View source for Harmonic

{{Short description|Wave with frequency an integer multiple of the fundamental frequency}}
{{About|the components of periodic signals}}
{{More citations needed|date=August 2020}}
[[Image:Moodswingerscale.svg|thumb|250px|The [[node (physics)|nodes]] of a vibrating string are harmonics.]]
[[Image:Cello natural harmonics.png|thumb|Two different notations of natural harmonics on the cello. First as sounded (more common), then as fingered (easier to sightread).]]

In [[physics]], [[acoustics]], and [[telecommunications]], a '''harmonic''' is a [[sinusoidal wave]] with a [[frequency]] that is a positive [[integer]] multiple of the ''[[fundamental frequency]]'' of a [[periodic signal]]. The fundamental frequency is also called the '''''1st harmonic'''''; the other harmonics are known as '''''higher harmonics'''''. As all harmonics are [[Periodic function|periodic]] at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a ''[[harmonic series (music)|harmonic series]]''.

The term is employed in various disciplines, including music, physics, [[acoustics]], electronic power transmission, radio technology, and other fields. For example, if the fundamental frequency is 50&nbsp;[[Hertz|Hz]], a common [[alternating current|AC]] power supply frequency, the frequencies of the first three higher harmonics are 100&nbsp;Hz (2nd harmonic), 150&nbsp;Hz (3rd harmonic), 200&nbsp;Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50&nbsp;Hz.

{{Quote|An ''n''<sup>th</sup> characteristic mode, for ''n'' > 1, will have nodes that are not vibrating. For example, the 3rd characteristic mode will have nodes at <math>\tfrac{1}{3}</math>''L'' and <math>\tfrac{2}{3}</math>''L'', where ''L'' is the length of the string. In fact, each ''n''<sup>th</sup> characteristic mode, for ''n'' not a multiple of 3, will '''not''' have nodes at these points. These other characteristic modes will be ''vibrating'' at the positions <math>\tfrac{1}{3}</math>''L'' and <math>\tfrac{2}{3}</math>''L''. If the player ''gently touches'' one of these positions, then these other characteristic modes will be suppressed. The tonal harmonics from these other characteristic modes will then also be suppressed. Consequently, the tonal harmonics from the ''n''<sup>th</sup> characteristic modes, where ''n'' is a multiple of 3, will be made relatively more prominent.<ref>{{Cite journal|first=Russell|last=Walker|date=2019-06-14|title=Russell Walker|url=http://dx.doi.org/10.1287/7648739e-8e59-466e-82cb-3ded22bbebf6|access-date=2020-12-21|website=Authors group|doi=10.1287/7648739e-8e59-466e-82cb-3ded22bbebf6|s2cid=241172832 |url-access=subscription}}</ref>}}

In music, harmonics are used on string instruments and wind instruments as a way of producing sound on the instrument, particularly to play higher notes and, with strings, obtain notes that have a unique sound quality or "tone colour". On strings, bowed harmonics have a "glassy", pure tone. On stringed instruments, harmonics are played by touching (but not fully pressing down the string) at an exact point on the string while sounding the string (plucking, bowing, etc.); this allows the harmonic to sound, a pitch which is always higher than the fundamental frequency of the string.

==Terminology==
Harmonics may be called "overtones", "partials", or "upper partials", and in some music contexts, the terms "harmonic", "overtone" and "partial" are used fairly interchangeably. But more precisely, the term "harmonic" includes ''all'' pitches in a harmonic series (including the fundamental frequency) while the term "overtone" only includes pitches ''above'' the fundamental. 

==Characteristics==
{{Quote|A whizzing, whistling tonal character, distinguishes all the harmonics both natural and artificial from the firmly stopped intervals; therefore their application in connection with the latter must always be carefully considered.{{cn|date=January 2024}} {{right|— Richard Scholz ({{circa|1888-1912}})<ref>{{cite web |title=Category:Scholz, Richard |type=site sub-index & mini-bio for Scholz |website=Petrucci Music Library / International Music Score Library Project (IMSLP) (imslp.org) |place=Canada |url=https://imslp.org/wiki/Category:Scholz,_Richard |access-date=2020-12-21}}</ref>}} }}

Most acoustic instruments emit complex tones containing many individual partials (component simple tones or sinusoidal waves), but the untrained human ear typically does not perceive those partials as separate phenomena. Rather, a musical note is perceived as one sound, the quality or [[timbre]] of that sound being a result of the relative strengths of the individual partials. Many acoustic [[oscillator]]s, such as the [[human voice]] or a [[Bow (music)|bowed]] [[violin]] string, produce complex tones that are more or less [[Periodic function|periodic]], and thus are composed of partials that are nearly matched to the integer multiples of fundamental frequency and therefore resemble the ideal harmonics and are called "harmonic partials" or simply "harmonics" for convenience (although it's not strictly accurate to call a &thinsp;''partial''&thinsp; a &thinsp;''harmonic'',&thinsp; the first being actual and the second being theoretical).

