both with the prime note and among themselves. At the same time the smaller the aliquot parts become in the ascending series, the less easily are they set in a state of separate vibration. Consequently these high dissonant harmonics are distinctly audible only on highly resonant metallic instruments, such as the cymbals, bell, and triangle, and for practical purposes the old term harmonic answers as well as the term 'partial.'
A few instruments, such as the tuning-fork and the wide stopped organ pipe, practically yield no harmonics. The human voice, the harmonium, and all orchestral instruments, are rich in them—the human voice probably the richest of all; but nature has so admirably compounded them that it is very difficult to analyse them scientifically. Rameau distinguished harmonics in the human voice as early as the beginning of the last century.
Harmonics naturally reinforce the fundamental sound, in which case their extent and distribution largely influence the intensity and the quality of the sound. They may, however, in many instances, be produced singly by mechanically checking the vibration of the fundamental note. In this relation they constitute an important practical department in most orchestral instruments.
Law of Harmonics. A sonorous body not only vibrates as a whole but in each of its several fractions or aliquot parts, 1⁄2, 1⁄3, 1⁄4, 1⁄5, 1⁄6, 1⁄7, and so on at the same time; and each of these parts gives a separate note, the 1⁄2 yielding the octave, the 1⁄3 the fifth, the 1⁄4 the double octave, the 1⁄5 the third above the double octave, and so on. The following scheme or diagram, taken from Moinigny, shows the harmonics of the open string G on the violoncello up to thirteen places:—
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Here the bottom G is produced by the vibration of the whole string. The two Gs next above are produced by the vibration of the two halves. The three Ds next above by the vibration of the three thirds; and so on. Thus the diagram represents the whole of the notes produced by the vibrations of the whole string and its various sections up to its one-fourteenth part.
In this scheme the first F (counting upwards), the C a fifth above it, and the topmost notes E and F, are more or less faulty. In practically deducing the diatonic scale from this scheme, these intervals have to be corrected by the ear. By inspection of this scheme we discover the intervals of the diatonic scale in the following order:—
From this scale may obviously be deduced the chords of the third, fifth, seventh, and ninth. By combining and transposing these notes into one octave we get the following scale:—
which is the scale of C major ascending from dominant to dominant. As the same thing happens in other keys, we have thus proved the law that the intervals of each scale are generated by its dominant. The dominant, not the tonic, is therefore the true root of the whole scale.
Practical effect of Harmonics heard simultaneously with the fundamental note. The harmonics not only determine the diatonic intervals, but to some extent the intensity and, as has been lately proved by Helmholtz, the quality of musical tones. On applying the ear to the soundhole of a violin during a long crescendo on one note, the reinforcement of the tone by the gradual addition of the higher and more piercing harmonics is distinctly perceptible. The principle and the effect are precisely the same in a crescendo produced by the addition of the mixture stops on an organ. The loudest musical instruments, cœteris paribus, are those in which the highest harmonics predominate, e.g. the cymbals, triangle, bell, and gong.
The effect of harmonics on the quality of musical sounds is easily tested by carefully comparing the tones of an old and a new violin. In the former the strong vibrations of the fundamental note and the lower harmonics leave but little force to be expended on the higher and noisier harmonics: in the latter the fundamental note and lower harmonics are capable of absorbing less of the force, which is transmitted to the upper harmonics, and produces a harsh quality of sound. When the fundamental note and lowest harmonics predominate in the tone, the quality is soft and flute-like; when the combination is well balanced by the addition of the intermediate harmonics up to the sixth, the quality is rich and sonorous; when the highest harmonics, above the sixth and seventh, predominate, the quality is harsh and screaming. When the high dissonant harmonics are produced in a tolerably even and continuous stream of sound, the quality is said to be 'metallic.' If an instrument is ill-strung or out of order the harmonic scale is disturbed; and the harsh, uncertain, and irregular tones which it yields consist of harmonics out of their true place. Less varied comparisons may be obtained on the stops of an organ. Wide pipes, yielding a dull, heavy tone, have virtually no harmonics. In the tone of narrower open pipes the harmonics up to the sixth can be detected by the aid of Helmholtz's resonators. Pipes conically narrowed at the upper end, such as compose the stops called Gemshorn, Salicional, and Spitz-flute, yield strong intermediate harmonics, which render the tone bright, though perceptibly thin. The Rohr-flute is so constructed as greatly to reinforce the fifth harmonic (2½ octaves above the prime note). The nasal quality of sound, such as is yielded by the softer