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'''Harmonic series (music)'''

[[Hertz]]

Harmonic Series

The harmonic series describes a pattern of frequencies naturally occurring as a real, physical (not theoretical or psychoacoustic) phenomenon. This phenomenon can be observed in many pitched sounds.

<small>(For a diagram of the harmonic series up to the 49th partial, see [https://www.diva-portal.org/smash/get/diva2:1869939/FULLTEXT03.pdf The Helmholtz-Ellis JI Pitch Notation Legend and Series])</small>

[[File:Cello natural harmonics.png|left|thumb|Cello Harmonics]]

Many musicians are already familiar with the harmonic series, even if they may not realize it. For example: the natural harmonics of a string instrument (bowed or strummed) and the open notes on a French horn are two manifestations of the harmonic series. Using the cello as an example, the low, open C string acts as the fundamental of its harmonic series. In this case, the first available natural harmonic is C one octave up, then G, C, E, G etc. To play these harmonics, one effectively shortens the length of the string, at ratios that match those in the harmonic series.

Vocalist sings on alternating vowels as harmonic partials are gradually reintroduced

[[File:(a) (e) (i) (o) (u) Video.mov|thumb|Singer singing on different vowels as partials are gradually reintroduced|alt=|left|0x0px]]

The harmonic series is also responsible for timbre: one can easily identify the sound of a trumpet, violin, or electric guitar, even if they play the same pitch. This is because while the same set of frequencies is (mostly) present in all these sounds, some partials will be more prominent than others depending on the sound source. This also applies to speech and singing, as different vowel formants are distinguished by their unique harmonic identity.

<small>This audio was created using Spear: a free, downloadable spectral analysis software allowing users to explore and edit individual frequencies within recorded sounds.</small>

The mathematical formula for the harmonic series is simple: each positive-integer multiple of the fundamental frequency represents one overtone. For example, if the fundamental frequency is 100Hz, the partials, in ascending order, will be 100Hz, 200Hz, 300Hz, 400Hz, etc...

Because frequency is exponential, the linear relationship between each partial (as demonstrated in the above example) results in partials becoming increasingly dense/close together (like the frets on a guitar). An octave represents a doubling in frequency: If the fundamental is, again, 100Hz, its first octave will be at 200Hz, the second one at 400Hz, the third at 800Hz, etc. With a new partial at every interval of 100Hz, the number of partials will double with each consecutive octave. For more information on the exponential nature of frequency, see [[Hertz]].

Beginning with the seventh partial, intervals in the harmonic series begin to deviate significantly from those used in 12EDO. 12EDO interval nomenclature (minor third, perfect fifth, etc.) fails to accurately represent these relationships because, for instance, the harmonic series contains an infinite number of ''different'' minor thirds. Therefore, it is standard in [[Just intonation|Just Intonation]] to use ratios (such as 6:5) to refer to a specific ''type'' of minor third. There are two main ways in which one might think of these ratios:

Interestingly, if one were to record themself clapping a 6:5 polyrhythm (6 eighth notes over an eighth note quintuplet), then speed up the audio by a factor of 1000, one would hear the same 6:5 minor third described above. For more information about how pitch is perceived, see [psychoacoustics page that does not exist yet].

Note that the terms ''overtone'' and ''overtone series'' are not quite synonymous with ''harmonic'' and ''harmonic series'', respectively, although interchangeable usage is also attested. Technically speaking, ''overtone series'' excludes the starting fundamental, so the 2nd harmonic is the 1st overtone. Because of that distinction, the math of the "overtone series" is off by one. So, "harmonic series" is arguably the preferred standard. Likewise, the standard way to refer to frequencies in the harmonic series is by using numbered '''partials''' (beginning with the fundamental as the first partial), not overtones.

The [[subharmonic series]] (or undertone series) is the inversion of the harmonic series: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7... ad infinitum. The words otonal (referring to overtones) and utonal (referring to undertones) are used to refer to the harmonic and subharmonic series, respectively.

The harmonic series contains an infinite number of harmonic series within it. By isolating every numbered partial with a given factor, one finds that an entire harmonic series manifests within this smaller subset of the original harmonic series. [Add example and image]

Prime-numbered partials are sometime referred to simply as "primes". These partials are of interest because each new prime produces a unique interval not present in any of the lower partials. For more information on this, see: [[Prime interval]]

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* [[Subharmonic series]]

* [[8th Octave Overtone Tuning]]

* Spectral music article on Wikipedia