Oscillators that produce harmonic partials behave somewhat like one-dimensional [[resonator]]s, and are often long and thin, such as a guitar string or a column of air open at both ends (as with the metallic modern orchestral [[transverse flute]]). Wind instruments whose air column is open at only one end, such as [[trumpet]]s and [[clarinet]]s, also produce partials resembling harmonics. However they only produce partials matching the ''odd'' harmonics—at least in theory. In practical use, no real acoustic instrument behaves as perfectly as the simplified physical models predict; for example, instruments made of [[nonlinear acoustics|non-linearly]] [[elasticity (physics)|elastic]] wood, instead of metal, or strung with [[catgut|gut]] instead of [[string (music)#string_core_anchor|brass or steel strings]], tend to have not-quite-integer partials.

Partials whose frequencies are not integer multiples of the fundamental are referred to as [[Inharmonicity|inharmonic partials]]. Some acoustic instruments emit a mix of harmonic and inharmonic partials but still produce an effect on the ear of having a definite fundamental pitch, such as [[piano]]s, strings plucked [[pizzicato]], vibraphones, marimbas, and certain pure-sounding bells or chimes. Antique [[singing bowl]]s are known for producing multiple harmonic partials or [[multiphonics]].
<ref>
{{cite press release
 |first1=Alexander |last1=Galembo
 |first2=Lola L. |last2=Cuddly
 |date=2 December 1997
 |title=Large grand and small upright pianos
 |publisher=[[Acoustical Society of America]]
 |website=acoustics.org |url=http://www.acoustics.org/press/134th/galembo.htm
 |url-status=dead |access-date=13 January 2024
 |archive-url=https://web.archive.org/web/20120209050929/http://www.acoustics.org/press/134th/galembo.htm
 |archive-date=2012-02-09
 |quote=There are many ways to make matters worse, but very few to improve.
}} — Minimally technical summary of string acoustics research given at conference; discusses listeners' perceptions of pianos' inharmonic partials.
</ref><ref>
{{cite journal
 |first=Sophie R.A. |last=Court
 |date=April 1927
 |title=''Golo und Genovefa'' {{grey|[by]}} Hanna Rademacher
 |type=book review
 |journal=Books Abroad
 |volume=1 |issue=2 |pages=34–36
 |doi=10.2307/40043442 |jstor=40043442 |issn=0006-7431
}}
</ref>
Other oscillators, such as [[cymbals]], drum heads, and most percussion instruments, naturally produce an abundance of inharmonic partials and do not imply any particular pitch, and therefore cannot be used melodically or harmonically in the same way other instruments can.

