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162                    HAND-BOOK OF ACOUSTICS.
Thus all dissonance between simple tones will be found on examination to be due to beats, generated, either by the simple tones themselves, by one of the simple tones and a Differential, or by two Differentials.
Before inquiring into the causes of dissonance between Compound Tones, it will be as well to call to mind the fact, that a single compound tone may and often does contain dissonant elements in itself. Let us take the compound tone C2, for example : Inasmuch as its funda­mental has the vibration number 64, the difference between the vibration numbers of any two successive partials must be 64. By reference to the accompanying table of partials, and to the table on page 159, we see that the intervals between the first 7 partials are greater than the Beating Distance, but that the intervals between the partials above the 7th are less than the Beating Distance. For, take the 8th and 9th partials, which are C1 and D1 respectively, the number of beats produced by these two simple tones is 64 and we know by the table on page 159 that the number of beats necessary to concord, in this part of the musical scale is 78; therefore a certain amount of roughness, due to these 64 beats will result. The dissonance gets worse as we ascend; for example, the number of beats per second between the 15th partial, B1, and the 16th, C2, is of course 64, which forms a very harsh dissonance in this part of the scale. As we have already seen, the partials of the tones of most instruments, become weaker and weaker, the farther they are from the fundamental; so that in general, these very high partials are not strong enough to produce any appreciable roughness, but this is by no means always the case. If the note 02 be sounded on the Harmonium or American Organ, especially with such a stop as the bassoon, it is quite easy to detect the jarring of these higher partials, and by means of a resonator tuned to a note intermediate between any two of them, the beating of those two is perceptibly increased. The same jarring effect may be readily perceived in the tones of the Trombone and Trumpet; in fact, it is this beating that gives to the tones of these instruments, their peculiar penetrating or braying character; a discontinuous sensation, as before observed, producing a much more intense