of between E and F, and B and C. To accomplish this, a tone must be substituted for F and another for C'. These are called F sharp and C' sharp respectively, and their vibration numbers are determined by multiplying the vibration numbers of the tones which they replace by $$\tfrac{25}{24}$$. The introduction of five such extra tones, making twelve in the octave, enables us to preserve the proper sequence of whole tones and semitones, whatever tone is