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In music, inharmonicity is the degree to which the
frequencies 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 ...
of
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 ...
s (also known as partials or partial tones) depart from whole multiples of the fundamental frequency ( harmonic series). Acoustically, a note perceived to have a single distinct pitch in fact contains a variety of additional overtones. Many
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
cymbal 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 ...
s,
tam-tam A gongFrom Indonesian and ms, gong; jv, ꦒꦺꦴꦁ ; zh, c=鑼, p=luó; ja, , dora; km, គង ; th, ฆ้อง ; vi, cồng chiêng; as, কাঁহ is a percussion instrument originating in East Asia and Southeast Asia. Gongs ...
s, and
chimes Tubular bells (also known as chimes) are musical instruments in the percussion family. Their sound resembles that of church bells, carillon, or a bell tower; the original tubular bells were made to duplicate the sound of church bells within a ...
, create complex and inharmonic sounds. Music harmony and intonation depends strongly on the harmonicity of tones. An ideal, homogeneous, infinitesimally thin or infinitely flexible string or column of air has exactly harmonic modes of vibration.How harmonic are harmonics? by Joe Wolfe, accessed 29 June 2008
/ref> In any real musical instrument, the resonant body that produces the music tone—typically a string, wire, or column of air—deviates from this ideal and has some small or large amount of inharmonicity. For instance, a very thick string behaves less as an ideal string and more like a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
(a tube of mass), which has natural resonances that are not whole number multiples of the fundamental frequency. However, in stringed instruments such as 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 ...
,
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 regular ...
, and
guitar The guitar is a fretted musical instrument that typically has six strings. It is usually held flat against the player's body and played by strumming or plucking the strings with the dominant hand, while simultaneously pressing selected stri ...
, or in some Indian drums such as
tabla A tabla, bn, তবলা, prs, طبلا, gu, તબલા, hi, तबला, kn, ತಬಲಾ, ml, തബല, mr, तबला, ne, तबला, or, ତବଲା, ps, طبله, pa, ਤਬਲਾ, ta, தபலா, te, తబల ...
, the overtones are close to—or in some cases, quite exactly—whole number multiples of the fundamental frequency. Any departure from this ideal harmonic series is known as inharmonicity. The less elastic the strings are (that is, the shorter, thicker, smaller tension or stiffer they are), the more inharmonicity they exhibit. When a string is bowed or tone in a
wind instrument A wind instrument is a musical instrument that contains some type of resonator (usually a tube) in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at or near the end of the resonator. The pitc ...
initiated by vibrating
reed Reed or Reeds may refer to: Science, technology, biology, and medicine * Reed bird (disambiguation) * Reed pen, writing implement in use since ancient times * Reed (plant), one of several tall, grass-like wetland plants of the order Poales * ...
or lips, a phenomenon called
mode-locking Mode locking is a technique in optics by which a laser can be made to produce pulses of light of extremely short duration, on the order of picoseconds (10−12 s) or femtoseconds (10−15 s). A laser operated in this way is sometimes r ...
counteracts the natural inharmonicity of the string or air column and causes the overtones to lock precisely onto integer multiples of the fundamental pitch, even though these are slightly different from the natural resonance points of the instrument. For this reason, a single tone played by a bowed string instrument, brass instrument, or reed instrument does not necessarily exhibit inharmonicity. However, when a string is struck or plucked, as with a piano string that is struck by a hammer, a violin string played
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 bowed ...
, or a guitar string that is plucked by a finger or
plectrum A plectrum is a small flat tool used for plucking or strumming of a stringed instrument. For hand-held instruments such as guitars and mandolins, the plectrum is often called a pick and is held as a separate tool in the player's hand. In harpsic ...
, the string will exhibit inharmonicity. The inharmonicity of a string depends on its physical characteristics, such as
tension Tension may refer to: Science * Psychological stress * Tension (physics), a force related to the stretching of an object (the opposite of compression) * Tension (geology), a stress which stretches rocks in two opposite directions * Voltage or el ...
,
stiffness Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a b ...
, and length. For instance, a stiff string under low tension (such as those found in the bass notes of small upright pianos) exhibits a high degree of inharmonicity, while a thinner string under higher tension (such as a treble string in a piano) or a more flexible string (such as a gut or nylon string used on a guitar or harp) will exhibit less inharmonicity. A wound string generally exhibits less inharmonicity than the equivalent solid string, and for that reason wound strings are often preferred.


