A harmonic is a

wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...

with a

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

that is a positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

multiple of the ''

fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...

'', the

of the original

periodic signal A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...

, such as a

sinusoidal wave A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...

. The original signal is also called the ''1st harmonic'', the other harmonics are known as ''higher harmonics''. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a '' 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 Hz, a common AC 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. 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 also be called "overtones", "partials" or "upper partials". The difference between "harmonic" and "overtone" is that the term "harmonic" includes all of the notes in a series, including the fundamental frequency (e.g., the open string of a guitar). The term "overtone" only includes the pitches above the fundamental. In some music contexts, the terms "harmonic", "overtone" and "partial" are used fairly interchangeably.

Characteristics

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 In music, timbre ( ), also known as tone color or tone quality (from psychoacoustics), is the perceived sound quality of a musical note, sound or tone. Timbre distinguishes different types of sound production, such as choir voices and musica ...

of that sound being a result of the relative strengths of the individual partials. Many acoustic oscillators, such as the human voice or a bowed

violin The violin, sometimes known as a '' fiddle'', is a wooden chordophone ( string instrument) in the violin family. Most violins have a hollow wooden body. It is the smallest and thus highest-pitched instrument ( soprano) in the family in regu ...

string, produce complex tones that are more or less periodic, and thus are composed of partials that are near matches to integer multiples of the 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 real and the second being ideal). Oscillators that produce harmonic partials behave somewhat like one-dimensional

resonator A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a 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 modern orchestral

transverse flute A transverse flute or side-blown flute is a flute which is held horizontally when played. The player blows across the embouchure hole, in a direction perpendicular to the flute's body length. Transverse flutes include the Western concert flut ...

). Wind instruments whose air column is open at only one end, such as

trumpet The trumpet is a brass instrument commonly used in classical and jazz ensembles. The trumpet group ranges from the piccolo trumpet—with the highest register in the brass family—to the bass trumpet, pitched one octave below the standard ...

s and clarinets, also produce partials resembling harmonics. However they only produce partials matching the ''odd'' harmonics, at least in theory. The reality of acoustic instruments is such that none of them behaves as perfectly as the somewhat simplified theoretical models would predict. Partials whose frequencies are not integer multiples of the fundamental are referred to as 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 The piano is a stringed keyboard instrument in which the strings are struck by wooden hammers that are coated with a softer material (modern hammers are covered with dense wool felt; some early pianos used leather). It is played using a keybo ...

s, strings plucked

pizzicato Pizzicato (, ; translated as "pinched", and sometimes roughly as "plucked") is a playing technique that involves plucking the strings of a string instrument. The exact technique varies somewhat depending on the type of instrument : * On bowe ...

, vibraphones, marimbas, and certain pure-sounding bells or chimes. Antique

singing bowl A standing bell or resting bell is an inverted bell (instrument), bell, supported from below with the rim uppermost. Such bells are normally bowl-shaped, and exist in a wide range of sizes, from a few centimetres to a metre in diameter. They are ...

s are known for producing multiple harmonic partials or

multiphonics A multiphonic is an extended technique on a monophonic musical instrument (one that generally produces only one note at a time) in which several notes are produced at once. This includes wind, reed, and brass instruments, as well as the human voi ...

. Other oscillators, such as

cymbals A cymbal is a common percussion instrument. Often used in pairs, cymbals consist of thin, normally round plates of various alloys. The majority of cymbals are of indefinite pitch, although small disc-shaped cymbals based on ancient designs soun ...

, drum heads, and other 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.

Dynamic tonality Dynamic tonality is a paradigm for tuning and timbre which generalizes the special relationship between just intonation and the harmonic series to apply to a wider set of pseudo-just tunings and related pseudo-harmonic timbres.Duffin, R.W., 2006 ...

, building on the work of

William Sethares William A. Sethares (born April 19, 1955) is an American music theorist and professor of electrical engineering at the University of Wisconsin. In music, he has contributed to the theory of Dynamic Tonality and provided a formalization of conson ...

, 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 In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness, unpl ...

of that pseudo-harmonic timbre with notes of that pseudo-just tuning. Alt URL

Partials, overtones, and harmonics

overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...

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 Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...

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): In many

musical instruments A musical instrument is a device created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object becomes a musical instrument. A person who pl ...

, 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 an octave, but in more complex cases many other pitch variations are obtained. In some cases it also changes the

of the note. This is part of the normal method of obtaining higher notes in wind instruments, where it is called ''

overblowing Overblowing is the manipulation of supplied air through a wind instrument that causes the sounded pitch to jump to a higher one without a fingering change or the operation of a slide. Overblowing may involve a change in the air pressure, in the ...

''. The

extended technique In music, extended technique is unconventional, unorthodox, or non-traditional methods of singing or of playing musical instruments employed to obtain unusual sounds or timbres.Burtner, Matthew (2005).Making Noise: Extended Techniques after Exper ...

of playing multiphonics also produces harmonics. On string instruments it is possible to produce very pure sounding notes, called

harmonics A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', ...

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 In music, unison is two or more musical parts that sound either the same pitch or pitches separated by intervals of one or more octaves, usually at the same time. ''Rhythmic unison'' is another term for homorhythm. Definition Unison or per ...

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 The cello ( ; plural ''celli'' or ''cellos'') or violoncello ( ; ) is a bowed (sometimes plucked and occasionally hit) string instrument of the violin family. Its four strings are usually tuned in perfect fifths: from low to high, C2, G ...

produces the same pitch as lightly fingering the node of the way down the second highest string. For the human voice see

Overtone singing Overtone singing – also known as overtone chanting, harmonic singing, polyphonic overtone singing, and diphonic singing – is a set of singing techniques in which the vocalist manipulates the resonances of the vocal tract, in order to arous ...

, 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. This is especially true of instruments other than stringed or brass/woodwind ones, e.g., xylophone, drums, bells etc., where not all the overtones have a simple whole number ratio with the fundamental frequency. The

is the

reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...

of the

period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...

of the periodic phenomenon.

On stringed instruments

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 (

) when used and heard in orchestration. 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.

Artificial harmonics

Occasionally a score will call for an

artificial harmonic Playing a string harmonic (a flageolet) is a string instrument technique that uses the nodes of natural harmonics of a musical string to isolate overtones. Playing string harmonics produces high pitched tones, often compared in timbre to a wh ...

, 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 Arnold Dreyblatt (born 1953) is an American composer, performance artist and visual artist. Biography Arnold Dreyblatt was born in 1953 in New York City. His mother, Lucille Wallenrod (1918–1998), was a painter. He started his studies at W ...

is able to bring out different harmonics on the single string of his modified

double bass The double bass (), also known simply as the bass () (or by other names), is the largest and lowest-pitched bowed (or plucked) string instrument in the modern symphony orchestra (excluding unorthodox additions such as the octobass). Similar i ...

by slightly altering his unique

bowing Bowing (also called stooping) is the act of lowering the torso and head as a social gesture in direction to another person or symbol. It is most prominent in Asian cultures but it is also typical of nobility and aristocracy in many European c ...

technique halfway between hitting and bowing the strings. Composer Lawrence Ball uses harmonics to generate music electronically.

References

External links

The Feynman Lectures on Physics: Harmonics

*
Harmonics

Hear and see harmonics on a Piano
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