Harmonic series: Difference between revisions

Latest revision as of 13:28, 11 October 2026

English Wikipedia has an article on:

Harmonic series (music)

The harmonic series is a sequence of tones generated by whole-number frequency ratios over a fundamental: 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, … ad infinitum. Each member of this series is a harmonic (which is short for "harmonic partial").

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.

In just intonation theory, the harmonic series is often treated as the foundation of consonance.

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.

Chord of nature

English Wikipedia has an article on:

Klang (music)

Treated as a chord, the harmonic series is sometimes called the chord of nature; in German this has been called the Klang.

The q-limit chord of nature is 1:2:3:4:…:q up to some odd number q, and is the basic q-limit otonality which can be equated via octave equivalence to other versions of the complete q-limit otonal chord.

Segments

A segment of the harmonic series is a run of consecutive harmonics. It is written a::b for the harmonics a through b. For example, 4::8 consists of the harmonics 4, 5, 6, 7 and 8, which as a chord is 4:5:6:7:8. Relative to its lowest member, the segment a::b has the frequency ratios a/a, (a+1)/a, …, b/a; the steps between its members are superparticular ratios.

A segment whose last harmonic is twice its first, n::2n, spans one octave and can be repeated at the octave to form a scale. These are the overtone scales. Several have their own pages, for example 8::16, 12::24 and 16::32. Used as a scale, the segment n::2n is also called mode n of the harmonic series.

Music based on the harmonic series

The chord of nature is the name sometimes given to the harmonic series, or the series up to a certain stopping point, regarded as a chord.

Steps between adjacent members of the harmonic series are called "superparticular," and they appear in the form (n+1)/n (e.g. 4/3, 28/27, 33/32).

One might compose with the harmonic series by, for instance:

Tuning to the first several harmonics over one fundamental;
Tuning to an octave-repeating segment of the harmonic series for use as a scale. For instance: 8::16, 12::24, 20::40, ... See overtone scale.
Tuning to the overtones of the overtones & the undertones of the undertones. (This can produce complex scales such as Harry Partch's 43-tone Monophonic; this kind of thing is more often called "just intonation" than "overtone music".)

Music

Richard Burdick

Planetary Ripples^{[dead link]}

Folkart Slovakia (site)

Various played with Fujara (slovak overtone flute)

Georg Friedrich Haas

String Quartet No.2, mm. 1~57

Dave Hill

Chord Progression on the Harmonic Overtone Series^{[dead link]} play^{[dead link]}

Norbert Oldani

Drone Inside An Harmonic Series^{[dead link]}

Dave Seidel

William Sethares

Immanent Sphere – detail | play

SoundWell (site)

Various ("Snake" overtone flute)

Spectral Voices (site)

Various (meditative new age with overtone singing)

Stimmhorn (site)

Various (experimental alphorn and yodeling combined with overtone singing)

Karlheinz Stockhausen

Stimmung (1968)
Sternklang (1971)

Cam Taylor

Harmonic series 4-8, 8-16 and 16-32 on the Lumatone (2022)

Chris Vaisvil

Rock Trio in Harmonic Series (2016) – blog | play

Glenn Branca (site)

Symphony No. 3 "Gloria" (1983)

External links

Wikipedia: Spectral music
www.naturton-musik.de^{[dead link]} - web site dedicated to overtone music (by Austrian composer Johannes Kotschy) - a lot of theory material and practical guides to write music based on the overtone series
Overtone music network - a portal for overtone music.
Oberton-Netzwerk (Xing)^{[dead link]} - German-speaking group dedicated to overtone music on the social network platform Xing. Microtonal music in general is welcome, too.
Noah Jordan videos:
- All Harmonics Within One Octave (1–64) | Harmonic Series Explained Visually (2026)
- How Prime Harmonics Fill the Octave (≤1024) — Microtonal & Just Intonation Visualization (2026)

