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A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose
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 ...
is an
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 a ''
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 ...
''. Pitched
musical instrument 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 ...
s are often based on an acoustic resonator such as a string or a column of air, which
oscillates Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
at numerous
modes Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
simultaneously. At the frequencies of each vibrating mode, waves travel in both directions along the string or air column, reinforcing and canceling each other to form
standing wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
s. Interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the typical spacing of the
resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscil ...
s, these frequencies are mostly limited to integer multiples, or
harmonic 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'', t ...
s, of the lowest frequency, and such multiples form the harmonic series. The musical pitch of a note is usually perceived as the lowest
partial Partial may refer to: Mathematics *Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial d ...
present (the fundamental frequency), which may be the one created by
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, su ...
over the full length of the string or air column, or a higher harmonic chosen by the player. The musical timbre of a steady tone from such an instrument is strongly affected by the relative strength of each harmonic.


Terminology


Partial, harmonic, fundamental, inharmonicity, and overtone

A "complex tone" (the sound of a note with a timbre particular to the instrument playing the note) "can be described as a combination of many simple periodic waves (i.e.,
sine 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 ...
s) or ''partials,'' each with its own frequency of
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, su ...
,
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
, and phase". (See also, Fourier analysis.) A partial is any of the sine waves (or "simple tones", as Ellis calls them when translating
Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...
) of which a complex tone is composed, not necessarily with an integer multiple of the lowest harmonic. A harmonic is any member of the harmonic series, an ideal set of frequencies that are positive integer multiples of a common
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 fundamental is a harmonic because it is one times itself. A harmonic partial is any real partial component of a complex tone that matches (or nearly matches) an ideal harmonic. An inharmonic partial is any partial that does not match an ideal harmonic. '' Inharmonicity'' is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial. Many pitched
acoustic instruments Acoustic music is music that solely or primarily uses instruments that produce sound through acoustic means, as opposed to electric or electronic means. While all music was once acoustic, the retronym "acoustic music" appeared after the adve ...
are designed to have partials that are close to being whole-number ratios with very low inharmonicity; therefore, in
music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the " rudiments", that are needed to understand music notation (k ...
, and in instrument design, it is convenient, although not strictly accurate, to speak of the partials in those instruments' sounds as "harmonics", even though they may have some degree of inharmonicity. The
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 keyboa ...
, one of the most important instruments of western tradition, contains a certain degree of inharmonicity among the frequencies generated by each string. Other pitched instruments, especially certain
percussion A percussion instrument is a musical instrument that is sounded by being struck or scraped by a beater including attached or enclosed beaters or rattles struck, scraped or rubbed by hand or struck against another similar instrument. Exc ...
instruments, such as
marimba The marimba () is a musical instrument in the percussion family that consists of wooden bars that are struck by mallets. Below each bar is a resonator pipe that amplifies particular harmonics of its sound. Compared to the xylophone, the timbre ...
,
vibraphone The vibraphone is a percussion instrument in the metallophone family. It consists of tuned metal bars and is typically played by using mallets to strike the bars. A person who plays the vibraphone is called a ''vibraphonist,'' ''vibraharpist ...
, tubular bells,
timpani Timpani (; ) or kettledrums (also informally called timps) are musical instruments in the percussion family. A type of drum categorised as a hemispherical drum, they consist of a membrane called a head stretched over a large bowl traditionall ...
, and singing bowls contain mostly inharmonic partials, yet may give the ear a good sense of pitch because of a few strong partials that resemble harmonics. Unpitched, or indefinite-pitched instruments, such as cymbals and
tam-tams The Tam-Tams is the informal name of a weekly free festival around the George-Étienne Cartier Monument in Mount Royal Park in Montreal, Quebec, Canada. Its name imitates the sound of drums and refers to the drum circles that form the focal poi ...
make sounds (produce spectra) that are rich in inharmonic partials and may give no impression of implying any particular pitch. An overtone is any partial above the lowest partial. The term overtone does not imply harmonicity or inharmonicity and has no other special meaning other than to exclude the fundamental. It is mostly the relative strength of the different overtones that give an instrument its particular timbre, tone color, or character. When writing or speaking of overtones and partials numerically, care must be taken to designate each correctly to avoid any confusion of one for the other, so the second overtone may not be the third partial, because it is the second sound in a series. Some electronic instruments, such as
synthesizer A synthesizer (also spelled synthesiser) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis a ...
s, can play a pure frequency with no overtones (a sine wave). Synthesizers can also combine pure frequencies into more complex tones, such as to simulate other instruments. Certain flutes and ocarinas are very nearly without overtones.


