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HARMONIC

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Originally appearing in Volume V12, Page 956 of the 1911 Encyclopedia Britannica.
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HARMONIC . In See also:

acoustics, a harmonic is a secondary See also:tone which accompanies the fundamental or See also:primary tone of a vibrating See also:string, See also:reed, &c.; the more important are the 3rd, 5th, 7th, and See also:octave (see See also:SOUND; See also:HARMONY). A harmonic proportion, in See also:arithmetic arid See also: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 See also:series consists of terms whose reciprocals See also:form an arithmetical progression; the simplest example is: I +1+ 1+1 +... (See ALGEBRA and ARITHMETIC). The occurrence of a similar proportion between segments of lines is the See also:foundation of such phrases as harmonic See also:section, harmonic ratio, harmonic conjugates, &c. (see See also:GEOMETRY: II. Projective). The connexion between acoustical and mathematical harmonicals is most probably to be found in the See also:Pythagorean See also:discovery that a vibrating string when stopped at z and of its length yielded the octave and 5th of the See also:original tone, the See also:numbers, 1 , being said to be, probably first by See also:Archytas, in harmonic See also:pro-portion. The mathematical investigation of the form of a vibrating string led to such phrases as harmonic See also:curve, harmonic See also:motion, harmonic See also:function, harmonic See also:analysis, &c. (see See also:MECHANICS and SPHERICAL HARMONICS).

End of Article: HARMONIC

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