Building on of [[William Sethares|Sethares]] (2004),<ref name=TTSS>
{{cite book
 |last=Sethares |first=W.A. |author-link=William Sethares
 |year=2004
 |title=Tuning, Timbre, Spectrum, Scale
 |publisher=Springer
 |isbn=978-1852337971
 |url=https://books.google.com/books?id=KChoKKhjOb0C |via=Google books
}}
</ref> [[dynamic tonality]] introduces the notion of pseudo-harmonic partials, in which the frequency of each partial is aligned to match the pitch of a corresponding note in a pseudo-just tuning, thereby maximizing the [[consonance]] of that pseudo-harmonic timbre with notes of that pseudo-just tuning.<ref name=Spectral_Tools>
{{cite journal
 | last1=Sethares | first1=W.A. | author1-link=William Sethares
 | last2=Milne    | first2=A.       | last3=Tiedje    | first3=S.
 | last4=Prechtl  | first4=A.       | last5=Plamondon | first5=J. 
 | year = 2009
 | title = Spectral tools for dynamic tonality and audio morphing
 | journal = [[Computer Music Journal]]
 | volume = 33  | issue = 2  | pages = 71–84
 | s2cid = 216636537  | doi = 10.1162/comj.2009.33.2.71  |doi-access=free
 | url = http://muse.jhu.edu/journals/computer_music_journal/summary/v033/33.2.sethares.html
 | access-date = 2009-09-20
}}</ref><ref name=Continua>
{{cite journal
 |first1=Andrew  |last1=Milne
 |first2=William |last2=Sethares  | author2-link=William Sethares
 |first3=James   |last3=Plamondon
 |date=29 August 2008
 |title=Tuning continua and keyboard layouts
 |journal=[[Journal of Mathematics and Music]]
 |volume=2  |issue=1  |pages=1–19
 |doi=10.1080/17459730701828677 |s2cid=1549755
 |url=http://oro.open.ac.uk/21504/1/tuningcontinua.pdf
 |url-status=live
 |archive-url=https://ghostarchive.org/archive/20221009/http://oro.open.ac.uk/21504/1/tuningcontinua.pdf
 |archive-date=2022-10-09
}} {{cite web
 |title=Alt URL
 |series=[[William Sethares|Sethares]] pers. academic site
 |publisher=[[University of Wisconsin]]
 |url=http://sethares.engr.wisc.edu/paperspdf/tuningcontinua.pdf
}}
</ref><ref name=Fingering>
{{cite journal
 |first1=A.   |last1=Milne
 |first2=W.A. |last2=Sethares  | author2-link=William Sethares
 |first3=J.   |last3=Plamondon
 |date=Winter 2007
 |title=Invariant fingerings across a tuning continuum
 |journal=[[Computer Music Journal]]
 |volume=31  |issue=4  |pages=15–32
 |doi=10.1162/comj.2007.31.4.15
 |s2cid=27906745
 |url=http://oro.open.ac.uk/21503/1/comj.2007.31.4.15
 |doi-access=free
}}</ref><ref name=X_System>
{{cite report
 |first1=A.   |last1=Milne
 |first2=W.A. |last2=Sethares  | author2-link=William Sethares
 |first3=J.   |last3=Plamondon
 | year = 2006
 | title = X System
 | publisher = Thumtronics Inc.
 | type = technical report
 | url = http://oro.open.ac.uk/21510/1/X_System.pdf
 | access-date = 2020-05-02
}}
</ref>

==Partials, overtones, and harmonics==
<!-- [[Flageolet (disambiguation)]] links to this section -->

An [[overtone]] is any partial higher than the lowest partial in a compound tone. The relative strengths and frequency relationships of the component partials determine the timbre of an instrument. The similarity between the terms overtone and partial sometimes leads to their being loosely used interchangeably in a [[music]]al context, but they are counted differently, leading to some possible confusion. In the special case of instrumental timbres whose component partials closely match a harmonic series (such as with most strings and winds) rather than being inharmonic partials (such as with most [[pitched percussion]] instruments), it is also convenient to call the component partials "harmonics", but not strictly correct, because harmonics are numbered the same even when missing, while partials and overtones are only counted when present. This chart demonstrates how the three types of names (partial, overtone, and harmonic) are counted (assuming that the harmonics are present):

{| class="wikitable"
|- class="hintergrundfarbe5"
! Frequency
! Order<br/>({{mvar|n}})
! Name&nbsp;1
! Name&nbsp;2
! Name&nbsp;3
! [[Standing wave]] representation
! [[Longitudinal wave]] representation
|-
| 1 × {{mvar|f}} = {{0}}440&nbsp;Hz
| {{mvar|n}} = 1
| 1st partial
| [[fundamental tone]]
| 1st harmonic
| [[File:Pipe001.gif|220px]]
| [[File:Molecule1.gif|220px]]
|-
| 2 × {{mvar|f}} = {{0}}880&nbsp;Hz
| {{mvar|n}} = 2
| 2nd partial
| 1st overtone
| 2nd harmonic
| [[File:Pipe002.gif|220px]]
| [[File:Molecule2.gif|220px]]
|-
| 3 × {{mvar|f}} = 1320&nbsp;Hz
| {{mvar|n}} = 3
| 3rd partial
| 2nd overtone
| 3rd harmonic
| [[File:Pipe003.gif|220px]]
| [[File:Molecule3.gif|220px]]
|-
| 4 × {{mvar|f}} = 1760&nbsp;Hz
| {{mvar|n}} = 4
| 4th partial
| 3rd overtone
| 4th harmonic
| [[File:Pipe004.gif|220px]]
| [[File:Molecule4.gif|220px]]
|}