Frequencies, wavelengths, and musical intervals in example systems

One of the simplest cases to visualise is a vibrating string, as in the illustration; the string has fixed points at each end, and each harmonic
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
divides it into an integer number (1, 2, 3, 4, etc.) of equal-sized sections resonating at increasingly higher frequencies. Similar arguments apply to vibrating air columns in wind instruments (for example, "the French horn was originally a valveless instrument that could play only the notes of the harmonic series"), although these are complicated by having the possibility of anti-nodes (that is, the air column is closed at one end and open at the other),
conical A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines c ...
as opposed to cylindrical bores, or end-openings that run the gamut from no flare, cone flare, or exponentially shaped flares (such as in various bells). In most pitched musical instruments, the fundamental (first harmonic) is accompanied by other, higher-frequency harmonics. Thus shorter-wavelength, higher-frequency
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 ...
s occur with varying prominence and give each instrument its characteristic tone quality. The fact that a string is fixed at each end means that the longest allowed wavelength on the string (which gives the fundamental frequency) is twice the length of the string (one round trip, with a half cycle fitting between the nodes at the two ends). Other allowed wavelengths are reciprocal multiples (e.g. , , times) that of the fundamental. Theoretically, these shorter wavelengths correspond to
vibration Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, su ...
s at frequencies that are integer multiples of (e.g. 2, 3, 4 times) the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator it vibrates against often alter these frequencies. (See inharmonicity and
stretched tuning Stretching is a form of physical exercise in which a specific muscle or tendon (or muscle group) is deliberately flexed or stretched in order to improve the muscle's felt elasticity and achieve comfortable muscle tone. The result is a feeling ...
for alterations specific to wire-stringed instruments and certain electric pianos.) However, those alterations are small, and except for precise, highly specialized tuning, it is reasonable to think of the frequencies of the harmonic series as integer multiples of the fundamental frequency. The harmonic series is an arithmetic progression (''f'', 2''f'', 3''f'', 4''f'', 5''f'', ...). In terms of frequency (measured in cycles per second, or
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
, where ''f'' is the fundamental frequency), the difference between consecutive harmonics is therefore constant and equal to the fundamental. But because human ears respond to sound nonlinearly, higher harmonics are perceived as "closer together" than lower ones. On the other hand, the
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
series is a geometric progression (2''f'', 4''f'', 8''f'', 16''f'', ...), and people perceive these distances as "
the same The Same was a punk band from Sundsvall. Members were among others Magnus Holmén, Per Kraft, Peter Byström and Tomas Broman. Their most popular song was "Kuken i styret". This song also resulted in that the P3 radio show Ny våg was convic ...
" in the sense of musical interval. In terms of what one hears, each octave in the harmonic series is divided into increasingly "smaller" and more numerous intervals. The second harmonic, whose frequency is twice the fundamental, sounds an octave higher; the third harmonic, three times the frequency of the fundamental, sounds a perfect fifth above the second harmonic. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third harmonic (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher).
Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
wrote: "The order of the Consonances is natural, and ... the way we count them, starting from unity up to the number six and beyond is founded in nature." However, to quote
Carl Dahlhaus Carl Dahlhaus (10 June 1928 – 13 March 1989) was a German musicologist who was among the leading postwar musicologists of the mid to late 20th-century. A prolific scholar, he had broad interests though his research focused on 19th- and 20th ...
, "the interval-distance of the natural-tone-row vertones .. counting up to 20, includes everything from the octave to the quarter tone, (and) useful and useless musical tones. The natural-tone-row armonic seriesjustifies everything, that means, nothing."Sabbagh, Peter (2003). ''The Development of Harmony in Scriabin's Works'', p. 12. Universal. . Cites: Dahlhaus, Carl (1972). "Struktur und Expression bei Alexander Skrjabin", ''Musik des Ostens'', Vol. 6, p. 229.


