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A glance at fig. 80, shows that the Fourth is not so perfect an interval as the Fifth. Its roughness arises chiefly from the beats generated between the usually powerful 2nd partial of the upper tone, and the almost equally loud 3rd partial of the lower one. To this may be added the much softer semitone and tone dissonances between 4th, 5th, and 6th partials; and a still slighter disturbing element may be sometimes present in the Summation Tone midway between the 2nd partials. It may be noted also that the 2nd and 3rd pairs of dissonating partials, the 4th and oth pairs, and also the Summation Tone, give rise to precisely the same number of beats.
As in the case of the Fifth, not only will the roughness of this interval vary with the varying intensities of dissonating partials, but other things being equal, with its position in the Musical Scale.
which (see table, page 159) is only just within beating distance. It may be noticed that this number is the same that we obtained in the case of the Fifth j q, showing that in order to obtain a Fourth of approximately equal smoothness with a Fifth, we must take the former a Fifth higher in pitch. Thus using the term in the same sense as before, we might call this the lower limit of a perfectly consonant Fourth.
Coming now to the Thirds, we find in both Major and Minor, that the Third partial of the Upper Tone dissonates with the 4th partial of tbe lower one; but while in the latter they form only a tone dissonance, in the former they dissonate at the much more unpleasant interval of a semitone. On the other hand, while softer 4th and 5th partials respectively of the Minor Third beat at a semitone distance, the corresponding partials of the Major Third do not beat at all. Further, the Summation tones when present will add to the roughness ; that of the Minor Third being slightly more detrimental than that of the Major.
The Thirds, in respect to their harmoniousness vary very greatly according to their position in the Musical Scale. They cannot be used very low in pitch, even when they are formed between Simple Tones: for as we have already seen the Thirds |q? and j ^?' between Simple Tones are at the beating distance, that is, C2 and C|