HARMONICON. 571 HARMONICS. from the successive explosions formed by the periodic combinations of the oxygon in the air with the liydrogen. The note Jojicnds on the size of the liame and the length of the tube., and with a long tube, by varying the position of the jet in the tube, a series of notes may be produced. This phenomenon, which was discovered by Lam- ])adius, has been investigated by Tyndall and described in his Lectures on .S'ohh<7. Before the jet is lighted, in an actual experiment, the for- mation ot hydrogen must be allowed to go on until the air is completely driven out of the generating liottlc; otherwise a dangerous explo- sion may lake place. HARMONIC PROGRESSION. See Series. HARMONIC PROPORTION. See Hak- JtO.MC UlVI.'SIUN. HARMONICS (from Lat. harmonicus, har- monious, musical ) . The accessory or concomitant tones ])roduced by an}- fundamental tone. When a string tuned to C. for instance, is set in mo- tion, it produces not only that one note, but a number of other tones less intense. These sec- ondary tones are so much weaker than the funda- mental that it requires even the closest attention of a tine ear to detect their presence. Mersenne, a Franciscan monk vho died in Paris in 1048, was the first who discovered this. But it was not until 1701 that Sauveur gave a scientific explana- tion. Ranieau availed himself of Sauveur's in- vestigations, and founded his new system of harmony upon that basis. When the string C vibrates, it swings not only in its entire length, but at the same time smaller oscillations are produced by the two halves, the three thirds, the four fourths, etc., of the string. Tones can be distinguished for every division of such a string up to one-sixteenth. The two halves vibrating produce each the octave; the three thirds each the fiftli above the octave; the four fourths each the fourth above the fifth, etc., of the fundamen- tal tone. This is illustrated by the following table ( 1 denoting the entire length of the string, 2 one-half, 3 one-third, etc.) : 1 2 3 4 5 6 7 8 9 10 fl 12 13 14 15 IG Co, g, cSe',gSbb',c=,d% e=, fr, g=, ab^bb^ b=, c'. The tones marked X are only approximately correct. The eleventh tone, for instance, is neither / natural nor f sharp, but one between these, and nearer to the latter than the former. As all these tones lie above the fundamental, they are called occrtones : because they are produced by vibrations of only a part of the string, they are alsO' called partial tones, or aliquot tones. The name hannonic tones is given to them because all the tones that are given in their true pitch, except the ninth (rf"), are elements of the funda- mental triad (C, E, G). The intensity ot the harmonic tones decreases in inverse ratio to their pitch. The lowest tone, C, is called the generator. It will easily be seen how from these overtones the consonance of the major triad is estal)lished, for each of the elements occurs more than once in the series. FJ, Ab, Bb, and B are not given in their true pitch, and D occurs onl_y once in the whole series. Taking a high tone, c', for instance, a series of tones lying below, and bearing the same rela- tion to the highest tone as the overtones to the fundamental or generator, is produced. These tones are called undertones or lower partials. Vol. IX.— 37. The following table illustrates this (notes pro- duced only with approximate correctness being again marked X ) : X X XXX 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 IG c c=, t c',ab,f, d, c, Bb,.Vb, Gb, F, lib. U, Db, C. In this series Db, D, Eb, and Gb 'irc not given in true pitch, and Bb occurs only once. F, Ab, C occur each more than once. They are the ele- ments of the F minor triad ; hence the conso- nance of the minor triad is established by the series of undertones in the same manner as the consonance of the major triad is established by the series of overtones. From the consonance of these chords the consonance of intervals is also determined, for all intervals found in these chords or their inversions are regarded as consonant. The second inversion of either the major or minor triad — i.e. w^ith the fifth as the bass tone (see IxvERSiox) — yields the intervals of the fourth and major and minor sixth respectively. The only intervals not derivable from the above two series are the second and seventh, which are, therefore, regarded as dissonant, as well as any chord into the formation of which they enter. See Interv.^l.
hat has been said about the vibrations of a
string applies equally to the vibrations of a column of air. On any stringed instrument harmonics can be produced by lightly touching with the finger-tip any nodal point of the string. (See Node.) The string then cannot vibrate in its entire length, but only in sections, the length ot which is determined by the length of that part of the string lying between the neck of the instru- ment and the nodal point touched by the finger. For instance, it the lowest string of the violon- cello (C) is lightly touched at the middle point between the neck and the bridge, the two sections will vibrate independently, and each half will produce the octave above the tone to which the string is tuned. Thus, instead of C, its next higher octave, c, will be heard. Touching the string at one-third of the distance between the neck and the bridge, it will vibrate in three inde- ])cndent sections, each producing the twelfth (or fifth above the octave) of the fundamental note. All the partial tones given in the above table can thus lie produced by touching a point at one- fourth, one-fifth, etc., the length of the string. Tones thus obtained from an open string are called natural harmonics. Harmonics can also by obtained by first stopping a string, i.e. press- ing it firmly to the finger-board, by one finger and toucliing it lightly with another finger at some nodal point between the pressod-down finger and the bridge. Tones produced in this manner are called artificial harmonics. Harmonics are dis- tinguished from tones produced in the ordinary manner by a peculiarly sweet and ethereal qual- itv. The opening bars of the prelude to Lohen- ririn are a splendid example ot the elTcct produced by harmonics. In musical notation harmonic tones are indicated by a ° placed over the note, or by open square notes Q. On wind instruments (both brass and wood) harmonic tones arc pro- duced by varying the intensity and direction of the air-current. The pressing down of a valve has the same efTect as the iircssing down of a finger on a stringed instrument. Consult Ilaupt- mann. Die Katur der Harmonik and der ilefrik (Leipzig, 1853).