In many [[musical instruments]], it is possible to play the upper harmonics without the fundamental note being present. In a simple case (e.g., [[Recorder (musical instrument)|recorder]]) this has the effect of making the note go up in pitch by an [[octave]], but in more complex cases many other pitch variations are obtained. In some cases it also changes the [[timbre]] of the note. This is part of the normal method of obtaining higher notes in [[wind instruments]], where it is called ''[[overblowing]]''. The [[extended technique]] of playing [[multiphonic]]s also produces harmonics. On [[string instruments]] it is possible to produce very pure sounding notes, called [[string harmonic|harmonics]] or ''flageolets'' by string players, which have an eerie quality, as well as being high in pitch. Harmonics may be used to check at a [[unison]] the tuning of strings that are not tuned to the unison. For example, lightly fingering the node found halfway down the highest string of a [[cello]] produces the same pitch as lightly fingering the node {{sfrac| 1 | 3 }} of the way down the second highest string. For the human voice see [[Overtone singing]], which uses harmonics.

While it is true that electronically produced periodic tones (e.g. square waves or other non-sinusoidal waves) have "harmonics" that are whole number multiples of the fundamental frequency, practical instruments do not all have this characteristic. For example, higher "harmonics" of piano notes are not true harmonics but are "overtones" and can be very sharp, i.e. a higher frequency than given by a pure [[harmonic series (music)|harmonic series]]. This is especially true of instruments other than [[stringed instrument|strings]], [[brass instrument|brass]], or [[woodwinds]]. Examples of these "other" instruments are xylophones, drums, bells, chimes, etc.; not all of their overtone frequencies make a simple whole number ratio with the fundamental frequency. (The [[fundamental frequency]] is the [[reciprocal (mathematics)|reciprocal]] of the longest [[period (physics)|time period]] of the collection of vibrations in some single periodic phenomenon.<ref>{{FS1037C}}</ref>)

==On stringed instruments==
[[Image:Flageolette.svg|thumb|200px|Playing a harmonic on a string]]{{Main|String harmonic}}
{{Quote|Harmonics may be singly produced [on stringed instruments] (1) by varying the point of contact with the bow, or (2) by slightly pressing the string at the nodes, or divisions of its aliquot parts (<math>\tfrac{\ 1\ }{2}</math>, <math>\tfrac{\ 1\ }{3}</math>, <math>\tfrac{\ 1\ }{4}</math>, etc.). {{nobr|(1) In the first}} case, advancing the bow from the usual place where the fundamental note is produced, towards the bridge, the whole scale of harmonics may be produced in succession, on an old and highly resonant instrument. The employment of this means produces the effect called '[[sul ponticello]].' {{nobr|(2) The production}} of harmonics by the slight pressure of the finger on the open string is more useful. When produced by pressing slightly on the various nodes of the open strings they are called 'natural harmonics'. ... Violinists are well aware that the longer the string in proportion to its thickness, the greater the number of upper harmonics it can be made to yield.|''[[Grove's Dictionary of Music and Musicians]]'' (1879)<ref>{{cite journal |last1=George alike |date=2011-01-21 |title=Free Online Games for Kids |website=MyFavoritegamez.com / SciVee.tv |url=http://doi.org/10.4016/26742.01 |access-date=2020-12-21 |doi=10.4016/26742.01 |url-status=dead |archive-url=https://web.archive.org/web/20210214224413/http://www.scivee.tv/node/26742/ |archive-date=2021-02-14}}</ref>|source=}}

The following table displays the stop points on a stringed instrument at which gentle touching of a string will force it into a harmonic mode when vibrated. String harmonics (flageolet tones) are described as having a "flutelike, silvery quality" that can be highly effective as a special color or tone color ([[timbre]]) when used and heard in [[orchestration]].<ref name=Marroco-2001>{{cite dictionary |last=Marrocco |first=W. Thomas |chapter=Kennan, Kent |date=2001 |dictionary=Oxford Music Online |publisher=Oxford University Press |doi=10.1093/gmo/9781561592630.article.14882 |chapter-url=http://dx.doi.org/10.1093/gmo/9781561592630.article.14882 |access-date=2020-12-21}}</ref> It is unusual to encounter natural harmonics higher than the fifth partial on any stringed instrument except the double bass, on account of its much longer strings.<ref name=Marroco-2001/>

