HARMONIC  . In See also:

• ACOUSTICS (from the Gr. &eobav to hear)

acoustics, a harmonic is a secondary See also:

• TONE, THEOBALD WOLFE (1763-1798)

tone which accompanies the fundamental or See also:

• PRIMARY

primary tone of a vibrating See also:

• STRING

string, See also:

• REED
• REED, ANDREW (1787-1862)
• REED, ISAAC (1742—1807)
• REED, JOSEPH (1741—1785)
• REED, THOMAS BRACKETT (1839—1902)

reed, &c.; the more important are the 3rd, 5th, 7th, and See also:

• OCTAVE (from Lat. octavus, eighth, octo, eight)

octave (see See also:

• SOUND
• SOUND, THE (Danish Oresund)

SOUND; See also:

• HARMONY (Gr. dpµovia, a concord of musical sounds, dpA'ew to join; &plan midi (sc. TeXvil)

HARMONY) . A harmonic proportion, in See also:

• ARITHMETIC (Gr. apeOµ7run7, sc. TEXVn, the art of counting, from (ipLBµos, number)

arithmetic arid See also:

• ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

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 (a Latin word from serere, to join)

series consists of terms whose reciprocals See also:

• FORM (Lat. forma)

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 (Lat. fundatio, from fundare, to found)

foundation of such phrases as harmonic See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section, harmonic ratio, harmonic conjugates, &c . (see See also:

• GEOMETRY

GEOMETRY: II . Projective) . The connexion between acoustical and mathematical harmonicals is most probably to be found in the See also:

• PYTHAGOREAN

Pythagorean See also:

• DISCOVERY

discovery that a vibrating string when stopped at z and of its length yielded the octave and 5th of the See also:

• ORIGINAL

original tone, the See also:

• NUMBERS, BOOK OF
• NUMBERS, PARTITION OF

numbers, 1 , being said to be, probably first by See also:

• ARCHYTAS (c. 428—347 B.C.)

Archytas, in harmonic See also:

• PRO

pro-portion . The mathematical investigation of the form of a vibrating string led to such phrases as harmonic See also:

• CURVE (Lat. curvus, bent)

curve, harmonic See also:

• MOTION (Lat. motio, from movere, to move)
• MOTION, LAWS OF

motion, harmonic See also:

• FUNCTION

function, harmonic See also:

• ANALYSIS (Gr. avci and Meta, to break up into parts)

analysis, &c . (see See also:

• MECHANICS

MECHANICS and SPHERICAL HARMONICS) .