EVERY musician is painfully familiar with the fact that two notes nearly, but not quite exactly, in unison with one another, produce, when sounded together, a throbbing sound commonly described as the phenomenon of “beats.” In the elementary theory of acoustics the cause of beats is shown to be the mutual interference of the two vibrations, one sound interfering with the other and silencing it, when one set of waves is half a vibration behind the other. Just as at certain points on the earth's surface there are no tides when a high tide and a low tide coming from different seas meet, so there is no sound when two sets of sound-waves meet in opposite phases. If the two notes differ just a little in pitch they will alternately reinforce and interfere with one another, and produce the throbbing sound of beats, the number of beats (or maxima of sound) per second being the same as the difference in the number of vibrations per second. If one tone makes m vibrations per second and the other n (a slightly smaller number, being a slightly flatter tone) there will be m – n beats per second heard. If this number be not more than 3 or 4 per second the beats can easily be counted. When they get as rapid as 12 or 14 per second they come too fast to be counted, and are very harsh and grating. They are most disagreeable at about 33 per second; and if yet more rapid, are heard as a harsh, disagreeable, rattling sound quite different from a true note. Imperfect octaves and imperfect twelfths likewise cause beats; in fact there are beats heard for any imperfectly tuned consonance in which the frequency of the higher note is 1, 2, 3, 4, 5, . . . or any integer number of times that of the lower.