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:

CDEFGABC
18⁄94⁄53⁄42⁄33⁄58⁄151⁄2
Keynote2d3d4th5th6th7thOctave

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.

Transcriber’s Note: For note 15, 960Hz is slightly below B-natural, not B-sharp.

[[Listen]]

NAME OF NOTEC1C2G2C3E3G3B♭3C4D4E4F♯4G4A4B♭4B♮4C5
ORDER OF SUCCESSION12345678910111213141516
VIBRATION PER SECOND641281922563203844485125766407047688328969601024