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Nonlinearity and the Sounds of Musical Instruments

Neville Fletcher -
Research School of Physical Sciences and Engineering
Australian National University
Canberra 0200, Australia

Popular version of paper 2pMUa1
Presented Tuesday afternoon 14 May 1996 at 2PM
ASA May meeting, Indianapolis IN
Embargoed until 14 May 1996

Musical instruments are not as simple as they seem --- indeed complex nonlinear processes are essential to produce the clear singing sound of a violin or a clarinet, and the same processes produce the chaotic vibrations responsible for the shimmering crash of a cymbal or a Chinese gong.

In the sixth century BC, Pythagoras showed that musical notes sound pleasantly together only when they are produced by strings whose lengths are in the ratio of two whole numbers. This simple fact also applies to the sounds of single notes played on musical instruments, which sound well only if the frequencies of the overtones contained within that note are related by the same whole-number rule, leading to what is called a harmonic relationship. Tensioned strings and cylindrical pipes are of great importance in the making of musical instruments because simple theory shows that these both have overtones that are exact harmonics.

Unfortunately, real strings and real pipes do not behave in this ideal manner, but still the sounds that they produce are exactly harmonic. How does this come about? The answer is in the way violin strings are excited by the friction of the bow, clarinets by the air flow through the vibrating reed, and trumpets by the player's vibrating lips. All these driving mechanisms are inherently nonlinear, by which we mean that doubling the input does not simply double the output, but produces all sorts of distortion products as well. In a music amplifier such distortion would be most unpleasant but, paradoxically, in a musical instrument it couples together the slightly out-of-tune natural modes of the string or pipe to produce an exactly harmonic total sound. The same is true of the human singing voice or the songs of birds. The mathematics of this paper shows how this comes about.

In instruments such as bells, gongs, and cymbals, in contrast, nonlinearity affects the natural vibrations in quite a different way. Bells are nearly linear in behavior because of the thickness of the metal from which they are made, and their sound has a "clanging" quality because their overtones are not harmonic. In a cymbal, however, the shimmering "swish" is produced by chaotic vibrations brought on by nonlinearity in the thinner metal, and the stupendous sound of a large Chinese gong is dominated by nonlinear frequency-multiplication effects. These sounds can be measured and analyzed, and the same theory explains what is going on.

Without the dominant effects of nonlinearity, our musical sounds would be very different from those with which we are now familiar!

Some References

"The Physics of Musical Instruments" by N.H. Fletcher and T.D. Rossing (Springer-Verlag, New York 1992)

"Nonlinearity, chaos and the sound of shallow gongs" by K.A. Legge and N.H. Fletcher, Journal of the Acoustical Society of America 86, 2439-2443 (1989)

"Nonlinear theory of musical wind instruments" by N.H. Fletcher, Applied Acoustics 30, 85-115 (1990)

"Nonlinear dynamics and chaos in musical instruments" by N.H. Fletcher, in "Complex Systems: From Biology to Computation" ed D.G. Green and T. Bossomaier (IOS Press, Amsterdam, 1993) pp. 106-117

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