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VIII. § 82.] COINCIDENCE OF PARTIAL-TONES. 171
fact tha,t many people who knew nothing either about vibrations or the delights of simple numerical relations, could tell a perfect Octave from an imperfect one a great deal better than most men of science. The true explanation, which it was left for Helmholtz to discover, lies in the fact that only by exactly satisfying the assigned numerical relation can the partial-tones of the higher clang be brought into exact coincidence with partial-tones of the lower, and thus all beats and consequent dissonance prevented.
82. No narrower interval than an Octave can be found which gives an absolutely perfect concord. The nearest approach to such a concord is the Fifth :—
Here we get two pairs of coincidences 3—2 and 6—4, but a certain roughness is caused by 3 of the higher clang being within beating-distance of both 4 and 5 of the lower clang. It is true that, since 4 and 5 are generally weak, and the beating intervals whole Tones, this roughness will be but slight: still the dissonance thus caused prevents our classing the Fifth as an interval quite equally smooth with the