Page:Elementary Text-book of Physics (Anthony, 1897).djvu/197

§ 156] of vibrations are $$n$$ and $$2n \pm m$$. Fig. 64 shows traces obtained mechanically. In I the numbers of the component vibrations were $$n$$ and $$n + m$$, in II and III $$n$$ and $$2n \pm m$$, and in IV $$n$$ and $$3n + m$$. In all these cases a variation of amplitude occurs during the same intervals, and it seems reasonable to suppose that those variations of amplitude should cause variations in intensity in the sound perceived.

Cross has shown that the beating of two tones is perfectly well perceived when the tones themselves are heard separately by the two ears; one tone being heard directly by one ear, while the other, produced in a distant room, is heard by the other ear by means of a telephone. Beats are also perceived when tones are produced at a distance from each other and from the listener, who hears them by means of separate telephones through separate lines. In this case there is no possibility of the formation of a resultant wave, or of any combination of the two sounds in the ear.

156. Resultant Tones.—Resultant tones are produced by combinations of two tones. Those most generally recognized have a vibration number equal to the sum or difference of the vibration numbers of their primaries. For instance, ut6, making 2048 vibrations, and re6, making 2304 vibrations, when sounded together give ut3, making 256 vibrations. These tones are only heard well when the primaries are loud, and it requires an effort of the attention and some experience to hear them at all. Summation tones are more difficult to recognize than difference tones, nevertheless they have an influence in determining the general effect produced when musical tones are sounded together. Other resultant tones may be heard under favorable conditions. As described above, two tones making $$n$$ and $$n + m$$ vibrations respectively, when $$m$$ is considerably less than $$n$$, give a resultant tone making $$m$$ vibrations; but a tone making $$n$$ vibrations in combination with one making $$2n m, 3n + m,$$ or $$xn + m$$ vibrations, gives the same resultant. This has sometimes been explained by assuming that internnediate resultants are produced, which, with one of the primaries, produce resultants of a higher order. In the case of the two tones making $$n$$ and