DEFINITION OF THE CONSONANT INTERVALS. 183 



2 beats per second will be heard from the pair marked (2), 4 beats per second from that marked (4), and 6 from that marked (6).
In tuning Fifths, Thirds, &c, between Compound Tones with perfect exactness, a resonator tuned to the pitch of the coincident partials will be found of great service; for these partials being thus reinforced, it will be easy to discriminate any beats between them, from the beats of other partials; and furthermore the disturbing effect of any dissonating partials which may be present, will be much lessened.
It will be seen from the above, that the particular partials which coincide in any interval are given by the figures which denote its vibration ratio. Thus, the vibration ratio of the octave is 2 : 1, and the coincident partials are the 2nd and 1st; the vibration ratio of of the Fifth is 3 : 2, and the coincident partials are the 3rd and 2nd, and so on.
Furthermore, the preceding illustrations show, that in any particular interval, if the lower of the two tones is one vibration too sharp or too flat, the number of beats produced by the lowest pair of coincident partials is the same as the greater of the two numbers which denote its vibration ratio. Thus, in the case of the Major Third taken above,—401 and 500,—we found the number ot beats per second to be 5, and that is the greater of the two numbers 5 : 4 which give its vibration ratio. Similarly, if the higher of the two tones of an interval be one vibration too sharp or too flat, the number of beats per second will be the smaller of the two numbers which denote its vibration ratio. For example, let the vibration numbers of a mistuned Fifth be 200 and 301 : Then 