::{| class="wikitable"
|-
! Harmonic
! [[Stop note]]
! Note sounded<br/><small>(relative to<br/>open string)</small>
!width="50"| [[Audio frequency]] ([[Hertz (unit)|Hz]])
!width="50"| [[Cent (music)|Cents]] above<br/> fundamental <small>(offset by octave)</small>
! Audio<br/> <small>(octave shifted)</small>
|-
|style="text-align:center;"| 1
| [[fundamental frequency|fundamental]],<br/> perfect [[unison]] 
|style="text-align:center;"| P1
| style="text-align:right;"| 600
| style="text-align:right;"| 0.0
| {{audio|Unison on C.mid|Play}}
|-
|style="text-align:center;"| 2
| first perfect [[octave]] 
|style="text-align:center;"| P8 
| style="text-align:right;"| 1,200
| style="text-align:right;"| 0.0
| {{audio|Perfect octave on C.mid|Play}}
|-
|style="text-align:center;"| 3
| [[perfect fifth]]
|style="text-align:center;"| P8 + P5 
| style="text-align:right;"| 1,800
| style="text-align:right;"| 702.0
| {{audio|Just perfect fifth on C.mid|Play}}
|-
|style="text-align:center;"| 4
| doubled perfect [[octave]]
|style="text-align:center;"| 2·P8
| style="text-align:right;"| 2,400
| style="text-align:right;"| 0.0
| {{audio|Just perfect fourth on C.mid|Play}}
|-
|style="text-align:center;"| 5
| [[just major third]],<br/> [[major third]]
|style="text-align:center;"| 2·P8 + M3 
| style="text-align:right;"| 3,000
| style="text-align:right;"| 386.3
| {{audio|Just major third on C.mid|Play}}
|-
|style="text-align:center;"| 6
| [[perfect fifth]]
|style="text-align:center;"| 2·P8 + P5
| style="text-align:right;"| 3,600
| style="text-align:right;"| 702.0
| {{audio|Just perfect fifth on C.mid|Play}}
|-
|style="text-align:center;"| 7
| [[harmonic seventh]],<br/> [[harmonic seventh|septimal minor seventh]] <br/> <small>(‘the lost chord’)</small>
|style="text-align:center;"| 2·P8 + m7↓
| style="text-align:right;"| 4,200
| style="text-align:right;"| 968.8
| {{audio|Harmonic seventh on C.mid|Play}}
|-
|style="text-align:center;"| 8
| third perfect [[octave]] 
|style="text-align:center;"| 3·P8
| style="text-align:right;"| 4,800
| style="text-align:right;"| 0.0
| {{audio|Perfect octave on C.mid|Play}}
|-
|style="text-align:center;"| 9
| [[Pythagorean interval|Pythagorean]] [[major second]] <br/> harmonic ninth
|style="text-align:center;"| 3·P8 + M2 
| style="text-align:right;"| 5,400
| style="text-align:right;"| 203.9
| {{audio|Major tone on C.mid|Play}}
|-
|style="text-align:center;"| 10
| [[just major third]]
|style="text-align:center;"| 3·P8 + M3 
| style="text-align:right;"| 6,000
| style="text-align:right;"| 386.3
| {{audio|Just major third on C.mid|Play}}
|-
|style="text-align:center;"| 11
| lesser undecimal [[tritone]],<br/> undecimal semi-[[augmented fourth]]
|style="text-align:center;"| 3·P8 + a4{{music|d}}
| style="text-align:right;"| 6,600
| style="text-align:right;"| 551.3
| {{audio|Eleventh harmonic on C.mid|Play}}
|-
|style="text-align:center;"| 12
| [[perfect fifth]] 
|style="text-align:center;"| 3·P8 + P5
| style="text-align:right;"| 7,200
| style="text-align:right;"| 702.0
| {{audio|Just perfect fifth on C.mid|Play}}
|-
|style="text-align:center;"| 13
| tridecimal [[neutral sixth]]
|style="text-align:center;"| 3·P8 + n6{{music|d}}↓
| style="text-align:right;"| 7,800
| style="text-align:right;"| 840.5
| {{audio|Tridecimal neutral sixth on C.mid|Play}}
|-
|style="text-align:center;"| 14
| [[harmonic seventh]],<br/> septimal [[minor seventh]] <br/> <small>(‘the lost chord’)</small>
|style="text-align:center;"| 3·P8 + m7⤈
| style="text-align:right;"| 8,400
| style="text-align:right;"| 968.8
| {{audio|Harmonic seventh on C.mid|Play}}
|-
|style="text-align:center;"| 15
| [[just intonation|just]] [[major seventh]]
|style="text-align:center;"| 3·P8 + M7
| style="text-align:right;"| 9,000
| style="text-align:right;"| 1,088.3
| {{audio|Just major seventh on C.mid|Play}}
|-
|style="text-align:center;"| 16
| fourth perfect [[octave]] 
|style="text-align:center;"| 4·P8
| style="text-align:right;"| 9,600
| style="text-align:right;"| 0.0
| {{audio|Perfect octave on C.mid|Play}}
|-
|style="text-align:center;"| 17
| septidecimal [[semitone]]
|style="text-align:center;"| 4·P8 + m2⇟
| style="text-align:right;"| 10,200
| style="text-align:right;"| 105.0
| {{audio|Just major semitone on C.mid|Play}}
|-
|style="text-align:center;"| 18
| [[Pythagorean interval|Pythagorean]] [[major second]]
|style="text-align:center;"| 4·P8 + M2
| style="text-align:right;"| 10,800
| style="text-align:right;"| 203.9
| {{audio|Major tone on C.mid|Play}}
|-
|style="text-align:center;"| 19
| nanodecimal [[minor third]]
|style="text-align:center;"| 4·P8 + m3{{music|d}}
| style="text-align:right;"| 11,400
| style="text-align:right;"| 297.5
| {{audio|19th harmonic on C.mid|Play}}
|-
|style="text-align:center;"| 20
| just [[major third]]
|style="text-align:center;"| 4·P8 + M3
| style="text-align:right;"| 12,000
| style="text-align:right;"| 386.3
| {{audio|Just major third on C.mid|Play}}
|}

