Editing Harmonic series (music) (section)

==Frequencies, wavelengths, and musical intervals in example systems==
[[File:Harmonic series to 32.png|250px|thumb|Even-numbered string harmonics from 2nd up to the 64th (five octaves)]]
One of the simplest cases to visualise is a [[String vibration|vibrating string]], as in the illustration; the string has fixed points at each end, and each harmonic [[normal mode|mode]] divides it into an integer number (1, 2, 3, 4, etc.) of equal-sized sections resonating at increasingly higher frequencies.<ref>{{cite book |title=The Physics and Psychophysics of Music |first=Juan G. |last=Roederer |isbn=0-387-94366-8 |page=106 |year=1995 |publisher=Springer }}</ref>{{failed verification|date=October 2022}} Similar arguments apply to vibrating air columns in [[wind instrument]]s (for example, "the French [[Horn (instrument)|horn]] was originally a valveless instrument that could play only the notes of the harmonic series"<ref>{{cite book|last1=Kostka|first1=Stefan|author1-link=Stefan Kostka|last2=Payne|first2=Dorothy|year=1995|title=Tonal Harmony|page=102|edition=3rd|publisher=McGraw-Hill|isbn=0-07-035874-5}}</ref>), 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), [[Cone (geometry)|conical]] as opposed to [[Cylinder (geometry)|cylindrical]] [[bore (wind instruments)|bore]]s, or end-openings that run the [[Diatonic and chromatic|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 [[wave]]s occur with varying prominence and give each instrument its characteristic [[Timbre|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 reciprocal multiples (e.g. {{frac|1|2}}, {{frac|1|3}}, {{frac|1|4}} times) that of the fundamental.

Theoretically, these shorter wavelengths correspond to [[vibration]]s at frequencies that are integer multiples of (e.g. 2, 3, 4 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 piano]]s.) 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 progression]] (''f'', 2''f'', 3''f'', 4''f'', 5''f'',&nbsp;...). In terms of frequency (measured in [[Cycle per second|cycles per second]], or [[hertz]], 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 [[Nonlinear system|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'',&nbsp;...), and people perceive these distances as "[[Octave#Equivalence|the same]]" in the sense of [[Interval (music)|musical interval]]. In terms of what one hears, each successively higher [[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).

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[[File:Harmonic Series.png|thumb|center|650px|An illustration in musical notation of the harmonic series (on C) up to the 20th&nbsp;harmonic. The numbers above the harmonic indicate the difference &ndash; in [[Cent (music)|cents]] &ndash; from [[equal temperament]] (rounded to the nearest integer). Blue notes are very flat and red notes are very sharp. Listeners accustomed to more [[Consonance and dissonance|tonal]] tuning, such as [[meantone temperament|meantone]] and [[well temperament]]s, notice many other notes are "off".]]

[[File:Harmonics to 32.png|thumb|650px|center|Harmonics on C, from 1st (fundamental) to 32nd&nbsp;harmonic (five octaves higher). Notation used is based on the [[Ben Johnston (composer)#Staff notation|extended just notation]] by [[Ben Johnston (composer)|Ben Johnston]][[File:Harmonics to 32.mid]]]]

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[[File:Harmonic series intervals.png|thumb|center|550px|Harmonic series as musical notation with intervals between harmonics labeled. Blue notes differ most significantly from equal temperament. One can listen to [[Media:Harmonics 110x16.ogg|A{{sub|2}} (110&nbsp;Hz) and 15 of its partials]] ]]

[[File:Notation of partials 1-19 for 1-1.png|thumb|400px|center|Staff notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C. These are "[[Prime number|prime]] harmonics".<ref>{{cite journal|last=Fonville|first=John|author-link=John Fonville|date=Summer 1991|title=Ben Johnston's Extended Just Intonation: A guide for interpreters|journal=[[Perspectives of New Music]]|volume=29|issue=2|pages=106–137 (121)|doi=10.2307/833435|jstor=833435}}</ref>[[File:Notation of partials 1-19 for 1-1.mid]]]]

[[Marin Mersenne]] wrote: "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."<ref>{{cite book|last=Cohen|first=H. F.|author-link=Floris Cohen|year=2013|title=Quantifying Music: The science of music at the first stage of scientific revolution 1580–1650|page=103|publisher=Springer|isbn=9789401576864}}</ref> However, to quote [[Carl Dahlhaus]], "the interval-distance of the natural-tone-row [<nowiki/>[[overtone]]s] [...], 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."<ref>Sabbagh, Peter (2003). ''The Development of Harmony in [[Alexander Scriabin|Scriabin]]'s Works'', p. 12. Universal. {{ISBN|9781581125955}}. Cites: [[Carl Dahlhaus|Dahlhaus, Carl]] (1972). "Struktur und Expression bei Alexander Skrjabin", ''Musik des Ostens'', Vol. 6, p. 229.</ref>