# Harmonic Series

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Related to Harmonic Series: geometric series, Harmonic sequence

## harmonic series

[här¦män·ik ′sir‚ēz]
(mathematics)
A series whose terms form a harmonic progression.

## 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):

Sn = ln n + C + єn

where C = 0.577215 … is Euler’s constant and єn→ 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.

References in periodicals archive ?
The other volumes will be: lit'ap to Expand the Harmonic Series, covering instruments that use methods such as slides, finger holes, and keys to fill in notes outside the harmonic series; Valves Evolve, covering the development of various valve mechanisms in the nineteenth century; Heyday of the Cornet; and The Modern Trumpet, covering twentieth-century trumpet development.
Alternatively, the external gravitational potential can be modeled by harmonic series expansions (e.g., Rummel et al., 1988).
The piece gets its unmistakable fanfare sound by confining its palette to the capabilities of valveless brass instruments such as bugles, which can play only the limited pitches of the harmonic series. The melody leaping from pitch to pitch accompanied by simple, folk-like harmonies, is reminiscent of bugles on the battlefield, giving the fanfare an air of pride and nobility.
Coverage includes elementary and advanced tests for convergence or divergence as well as information about the harmonic series, the alternating harmonic series, and closely related series.
He explains the tuning process and related issues, historical factors, and theories including frequency, ratios, cents, fixed intonation, and the harmonic series. Sixteen temperaments are discussed and compared in detail.
As in the Green functions approach, the problem of computing the infinite harmonic series [34] and [36] may be considered.
In the new tuning system the major third was an imperfect consonance that could be directly equated with the harmonic series.
Additionally, ratios are an appropriate descriptive language for the pitch relationships of the harmonic series, which Helmholtz discussed at length in his influential book On the Sensations of Tone.(5) Thanks to his encounter with Helmholtz's work, the harmonic series formed a major source of Partch's investigations in the years 1923-28, when his theoretical ideas were crystallizing.
The slide trumpet (illus.2) filled in the gaps in its harmonic series with its single slide, which offered a pitch-change of up to one and a half tones (more with some note-bending and imagination).
The title refers to my association between the natural harmonic series, which I use as the harmonic basis of the work, and the full spectrum of light as one would observe refracted by a prism.
The symbol [f.sub.o] was agreed upon, written with a lower case f and a subscript "o," not "zero." The reason given was that a "zero" frequency does not ft into the harmonic series of the sound source.
Not to be missed catch Deerhunter tonight at The Button Factory as part of a new Harmonic series. Tickets: EUR20.

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