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166                    HAND-BOOK OF ACOUSTICS.
overtone of the other (S|). The harshness of this dissonance will consequently chiefly depend on the intensity of this overtone, which will vary in different instruments, and even in different parts of the same instrument. Thus, in the lower notes of the piano, the 1st overtone is not unfrequently louder than the fundamental itself. On other instruments, however, and in general, its intensity is not so great; in a well bowed violin, for example, it is only about one fourth as loud. Further, the beating between the 3rd pair of partials, and between the oth pair of No. 1, is wanting in No. 2. Thus, on the whole, this latter dissonance is much less harsh than No. 1. The dissonance in No. 3 is of a very mild character, for the Fundamental (f) beats only against the 4th partial (s) and as a general rule, the 4th partial is comparatively weak. In No. 4 there is no beating whatever, unless the 7th or 8th partial is audible, and even then it would be very slight.
The late Mr. Curwen proposed to distinguish dissonances such as Nos. 1, 2, 3, and 4 above, by terming them respectively Primary, Secondary, Tertiary, and Quaternary dissonances. Thus, in Primary dissonances the fundamentals themselves beat, while in Secondary, Tertiary, and Quaternary dissonances, the Fundamental of the one tone beats respectively with the 2nd, 4th, and 8th partials of the other.
The above conclusions must of course be modified for the tones of instruments, which have not the complete series of partials up to the 6th. For example, the tones of stopped organ pipes, and of clarionets are wanting in the even partials, and therefore a secondary dissonance between such tones, is of a very mild character indeed, the only beating which occurs, arising from a 5th and a third partial, as shown in the accompanying sketch.
(3) Beats between the overtones of Compound Tones. In studying these beats, we shall for the reasons stated above, take into considera­tion, the first six, and only the first six partials; and the student must continually bear in mind the fact, that, in general, the intensity of these partials rapidly diminishes. as we go farther and farther from the funda­mental.