When you tune your violin to another instrument, or when you tune your
violin strings to each other, you can rely on the interaction,
favorably or otherwise, between the fundamental and harmonics of the
two sound sources that you are tuning to each other. If t the
frequencies of fundamental or harmonics don't match when they should,
you may well hear a strong clash that you can solve by adjusting
the string that you are tuning.
Here we explain the cause of these harmonic clashes. A string can
have resonate at more than one frequency at a time. The end
points do not move sideways, but other parts of the string will.
Pure sine waves are the building blocks for all these sounds, and each
of these harmonics produced by itself would be a pure sine wave.
But a
note played on the violin, or most other instruments,. is not a pure
sine wave. It contains harmonics (sine waves whose frequency is
some
integer multiple of the fundamental) and other impurities.
The stirng's fundamental (slowest) vibration has no sideways motion a
the ends but one spot in the middle with maximum sideways motion.
The string's second harmonic (twice the frequency of the fundamental)
has no motion at the ends and none in the middle, but maximum
displacement at 1/4 and 3/4 along the string.
The string's third harmonic (three times the frequency of the
fundamental) has no motion at the ends and no motion at points 1/3 and
2/3 along the string, but maximum displacement at points 1/6, 1/2, and
5/6 along the string.
The string's fourth harmonic (four times the frequency of the
fundamental) has no motion at the ends and no motion at points 1/4, 1/2
and 3/4 along the string, but maximum displacement at points 1/8, 3/8,
5/8 and 7/8 along the string.
The string's fifth harmonic (five times the frequency of the
fundamental) has no motion at the end points and no motion at points
1/5, 2/5, 3/5, and 4/5 along the string, but maximum displacement at
points 1/10, 3/10, 1/2, 7/10, and 9/10 along the string.
Below is a series of waveform displays made by recording audio signals
with Cakewalk pro audio 9, then displaying their waveforms with the
Waveform editor in Cakewalk.
Take a look at this first example, a display of an open G bowed on a
steel string (top) and of a sine wave G from a Hewlett Packard signal
generator,.
Note the purity of the sine wave compared to the bowed G. If the
bowed G were producing a waveform with no harmonics, it would look like
the sine wave.
Now check out this second waveform editor display. It's the same
G comparison, but after the bow comes up off the string and the steel G
is still resonating on its own.
The G from the resonating string has many fewer impurities. The
bow hair rubbing on the string must have caused those impurities.
Now, there are only a few deviations from a sine wave, and I contend
those are some of the harmonics (sine waves themselves) of the
resonating string added to the sine wave fundamental of the
string. The trig course we had in college would bear me out, if I
could remember enough of it.
Now, we use a technique to play harmonics on the string, well known to
guitar players and fiddle players. With this technique, you lay
your finger on the string, without pressing down on the string, in
certain places., to block out one or more of the possible harmonics
(and fundamental) that the string will ordinarily produce. In the
chart below, we show with an 'X' which harmonics will still sound when
you lay your finger at various points along the string.
F
2nd
3rd
4th
5th
6th
8th
10th
1/2
x
x
x
x
x
1/3,2/3
x
x
1/4,3/4
x
x
1/5,2/5,3/5,4/5
x
x
This chart is for transverse vibration of a string. The fundamental has
half a wavelength vibration in the string, such that the greatest
displacement is in the middle of the string. Putting your finger
there blocks the fundamental. The second harmonic has no
displacement in the middle but greatest displacement at 1/4 and 3/4
along the string, so your finger at the halfway point does not affect
the second harmonic.... or the fourth, or the sixth or the eighth, that
also have zero displacement in the middle of the string. Putting
your finger at the 1/3 or 2/3 point allows any harmonic to still
sound that has no displacement at those points. That would be the
third and sixth harmonics. This is how you can generate this
chart.
So, now looking at the waveform produced with your finger at the
halfway point (and after you stop bowing) there is a much simpler
waveform compared to an open G, after bowing.
The string is adding together all the even-numbered harmonics,
but omitting the fundamental, 3rd, 5th, 7th, etc. This waveform
is much simpler than before. Also, note how there are two
waveforms in the upper graph per every one in the lower graph.
Placing your finger at the 1/3 or 2/3 point along the string produces
this waveform, compared with a sine wave.
This upper waveform would contain the 3rd harmonic, 6th, and 9th I
assume. But the higher you go, the weaker the harmonics
become. Also note that there are three waveforms in the upper
graph for every waveform in the lower graph.
This chart shows a waveform produced by placing your finger at the 1/4
or 3/4 points along the string.
This upper waveform is looking fairly similar to a sine wave. It
has primarily the fourth harmonic and the eighth harmonic, which is a
simpler combination of overtones. And, note how there are four
waveforms in the upper graph per waveform in the lower graph.
Here is one final waveform, produced by placing your finger at the 1/5,
2/5, 3/5 or 4/5 positions on the string, compared to a sine wave.
This upper waveform contains mainly the fifth harmonic plus the tenth
harmonic. And here, there are five (jaggy) waveforms in the upper
graph per wave form in the lower graph.
Comparing the waveforms for fundamental and second harmonic, note how
the latter was simpler than the former. If you accept that a
string free to vibrate produces fundamental, 2nd, 3rd, 4th, 5th, sixth,
(7th), and 8th harmonics, and that the halfway point dampens 3rd, 5th
and 7th but passes 2nd, 4th 6th and 8th, then the
difference between the two waveforms is thae absence of the
fundamental, 3rd, 5th and 7th in the second waveform. So
basically we've removed all the odd-numbered harmonics in the second
waveform.
Here is another comparison. Comparing the waveforms for second
and fourth harmonics, note how the latter was simpler than fhr former.
If you accept that the halfway point dampens fundamental, 3rd, 5th, and
7th, but passes 2nd, 4th and 8th harmonics, and that the 1/4 and 3/4
points will dampen fundamental, 2nd, 3rd, 5th and 7th, then the
difference between the two waveforms is the absense of the second
harmonic in the second chart.
This is just an attempt to demonstrate that a string left to its own
devices will generate harmonics whose frequency is some integer
multiple of the fundamental.
If you can accept that concept, then you are ready to see how you can
use these harmonics to help tune an instrument. But first you
need to understand intervals and their frequency ratios, so you can
determine which intervals would have harmonics that should match up.
