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182 BEATS OF COMBINATION-TONES. [VIII. § 91.
order becomes higher. Those of the first order are usually distinct enough, and those of the second to be heard with a little trouble. The third order is only recognisable when entire stillness is secured, and the greatest attention paid. It is a moot point whether the fourth-order tones can be heard at all.
91. We can now show that the existence of combination-tones prevents intervals formed by two simple tones from altogether lacking the characteristic differences of consonance and dissonance, though those differences are far less marked than in the case of composite sounds. To begin with the Octave. Let us suppose that we have two simple tones forming nearly this interval, but that the higher of them is a little sharp, so that the Octave is not strictly in tune, is what is called slightly impure. Let the lower tone make 100, the higher 201, vibrations per second. They will give rise to a combination-tone making 101 vibrations per second (p. 179), and this with the lower primary will produce one beat pei' second. If the higher primary had been flat instead of sharp, making say 199 vibrations per second, we should have had 99 as combination-tone, giving rise, with 100, to beats of the same rapidity as before. These beats cannot be got rid of except by making the vibration-ratio exactly 1 : 2, i.e. the Octave perfectly pure. The roughness must increase both