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202 .                RAND-BOOK OF ACOUSTICS.
roughness of the intervals is shown hy the vertical distances of the curved line from the corresponding points E2, E, &c, on the horizontal line. For example ; the roughness of the interval is represented hy the length of the vertical line over the point E2 ; the roughness of the interval hy the short vertical line over the point E, and so on. Thus, if we liken the curve to the outline of a mountain chain, the dissonances are represented by peaks, while the consonances correspond to passes.
According to this figure, the consonances in the order of their relative harmoniousness, are,
In making use of the figure, however, the student must 3ontinually bear in mind, the assumptions on which it was calculated; viz., that the roughness vanishes when there are no beats ; that it increases from this to a maximum for 33 beats per second ; that it diminishes from this point as the number of beats per second increases; and lastly, that the intensity of the partial tones diminishes inversely as the square of their order. The conclusions expressed in the diagram, are therefore only true in those cases in which these assumptions are true, or approximately true.
All the consonant intervals between Simple Tones are equally smooth or harmonious.
Intervals, whether between Simple or Compound Tones, having the following vibration ratios,
are perfect in their smoothness; they bave no elements of roughness whatever.