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.

Carrying the experiment further, we may, by dividing the given string at other points on its surface, obtain all the other notes of the musical scale. It will not be necessary to repeat the explanation in each case, and the reader will have no difficulty in comprehending the following table, which gives the relative string length required to produce the eight notes of the diatonic scale of C major, taking the length of the complete string that gives the keynote as 1, and considering all other pertinent conditions to remain equal:

C D E F G A B C 1 8/9 4/5 3/4 2/3 3/5 8/15 1/2 Keynote 2d 3d 4th 5th 6th 7th Octave

At first sight it might appear that the above data ought to give us all necessary information in regard to the phenomena of vibrating strings. Undoubtedly, the difficulties that surround the pianoforte designer would have little power to cause worry if there were nothing more to learn. Our troubles, however, are but just now beginning, and the difficulties that still exist are greater than any that we have yet investigated. These difficulties have their origin in the _nature_ of the sounds that are emitted by musical strings.

While we have been investigating the relative vibration speeds and pitches that pertain to the strings under various conditions, we have not as yet paid attention to any other difficulties that might have their origin under entirely different circumstances. There are, however, certain highly important phenomena which are determined by the nature of the strings themselves, irrespective of all other conditions. These phenomena affect the constitution of such sounds as any musical string may produce. Sounds produced through the agency of musical strings are not and cannot be simple sounds. And this peculiarity arises from the fact that such strings in common with most other agencies for the production of musical sounds are incapable of performing perfectly simple vibrations. If a string vibrated as a whole uniformly and all the time, its motions might be compared to the rhythmic swing of a pendulum, and the sounds that it emitted would be absolutely simple and absolutely pure. The fact, however, is that this never occurs. No string ever vibrates as a whole without simultaneously vibrating in segments, which are aliquot parts of the whole. These segments, when thus vibrating, give out the sounds that pertain to them according to their relative lengths; while the vibration of the whole length of the string, at the same time, causes the production of the sound proper to it, which is called the “prime” or “fundamental” tone. The sounds produced by the simultaneously vibrating segments are called “partial tones” or “upper partials.” In the case of sounding strings, such as we are now investigating, the partials follow each other in arithmetical progression and are produced by the vibrating of segments the proportions of which may be expressed by the harmonic series 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and so on ad infinitum. Now, if we examine this series we shall see that the lower of the partial tones that are represented by the various fractions must bear distinct harmonic relations to the fundamental tone. It will simultaneously be observed, however, that as the series is continued, the fractional quantities become uniformly smaller, and the difference between any pair of them (for the same reason) is smaller as the position of the given pair is more remote from unity. Naturally, this means that the partial tones represented by such fractional quantities are separated by continually decreasing intervals. If the process is carried far enough, the time comes when the interval of separation is less than a semitone. Clearly, then, partial tones in this condition can bear no proper harmonic relation to the fundamental tone. They are, in fact, dissonant.

Here, then, we come upon a fact that has a very wide bearing. It is a demonstrated acoustical truth that tone quality depends upon the number and intensity of the partial tones that accompany the fundamental during the sounding of any musical note. If, through any cause, these high and dissonant partials are excited into undue prominence, they may, and do, exercise a profound and maleficent influence upon the quality of musical sounds. We shall later have occasion to confirm the truth of this statement, and we shall learn, in the course of our investigation, fully to appreciate its importance in the practical problems of pianoforte design.

For the purpose of assisting the reader in the comprehension of the above argument, the following table is given, showing the order of succession and pitch of the partial tones of the note C (second line below the staff in the bass clef). Taking the pitch of the octave above middle C, for convenience of calculation, as 512 vibrations per second, this gives us 64 for the C in question.

It should be observed that the seventh, eleventh and fourteenth partials and their multiples cannot more than approximately be indicated in musical notation, as they do not exactly correspond to the notes that are written to represent them. We are obliged to be content with an approximation to the true pitch of these partials and the notation given here is as near as it is possible to approach.

A brief consideration of the facts thus presented will convince the reader that a combination of any fundamental tone with its first eight partials will produce a relatively harmonious effect. At the same time we must observe that this harmoniousness is more and more obliterated as the higher partials are permitted to sound simultaneously with the others. In fact, it may be said that, although we can not and must not eliminate the dissonant partials altogether, we should attempt to cause the strings to vibrate with entire freedom only as far as concerns the first eight partials, and less freely as far as concerns the others.

Now, in what manner can this desirable end be attained? To answer this question we must first discover what pre-disposing causes, if any, exist towards the favoring of any combination of partials at the expense of any other.

In speaking of the automatic sub-division of a string into vibrating segments, we omitted, at the time, to make mention of a fact which should, however, be obvious to the reader; namely, that the various points at which the sub-division occur are themselves motionless.

It would be more correct, perhaps, to say “apparently motionless”; for, of course, if these dividing points or “nodes,” as they are called, were entirely without motion, the formation of the vibrating segments would be impossible. In most cases, however, the “amplitude” or length of swing of the nodes when in motion is very much smaller than the amplitude of vibration of the segments. Consequently, as the vibration of the segments of a string is itself ordinarily invisible, the motion of the nodes may be considered as inappreciable.