Harmonics and tuning

If the harmonics are octave displaced and compressed into the span of one
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
, some of them are approximated by the notes of what the
West West or Occident is one of the four cardinal directions or points of the compass. It is the opposite direction from east and is the direction in which the Sun sets on the Earth. Etymology The word "west" is a Germanic word passed into some ...
has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equal semitones, which is slightly out of tune with many of the harmonics, especially the 7th, 11th, and 13th harmonics. In the late 1930s, composer
Paul Hindemith Paul Hindemith (; 16 November 189528 December 1963) was a German composer, music theorist, teacher, violist and conductor. He founded the Amar Quartet in 1921, touring extensively in Europe. As a composer, he became a major advocate of the ' ...
ranked musical intervals according to their relative dissonance based on these and similar harmonic relationships. Hindemith, Paul (1942)
''The Craft of Musical Composition: Book 1 – Theoretical Part''
pp. 15ff. Translated by
Arthur Mendel Arthur Mendel (June 6, 1905 – October 14, 1979) was an American musicologist, known as a Bach scholar. He was born in Boston and died in Newark, New Jersey. Education He graduated from Harvard University in 1925 before going to study with ...
(London: Schott & Co; New York: Associated Music Publishers. ). .
Below is a comparison between the first 31 harmonics and the intervals of
12-tone equal temperament Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That result ...
(12TET), octave displaced and compressed into the span of one octave. Tinted fields highlight differences greater than 5 cents ( of a semitone), which is the human ear's " just noticeable difference" for notes played one after the other (smaller differences are noticeable with notes played simultaneously). The frequencies of the harmonic series, being integer multiples of the fundamental frequency, are naturally related to each other by whole-numbered ratios and small whole-numbered ratios are likely the basis of the consonance of musical intervals (see just intonation). This objective structure is augmented by psychoacoustic phenomena. For example, a perfect fifth, say 200 and 300 Hz (cycles per second), causes a listener to perceive a combination tone of 100 Hz (the difference between 300 Hz and 200 Hz); that is, an octave below the lower (actual sounding) note. This 100 Hz first-order combination tone then interacts with both notes of the interval to produce second-order combination tones of 200 (300 − 100) and 100 (200 − 100) Hz and all further nth-order combination tones are all the same, being formed from various subtraction of 100, 200, and 300. When one contrasts this with a dissonant interval such as a tritone (not tempered) with a frequency ratio of 7:5 one gets, for example, 700 − 500 = 200 (1st order combination tone) and 500 − 200 = 300 (2nd order). The rest of the combination tones are octaves of 100 Hz so the 7:5 interval actually contains four notes: 100 Hz (and its octaves), 300 Hz, 500 Hz and 700 Hz. Note that the lowest combination tone (100 Hz) is a seventeenth (two octaves and a major third) below the lower (actual sounding) note of the tritone. All the intervals succumb to similar analysis as has been demonstrated by
Paul Hindemith Paul Hindemith (; 16 November 189528 December 1963) was a German composer, music theorist, teacher, violist and conductor. He founded the Amar Quartet in 1921, touring extensively in Europe. As a composer, he became a major advocate of the ' ...
in his book ''The Craft of Musical Composition'', although he rejected the use of harmonics from the seventh and beyond. The
Mixolydian mode Mixolydian mode may refer to one of three things: the name applied to one of the ancient Greek ''harmoniai'' or ''tonoi'', based on a particular octave species or scale; one of the medieval church modes; or a modern musical mode or diatonic sc ...
is consonant with the first 10 harmonics of the harmonic series (the 11th harmonic, a tritone, is not in the Mixolydian mode). The Ionian mode is consonant with only the first 6 harmonics of the series (the seventh harmonic, a minor seventh, is not in the Ionian mode).