Pianos


Sound quality of inharmonicity

In 1943, Schuck and Young were the first scientists to measure the spectral inharmonicity in piano tones. They found that the spectral partials in piano tones are progressively stretched—that is to say, the lowest partials are stretched the least and higher partials are progressively stretched further. Inharmonicity is not necessarily unpleasant. In 1962, research by Harvey Fletcher and his collaborators indicated that the spectral inharmonicity is important for tones to sound piano-like. They proposed that inharmonicity is responsible for the "warmth" property common to real piano tones. According to their research synthesized piano tones sounded more natural when some inharmonicity was introduced. In general, electronic instruments that duplicate acoustic instruments must duplicate both the inharmonicity and the resulting stretched tuning of the original instruments.


Inharmonicity leads to stretched tuning

When pianos are tuned by piano tuners, the technician sometimes listens for the sound of "
beating Beat, beats or beating may refer to: Common uses * Patrol, or beat, a group of personnel assigned to monitor a specific area ** Beat (police), the territory that a police officer patrols ** Gay beat, an area frequented by gay men * Battery ...
" when two notes are played together, and tunes to the point that minimizes roughness between tones. Piano tuners must deal with the inharmonicity of piano strings, which is present in different amounts in all of the ranges of the instrument, but especially in the bass and high treble registers. The result is that octaves are tuned slightly wider than the harmonic 2:1 ratio. The exact amount octaves are stretched in a piano tuning varies from piano to piano and even from register to register within a single piano—depending on the exact inharmonicity of the strings involved. Because of the problem of inharmonicity, electronic piano tuning devices used by piano technicians are not designed to tune according to a simple harmonic series. Rather, the devices use various means to duplicate the stretched octaves and other adjustments a technician makes by ear. The most sophisticated devices allow a technician to make custom inharmonicity measurements—simultaneously considering all partials for pitch and volume to determine the most appropriate stretch to employ for a given instrument. Some include an option to simply record a tuning that a technician has completed by ear; the technician can then duplicate that tuning on the same piano (or others of similar make and model) more easily and quickly. The issues surrounding setting the stretch by ear vs machine have not been settled; machines are better at deriving the absolute placement of semitones within a given chromatic scale, whereas non-machine tuners prefer to adjust these locations preferentially due to their temptation to make intervals more sonorous. The result is that pianos tuned by ear and immediately checked with a machine tend to vary from one degree to another from the purely theoretical semitone (mathematically the 12th root of two) due to human error and perception. (If pleasing the ear is the goal of an aural tuning, then pleasing the math is the goal of a machine tuning.) This is thought to be because strings can vary somewhat from note to note and even from neighbors within a unison. This non-linearity is different from true falseness where a string creates false harmonics and is more akin to minor variations in string thickness, string sounding length or minor bridge inconsistencies. Piano tuning is a compromise—both in terms of choosing a
temperament In psychology, temperament broadly refers to consistent individual differences in behavior that are biologically based and are relatively independent of learning, system of values and attitudes. Some researchers point to association of temperam ...
to minimize out-of-tuneness in the intervals and chords that will be played, and in terms of dealing with inharmonicity. For more information, ''see
Piano acoustics Piano acoustics is the set of physical properties of the piano that affect its sound. It is an area of study within musical acoustics. String length, mass and tension The strings of a piano vary in thickness, and therefore in mass per length, w ...
'' and ''
Piano tuning Piano tuning is the act of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune. The meaning of the term 'in tune', in the context of piano tuning, is not simply a particular fixed ...
''. Another factor that can cause problems is the presence of rust on the strings or dirt in the windings. These factors can slightly raise the frequency of the higher modes, resulting in more inharmonicity.


