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Harmonic

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Wave with frequency an integer multiple of the fundamental frequency

This article is about the components of periodic signals. For other uses, seeHarmonic (disambiguation).

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Thenodes of a vibrating string are harmonics.

Two different notations of natural harmonics on the cello. First as sounded (more common), then as fingered (easier to sightread).

Inphysics,acoustics, andtelecommunications, aharmonic is asinusoidal wave with afrequency that is a positiveinteger multiple of thefundamental frequency of aperiodic signal. The fundamental frequency is also called the1st harmonic; the other harmonics are known ashigher harmonics. As all harmonics areperiodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms aharmonic 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 Hz, a commonAC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz.

An $\ n$ ^th characteristic mode, for $\ n>1\ ,$ will have nodes that are not vibrating. For example, the3^rd characteristic mode will have nodes at $\ {\tfrac {1}{3}}\ L\$ and $\ {\tfrac {2}{3}}\ L\ ,$ where $\ L\$ is the length of the string. In fact, each $\ n$ ^th characteristic mode, for $\ n\$ not a multiple of 3, willnot have nodes at these points. These other characteristic modes will bevibrating at the positions $\ {\tfrac {1}{3}}\ L\$ and $\ {\tfrac {2}{3}}\ L~.$ If the playergently 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 $\ n$ ^th characteristic characteristic modes, where $\ n\$ is a multiple of 3, will be made relatively more prominent.^[1]

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

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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" includesall pitches in a harmonic series (including the fundamental frequency) while the term "overtone" only includes pitchesabove the fundamental.

Characteristics

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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.^{[citation needed]}
— Richard Scholz (c. 1888–1912)[2]

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 ortimbre of that sound being a result of the relative strengths of the individual partials. Many acousticoscillators, such as thehuman voice or abowed violin string, produce complex tones that are more or lessperiodic, 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 partial a harmonic, the first being actual and the second being theoretical).

Oscillators that produce harmonic partials behave somewhat like one-dimensionalresonators, 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 orchestraltransverse flute). Wind instruments whose air column is open at only one end, such astrumpets andclarinets, also produce partials resembling harmonics. However they only produce partials matching theodd 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 ofnon-linearly elastic wood, instead of metal, or strung withgut instead ofbrass or steel strings, tend to have not-quite-integer partials.

Partials whose frequencies are not integer multiples of the fundamental are referred to asinharmonic 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 aspianos, strings pluckedpizzicato, vibraphones, marimbas, and certain pure-sounding bells or chimes. Antiquesinging bowls are known for producing multiple harmonic partials ormultiphonics.^[3]^[4]Other oscillators, such ascymbals, 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 ofSethares (2004),^[5]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 theconsonance of that pseudo-harmonic timbre with notes of that pseudo-just tuning.^[6]^[7]^[8]^[9]

Partials, overtones, and harmonics

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Anovertone 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 amusical 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 mostpitched 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):

Frequency	Order (n)	Name 1	Name 2	Name 3
1 ×f =440 Hz	n = 1	1st partial	fundamental tone	1st harmonic
2 ×f =880 Hz	n = 2	2nd partial	1st overtone	2nd harmonic
3 ×f = 1320 Hz	n = 3	3rd partial	2nd overtone	3rd harmonic
4 ×f = 1760 Hz	n = 4	4th partial	3rd overtone	4th harmonic

In manymusical instruments, it is possible to play the upper harmonics without the fundamental note being present. In a simple case (e.g.,recorder) this has the effect of making the note go up in pitch by anoctave, but in more complex cases many other pitch variations are obtained. In some cases it also changes thetimbre of the note. This is part of the normal method of obtaining higher notes inwind instruments, where it is calledoverblowing. Theextended technique of playingmultiphonics also produces harmonics. Onstring instruments it is possible to produce very pure sounding notes, calledharmonics orflageolets by string players, which have an eerie quality, as well as being high in pitch. Harmonics may be used to check at aunison the tuning of strings that are not tuned to the unison. For example, lightly fingering the node found halfway down the highest string of acello produces the same pitch as lightly fingering the node⁠ 1 / 3⁠ of the way down the second highest string. For the human voice seeOvertone 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 pureharmonic series. This is especially true of instruments other thanstrings,brass, orwoodwinds. 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. (Thefundamental frequency is thereciprocal of the longesttime period of the collection of vibrations in some single periodic phenomenon.^[10])

On stringed instruments

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Main article:String harmonic

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 ( ${\tfrac {1}{2}}$ , ${\tfrac {1}{3}}$ , ${\tfrac {1}{4}}$ , etc.).(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.'(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)^[11]

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 inorchestration.^[12] 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.^[12]