Now, these nodes exercise considerable influence upon the problems that we are considering. For example, according to the researches of Young, it appears that when a string is struck at any point all those partials are obliterated that have their nodes at that point. Curiously enough, however, it has since been found in the case of the pianoforte, that those upper partials are not _necessarily_ eliminated that have their nodes at the striking point. Undoubtedly, however, a properly chosen node provides the best possible striking point, since its selection permits the operation of its tendency to suppress those particular partials that have their nodes at the same place.

A consideration of the phenomena already observed has caused us to perceive that the highest partials of the compound tone produced by a musical string do not bear precise harmonic relations to the prime tone. As the successive sub-divisions of the string approach closer and closer to each other, the tones thus generated are seen to be distant by proportionately less intervals, until at length they cease to have a close similarity to any tone of the musical scale. Consequently, as was said before, they exercise a generally harsh and dissonant influence upon the nature of the compound tone. We have already concluded that, broadly speaking, we should aim to eliminate these dissonant partials and, conversely, to favor the prominence of those which are more nearly harmonic. The reasoning which has served to lead us to this conclusion may profitably be carried a step further. If the highest partials are non-harmonic, it is obvious that their presence or absence, their prominence or the reverse, must necessarily exercise much influence upon the actual quality of a musical sound; upon the individual color which different generating agencies impart to the same musical note; in effect, upon all the numerous gradations of what we are accustomed to call harshness, hollowness or mellowness of tonal quality.

This inevitable conclusion has been fully substantiated by the results of experiment. The labors of Helmholtz and Koenig have demonstrated conclusively that the quality of a musical sound depends upon the number and intensity of the partial tones that accompany the fundamental. Thus the mystery of the individual tone coloring that distinguishes the voices of different musical instruments or of different persons is transferred from the realm of psychology to that of science. In fine, it becomes clear that if we can govern the number of the segments into which a vibrating string divides itself, and if we can also control the amplitude of vibration of these segments, we shall find it possible to alter the tone quality of a musical instrument at our pleasure.

It has already been observed that the generation of certain partial tones is assisted or retarded as the position of the striking point on a string is changed. It may not be out of place to note that the various other methods of exciting a string, such as plucking, bowing, etc., permit the production of equally variable effects as the points at which they operate are changed. Our inquiry, however, is confined to the pianoforte, and we shall therefore continue to limit ourselves to the cases of pianoforte strings as struck by the usual hammers.

The matter of choosing a proper striking point was first systematically investigated by John Broadwood, founder of the celebrated house of that name, in the early part of the nineteenth century. Until that time the pianoforte makers had, apparently, paid no attention to this important problem and had been content to follow in the steps of the builders of harpsichords and spinets. Examination of any of the instruments that are direct ancestors of the pianoforte will show that the strings are struck, indifferently, at any point from one-tenth to one-half of the speaking length. The only exceptions appear to be those clavichords in which the strings are all of the same length and in which the tangents on the keys impinge upon the strings at different fixed points to give the corresponding notes of the scale. Since the time of Broadwood, however, the vast importance of correctness in this particular has come to be recognized with more or less unanimity.

The investigations undertaken by this eminent maker convinced him that the ideal striking point lay between one-seventh and one-ninth of the speaking length of the string. Now, our investigations have shown us that the most harmonious and agreeable compound tone is that which is formed by the combination of the first eight partials. It would seem, therefore, that one-eighth of the speaking length would be more correct than the approximation that was arrived at by Broadwood. Theoretically, indeed, the latter is nearer to the ideal point; is, in fact, the ideal point. For obvious mechanical reasons, however, it is usually impossible to hit this point with exactitude, and the approximation suggested and used by Broadwood has been proved, by the practice of the best makers, to offer the nearest practical solution.

We may, then, lay it down as a rule to be followed that a point as nearly as possible midway between one-seventh and one-ninth of the speaking length of the string should be chosen and adhered to as the proper place where the blow of the hammer should be struck. If this rule be faithfully followed the greatest obstacle to purity of tone is removed and the most harmonious and agreeable combination of partials is in a fair way to be secured. Nevertheless, it is necessary to make an exception for the highest notes on the piano. Practical experience has shown that one-tenth is a better striking point for the very highest and shortest strings.

Thus we have been able to enunciate and discuss the principal laws that govern the activities of sounding strings, particularly those of the pianoforte. As the argument is developed, it will often appear that the theoretical exactitude of the rules here laid down must be modified in practice. Such a condition is always inevitable as between a body of laws and the application thereof. It will be found, however, that the variations to be recorded are not generally very important, and the reader will be well advised to make the rules enunciated in this chapter his continual leaning post and guide.

The most conspicuous difference is, perhaps, that which exists between the theoretical and practical results of halving string lengths to obtain octaves. In practice it is found that pianoforte strings generally sound a little flat of the octave when divided at exactly the middle point. But the variation is the fault of the steel wire and not of the rule.