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The Theory Of Sound Which Constitutes The Physical Basis Of The Art Of Music.

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170                                      OCTAVE                          [VIII. § 81.
others give respectively 32 and 48 beats per second. A semi-Tone corresponds to about maximum rough­ness in the middle region of the scale, so that we have before us an exceedingly harsh discord. As the pitch of the higher note is gradually corrected, the rapidity of the beats diminishes, but the tuning must be rigorously accurate to make them entirely vanish. If the note makes but one vibration per second too many or too few, which corresponds to a difference in pitch of only about a thirtieth part of a whole Tone, the defect of tuning makes itself felt by three sets of beats, of 1, 2 and 3 per second respec­tively. The tuner must keep slightly altering the pitch until he at length hits on that which com­pletely extinguishes the beats. We saw in an ear­lier part of this inquiry (p. 68) that, when two sounds form with each other the interval of an Octave, their vibration-numbers must be in the ratio of 2:1. Long after it had been experimentally ascertained that rigorous compliance with this arithmetical con­dition was essential for securing a perfectly smooth Octave, the reason for this necessity remained entirely unknown, and nothing but the vaguest and most fanciful suggestions were offered to account for it—such as, for instance, that " the human mind delights in simple numerical relations." This attempt at explanation overlooked the obvious
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