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the 1st order; differentials of the 2nd order seldom or never occur-ing between overtones. Let us consider the differentials of the 1st order generated between the partials of two compound tones, the fundamentals of which have, as we will suppose, the vibration numbers 200 and 304. Then the numbers in the first horizontal line of the following table are the partials of the former tone, and those in the first vertical column those of the latter. At their intersections are found the differences of these numbers; that is, the vibration numbers of the differentials due to them.
If we arrange these tones in the order of their pitch, omitting the 8, which of course produces no tone at all, we have the groups:—
The difference hetween each pair is 8, and this is the only number of heats per second, which will be produced by all these differential tones; for the difference between any other two of the above numbers, gives too great a number of beats per second to be perceptible at the pitch of the tones that produce them.
Now let us ascertain the number of beats per second that will be generated by direct action between these same partials :
Ninety-six beats per second, at the pitch of 400, is far beyond beating distance ; 8 beats per second therefore is the number due to the direct action of overtones, that is, the same number which we found to arise from the action of the differentials. A like result will be obtained whatever numbers are selected for the funda­mentals, so that in general, " dissonance due to combination tones