harmonic series

n.1. Mathematics A series whose terms are in harmonic progression, especially the series 1 + ¹/₂ + ¹/₃ + ¹/₄ + ....2. Music A series of tones consisting of a fundamental tone and the consecutive harmonics produced by it.

harmonic series

n 1. (Mathematics) maths a series whose terms are in harmonic progression, as in 1 + + + … 2. (Music, other) acoustics the series of tones with frequencies strictly related to one another and to the fundamental tone, as obtained by touching lightly the node points of a string while playing it. Its most important application is in the playing of brass instruments

harmon′ic se′ries

n. a series in which the reciprocals of the terms form an arithmetic progression. [1865–70]

Harmonic Series

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):

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.

单词	harmonic series
释义	harmonic series harmonic series n.1. Mathematics A series whose terms are in harmonic progression, especially the series 1 + ¹/₂ + ¹/₃ + ¹/₄ + ....2. Music A series of tones consisting of a fundamental tone and the consecutive harmonics produced by it. harmonic series n 1. (Mathematics) maths a series whose terms are in harmonic progression, as in 1 + + + … 2. (Music, other) acoustics the series of tones with frequencies strictly related to one another and to the fundamental tone, as obtained by touching lightly the node points of a string while playing it. Its most important application is in the playing of brass instruments harmon′ic se′ries n. a series in which the reciprocals of the terms form an arithmetic progression. [1865–70] Harmonic Series 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): 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.
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