The Voice in Singing

by Emma Seiler

TONE, AND ITS LAWS OF VIBRATION

A tone is produced by a periodical motion of the sounding body—a noise by motions not periodical. We can see and feel the sounding vibrations of stationary bodies. The eye can perceive the vibrations of a string, and a person playing on a clarionet, oboe, or any similar instrument, feels the vibration of the reed of the mouthpiece. How the movements of the air, agitated by the vibrations of the stationary body, are felt by the ear as tone (Klang), Helmholtz illustrates by the motion of waves of water in the following way: Imagine a stone thrown into perfectly smooth water. Around the point of the surface struck by the stone there is instantly formed a little ring, which, moving outwards equally in all directions, spreads to an ever-enlarging circle. Corresponding to this ring, sound goes out in the air from an agitated point, and enlarges in all directions as far as the limits of the atmosphere permit. What goes on in the air is essentially the same that takes place on the surface of the water; the chief difference only is that sound spreads out in the spacious sea of air like a sphere, while the waves on the surface of the water can extend only like a circle. At the surface the mass of the water is free to rise upward, where it is compressed and forms billows, or crests. In the interior of the aerial ocean the air must be condensed, because it cannot rise. For, “in fact, the condensation of the sound-wave corresponds to the crest, while the rarefaction of the sound-wave corresponds to the sinus of the water-wave.”[ 7 ]

The water-waves press continually onwards into the distance, but the particles of the water move to and fro periodically within narrow limits. One may easily see these two movements by observing a small piece of wood floating on water; the wood moves just as the particles of water in contact with it move. It is not carried along with the rings of the wave, but is tossed up and down, and at last remains in the same place where it was at the first. In a similar way, as the particles of water around the wood are moved by the ring only in passing, so the waves of sound spread onwards through new strata of air, while the particles of air, tossed to and fro by these waves as they pass, are never really moved by them from their first place. A drop falling upon the surface of the water creates in it only a single agitation; but when a regular series of drops falls upon it, every drop produces a ring on the water. Every ring passes over the surface just like its predecessor, and is followed by other rings in the same way. In this way there is produced on the water a regular series of rings ever expanding. As many drops as fall into the water in a second, so many waves will in a second strike a floating piece of wood, which will be just so many times tossed up and down, and thus have a periodical motion, the period of which corresponds with the interval at which the drops fall. In like manner a sounding body, periodically moved, produces a similar periodic movement, first of the air, and then of the drum in the ear; the duration of the vibrations constituting the movement must be the same in the ear as in the sounding body.

THE PROPERTIES OF TONE (KLANG)

The sounds produced by such periodic agitations of the air have three peculiar properties: 1. Strength, 2. Pitch, 3. Timbre.

The strength of the tone depends on the greater or less breadth of its vibrations, that is, of the waves of sound, the higher or lower pitch of the tones upon the number of the vibrations; that is, the tones are always higher the greater the number of the vibrations, or lower the less the number of the vibrations. A second is used as the unit of time, and by number of vibrations is understood the number of vibrations which the sounding body gives forth in a second of time. The tones used in music lie between 40 and 4000 vibrations per second, in the extent of seven octaves. The tones which we can perceive lie between 16 and 38,000 vibrations to the second, within the compass of eleven octaves. The later pianos usually go as low as C₁ with 33, or even to A₂ with 27½ vibrations; mostly as high as a⁴ or c⁵, with 3520 and 4224 vibrations. The one lined a¹, from which all instruments are tuned, has now usually 440 to 450 vibrations to the second in England and America. The French Academy, however, has recently established for the same note 435 vibrations, and this lower tuning has already been universally introduced in Germany.[ 8 ]

The high octave of a tone has in the same time exactly double the number of vibrations of the tone itself. Suppose, therefore, that a tone has 50 vibrations in a second, its octave has 100 in the same time; i. e., twice as many. The octave above this has 200 vibrations, &c. The Pythagoreans knew this acoustic law of the ascending tones, and that the octave of a tone had twice as many vibrations in a second as the tone itself, and that the fifth above the first octave had three times as many; the second octave, four times; the major third above the second octave, five times as many; the fifth of the same octave, six times; the small seventh of the same octave, seven times. In notation it would be thus, if we take as the lowest note C, for example:

The figures below the lines denote how many times greater the number of vibrations is than that of the first tone. In the first octave we find only one tone; in the second, two; in the third, all the tones of the major chord with the minor seventh. In the fourth octave we find sixteen tones (which, however, we divide in our system of music into twelve). Likewise, we find in the fifth octave thirty-two tones, which number is doubled in the sixth. Hence, the Greeks had quarter and eighth tones, which we in our equal-tempered tuning have done away with.[ 9 ]

The production of a higher pitch in a tone rests in all sounding bodies upon the uniform law which we may observe in the strings of musical instruments, whose tones ascend either by greater tension, by shortening, or through a diminution of the density of the strings.