# Harmonic Series

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Related to Harmonic Series: geometric series, Harmonic sequence

## harmonic series

[här¦män·ik ′sir‚ēz]## Harmonic Series

the numerical series

Each term of the series (starting with the second) is the harmonic mean of the two adjacent terms (hence the name “harmonic series”). The terms of a harmonic series approach zero, but the series is divergent (G. Leibniz, 1673). The sum of the first *n* terms has the following asymptotic expression (L. Euler, 1740):

*S _{n}* = ln

*n*+

*C*+

*є*

_{n}where *C* = 0.577215 … is Euler’s constant and *є _{n}*→ as

*n*→∞.

## Harmonic Series

in musical acoustics, the ascending series of overtones, or partial tones: the series of harmonics beginning with the fundamental tone.

The relationship of the frequencies of the overtones is expressed by a series of prime numbers. In order that these prime numbers correspond to the ordinal numbers of the overtones, the fundamental tone of the harmonic series by convention is called the first overtone.

Because of a difference in frequencies, the pitch of some overtones differs slightly from the corresponding tempered pitch. Overtones whose pitch is higher than the tempered pitch are marked with a plus sign. The six lowest overtones are part of the major triad, which thus conforms to acoustic laws.