“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…”

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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Ancient Greek mathematician 60–120

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“It was Pythagoras who discovered that the 5th and the octave of a note could be produced on the same string by stopping at 2⁄3 and ½ of its length respectively. Harmony therefore depends on a numerical proportion. It was this discovery, according to Hankel, which led Pythagoras to his philosophy of number. It is probable at least that the name harmonical proportion was due to it, since1:½ :: (1-½):(2⁄3-½).Iamblichus says that this proportion was called ύπ eναντία originally and that Archytas and Hippasus first called it harmonic.”

James Gow (scholar) (1854–1923) scholar

Nicomachus gives another reason for the name, viz. that a cube being of 3 equal dimensions, was the pattern άρμονία: 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.
A Short History of Greek Mathematics (1884)

“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

“It is picked out from numbers progressing in continuous proportion. Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical one which advances by unequal and proportionally increasing or decreasing intervals. Arithmetical progressions: 1, 2, 3, 4, 5, 6, 7, &c.; or 2, 4, 6, 8, 10, 12, 14, 16, &c, Geometrical progressions: 1, 2, 4, 8, 16, 32, 64, &c.; or 243, 81, 27, 9, 3, 1.”

John Napier (1550–1617) Scottish mathematician

The Construction of the Wonderful Canon of Logarithms (1889)

“To the Greek harmony was of supreme importance, and if the triglyphs represented the harmonic scale… their meaning and use is abundantly explained. …The universal admiration which Greek proportions have always excited proves that the method of obtaining them was correct.”

Ernest Flagg (1857–1947) American architect

Small Houses: Their Economic Design and Construction (1922)

“The problem of doubling the cube was henceforth tried exclusively in the form of the problem of the two mean proportionals.”

Thomas Little Heath (1861–1940) British civil servant and academic

p, 125
Achimedes (1920)

“A Greek architect of the great epoch would no more have thought of omitting the mark of the harmonic scale of proportion, on which the design was based, than would the composer of music think of omitting the harmonic scale of his composition.”

Ernest Flagg (1857–1947) American architect

Source: Small Houses: Their Economic Design and Construction (1922), Ch. II

“Hippocrates also attacked the problem of doubling the cube. …Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i. e. finding x, y such that a:x=x:y=y:b, where a, b are the two given straight lines. It is easy to see that, if a:x=x:y=y:b, then b/a = (x/a)3, and, as a particular case, if b=2a, x3=2a3, so that the side of the cube which is double of the cube of side a is found.”

Thomas Little Heath (1861–1940) British civil servant and academic

Achimedes (1920)

“The bases of music are rhythm and harmony. Rhythm is ordered recurrence in time… As the planets move around the sun, they repeat their orbits periodically; thus there is already a primitive kind of rhythm in their motion…. Harmony… can be considered a special kind of rhythm…. pure musical tones are produced when the vibrations are… periodic or… repeat themselves regularly in time. Two tones harmonize if their intervals of repetition are in rhythm—or, in mathematical language, if their periods are in proportion. Kepler… in the third book of Harmonice mundi… attempted to make other… related, connections between musical harmony and mathematical proportion.”

Frank Wilczek (1951) physicist

Longing for the Harmonies: Themes and Variations from Modern Physics (1987)

“The Rule of Three, or Golden Rule of Arithmeticall whole Numbers. Be the three termes given 2 3 4. …To finde their fourth proporcionall Terme: that is to say, in such Reason to the third terme 4, as the second terme 3, is to the first terme 2 [Modern notation: \frac{x}{4} = \frac{3}{2}]. …Multiply the second terme 3, by the third terme 4, & giveth the product 12: which dividing by the first terme 2, giveth the Quotient 6: I say that 6 is the fourth proportional terme required.”

Simon Stevin (1548–1620) Flemish scientist, mathematician and military engineer

Disme: the Art of Tenths, Or, Decimall Arithmetike (1608)

“But, good God, what is this when compared with that deep and true music of the wise, whereby the proportions of natural things are investigated, the harmonical concord and the qualities of the whole world are revealed, by which also connected things are bound together, peace established between conflicting elements, and whereby each star is perpetually suspended in its appointed place by its weight and strength, and by the harmony of its lucent spirit”

Robert Fludd (1574–1637) British mathematician and astrologer

Robert Fludd, cited in: Waite (1887, p. 291)

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