BEATS (DIE SCHWEBUNGEN)
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.
Those intervals in which few or no beats occur are called consonances. As the combination or interfering tones, as well as the beats, have importance and interest only in harmonizing several voices, in tuning pianos, as well as in composition in general, and as we have in view in these pages only the culture of single voices, we cannot further enlarge on these discoveries, interesting as they are. According to the purpose of this little work, I introduce only so much of the latest investigations and discoveries as will help to show the prevailing evils of our mode of teaching singing, and, by their practical application to the business of instruction, serve to improve the vocal art. But whoever has an interest in this branch of science will find in the invaluable work of Helmholtz, “Die Lehre von den Tonempfindungen,” an abundance of most interesting observations and of the most thoroughly scientific illustrations of the theory of music, and of those processes in the domain of tone which we have hitherto always felt, but never understood.