
Brazilian guitarist Badi Assad
The guitar is the most common stringed instrument, and shares
many characteristics
with other stringed instruments. For example, the
overtones potentially available on
any stringed instrument are the same. Why, then, does a guitar sound so much
different from, say, a violin? The answer lies in which overtones are
emphasized in a particular instrument, due to the shape and materials
in the resonator (body), strings, how it's played, and other
factors. In the course of studying the overtones, or harmonics
of a string fixed at both ends, we will uncover the
overtone series for strings, which is the basis of
Western harmony.
Waves on a String
A guitar string is a common example of a string fixed at both ends
which is elastic and can vibrate. The vibrations of such a
string are called standing waves, and they satisfy the
relationship between wavelength and frequency that comes from
the definition of waves:
v = f ,
where
v is the speed of the wave, f is the
frequency (measured in cycles/second or Hertz, Hz) and
is the wavelength.
The speed v of waves on a string depends on the
string tension T and linear mass density (mass/length)
µ, measured in kg/m. Waves travel
faster on a tighter string and the frequency is therefore higher
for a given wavelength. On the other hand, waves travel
slower on a more massive string and the frequency is therefore
lower for a given wavelength. The relationship between speed,
tension and mass density is a bit difficult to derive, but
is a simple formula:
v = T/µ
Since the fundamental wavelength of a standing wave on a
guitar string is twice the distance between the bridge and the
fret, all six strings use the same range of wavelengths. To
have different pitches (frequencies) of the strings, then, one must
have different wave speeds. There are two ways to do
this: by having different tension T or by having
different mass density µ (or a combination of the two).
If one varied pitch only by varying tension, the high strings
would be very tight and the low strings would be very loose
and it would be very difficult to play. It is much easier to
play a guitar if the strings all have roughly the same
tension; for this reason, the lower strings have higher mass
density, by making them thicker and, for the 3 low strings,
wrapping them with wire. From what you have learned so far,
and the fact that the strings are a perfect fourth apart in
pitch (except between the G and B strings in standard tuning),
you can calculate
how much µ increases between strings for T to
be constant.
String Harmonics (Overtones)
If a guitar string had only a single frequency vibration on it, it
would sound a bit boring (you can listen to a single frequency
sound with the
Overtones Applet).
What makes a guitar or any stringed instrument interesting is
the rich variety of harmonics that are present. Any
wave that satisfies the condition that it has nodes at the
ends of the string can exist on a string. The
fundamental, the main pitch you hear, is the lowest
tone, and it comes from the string vibrating with one big arc
from bottom to top:
fundamental (l = /2)
The fundamental satisfies the condition
l = /2, where
l is the length of the freely vibrating portion of
the string. The first harmonic or overtone comes from
vibration with a node in the center:
1st overtone (l = 2 /2)
The 1st overtone satisfies the condition
l = . Each higher
overtone fits an additional half wavelength on the string:
2nd overtone (l = 3 /2)
3rd overtone (l = 4 /2)
4th overtone (l = 5 /2)
Since frequency is inversely proportional to wavelength, the
frequency difference between overtones is the fundamental
frequency. This leads to the overtone series for a string:
overtone | f/f0 | freq/tonic | approx interval |
fundamental | 1 | 1=1.0 | tonic |
1st | 2 | 1=1.0 | tonic |
2nd | 3 | 3/2=1.5 | perfect 5th |
3rd | 4 | 1=1.0 | tonic |
4th | 5 | 5/4=1.25 | major 3rd |
5th | 6 | 6/4=1.5 | perf 5th |
6th | 7 | 7/4=1.75 | dominant 7th |
7th | 8 | 1=1.0 | tonic |
8th | 9 | 9/8=1.125 | major 2nd |
9th | 10 | 10/8=1.25 | major 3rd |
10th | 11 | 11/8=1.375 | between 4th and dim 5th |
11th | 12 | 12/8=1.5 | perfect 5th |
12th | 13 | 13/8=1.625 | between aug 5th and maj 6th |
Most of the first 12 overtones fall very close to tones of the Western
musical scale, and one can argue that this is not coincidence:
it is natural to use a musical scale which incorporates the
overtones of stringed instruments.
The equal-tempered scale has 12 intervals (half-steps) making
up an octave (factor of two). The ratio, r, of
frequencies for a half-step therefore satisfies
r12=2, which means r=1.0595.
The scale, notated with interval names,
then corresponds to frequency multiples of:
tonic maj2nd maj3rd 4th 5th maj6th maj7th octave
1.000 1.1225 1.2599 1.3348 1.4983 1.6818 1.8877 2.0000
min2nd min3rd dim5th aug5th dom7th
1.0595 1.1892 1.4142 1.5874 1.7818
The top row shows the intervals of the major scale. The
equal-tempered scale and overtone series don't match
perfectly, of course, but the difference between, say, a major
3rd of the equal-tempered scale (1.2599) and the 4th overtone
(1.2500) is pretty hard to hear.
In fact, I often tune my guitar
using harmonics. I strike a B at the 7th fret (2nd overtone)
of the bottom E string to tune the B string. This means that
my B string is at a pitch of 1.500 above E rather than the
equal-tempered value of 1.4983, ie. the B string is slightly
sharp. I tune the A string by striking at the 5th fret of E
(3rd overtone) to get an E which matches the E I make on the A
string by striking the 2nd overtone at the 7th fret. This
means my A string is 4/3=1.3333 above E rather than 1.3348 of
the equal-tempered scale, ie. it's slightly flat. Then I do
the same match to get the D from the A string, which means my
D is 4/3 above A or 16/9=1.7778 above E rather than 1.7818 of
the equal-tempered scale, ie. it's even more flat. That
leaves the G string, which becomes a problem. The B string
above it is sharp and the D string below it is flat, so
there's a mismatch: if I tune the G string from the D string,
it is really too flat, and if I tune the G string from the B
string it is sharp.
Alternatively, I could use the 4th overtone of E, which is
hard to make loud, to get a G at 5/4=1.2500, which is slightly
below
the equal-tempered value of 1.2599. This deviation from
equal-tempering when you use harmonics to tune is a pain,
and sometimes you're better off just matching the next string up to
that note on the lower string since the frets are spaced to
produce the equal-tempered scale.
Guitar Overtones
The thing that makes a guitar note "guitarry" is the overtone
content and how the note rises and decays in time. This
varies with how you play it, such as with a
pick vs. a finger, or near the bridge vs. in the
middle. (This, of course, isn't counting all the electronic
methods for emphasizing different overtones such as the
bass/treble control on electric guitars.)
As an example, I sampled the A string on my nylon-string
guitar played two different ways: plucking in the middle,
which emphasizes the fundamental and odd-multiple overtones
which have a peak in the center of the string, and plucking
near the bridge, which produces more of the even-multiple
overtones with nodes in the center of the string to make a
more "twangy" sound. Here are sample waveforms taken about a
half second after the string was struck:

