Text: 13) A harmonic scale is formed by taking a series of notes produced by vibration whose numbers in a given time are respectively as 1, 2, 3, 4, et. If we take as fundamental tone the open C string of the violoncello, the series of tones which with it forms a harmonic scale will be as pictured: The notes marked with an asterisk do not exactly represent the corresponding tones; but are the nearest representives which the modern notation supplies. All the notes of the harmonic scale can theoretically be produced by either a single string, or by a simple tube used as a trumpet. If we lightly touch the string of a violin, without causing it to come in contact with the finger baord, at any one of a series of points dividing it into a number of a equal parts, and excite it by means of a bow, it no longer vibrates as a whole, but separates into the number of equal vibrating segments which is the least possible consistent with that point forming one of their points of diversion; the latter remain stationary, or very nearly so, and are called nodes, their number being evidently just one less than that of the segments. It is plain that if the point of appication of the bow be one of a series of nodes, no sound will be produced, provided, of course, the finger remains on any other of the same series, and this may serve to explain why it is sometimes difficult to bring out the higher harmonics of a violin, as the bow may, unconsciously to the performer, be passing exactly over one of the corresponding nodes. The first harmonic, as it is called, of the open string is produced by touching it while on a state of vibration at its middle point, and thereby dividing it into two equal portions, both of which vibrate twice as fast as the whole, and accordingly give the octave. The second harmonic, or the twelfth of the fundamental, corresponds to a division of the string into three equal portions, and so on. And generally, in order to produce the nth harmonic the finger should touch the string at any one of the series of points which divide it into n equal portions. In practise, however, the finger should always touch the string at the point of division adjacent to either end.

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Source: 125