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-=Harmonic series=
+{{Interwiki
-[[Harmonic series#mw-head|Jump to navigation]][[Harmonic series#searchInput|Jump to search]]
+| en = Harmonic series
+| de = Obertonreihe
+| es =
+| ja =
+| ko = 배음렬
+| ro = Seria armonică
+| zh = 泛音列
+}}
+{{Wikipedia|Harmonic series (music)}}
+The '''harmonic series''' is a sequence of [[pitch|tones]] generated by whole-number frequency [[ratio]]s over a fundamental: [[1/1]], [[2/1]], [[3/1]], [[4/1]], [[5/1]], [[6/1]], [[7/1]], … ad infinitum. Each member of this series is a [[harmonic]] (which is short for "harmonic partial").
-English Wikipedia has an article on:
+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.
-'''Harmonic series (music)'''
+In [[just intonation]] theory, the harmonic series is often treated as the foundation of consonance.
-==Prerequisite Knowledge==
-[[Hertz]], basic 12EDO Intervals, basic understanding of linear vs. exponential relationships
-==Overview==
-[[File:Harmonic series intervals.png|alt=|thumb|600x600px|Harmonic Series|link=https://en.xen.wiki/w/File:Harmonic_series_intervals.png]]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 most sounds.
-The '''fundamental''' is the lowest frequency (or '''partial''') in a given harmonic series. While the fundamental is generally the main audible pitch of a given sound, the harmonic series contains an infinitely proliferating sequence of higher partials called '''overtones'''. With practice, one can learn to hear and identify specific 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.
-[https://www.youtube.com/watch?v=hDLhe-NkH2A&ab_channel=mannfishh Learn to hear Harmonics!! (Intros to Just Intonation) by Mannfish]
+[[File:HEJI harmonics 1-16.png|thumb|center|650px|Harmonic series on A, partials 1 to 16, notated in [[HEJI]].]]
-The approach of creating music based on intervallic relationships derived from the harmonic series relationships is called [[Just intonation|Just Intonation]].
+== Chord of nature ==
+{{Wikipedia|Klang (music)}}
+Treated as a [[chord]], the harmonic series is sometimes called the '''chord of nature'''; in German this has been called the '''Klang'''.
-<small>(For a diagram of the harmonic series up to the 49th partial, see [https://heji.plainsound.org/ The Helmholtz-Ellis JI Pitch Notation Legend and Series])</small>
+The ''q''-limit chord of nature is 1:2:3:4:…:''q'' up to some odd number ''q'', and is the basic ''q''-[[limit]] [[Otonality and utonality|otonality]] which can be equated via [[Octave reduction|octave equivalence]] to other versions of the complete ''q''-limit otonal chord.
-=== The Harmonic Series In Real Life ===
-[[File:Cello natural harmonics.png|left|thumb|Cello Natural Harmonics|alt=|200x200px|link=https://en.xen.wiki/w/File:Cello_natural_harmonics.png]]
-====Musical Instruments====
-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.[[File:(a) (e) (i) (o) (u) Video.mov|Vocalist sings on alternating vowels as harmonic partials are gradually reintroduced|alt=|thumb|left|link=https://en.xen.wiki/w/File:(a)_(e)_(i)_(o)_(u)_Video.mov]]
-====Timbre====
-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.
-In this video, individual harmonic partials are gradually re-introduced to a recording of a singer alternating between different vowels. A psychoacoustic illusion is created where at a certain point, the the individual frequencies are suddenly interpreted by the brain as one, timbrally unique and recognizable sound: the human voice. Also note how the intensity of each partial varies depending on the vowel being sung.
+== Segments ==
+A ''segment'' of the harmonic series is a run of consecutive harmonics. It is written ''a''::''b'' for the harmonics ''a'' through ''b''. For example, 4::8 consists of the harmonics 4, 5, 6, 7 and 8, which as a chord is 4:5:6:7:8. Relative to its lowest member, the segment ''a''::''b'' has the frequency ratios ''a''/''a'', (''a''+1)/''a'', …, ''b''/''a''; the steps between its members are [[superparticular]] ratios.
