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On the Relative Harmoniousness of the Consonant
We have now to examine into the causes of the relative smoothness of those intervals which are usually called consonant.
With regard to perfectly simple tones, there is, as we have already seen, no element of roughness in any of these intervals, except in the case of the Thirds, and in these only when very low in pitch ; consequently, there is found to be little or no difference in smoothness, between any of these intervals, when strictly Simple Tones are employed, and when the tones in question are in perfect tune.
With Compound Tones, however, the case is very different: not only do these intervals vary in smoothness—in harmoniousness— one with another, but the smoothness of any one particular interval varies according to the constitution, that is the quality, of its Compound Tones.
In the first instance, we shall consider these intervals as formed between Compound Tones, each consisting of the first six partials ; and as before, we shall suppose, as is generally the case, that the intensity of these partials rapidly diminishes as we ascend in the series. In fig. 80 we have the ordinary Consonant Intervals, together with a few others, drawn out so as to show the first five or six partials of each tone. To facilitate comparison, the lower of the two tones in each interval, is supposed to be of the same pitch throughout, so that tones on the same horizontal lines are of the same pitch. The symbols for the partials diminish in size, as they rise above the fundamental, in order to represent roughly their diminution in intensity. As before, partials forming a tone dissonance are connected by a single line, those that dissonate at a semitone are joined by a double one. In comparing the intervals of the figure, it must be borne in mind, not only that the beats of the semitone