Inharmonicity

In music, inharmonicity is the degree to which the frequencies of overtones (known as partials, partial tones, or harmonics) depart from whole multiples of the fundamental frequency. Acoustically, a note perceived to have a single distinct pitch in fact contains a variety of additional overtones. Many percussion instruments, such as cymbals, tam-tams, and chimes, create complex and inharmonic sounds. In stringed instruments such as the piano, the less elastic the strings are (that is, the shorter, thicker, and stiffer they are), the more inharmonicity they exhibit.

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. [http://www.phys.unsw.edu.au/jw/harmonics.html How harmonic are harmonics? by Joe Wolfe, accessed 29 June 2008] ] 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, when a string gets thick enough, compared to the length of the string, it begins to behave less as an ideal string and more like a cylinder (a tube of mass), which has natural resonances that are not whole number multiples of the fundamental frequency.

When a string is bowed or tone in a wind instrument initiated by vibrating reed or lips, a phenomenon called mode locking 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 than the natural resonance points of the instrument. For this reason, as single tone played by a bowed string instrument, brass instrument, or reed instrument does not exhibit inharmonicity.

However, when a string is struck or plucked, as with a piano string that is struck by a hammer or a guitar string that is plucked by a finger or plectrum, the string will exhibit inharmonicity. The inharmonicity of a string depends on its physical characteristics, such as tension, stiffness, 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 wrapped string generally exhibits less inharmonicity than the equivalent solid string.

Pianos

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. In 1962, Harvey Fletcher's research 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. [ [http://www.acoustics.org/press/134th/galembo.htm Acoustical Society of America - Large grand and small upright pianos ] ] . "Inharmonicity is not necessarily unpleasant. Fletcher, Blackham, and Stratton [1] pointed out that a slightly inharmonic spectrum added certain “warmth” into the sound. They found that synthesized piano tones sounded more natural when the partials below middle C were inharmonic." [http://www.acoustics.hut.fi/~hjarvela/publications/icmc99.pdf]

Pianos are tuned by ear by technicians called piano tuners who listen for the sound of "beating" when two notes are played together. 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. Piano strings are much heavier per unit length than the strings on a violin or guitar, and so piano strings must be wound to much greater tension to reach the same fundamental frequency range. Another factor that can cause problems is the presence of rust on the strings or dirt in the windings. [http://books.google.com/books?id=kEy1MRsnVHIC&pg=PA106&lpg=PA106&dq=inharmonicity&source=web&ots=1ERAuM9M-2&sig=rbIk1KfxFX21uMDLFzdhctwO6pYT] These elements can result in inharmonicity, which has the effect of slightly raising in frequency of the higher modes, which means that they cease to be exact integer multiples of the fundamental.

The harmonic series of strings does not fall exactly into whole-number multiples of a fundamental frequency, but rather each harmonic is slightly sharper than a whole-number ratio, and this sharpness increases as higher tones in the harmonic series are reached. This means that an aurally tuned octave will be a "stretched octave" which is slightly wider than the just 2:1 ratio. The amount of stretching depends on the style of piano and is determined mainly by the length of the strings. On a piano, the notes in the higher register will end up being tuned slightly sharper than those in the lower octave. This is less apparent on longer pianos which have proportionally thinner strings, because string inharmonicity is directly related to the ratio of string thickness to length. (for more information, "see Piano acoustics").

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" do. 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 tuners in the 1980s hit 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.

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.

Violin and other string family instruments

Other stringed instruments such as the violin, cello, and double bass also exhibit inharmonicity when notes are plucked using the pizzicato technique. However, the "inharmonicity disappears when the strings are bowed" because the "bow's stick-slip action is periodic, [so] it drives 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." [ [http://www.phys.unsw.edu.au/jw/harmonics.html How harmonic are harmonics? ] ] "Worn or dirty strings are also inharmonic and harder to tune", a problem that can be partially resolved by cleaning strings. [Ibid]

ee also

*Anharmonicity
*Electronic tuner
*Pseudo-octave
*Stretched tuning
*String resonance (music)

External links

* [http://www.ihear.com/Pitch/paradoxical.html Pitch Paradoxical]

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.” "Report to the 3rd US Conference on Music Perception and Cognition", MIT, Cambridge, MA, July - August 1997.

References