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178                    HAND-BOOK OF ACOUSTICS.
A Fifth between Compound Tones, therefore, though also guarded by Differentials, is chiefly defined by the coincidence of the 3rd partial of the lower, with the 2nd of the upper tone, and this coin­cidence can evidently only happen, when the vibration numbers of the Fundamentals are in the ratio of 2 : 3. If the vibration numbers vary from this ratio, beats will be heard between these partials. For example, let the vibration numbers be 201 and 300 respectively, then
and 603 — 600 = 3 beats per second will be heard. To tune a false Fifth, therefore, one of the tones must be altered, till these beats vanish.
Inasmuch as the definition of a Fifth depends upon the coincidence of 2nd and 3rd partials, while the definition of an Octave depends upon that of 1st and 2nd partials, we see that beats from a mistuned Fifth will not usually be so powerful as those from a mistuned Octave; that is to say, the same rigorous exacti­tude in tuning, which the octave demands, is not so essential in the case of the Fifth. As an illustration of this fact, it may be mentioned, that while the Octave is preserved intact, in all systems of temperament, the Fifth is always more or less tampered with.