“It is necessary that every thing which is harmonized, should be generated from that which is void of harmony, and that which is void of harmony from that which is harmonized. …But there is no difference, whether this is asserted of harmony, or of order, or composition… the same reason will apply to all of these.”

—  Aristotle , book Physics

Book I, Ch. VI, p. 57.
Physics

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Aristotle 230
Classical Greek philosopher, student of Plato and founder o… -384–-321 BC

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Context: Everything harmonizes with me, which is harmonious to thee, O Universe. Nothing for me is too early or too late, which is in due time for thee. There is one light of the sun, though it is interrupted by walls, mountains and infinite other things. There is one common substance, though it is distributed among countless bodies which have their several qualities. There is one soul, though it is distributed among several natures and individual limitations. There is one intelligent soul, though it seems to be divided.

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“Some… agreeing with Philolaus, believe that the proportion is called harmonic because it attends upon all geometric harmony, and they say that 'geometric harmony' is the cube because it is harmonized in all three dimensions, being the product of a number thrice multiplied together. For in every cube this proportion is mirrored; there are in every cube 12 sides, 8 angles and 6 faces; hence 8, the [harmonic] mean between 6 and 12, is according to harmonic proportion…”

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this harmonic proportion may be expressed as <math>\frac{12}{6}=\frac{12-8}{8-6}</math> or inversely.
Nicomachus of Gerasa: Introduction to Arithmetic (1926)

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Ernest Flagg photo

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Ernest Flagg photo

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Annie Besant photo

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St. 1.
A Song for St. Cecilia's Day http://www.englishverse.com/poems/a_song_for_st_cecilias_day_1687 (1687)
Context: From harmony, from heavenly harmony,
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Of jarring atoms lay,
And could not heave her head,
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'Arise, ye more than dead!'
Then cold, and hot, and moist, and dry,
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Nicomachus gives another reason for the name, viz. that a cube being of 3 equal dimensions, was the pattern &#940;&rho;&mu;&omicron;&nu;&#943;&alpha;: and having 12 edges, 8 corners, 6 faces, it gave its name to harmonic proportion, since:<center>12:6 :: 12-8:8-6</center>
Footnote, citing Vide Cantor, Vorles [Vorlesüngen über Geschichte der Mathematik ?] p 152. Nesselmann p. 214 n. Hankel. p. 105 sqq.
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