1911 Encyclopædia Britannica/Harmonic

HARMONIC. In acoustics, a harmonic is a secondary tone which accompanies the fundamental or primary tone of a vibrating string, reed, &c.; the more important are the 3rd, 5th, 7th, and octave (see Sound; Harmony). A harmonic proportion in arithmetic and algebra is such that the reciprocals of the proportionals are in arithmetical proportion; thus, if a, b, c be in harmonic proportion then 1/a, 1/b, 1/c are in arithmetical proportion; this leads to the relation 2/b＝ac/(a＋c). A harmonic progression or series consists of terms whose reciprocals form an arithmetical progression; the simplest example is: 1＋1/2＋1/3＋1/4＋. . . (see Algebra and Arithmetic). The occurrence of a similar proportion between segments of lines is the foundation of such phrases as harmonic section, harmonic ratio, harmonic conjugates, &c. (see Geometry: II. Projective). The connexion between acoustical and mathematical harmonicals is most probably to be found in the Pythagorean discovery that a vibrating string when stopped at 1/2 and 2/3 of its length yielded the octave and 5th of the original tone, the numbers, 1 2/3, 1/2 being said to be, probably first by Archytas, in harmonic proportion. The mathematical investigation of the form of a vibrating string led to such phrases as harmonic curve, harmonic motion, harmonic function, harmonic analysis, &c. (see Mechanics and Spherical Harmonics).