HARMONICS, the accessory, or concomitant sounds which are produced by a funda mental musical sound, either naturally, or by a division into aliquot parts. Every musical sound, although to the ordinary ea'r it appears to be only one sound, will, on close observation, be perceived to consist of a principal or fundamental sound, accom panied by other feeble acute sounds in perfect harmony (see HAnwONY). The existence of such accompanying sounds, which are called harmonics, can be best demonstrated by the vibrattims of a string stretched between two points, or bridges. Eight feet is a good length for such a string, although 16 ft., or even 32, would be better, from bridge to bridge. A scale or measure, accurately dividing the length of the string into aliquots from / up to alongside of it. When a is drawn across the. string it vibrates from end to end, and gives out its fundamental sound. Divide the string into halves by slightly touching it with the finger at the mark on the scale, or better, with a stretched thread lightly pressed upon it at that point; when sounded, it will be found to vibrate in two halves, each part vibrating as fast again as the entire string, and producing a sound an octave above the fundamental one, or as 2 to 1. Divide in the same manner at and the sound produced is the fifth above the last octave, being in the proportion of 8 to 2. It is not necessary to touch the string on more than one of the points of the division, for the long side of the string always divides of itself naturally, which can be seen by the eye. The parts where the string seems at rest, are called the nodal points. Divide as before at }, and the second octave above the lowest sound is heard, -being to the first octave as 4 to 2. At the major third above the last octave is found, being as 3 to 4. At the octave of the cornier fifth. 3 to 2. At + we find the true that seventh, or 7 to 4; at /, again the octave of the lowest; at the major second, or 9 to 8; and above this, at A, A., we find the octaves of the major third, the fifth, and the flat seventh; while at we obtain the sharp seventh, or 15 to 8; and at another octave of the fundamental sound. The following is the order in which the
harmonics arise, assuming that the string, at its full length, sounds the note C on the secuud ledger line below the bass staff, or lowest string on a violoncello, From these harmonics, the true ratios of all the intervals 'of the diatonic scale, in relation to a fundamental keynote, are found, and in the most perfect tune; they are as follows: Degrees of the scale. I 11. III. IV. V. VI. VII. VIII.
Notes of the scale..... ...... F GA B Mums to key--note ••• • 1 le I Assuming 24 as the number of vibrations of C in any given time, the other notes of the scale may be expressed in whole numbers thus: Notes of the scale C D E F G A BC In whole numbers..... 24 27 30 32 30 45 48 In the artificial division of the octave into a chromatic scale of twelve equal semi tones, all the intervals must necessarily be made somewhat imperfect, which is called temperament (seaTE3tPERAmENT). This must be especially attended to in keyed instru ments. Singers, and performers on stringed instruments, are guided by their car, being free from the fetters of fixed notes, to which keyed instruments are necessarily subject. Even in the natural diatonic scale as produced by the harmonics, it will be found, on analysis, that a certain degree of temperament is required to make the fifths within the octave equal. For example, the fifths from-F to C, and from E to B, will be found to be accurately the same as the fifth from C to G—viz., which is easily ascertained by reducing their respective numbers to the lowest fraction; thus, F to k.t is -4-A = to I; from E to B is = while from D to A, which in practical music must also betreated as a fifth, will be found to be too flat; thus, D to A is p, which cannot be reduced to l; but when both are brought to the fractions of a common denominator, which is done by multiplying by by 2 = and I by 27 = tt, it is shown that D to A, in the scale of nature, is flatter than a perfect fifth, in the proportion of 81 to 80; so that without tem perament A cannot at the same time be a perfect major sixth to C, as a key-note, and also a perfect fifth to D, the true major second of C.