Theory and Practice of Piano Construction / With a Detailed, Practical Method for Tuning

by William Braid White

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.