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1Music

A set of frequencies consisting of a fundamental and the harmonics related to it by an exact fraction.*‘The first lowers the harmonic series by a whole tone, the second by a semitone, and the third by a tone and a half.’**‘Tuned to the harmonic series, it sounds along with the rhythm of the waves, and its sound is more ambient than tuneful, although you do hear elements of tune and resolving discords.’**‘And it is fascinating how musical notes, although derived from something very material like the harmonic series, do not correlate with any material phenomena when they are in horizontal motion or vertical grouping.’**‘The organ works off the swell of the seawater at high tide, which creates air pressure in the pipes leading to a sequence of musical chords based on the harmonic series.’**‘Being a valveless instrument, the Alphorn produces only those pitches that occur in the natural overtone or harmonic series.’*

2Mathematics

A harmonic progression.*‘Euler asserts that the sum of the harmonic series equals the natural logarithm of infinity plus a quantity that is nearly a constant.’**‘He begins with the summation of geometric series, then shows that the harmonic series does not converge.’**‘Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series.’**‘By the study of the harmonic series and related sums, Euler systematically developed an important method of summation of series which is now known as Euler-Maclaurin summation.’**‘If s = 1, you have the plain old harmonic series, which also increases to infinity but does so more slowly.’*

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