What is "harmonic series"

Harmonic \Har*mon"ic\ (h[aum]r*m[o^]n"[i^]k), Harmonical \Har*mon"ic*al\ (-[i^]*kal), a. [L. harmonicus, Gr. "armoniko`s; cf. F. harmonique. See Harmony.]

1. Concordant; musical; consonant; as, harmonic sounds.

   Harmonic twang! of leather, horn, and brass.
   --Pope.

2. (Mus.) Relating to harmony, -- as melodic relates to melody; harmonious; esp., relating to the accessory sounds or overtones which accompany the predominant and apparent single tone of any string or sonorous body.

3. (Math.) Having relations or properties bearing some resemblance to those of musical consonances; -- said of certain numbers, ratios, proportions, points, lines, motions, and the like.

   Harmonic interval (Mus.), the distance between two notes of a chord, or two consonant notes.

   Harmonical mean (Arith. & Alg.), certain relations of numbers and quantities, which bear an analogy to musical consonances.

   Harmonic motion, the motion of the point A, of the foot of the perpendicular PA, when P moves uniformly in the circumference of a circle, and PA is drawn perpendicularly upon a fixed diameter of the circle. This is simple harmonic motion. The combinations, in any way, of two or more simple harmonic motions, make other kinds of harmonic motion. The motion of the pendulum bob of a clock is approximately simple harmonic motion.

   Harmonic proportion. See under Proportion.

   Harmonic series or Harmonic progression. See under Progression.

   Spherical harmonic analysis, a mathematical method, sometimes referred to as that of Laplace's Coefficients, which has for its object the expression of an arbitrary, periodic function of two independent variables, in the proper form for a large class of physical problems, involving arbitrary data, over a spherical surface, and the deduction of solutions for every point of space. The functions employed in this method are called spherical harmonic functions.
   --Thomson & Tait.

   Harmonic suture (Anat.), an articulation by simple apposition of comparatively smooth surfaces or edges, as between the two superior maxillary bones in man; -- called also harmonia, and harmony.

   Harmonic triad (Mus.), the chord of a note with its third and fifth; the common chord.

n. 1 (label en mathematics analysis) The divergent series whose terms are reciprocals of successive natural numbers starting with 1. In symbols: $sum\_\{n=1\}^infty\; frac\; 1\; n$. (attention en Is this a proper noun?) 2 (label en music) (rfdef: English)

A harmonic series is the sequence of sounds where the base frequency of each sound is an integral multiple of the lowest base frequency.

Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves. Interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the typical spacing of the resonances, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest frequency, and such multiples form the harmonic series (see harmonic series (mathematics)).

The musical pitch of a note is usually perceived as the lowest partial present (the fundamental frequency), which may be the one created by vibration over the full length of the string or air column, or a higher harmonic chosen by the player. The musical timbre of a steady tone from such an instrument is determined by the relative strengths of each harmonic.

In mathematics, the harmonic series is the divergent infinite series:

$\sum_{n=1}^\infty\,\frac{1}{n} \;\;=\;\; 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots.\!$

Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music.

Harmonic series may refer to either of two related concepts:

• Harmonic series (mathematics)
• Harmonic series (music)