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192                   HAND-BOOK OF ACOUSTICS.
are the limits respectively at, and below which, a Major and a Minor Third between Simple Tones, become dissonant. Thirds at or below these limits, between Compound Tones, contain of course these same elements of roughness, between their fundamentals; to which however must be added, the further roughnesses due to their beating overtones. Thirds, above these limits, must owe their roughness chiefly to beating overtones.
From fig. 80, we see that the smoothness of a Major or Minor Third between Compound Tones, above the limit just referred to, depends chiefly upon the loudness of the beats between 3rd, 4th, and 5th partials. Now observation shows that in the case of the Voice, Harmonium, and Piano, these partials generally become weak or even altogether absent above middle C; consequently, Thirds above this region, on these instruments will be as a rule sufficiently smooth. As we descend, however, from this region, Thirds rapidly deteriorate, for in the first place these partials begin to assert themselves, and secondly, the fundamentals are approxi­mating to the beating distance.
and 32 beats per second, in the region of C1 = 512 are very harsh if at all prominent.
which is within beating distance in the region of C = 512. To this must be added, first, the roughness due to the 153f— 128=25f, beats per second between the fundamentals, which are just about the beating distance, secondly, that due to the possible Summa­tion Tone, and thirdly, that arising from the dissonant 5th and 4th partials.