Harmonic order	Stop note	Note sounded (relative to open string)	Audio frequency (Hz)	Centsabove fundamental(offset by octave)	Audio (octave shifted)
1^st	fundamental, perfectunison	P 1	600Hz	0.0 ¢	Play^ⓘ
2^nd	first perfectoctave	P 8	1200Hz	0.0 ¢	Play^ⓘ
3^rd	perfect fifth	P 8 +P 5	1800Hz	702.0 ¢	Play^ⓘ
4^th	doubled perfectoctave	2 ·P 8	2400Hz	0.0 ¢	Play^ⓘ
5^th	just major third, major third	2 ·P 8 +^M3	3000Hz	386.3 ¢	Play^ⓘ
6^th	perfect fifth	2 ·P 8 +P 5	3600Hz	702.0 ¢	Play^ⓘ
7^th	harmonic seventh, septimal minor seventh (‘the lost chord’)	2 ·P 8 +m 7^↓	4200Hz	968.8 ¢	Play^ⓘ
8^th	third perfectoctave	3 ·P 8	4800Hz	0.0 ¢	Play^ⓘ
9^th	Pythagorean major second harmonic ninth	3 ·P 8 +^M2	5400Hz	203.9 ¢	Play^ⓘ
10^th	just major third	3 ·P 8 +^M3	6000Hz	386.3 ¢	Play^ⓘ
11^th	lesser undecimaltritone, undecimal semi-augmented fourth	3 ·P 8 +^A4	6600Hz	551.3 ¢	Play^ⓘ
12^th	perfect fifth	3 ·P 8 +P 5	7200Hz	702.0 ¢	Play^ⓘ
13^th	tridecimalneutral sixth	3 ·P 8 +n 6^↓	7800Hz	840.5 ¢	Play^ⓘ
14^th	harmonic seventh, septimalminor seventh (‘the lost chord’)	3 ·P 8 +m 7^⤈	8400Hz	968.8 ¢	Play^ⓘ
15^th	just major seventh	3 ·P 8 +^M7	9000Hz	1088.3 ¢	Play^ⓘ
16^th	fourth perfectoctave	4 ·P 8	9600Hz	0.0 ¢	Play^ⓘ
17^th	septidecimalsemitone	4 ·P 8 +m 2^⇟	10200Hz	105.0 ¢	Play^ⓘ
18^th	Pythagorean major second	4 ·P 8 +^M2	10800Hz	203.9 ¢	Play^ⓘ
19^th	nanodecimalminor third	4 ·P 8 +m 3	11400Hz	297.5 ¢	Play^ⓘ
20^th	justmajor third	4 ·P 8 +^M3	12000Hz	386.3 ¢	Play^ⓘ

**Notation key**
P	perfect interval
^A	augmented interval (sharpened)
^M	major interval
m	minor interval (flattened major)
n	neutral interval (between major and minor)
	half-flattened (approximate) (≈ −38 ¢ forjust, −50 ¢ for12TET)
^↓	flattened by asyntonic comma (approximate) (≈ −21 ¢)
^⤈	flattened by a half-comma (approximate) (≈ −10 ¢)
^⇟	flattened by a quarter-comma (approximate) (≈ −5 ¢)

Artificial harmonics

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Occasionally a score will call for anartificial 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

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Harmonics may be either used in or considered as the basis ofjust intonation systems. ComposerArnold Dreyblatt is able to bring out different harmonics on the single string of his modifieddouble bass by slightly altering his uniquebowing technique halfway between hitting and bowing the strings. ComposerLawrence Ball uses harmonics to generate music electronically.

References

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^Walker, Russell (14 June 2019)."Russell Walker".Authors' group (online magazine). Catonsville, MD: Institute for Operations Research and the Management Sciences.doi:10.1287/7648739e-8e59-466e-82cb-3ded22bbebf6.S2CID 241172832. Retrieved21 December 2020 – via informs.scienceconnect.io.
^"Category:Scholz, Richard".Petrucci Music Library / International Music Score Library Project (IMSLP) (imslp.org) (site sub-index & mini-bio for Scholz). Canada. Retrieved2020-12-21.
^Galembo, Alexander; Cuddly, Lola L. (2 December 1997)."Large grand and small upright pianos".acoustics.org (Press release).Acoustical Society of America. Archived fromthe original on 2012-02-09. Retrieved13 January 2024.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.
^Court, Sophie R.A. (April 1927). "Golo und Genovefa[by] Hanna Rademacher".Books Abroad (book review).1 (2):34–36.doi:10.2307/40043442.ISSN 0006-7431.JSTOR 40043442.
^Sethares, W.A. (2004).Tuning, Timbre, Spectrum, Scale. Springer.ISBN 978-1852337971 – via Google books.
^Sethares, W.A.; Milne, A.; Tiedje, S.; Prechtl, A.; Plamondon, J. (2009)."Spectral tools for dynamic tonality and audio morphing".Computer Music Journal.33 (2):71–84.doi:10.1162/comj.2009.33.2.71.S2CID 216636537.
^Milne, Andrew;Sethares, William; Plamondon, James (29 August 2008)."Tuning continua and keyboard layouts"(PDF).Journal of Mathematics and Music.2 (1):1–19.doi:10.1080/17459730701828677.S2CID 1549755.Archived(PDF) from the original on 2022-10-09."Alt URL"(PDF).Sethares pers. academic site.University of Wisconsin.
^Milne, A.;Sethares, W.A.; Plamondon, J. (Winter 2007)."Invariant fingerings across a tuning continuum".Computer Music Journal.31 (4):15–32.doi:10.1162/comj.2007.31.4.15.S2CID 27906745.
^Milne, A.;Sethares, W.A.; Plamondon, J. (2006).X System(PDF) (technical report). Thumtronics Inc. Retrieved2020-05-02.
^ This article incorporatespublic domain material fromFederal Standard 1037C.General Services Administration. Archived fromthe original on 2022-01-22.
^Grove, George (1879).A Dictionary of Music and Musicians (A.D. 1450–1889), Vol. 2, p. 665. Macmillan. [ISBN unspecified].
^^a ^bMarrocco, W. Thomas (2001)."Kennan, Kent".Oxford Music Online. Oxford University Press.doi:10.1093/gmo/9781561592630.article.14882. Retrieved2020-12-21.

External links

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The Feynman Lectures on Physics: Harmonics
Harmonics, partials and overtones from fundamental frequency
Chisholm, Hugh, ed. (1911)."Harmonic" .Encyclopædia Britannica (11th ed.). Cambridge University Press.
Harmonics
Hear and see harmonics on a Piano
Configurable animated SVG representation of base tone with harmonics

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