Inharmonicity

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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.

When a string gets thick enough, compared to the length of the string, it stops behaving as a string and starts acting more like a cylinder (a tube of mass), which have different harmonics than strings. On wind instruments, the harmonic overtones are even multiples of the main frequency. However, on stringed instruments the overtones are inharmonic, which is caused by a "fastening fault in the string endings"; the string endings are fastened at each end, which means that they cannot "vibrate all the way to its ends." As such, the "effective length of a string is shorter than its geometrical length," especially for shorter, stiffer strings.[1]

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[edit] 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. [2]. "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."[3]

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 under enormous tension compared with the strings on a violin or guitar, and as a result, piano strings are much harder and stiffer. Another factor that can cause problems is the presence of rust on the strings or dirt in the windings. [4]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).

[edit] Guitar

While piano tuning is normally done by trained technicians, 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." [5]

[edit] 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." [6] "Worn or dirty strings are also inharmonic and harder to tune", a problem that can be partially resolved by cleaning strings.[7]

[edit] See also

[edit] External links

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[edit] 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.

[edit] References

  1. ^ Definitioner
  2. ^ Acoustical Society of America - Large grand and small upright pianos
  3. ^ http://www.acoustics.hut.fi/~hjarvela/publications/icmc99.pdf
  4. ^ http://books.google.com/books?id=kEy1MRsnVHIC&pg=PA106&lpg=PA106&dq=inharmonicity&source=web&ots=1ERAuM9M-2&sig=rbIk1KfxFX21uMDLFzdhctwO6pYT
  5. ^ How to tune the guitar expertly by ear. by Brian Capleton http://www.amarilli.co.uk/guitar/howto.asp
  6. ^ How harmonic are harmonics?
  7. ^ Ibid