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 producea 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.
**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.
**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
stings 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.)
**Real strings** are also inharmonic 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** are inharmonic 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 Uncertainty Principle. 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 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), at least in their lowest register. This effect is discussed
in "pipes
and harmonics" and "flutes
vs clarinets". |