HARMONIC. In acoustics, a harmonic is a secondary tone which accompanies the fundamental or primary tone of a vibrating string, reed, etc.; 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 i/a, i/b, i/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:
j + j-j-j + j-f-... (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, etc. (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 and f of its length yielded the octave and 5th of the original tone, the numbers, i f , 3 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, etc. (see MECHANICS and SPHERICAL HARMONICS).
Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)