Timbre of musical instruments

The relative
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
s (strengths) of the various harmonics primarily determine the timbre of different instruments and sounds, though onset
transients Transience or transient may refer to: Music * ''Transient'' (album), a 2004 album by Gaelle * ''Transience'' (Steven Wilson album), 2015 * Transience (Wreckless Eric album) Science and engineering * Transient state, when a process variable or ...
, formants,
noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
s, and inharmonicities also play a role. For example, the
clarinet The clarinet is a musical instrument in the woodwind family. The instrument has a nearly cylindrical bore and a flared bell, and uses a single reed to produce sound. Clarinets comprise a family of instruments of differing sizes and pitch ...
and
saxophone The saxophone (often referred to colloquially as the sax) is a type of Single-reed instrument, single-reed woodwind instrument with a conical body, usually made of brass. As with all single-reed instruments, sound is produced when a reed (mouthpi ...
have similar mouthpieces and reeds, and both produce sound through
resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscil ...
of air inside a chamber whose mouthpiece end is considered closed. Because the clarinet's resonator is cylindrical, the ''even''-numbered harmonics are less present. The saxophone's resonator is conical, which allows the even-numbered harmonics to sound more strongly and thus produces a more complex tone. The inharmonic ringing of the instrument's metal resonator is even more prominent in the sounds of brass instruments. Human ears tend to group phase-coherent, harmonically-related frequency components into a single sensation. Rather than perceiving the individual partials–harmonic and inharmonic, of a musical tone, humans perceive them together as a tone color or timbre, and the overall pitch is heard as the fundamental of the harmonic series being experienced. If a sound is heard that is made up of even just a few simultaneous sine tones, and if the intervals among those tones form part of a harmonic series, the brain tends to group this input into a sensation of the pitch of the fundamental of that series, even if the fundamental is not present. Variations in the frequency of harmonics can also affect the ''perceived'' fundamental pitch. These variations, most clearly documented in the piano and other stringed instruments but also apparent in brass instruments, are caused by a combination of metal stiffness and the interaction of the vibrating air or string with the resonating body of the instrument.


Interval strength

David Cope (1997) suggests the concept of interval strength, Cope, David (1997). ''Techniques of the Contemporary Composer'', p. 40–41. New York, New York: Schirmer Books. . in which an interval's strength, consonance, or stability (see
consonance and dissonance 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 ...
) is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps–Meyer law. Thus, an equal-tempered perfect fifth () is stronger than an equal-tempered minor third (), since they approximate a just perfect fifth () and just minor third (), respectively. The just minor third appears between harmonics 5 and 6 while the just fifth appears lower, between harmonics 2 and 3.


See also

*
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
* Klang (music) *
Otonality and Utonality ''Otonality'' and ''utonality'' are terms introduced by Harry Partch to describe chord (music), chords whose pitch classes are the harmonics or subharmonics of a given fixed root (chord), tone (identity (tuning), identity), respectively. For ...
* Piano acoustics * Scale of harmonics *
Undertone series In music, the undertone series or subharmonic series is a sequence of notes that results from inverting the intervals of the overtone series. While overtones naturally occur with the physical production of music on instruments, undertones mus ...


Notes

Sources *


Further reading

* * * (see '' Sensations of Tone'') * * * * * * {{Strings (music)