page_title: Harmonic series (music)
section_title: Frequencies, wavelengths, and musical intervals in example systems
hierarchical_section_title: Harmonic series (music) / Frequencies, wavelengths, and musical intervals in example systems
caption_attribution_description: English: Harmonic series with intervals labeled. Blue notes differ significantly from equal temperament, though all notes (but 1, 2, 4, 8, and 16) differ some. Source for image with only subminor third, supermajor second, and greater and lesser tone labels: Leta E. Miller and Fredric Lieberman (Summer, 1999). "Lou Harrison and the American Gamelan", p.162, American Music, Vol. 17, No. 2, pp. 146-178. Other labels added by Hyacinth (talk). Created by Hyacinth (talk) using Sibelius. External use: http://www.sengpielaudio.com/calculator-harmonics.htm
context_page_description: A harmonic series is the sequence of frequencies, musical tones, or pure tones in which each frequency is an integer multiple of a fundamental.
  Pitched musical instruments are often based on an acoustic resonator such as a string or a column of air, which oscillates at numerous modes simultaneously. At the frequencies of each vibrating mode, 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.
  The musical pitch of a note is usually perceived as the lowest partial present, 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 strongly affected by the relative strength of each harmonic.
context_section_description: One of the simplest cases to visualise is a vibrating string, as in the illustration; the string has fixed points at each end, and each harmonic mode divides it into 1, 2, 3, 4, etc., equal-sized sections resonating at increasingly higher frequencies. Similar arguments apply to vibrating air columns in wind instruments (for example, "the French horn was originally a valveless instrument that could play only the notes of the harmonic series"), although these are complicated by having the possibility of anti-nodes (that is, the air column is closed at one end and open at the other), conical as opposed to cylindrical bores, or end-openings that run the gamut from no flare, cone flare, or exponentially shaped flares (such as in various bells).
  In most pitched musical instruments, the fundamental (first harmonic) is accompanied by other, higher-frequency harmonics. Thus shorter-wavelength, higher-frequency waves occur with varying prominence and give each instrument its characteristic tone quality. The fact that a string is fixed at each end means that the longest allowed wavelength on the string (which gives the fundamental frequency) is twice the length of the string (one round trip, with a half cycle fitting between the nodes at the two ends). Other allowed  wavelengths are ​¹⁄₂, ​¹⁄₃, ​¹⁄₄, ​¹⁄₅, ​¹⁄₆, etc. times that of the fundamental.
  Theoretically, these shorter wavelengths correspond to vibrations at frequencies that are 2, 3, 4, 5, 6, etc., times the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator it vibrates against often alter these frequencies. (See inharmonicity and stretched tuning for alterations specific to wire-stringed instruments and certain electric pianos.) However, those alterations are small, and except for precise, highly specialized tuning, it is reasonable to think of the frequencies of the harmonic series as integer multiples of the fundamental frequency.
  The harmonic series is an arithmetic series (1×f, 2×f, 3×f, 4×f, 5×f, ...). In terms of frequency (measured in cycles per second, or hertz (Hz) where f is the fundamental frequency), the difference between consecutive harmonics is therefore constant and equal to the fundamental. But because human ears respond to sound nonlinearly, higher harmonics are perceived as "closer together" than lower ones. On the other hand, the octave series is a geometric progression (2×f, 4×f, 8×f, 16×f, ...), and people perceive these distances as "the same" in the sense of musical interval. In terms of what one hears, each octave in the harmonic series is divided into increasingly "smaller" and more numerous intervals.
  The second harmonic, whose frequency is twice the fundamental, sounds an octave higher; the third harmonic, three times the frequency of the fundamental, sounds a perfect fifth above the second harmonic. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third harmonic (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher).
  As Mersenne writes, "the order of the Consonances is natural, and ... the way we count them, starting from unity up to the number six and beyond is founded in nature." However, to quote Carl Dahlhaus, "the interval-distance of the natural-tone-row [overtones] [...], counting up to 20, includes everything from the octave to the quarter tone, (and) useful and useless musical tones. The natural-tone-row [harmonic series] justifies everything, that means, nothing."
caption: Harmonic series as musical notation with intervals between harmonics labeled. Blue notes differ most significantly from equal temperament. One can listen to A2 (110 Hz) and 15 of its partials
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