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 + ½ + ⅓ + ¼ + ... (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 ½ and ⅔ of its length yielded the octave and 5th of the original tone, the numbers, 1 ⅔, ½ 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).