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COMBINATION TONES.
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between the vibration numbers of its generators; in the other, it is equal to their sum. The former are consequently termed Difference or Differential Tones, and the latter, Summation Tones.
DlFFERENTIAI TONES.
These tones have been known to musicians for more than a century. They appear to have been first noticed in 1740 by Sorge, a German organist. Subsequently, attention was drawn to them by Tartini, who called them " grave harmonics," and endeavoured to make them the foundation of a system of harmony.
As already stated, the vibration number of a differential tone, is the difference between the vibration numbers of its generators. It is easy therefore to calculate what differential any two given generators will produce. For example, two tones, having the vibration numbers 256 and 412 respectively are sounded simul­taneously, what will be the vibration number of the differential tone produced ? Evidently 412 — 256, that is 156.
Further, if the two generators form any definite musical interval, the differential tone may be easily ascertained, though their vibra­tion numbers may be unknown. For example, what differential will be produced by two generators at the interval of an octave ? "Whatever the actual vibration numbers of the generators, they must be in the ratio of 2 : 1. Therefore the difference between them must be the same as the vibration number of the lower of the two generators; that is, the differential will coincide with the lower generator. Or shortly it may be put thus:—
Again, what differential will be produced by two tones at the interval of a Fifth ? The vibration numbers of two tones at the interval of a Fifth are as 2 : 3, difference = 3 — 2 = 1. Therefore the vibration number of the differential will be to the vibration number of the lower of the two tones as 1 : 2; that is, the differential will be an octave below the lower generator, or briefly,
In the following Table, the last column shows the Differentials produced by the generators given in the second column.
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