Guitar

While piano tuning is normally done by trained technicians, guitars such as acoustic guitars, electric guitars, and electric bass guitars are usually tuned by the guitarist themselves. When a guitarist tunes a guitar by ear, they have to take both temperament and string inharmonicity into account. The inharmonicity in guitar strings can "cause stopped notes to stop sharp, meaning they will sound sharper both in terms of pitch and beating, than they "should". This is distinct from any temperament issue." Even if a guitar is built so that there are no "fret or neck angle errors, inharmonicity can make the simple approach of tuning open strings to notes stopped on the fifth or fourth frets" unreliable. Inharmonicity also demands that some of the "octaves may need to be compromised minutely." How to tune the guitar expertly by ear. by Brian Capleton http://www.amarilli.co.uk/guitar/howto.asp When strobe tuners became available in the 1970s, and then inexpensive
electronic tuner In music, an electronic tuner is a device that detects and displays the pitch of musical notes played on a musical instrument. "Pitch" is the perceived fundamental frequency of a musical note, which is typically measured in Hertz. Simple tune ...
s in the 1980s reached the mass market, it did not spell the end of tuning problems for guitarists. Even if an electronic tuner indicates that the guitar is "perfectly" in tune, some chords may not sound in tune when they are strummed, either due to string inharmonicity from worn or dirty strings, a misplaced fret, a mis-adjusted bridge, or other problems. Due to the range of factors in play, getting a guitar to sound in tune is an exercise in compromise. "Worn or dirty strings are also inharmonic and harder to tune", a problem that can be partially resolved by cleaning strings. Some performers choose to focus the tuning towards the key of the piece, so that the tonic and dominant chords will have a clear, resonant sound. However, since this compromise may lead to muddy-sounding chords in sections of a piece that stray from the main key (e.g., a bridge section that modulates a semitone down), some performers choose to make a broader compromise, and "split the difference" so that all chords will sound acceptable.


Mode-locking

Other stringed instruments such as the violin, viola, cello, and double bass also exhibit inharmonicity when notes are plucked using the
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 bowed ...
technique. However, this inharmonicity disappears when the strings are bowed, because the bow's stick-slip action is periodic,Neville H. Fletcher (1994)
"Nonlinear Dynamics and Chaos in Musical Instruments"
. Complexity International.
driving all of the resonances of the string at exactly harmonic ratios even if it has to drive them slightly off their natural frequency. As a result, the operating mode of a bowed string playing a steady note is a compromise among the tunings of all of the (slightly inharmonic) string resonances, which is due to the strong non-linearity of the stick-slip action. Mode locking also occurs in the
human voice The human voice consists of sound made by a human being using the vocal tract, including talking, singing, laughing, crying, screaming, shouting, humming or yelling. The human voice frequency is specifically a part of human sound production ...
and in
reed instrument Reed aerophones is one of the categories of musical instruments found in the Hornbostel-Sachs system of musical instrument classification. In order to produce sound with these Aerophones the player's breath is directed against a lamella or pair of ...
s such as 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 pitches ...
.