-<small>This audio was created using [https://www.klingbeil.com/spear/ Spear]: a free, downloadable spectral analysis software allowing users to explore and edit individual frequencies within recorded sounds.</small>
+A segment whose last harmonic is twice its first, ''n''::2''n'', spans one octave and can be repeated at the octave to form a scale. These are the [[overtone scale]]s. Several have their own pages, for example [[Harmonics 8–16|8::16]], [[Harmonics 12–24|12::24]] and [[Harmonics 16–32|16::32]]. Used as a scale, the segment ''n''::2''n'' is also called mode ''n'' of the harmonic series.
-==Mathematical Formula and Ratio Notation ==
-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]].
+== Music based on the harmonic series ==
-===Musical Intervals As Ratios===
+The [[chord of nature]] is the name sometimes given to the harmonic series, or the series up to a certain stopping point, regarded as a chord.
-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:
-*6:5 represents the 6th and 5th partials of the harmonic series
-*6:5 is the mathematical ratio of the two pitches' frequencies (600Hz and 500Hz, for example).
-These two frames of reference are fundamentally the same.
+Steps between adjacent members of the harmonic series are called "[[superparticular]]," and they appear in the form (''n''+1)/''n'' (e.g. [[4/3]], [[28/27]], [[33/32]]).
-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|psychoacoustics.]]
+One might compose with the harmonic series by, for instance:
-===Terminology===
+* Tuning to the first several harmonics over one fundamental;
-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.
+* Tuning to an octave-repeating segment of the harmonic series for use as a scale. For instance: [[Harmonics 8–16|8::16]], [[Harmonics 12–24|12::24]], [[Harmonics 20–40|20::40]], ... See [[overtone scale]].
+* Tuning to the overtones of the overtones & the undertones of the undertones. (This can produce complex scales such as [[Harry Partch]]'s 43-tone Monophonic; this kind of thing is more often called "just intonation" than "overtone music".)
-Also note that interval ratios are always notated with the greater number first. In order to change the order of the two pitches, one must use the ratio's '''reciprocal.''' For example, to represent a major sixth instead of a minor third, one must notate the interval as 10:6, not 5:6.
+== Music ==
+; [[Richard Burdick]]
+* [http://www.i-ching-music.com/FREE102.html ''Planetary Ripples'']{{dead link}}
-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.
+; [[Folkart Slovakia]] ([http://www.fujara.sk/audio_samples.htm site])
-==Advanced Concepts==
+* Various played with Fujara (slovak overtone flute)
-===The Harmonic Series As A Fractal===
-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.
+; {{W|Georg Friedrich Haas}}
-===Prime Partials===
+* String Quartet No.2, mm. 1~57
-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]]
-==Further Reading==
+; [[Dave Hill]]
-{| class="wikitable"
+* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Chord Progression on the Harmonic Overtone Series'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3 play]{{dead link}}
-|+
-!Title
+; [[Norbert Oldani]]
-!Author
+* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Oldani/DroneInsideAnHarmonicSeries.mp3 Drone Inside An Harmonic Series]''{{dead link}}
-|-
-|The Arithmetic Of Listening
+; [[Dave Seidel]]
-|Gann, Kyle
+* [http://mysterybear.net/article/18/threnody ''Threnody'']{{dead link}} [https://web.archive.org/web/20201127015923/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Seidel/Threnody.mp3 play]
-|-
+* [http://mysterybear.net/article/22/owllight ''Owllight'']{{dead link}} [https://web.archive.org/web/20201127012201/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Seidel/Owllight.mp3 play]
-|Harmonic Experience: Tonal Harmony From Its Natural Origins to Its Modern Expression
+* [http://mysterybear.net/article/23/palimpsest ''Palimsest'']{{dead link}} [https://web.archive.org/web/20201127012920/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Seidel/Palimpsest.mp3 play]
