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.