List of instruments

; Perfectly harmonic :*
Bowed string instrument Bowed string instruments are a subcategory of string instruments that are played by a bow rubbing the strings. The bow rubbing the string causes vibration which the instrument emits as sound. Despite the numerous specialist studies devoted to t ...
s (
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 regular ...
,
cello The cello ( ; plural ''celli'' or ''cellos'') or violoncello ( ; ) is a Bow (music), bowed (sometimes pizzicato, plucked and occasionally col legno, hit) string instrument of the violin family. Its four strings are usually intonation (music), t ...
,
erhu The ''erhu'' (; ) is a Chinese two-stringed bowed musical instrument, more specifically a spike fiddle, which may also be called a ''Southern Fiddle'', and is sometimes known in the Western world as the ''Chinese violin'' or a ''Chinese two-s ...
, ...) :*
Brass instrument A brass instrument is a musical instrument that produces sound by sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips. Brass instruments are also called labrosones or labrophones, from Latin a ...
s (
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 ...
,
horn Horn most often refers to: *Horn (acoustic), a conical or bell shaped aperture used to guide sound ** Horn (instrument), collective name for tube-shaped wind musical instruments *Horn (anatomy), a pointed, bony projection on the head of various ...
,
trombone The trombone (german: Posaune, Italian, French: ''trombone'') is a musical instrument in the Brass instrument, brass family. As with all brass instruments, sound is produced when the player's vibrating lips cause the Standing wave, air column ...
, ...) :*
Reed aerophone Reed aerophones is one of the categories of musical instruments found in the Hornbostel-Sachs system of musical instrument classification. In order to produce sound with these Aerophones the player's breath is directed against a lamella or pair of ...
s (
oboe The oboe ( ) is a type of double reed woodwind instrument. Oboes are usually made of wood, but may also be made of synthetic materials, such as plastic, resin, or hybrid composites. The most common oboe plays in the treble or soprano range. A ...
,
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 pitches ...
, ...) ; Nearly harmonic :*
Plucked string instrument Plucked string instruments are a subcategory of string instruments that are played by plucking the strings. Plucking is a way of pulling and releasing the string in such a way as to give it an impulse that causes the string to vibrate. Plucki ...
s (
guitar The guitar is a fretted musical instrument that typically has six strings. It is usually held flat against the player's body and played by strumming or plucking the strings with the dominant hand, while simultaneously pressing selected stri ...
,
harpsichord A harpsichord ( it, clavicembalo; french: clavecin; german: Cembalo; es, clavecín; pt, cravo; nl, klavecimbel; pl, klawesyn) is a musical instrument played by means of a keyboard. This activates a row of levers that turn a trigger mechanism ...
,
harp The harp is a stringed musical instrument that has a number of individual strings running at an angle to its soundboard; the strings are plucked with the fingers. Harps can be made and played in various ways, standing or sitting, and in orche ...
...) ; Approximately harmonic :*
Tuned percussion A pitched percussion instrument (also known as a melodic or tuned percussion instrument) is a percussion instrument used to produce note (music), musical notes of one or more pitch (music), pitches, as opposed to an unpitched percussion instrume ...
; Not harmonic :*
Untuned percussion An unpitched percussion instrument is a percussion instrument played in such a way as to produce sounds of indeterminate pitch, or an instrument normally played in this fashion. Unpitched percussion is typically used to maintain a rhythm or to ...


See also

*
Anharmonicity In classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmo ...
*
Pseudo-octave A pseudo-octave, pseudooctave,"Interview with Max Mathews", p.21. Author(s): C. Roads and Max Mathews. Source: ''Computer Music Journal'', Vol. 4, No. 4, (Winter, 1980), pp. 15–22. Published by: The MIT Press. or paradoxical octave"The Parad ...
*
Subharmonic 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 must ...


References


Further reading

* B. C. J. Moore, R.W. Peters, and B. C. Glasberg, “Thresholds for the detection of inharmonicity in complex tones,” ''Journal of the Acoust. Soc. Am.,'' vol. 77, no. 5, pp. 1861–1867, 1985. * F. Scalcon, D. Rocchesso, and G. Borin, “Subjective evaluation of the inharmonicity of synthetic piano tones,” in ''Proc. Int. Comp. Music Conf.'' ICMC’98, pp. 53–56, 1998. * A. Galembo and L. Cuddy, “String inharmonicity and the timbral quality of piano bass tones: Fletcher, Blackham, and Stratton (1962) revisited.” ''Proceedings of the Society for Music Perception and Cognition'', MIT, Cambridge, Massachusetts, July - August 1997. {{Acoustics Acoustics Musical tuning