-|Mathieu, W. A.
-|-
+; [[William Sethares]]
-|An Introduction To the Harmonic Series And Logarithmic Integrals-For High School Students Up To Researchers
+* ''Immanent Sphere'' – [https://sethares.engr.wisc.edu/mp3s/immanent.html detail] | [https://web.archive.org/web/20201127013148/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/ImmanentSphere.mp3 play]
-|Olaikhan, Ali
-|}
+; [[SoundWell]] ([http://www.soundwell.com/en/music site])
-==See also[edit | edit source]==
+* Various ("Snake" overtone flute)
-*[[Subharmonic series]]
-*[[Gallery of just intervals]]
+; [[Spectral Voices]] ([http://www.spectralvoices.com/ site])
-*[[Isoharmonic chords]]
+* Various (meditative new age with overtone singing)
-*[[First Five Octaves of the Harmonic Series]]
-*[[Overtone scales]]
+; [[Stimmhorn]] ([http://www.stimmhorn.ch/ site])
-*[[List of octave-reduced harmonics]]
+* Various (experimental alphorn and yodeling combined with overtone singing)
-*[[Prime harmonic series]]
-*[[Mike Sheiman's Very Easy Scale Building From The Harmonic Series Page]]
+; {{W|Karlheinz Stockhausen}}
-*[[8th Octave Overtone Tuning]]
+* {{W|Stimmung|''Stimmung''}} (1968)
-===External links[edit | edit source]===
+* {{W|Sternklang|''Sternklang''}} (1971)
-*Spectral music article on Wikipedia
-*www.naturton-musik.de <sup>[<nowiki/>[[:Category:Pages containing dead links|''dead link'']]]</sup> - web site dedicated to overtone music (by Austrian composer Johannes Kotschy) - a lot of theory material and practical guides to write music based on the overtone series<sup><nowiki/></sup>
+; [[Cam Taylor]]
-*Overtone music network - a portal for overtone music.
+* [https://www.youtube.com/watch?v=ECLEbVXoTvA Harmonic series 4-8, 8-16 and 16-32 on the Lumatone] (2022)
-*Oberton-Netzwerk (Xing) <sup>[<nowiki/>[[:Category:Pages containing dead links|''dead link'']]]</sup> - German-speaking group dedicated to overtone music on the social network platform Xing. Microtonal music in general is welcome, too.
+; [[Chris Vaisvil]]
+* ''Rock Trio in Harmonic Series'' (2016) – [http://chrisvaisvil.com/rock-trio-in-harmonic-series/ blog] | [http://micro.soonlabel.com/harmonic_series/Nevadatite20160226_harmonic_band.mp3 play]
+; {{W|Glenn Branca}} ([http://www.glennbranca.com/ site])
+* ''Symphony No. 3 "Gloria"'' (1983)
+== See also ==
+* [[Overtone scales]]
+* [[Gallery of just intervals]]
+* [[Isoharmonic chords]]
+* [[List of octave-reduced harmonics]]
+* [[Prime harmonic series]]
+* [[Kite's thoughts on the V-I cadence in higher prime limits]]
+* [[Harmonics 128–256|8th octave overtone tuning]]
+== External links ==
+* [[Wikipedia: Spectral music]]
+* [http://www.naturton-musik.de/ www.naturton-musik.de]{{dead link}} - web site dedicated to overtone music (by Austrian composer Johannes Kotschy) - a lot of theory material and practical guides to write music based on the overtone series
+* [http://www.overtone.cc Overtone music network] - a portal for overtone music.
+* [https://www.xing.com/net/overtonenetwork Oberton-Netzwerk (Xing)]{{dead link}} - German-speaking group dedicated to overtone music on the social network platform [http://www.xing.com Xing]. Microtonal music in general is welcome, too.
+* [[Noah Jordan]] videos:
+** [https://www.youtube.com/watch?v=iyceyPMJAb8 All Harmonics Within One Octave (1–64) | Harmonic Series Explained Visually] (2026)
+** [https://www.youtube.com/watch?v=DAUeGio1Fdg How Prime Harmonics Fill the Octave (≤1024) — Microtonal & Just Intonation Visualization] (2026)
 [[Category:Harmonic]]
 [[Category:Listen]]
 [[Category:Terms]]

Harmonic series: Difference between revisions

Latest revision as of 13:28, 11 October 2026

Contents

Chord of nature

Segments

Music based on the harmonic series

Music

See also

External links

Navigation menu

Harmonic series: Difference between revisions

Latest revision as of 13:28, 11 October 2026

Chord of nature

Segments

Music based on the harmonic series

Music

See also

External links

Navigation menu

Search