But beside the over-tones, which sound with every good, simple sound, there are other _partial tones_, which, like the long-known combination tones, do not usually present themselves to our consciousness. Combination tones were first discovered in 1745 by the organ-builder, _Sorge_. By an act of concentrated attention one hears these tones at the accord of two different tones. They lie always lower than the interval to which they belong, and arise from the meeting of the nodes of vibration of the tones producing the interval. The node of vibration is the name of that place where, after every completed vibration, the sounding body returns to its former position. When, for instance, we give the third c¹ e¹, we hear the c, lying an octave lower than the third, sounding at the same time as a combination tone. For the tone c¹ a string has two vibrations, while in the same space of time e¹ has three. The vibration node of the c¹ will thus, after two vibrations, coincide with the vibration node of the e¹. By the coincidence of these nodes of vibration is produced the number of vibrations requisite for the c below. Besides these combination tones there are summation tones, discovered by Helmholtz, which arise from the vibrations collectively (_Gesammtzahl_) belonging to the above interval, and are higher than the interval. Both kinds of partial tones have again their faint over-tones.
We have explained the movements of the waves of sound by the movements on the surface of water, and we know that, instead of the billows and hollows that we have in the water, the air is condensed and rarefied. We know further that if two different lines of waves run along with one another, their crests and hollows fall together, and their crests become as high again and their hollows as deep again. So two tones from different sources of sound are twice as strong when they are both equally high, and a new tone of the same height added to them will still further increase the sound. But when two agitations of the surface of the water so move that the crests of one fall into the hollows of the other, their movements neutralize each other. The same thing happens in tones when one is not struck until half the vibrations of the preceding tone are concluded. But if the sounding bodies vary in only a small part of a vibration sound, they will be alternately stronger and weaker, and this is termed beats (_Schwebungen_), which are only produced by tones _very near to each other_. Those intervals whose combination and over-tones so fall together that many beats are produced, sound harsh and disagreeable, and we call them dissonances.