150 HAND-BOOK OF ACOUSTICS.
that the number of beats generated by two sounds, is equal to the difference of their vibration numbers; therefore the vibration number of the fork under trial, must be either 512 + 3 or 512 — 3; that is, either 515 or 509, according as it is sharper or natter than the standard,—a matter, "which the ear of the musician can easily decide.
It is found by experience, that beats which occur at the rate of from 2 to 5 per second, are the most easily counted. Beyond five beats in a second, there is considerable difficulty in counting, owing to their rapidity ; and below two beats in a second, there is also a difficulty, owing to the length of time occupied by each loudness. For ascertaining the pitch of instruments in the way just described, cases of tuning-forks are constructed consisting each of twelve forks, the vibration numbers of which increase by four vibrations per second, from 412 to 456 for A, and from 500 to 544 for C. To show the method of using them, we will take the following case. It was desired to ascertain the pitch of a certain piano. In a preliminary trial, by sounding the with each of the forks, it was found that it produced with the 536 fork, from 2 to 3 beats per second, and with the 540 fork, beats at a somewhat slower rate. The former was first taken, and the beats produced by it with the pianoforte, carefully counted for 30 seconds. The number was found to be 75, which is at the rate of beats per second.
Therefore the vibration number of the note in question was 536 + To verify this, the 540 fork was sounded with
the pianoforte C; 44 beats were now counted in 30 seconds, that is beats per second, nearly. This gives the same result as before, viz.,
It is possible, however, to ascertain the vibration number of a musical sound by means of beats, independently of any previously ascertained standard. This will be seen from the following considerations. Suppose we have two forks, one of which gives the exact octave of the other. Let us further suppose, that it is possible to count the number of beats per second produced, when they are sounded together, and let the number be, say 100. What will be the vibration numbers of the forks ? Now, in the first place, it is evident that, whatever they are, the difference between them must be 100; since the number of beats per second, produced by two sounds, is equal to the difference of their vibration numbers. In the second place, the vibration number of the higher fork must be twice that of the lower, since they are an octave apart. Thus the