A complete view of Acoustical Science & its bearings on music, for musicians & music students.

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problem reduces itself to finding two numbers, one of which, is double the other, the difference between them being 100. Now 200 and 100 are the only numbers which satisfy these conditions, and therefore the vibration numbers of the forks will be 200, and 100, respectively.
To put this in a general way
Let x denote the vibration number of the lower fork, then 2x will denote the vibration number of the higher fork, therefore if n denote the number of beats per second produced by them
Therefore, if two sounds are exactly an octave apart, the number of beats they generate per second, will be the vibration number of the lower sound.
But when two sounds, at the interval of an octave, are heard together, no beats at all are perceived. How is this difficulty to be overcome ? Let us suppose we have two forks A and Z, an octave apart, A being the lower one. Tune another fork B slightly sharper than A, so that it produces with it, not more than 4 beats per second; tune another fork C sharper than B, and making with it about 4 beats per second; tune another fork D in the same manner, to beat with C; and so on, till we get a fork within 4 beats of Z. Now count accurately the number of beats between A and B, B and C, C and D, and so on up to Z; add these all together, and the total will evidently be the number of beats between A and Z.
Instruments constructed on the above principle are called Tonometers, of which there are two varieties: the Tuning-fork Tonometer and the Eeed Tonometer.
The Tuning-fork Tonometer was invented by Scheibler, who died in 1837. One of his instruments, which still exists, consists of 56 forks, each of which produces four beats per second with the succeeding one. Therefore, between the lowest and the highest forks, there are 55 sets of four beats; that is, 55 X 4 = 220, which, by the above, must be the vibration number of the lowest fork, 440 being that of the higher one.
In Appun's Tonometer, the tuning-forks are replaced by reeds. Although better adapted to all purposes of lecture illustration than the Tuning-fork Tonometer, the Beed Tonometer has two serious drawbacks, viz.: the reeds do not retain their pitch with accuracy, and their variation with temperature is unknown.