CHAPTER IV.
ACOUSTICAL LAWS OF SOUNDING STRINGS.

Sound is an impression produced upon the brain through the ear by the motion of air particles excited by an external body. In the transmission of sound from the vibrating or “sonorous” body to the ear it is motion that is transferred and not the substance of the air itself. In the same way there can be no sensation of sound without the interposition of an elastic fluid such as air or water, and the production of sound in a vacuum is, therefore, impossible.

Sound, in short, has no objective existence. We know it simply as a sensation, primarily caused by certain physical processes, the nature of which is comparatively familiar to us. We are aware of all that goes on between a sounding body and the ear, but we know nothing of the processes whereby these physical motions are transformed until they become, within the brain, sensations of musical sound or of noise.

While so much of mystery clouds our conception of the nature of sound, we may take comfort in the knowledge that to penetrate the enigma is by no means necessary. Not even the musician requires such transcendent knowledge. To the student of musical craftsmanship it is equally non-essential. It is well, however, to recognize the fact that as soon as we leave the sure ground of physical investigation, we become lost in impenetrable mystery and find ourselves face to face with the ancient, yet ever new, questions of our origin and destination. When we reflect upon the essentially spiritual and unearthly influence of music, we cannot but feel that, in the making of instruments to serve this art, we are ourselves assisting, however blindly, at a more than Eleusinian mystery.

The ear easily distinguishes between musical and non-musical sounds. Nor does it fail to recognize differences in relative loudness or softness of any given musical sound. Again, the relative degree of acuteness or gravity is distinguished, and, lastly, the quality of the same musical note when played upon two different instruments or when sung by two different voices is no less easily observed.

Now we have first to ask ourselves in what the difference between musical and non-musical sounds consists. We may say that a musical sound is produced by regularly recurring motions of the sounding body communicated to the air; or, more technically, a musical sound may be defined as a sound produced by periodical vibrations. This may be proved by holding a piece of cardboard against a rapidly revolving toothed wheel. As long as the revolutions of the wheel are performed at a comparatively slow speed the noise produced by the impact of the cardboard is broken and disjointed; but as the wheel is caused to revolve with greater rapidity the noise becomes gradually continuous and assumes a definite pitch. By increasing the speed of the wheel we cause a higher pitched musical sound to be produced. Now, if we arrange a second card and wheel and cause them to be set in motion together with the first we shall find that when the two wheels are revolved at the same speed, they produce sounds of the same pitch. Thus it is apparent that the pitch of a musical sound depends upon the speed of vibration, or upon the number of vibrations per second. Without going too deeply into technicalities it may be said that similar experiments have enabled investigators to determine the behavior of sonorous bodies in reference to all the other conditions that pertain to them. Thus, in the case of strings such as are used in the pianoforte, we are in possession of facts that make it possible for us to state accurately the pitches that will pertain to strings of given lengths, densities and thicknesses, which are stretched at given tensions. It is unnecessary to go into details of the precise methods employed to demonstrate these laws, and it will be quite sufficient to quote the laws themselves. The reader is therefore invited to note carefully that:

  1. The number of vibrations of a string is inversely proportional to the length of the string.
  2. The pitch of a musical sound is proportional to the number of vibrations per second; the greater the number of vibrations, the higher the pitch.
  3. The number of vibrations per second of a string is proportional to the square root of its tension. That is to say, if a string is stretched with a weight of one pound it will give forth a sound one octave lower than the sound that it would emit if stretched with a weight of four pounds.
  4. The number of vibrations of a string varies inversely as the thickness of the string. So that if there are two strings of the same material and length and subjected to the same tension, and if the diameter of the first is twice the diameter of the second, the first will produce one-half as many vibrations as the second.
  5. The number of vibrations per second of a string varies inversely as the square root of its density. Thus, if one string has four times the density of another, the first will produce one-half as many vibrations as the second.

In addition to these valuable laws, there are certain others which have reference to the actual musical sounds produced by strings. By means of them we know the relative proportions of the strings that will, other things being equal, give the various notes of the musical scale. If a perfect musical string be stretched and excited into vibration it will be found that an exact octave above the note that the whole string gives out may be produced by dividing the string at its precise middle point and causing one of the halves to vibrate. Now we have already noted that the number of vibrations of a string is proportional to its length, and it is therefore obvious that the halves of the given string each have double the number of vibrations of the whole, and that, consequently, the octave to a note is produced by either twice or half the number of vibrations that suffice to produce the given note.