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For the final part of the experiment, take a bow and bow the non-harmonic
string. If the mass is not too large, and particularly if you have good
bow control, you will be able to produce a good musical note with a clear
pitch. Why so? In the section Bows
and Strings, we explain how the stick-slip action of the bow on the
string produces a periodic vibration — that is a vibration that repeats
itself exactly each cycle, and so has a spectrum of exact harmonics.
The sum of harmonic vibrations is a periodic vibration. This is one half of Fourier's theorem, and is easy to see. Let the fundamental frequency f have a period T. The second harmonic with frequency 2f has a period of T/2, so, after one vibration of the fundamental (after time T), the second harmonic has had exactly two vibrations, so the two waves are ready to start again with exactly the same relative position to each other, so they will produce the same combination that they did for the first cycle of the fundamental. The same is true for each harmonic nf, where n is a whole number. After time T, exactly n cycles of the nth harmonic have passed, and so all the harmonics are ready to start again for a new cycle. This is explained in more detail, and with diagrams, in What is a Sound Spectrum? The harmonic series is special because any combination of its vibrations produces a periodic or repeated vibration at the fundamental frequency f. Now the converse is also true: a periodic vibration has a harmonic spectrum. This is the other half of Fourier's theorem, but it is harder to show. The resonances of strings and pipes are not inherently harmonic. An ideal, homogeneous, infinitely thin or infinitely flexible string has exactly harmonic modes of vibration. So does an ideal, homogeneous, infinitely thin pipe. Real strings and pipes do not. We saw in the experiment that adding a mass - making the string inhomogeneous - makes the string inharmonic. (By the way, worn or dirty strings are also inharmonic and harder to tune. Washing them can help.) For wind instruments, variations in bore diameter, the presence of tone holes and other perturbations mean that their resonances are inharmonic as well. Real strings also have inharmonic resonances because they are not infinitely thin or flexible, and so do not bend perfectly easily at the bridge and the nut. This bending stiffness affects the higher modes more than the lower, so the harmonics are stretched. Solid strings are worse than wound strings, steel strings are worse than others, pianos - especially little pianos - are worse than harps. The inharmonicity disappears when the strings are bowed, but is more noticeable when they are plucked or struck. Because the bow's stick-slip action is periodic, 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. Thus 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. (For the technically minded, this phenomenon is due to the strong non-linearity of the stick-slip action. It is called mode locking.) Real pipes have inharmonic resonances because of their finite diameter: the end effects are frequency dependent. The pipes of musical instruments are complicated by departures from cylindrical or conical shape (valves and tone holes). Some of these complications are there to improve the harmonicity, but the results are rarely perfect. For any one note, however, the lip or the reed performs the same (strongly non-linear) role as the bow: the lip or reed undergoes periodic vibration and so produces a harmonic spectrum. Again, the operating mode of a brass or woodwind instrument playing a steady* note is a compromise among the tunings of all of the (slightly inharmonic) pipe resonances (mode locking again.) * "steady" here means over a very long time. Measurements of frequency are ultimately limited by the Musician's Uncertainty Principle (which is almost the same as Heisenberg's Uncertainty Principle, see this explanation). If you play a note for m seconds, the frequency of its harmonics cannot be measured with an accuracy greater than about 1/m Hz. If your spectrum analyser measures over only k seconds, it cannot measure much more accurately than about 1/k Hz. One final remark: the sound spectra of clarinets tend to have strong odd harmonics (fundamental, 3rd, 5th etc) and weak even harmonics (2nd, 4th etc), in their lowest register (but not in high registers). This effect is discussed in "pipes and harmonics" and "flutes vs clarinets". Go back to Music Acoustics FAQ. |