===Artificial harmonics===

Occasionally a score will call for an [[artificial harmonic]], produced by playing an overtone on an already stopped string. As a performance technique, it is accomplished by using two fingers on the fingerboard, the first to shorten the string to the desired fundamental, with the second touching the node corresponding to the appropriate harmonic.

==Other information==

Harmonics may be either used in or considered as the basis of [[just intonation]] systems. Composer [[Arnold Dreyblatt]] is able to bring out different harmonics on the single string of his modified [[double bass]] by slightly altering his unique [[bow (music)|bowing]] technique halfway between hitting and bowing the strings. Composer [[Lawrence Ball]] uses harmonics to generate music electronically.

==See also==

{{listen
 | filename     = Violin harmonics.ogg
 | title        = Violin harmonics
 | description  = Violin natural harmonic stop points on the A string
 | format       = [[Ogg]]
 | filename2    = Harmonics 110x16.ogg
 | title2       = Harmonics 110x16
 | description2 = Demonstration of 16 harmonics using electronic sine tones, starting with 110 Hz fundamental, 0.5s each. Note that each harmonic is presented at the same signal level as the fundamental; the sample tones sound louder as they increase in frequency
 | format2      = [[Ogg]]
}}
* {{annotated link|Aristoxenus}}
* {{annotated link|Electronic tuner}}
* {{annotated link|Formant}}
* {{annotated link|Fourier series}}
* {{annotated link|Guitar harmonic}}
* {{annotated link|Harmonic analysis}}
* {{annotated link|Harmonics (electrical power)}}
* {{annotated link|Harmonic generation}}
* {{annotated link|Harmonic oscillator}}
* {{annotated link|Harmony}}
* {{annotated link|Pure tone}}
* {{annotated link|Pythagorean tuning}}
* {{annotated link|Scale of harmonics}}
* {{annotated link|Spherical harmonics}}
* {{annotated link|Stretched octave}}
* {{annotated link|Subharmonic}}
* {{annotated link|Xenharmonic music}}

==References==
{{Reflist}}

==External links==
* [https://feynmanlectures.caltech.edu/I_50.html The Feynman Lectures on Physics: Harmonics]
* [http://www.sengpielaudio.com/calculator-harmonics.htm Harmonics, partials and overtones from fundamental frequency]
* {{Cite EB1911|wstitle=Harmonic}}
* [https://web.archive.org/web/20091229052719/http://www.dranetz-bmi.com/pdf/harmonicsall.pdf Harmonics]
* [http://harmonicviewerforpiano.com/ Hear and see harmonics on a Piano]

{{Acoustics}}
{{Mental processes}}
{{Authority control}}

[[Category:Acoustics]]
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[[Category:Spectrum (physical sciences)]]