This figure above shows the waveform when the string is plucked in
the center. The fundamental is at A 110 Hz and is very large.
Note that the
odd-multiple overtones (330 Hz, 550 Hz, etc.) are much larger
than the even-multiple ones (220 Hz, 440 Hz, etc.). This is
very characteristic of a nylon string guitar played this way,
which is pretty far from "twangy". The waveform is
close to a triangle wave, which results when only the
odd-multiple overtones are present.

This is the same note plucked near the bridge to make a
"twangy" sound. The overtone content is much richer, with
plenty of even-multiple overtones present. Note also that the
waveform amplitude is smaller. A "twangy" note dies out much
more quickly than a note with a strong fundamental like the
previous one.
You can hear these two sounds in the
Guitar A String applet.
Fret Spacing
You've probably noticed that the frets on a guitar get closer
together towards the bridge. From the the fact that each
successive note is r=1.0595 higher in pitch, and
the fact that
v=f =constant on a
given string, we can figure out the fret spacing. Let's say
the open string length is l. Then the first fret
must be placed a distance l/1.0595 from the bridge,
the second fret a distance l/1.0595² from the
bridge, and so on. The twelfth fret, which makes an octave,
is at a distance l/1.059512=l/2
from the bridge. The diagram below shows the fret positions
(as does the photo at the top of this page for that matter!).

Equations
- wave velocity, frequency, wavelength:
v = f
- standing waves on a string of length l:
l = n
/2
(n is a positive integer)
- half-step frequency ratio in equal-tempered scale:
r12 = 2 -> r = 1.0595
Applets
Summary
- A guitar string sound consists of standing waves: the fundamental
and overtones. The fundamental wavelengh is twice the
length of the vibrating part of the string.
- The Western musical scale is based on the overtone series for a
string: all the overtones up to the 9th are close to notes
of the equal-tempered scale (and define the notes of the
perfect-tempered scale).
- The timber of a stringed instrument depends on the overtone
content of the sound: a "twangy" sound has both odd and
even multiples of the fundamental, while a "smooth" sound
tends to have only odd multiples.
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