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162 BEATS OF COMPOUND SOUNDS. [VII 78
clangs coexist, each of which contains the first partial-tones of the series audibly developed. Since the second, third, &c. partial-tones of each clang make twice, three times &c. as many vibrations per second as their respective fundamental-tones, [§ 43] it follows that the differences between the vibration-numbers of successive pairs of partial-tones belonging to the two clangs will be twice, three times, &c, the difference between the vibration-numbers of the two fundamental-tones. Accordingly, if the fundamental-tones give rise to beats, we may hear, in addition to the series so accounted for, five other sets of beats, respectively twice, three, four, five and six times as rapid as they. In order to determine the number of beats per second for any such set, we need only multiply the number of the fundamental beats by the order of the partial-tones concerned. The beats of two simple tones necessarily become more rapid if the higher tone be sharpened, or the lower flattened ; i. e. if the interval they form with each other be widened. The beats may, however, also be accelerated without altering the interval, by merely placing it in a higher part of the scale. Greater vibration-numbers are thus obtained with a proportionately larger difference between them, though their ratio to each other remains what it was before. Thus the rapidity of the beats due to an |
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