A Dictionary of Music and Musicians/Harmonics

by Edward John Payne

HARMONICS, tones of higher pitch which accompany every perfect musical sound in a regular series. As they ascend they diminish in intensity, and approximate in pitch. If the piano be opened and a note—say D in the bass—be struck smartly and kept down, on listening attentively a succession of faint sounds will be heard, apparently rising out of the principal sound and floating round it. These are the harmonics. They are really constituents of the main musical tone, and are produced by the concurrent vibration of the aliquot parts of the string. Hence Helmholtz proposes to call them 'partial tones' (Partialtöne). This term is no doubt more appropriate, inasmuch as above the tenth degree most of these notes form intervals dissonant from the prime note and also from each other, and thus become perceptibly inharmonic. On the best musical instruments, however, these high inharmonic tones are not reached, the vibratory impulse being exhausted on the prime note and the lower harmonics, which are consonant 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:—

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 reed-stops, by violins of a certain build, and by the clarinet, bassoon, etc., is produced by the predominance of the uneven harmonics (¹⁄₃, ¹⁄₅, ¹⁄₇, etc ). On the harmonium these uneven harmonies are stronger than the even ones. The peculiar tinkling tones of the zither arise from the high uneven harmonics yielded by its comparatively thick metal strings.

If a singer produces a low note crescendo against a reflecting surface, the harmonics become distinctly audible. If the note is produced partly through the nose, the uneven harmonics perceptibly predominate. The number of upper harmonics in the human voice is very great: and they are, according to Helmholtz, distinct and powerful in their whole range.

Practical use of single Harmonic tones on stringed instruments. Harmonics may be singly produced (1) by varying the point of contact with the bow, or (2) by slightly pressing the string at the nodes, or divisions of its aliquot parts (¹⁄₂, ¹⁄₃, ¹⁄₄, etc.). (1) In the first case, advancing the bow from the usual place where the fundamental note is produced, towards the bridge, the whole scale of harmonics may be produced in succession, on an old and highly resonant instrument. The employment of this means produces the effect called 'sul ponticello.' [See Ponticello.] (2) The production of harmonics by the slight pressure of the finger on the open string is more useful. When produced by pressing slightly on the various nodes of the open strings they are called 'Natural harmonics.' In the following example the lower notes represent the fingering, the upper ones the effect:—

Scotch Air.

Natural harmonics are occasionally employed pizzicato on the violin and violoncello, and are an important resource in harp music. Accurate violinists are disinclined to use them, because the player has no control over their exact intonation, which is rigidly determined by that of the open string; and the tones of the open strings, which are tuned by perfect fifths, are in certain scales slightly dissonant. In the key of G, for instance, the harmonics of the first or E string are slightly dissonant, though they are perfect in the key of A.

Artificial harmonics are produced by stopping the string with the first or second finger, and thus making an artificial 'nut,' and then slightly pressing the node with the fourth finger. By this means harmonics in perfect intonation can be produced in all scales. Example—

Carnaval de Venise.

etc.

For the entire theory of artificial harmonics in single and double scales see 'L'Art de Jouer du Violon de Paganini' by Guhr. They can however only be produced by using thin strings, and are little employed by the best writers. In modern music they are designated by an open note of this

form. (See the Andante of Joachim's Concerto, etc.)

Practical use of single harmonic tones on wind instruments. As in the case of stringed instruments, the harmonics of wind instruments naturally reinforce the prime note, but are separable from it by artificial means. In wind instruments this is done by varying the intensity or the direction of the air current from the mouth, which sets in vibration the air-column in the tube, so as to throw the air-column into vibrating portions of different lengths, as in the case of the aliquot parts of a string. The falsetto voice consists of harmonic octaves of the natural voice. All the notes of the flute above the lowest octave are harmonic octaves, twelfths, and double octaves of the lower notes. Like the corresponding harmonics on the oboe and clarinet, these tones are produced by overblowing. Brass instruments are richest in the practical employment of harmonics. Any brass instrument, such as the hunting horn or military bugle, yielding one fundamental note, yields the familiar harmonic scale

Violinists are well aware that the longer the string in proportion to its thickness, the greater the number of upper harmonics it can be made to yield. Similarly, the longer the tube of a brass instrument, the higher does the series of its practicable harmonic tones ascend. The old French horn consists simply of a conical tube of jreat length, which readily yields the scale of larmonic intervals. They are produced by gently varying the degree and direction of the current of air. The dissonant notes (¹⁄₇, ¹⁄₁₁, ¹⁄₁₃, ¹⁄₁₄) in the scale are to some extent corrected, and some of the missing tones are supplied by introducing the hand into the bell. Mechanical appliances lave been contrived for the same purposes. On,he trumpet the tube is extended for the same purposes by means of a slide. [See Horn, Trumpet, etc.]

[ E. J. P. ]