Download wave and sound: the guitar

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Using Everyday Examples in Engineering (E3)
Physics – Waves and Sound: The Guitar
Samuel Hokin, IMS LLC
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 = fλ
where v is the speed of the wave, f is the frequency (measured in cycles/second or Hertz,
Hz) and λ 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 = 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
This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 083306. Any
opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not
necessarily reflect the views of NSF.
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
(http://www.bsharp.org/physics/applets/overtones) 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 = λ/2)
The fundamental satisfies the condition l = λ/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 = 2λ/2)
The 1st overtone satisfies the condition l =λ. Each higher overtone fits an additional half
wavelength on the string:
2nd overtone (l = 3λ/2)
3rd overtone (l = 4λ/2)
4th overtone (l = 5λ/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
fundamental
1
1st
2
nd
2
3
3rd
4
th
4
5
th
5
6
6th
7
th
7
8
8th
9
th
9
10
th
10
11
11th
12
th
12
13
freq/tonic
1=1.0
1=1.0
3/2=1.5
1=1.0
5/4=1.25
6/4=1.5
7/4=1.75
1=1.0
9.8=1.125
10/8=1.25
11/8=1.375
12/8=1.5
13/8=1.625
approx interval
tonic
tonic
perfect 5th
tonic
major 3rd
perf 5th
dominant 7th
tonic
major 2nd
major 3rd
between 4th and dim 5th
perfect 5th
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
1.0595 1.1892
dim5th aug5th dom7th
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.
(http://www.bsharp.org/physics/applets/guitar)
Fret Spacing
You've probably noticed that the frets on a guitar get closer together towards the bridge.
From the fact that each successive note is r=1.0595 higher in pitch, and the fact that
v=fλ=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 = fλ
• standing waves on a string of length l: l = nλ/2 (n is a positive integer)
12
• half-step frequency ratio in equal-tempered scale: r = 2 -> r = 1.0595
Applets
• The Overtones Applet - http://www.bsharp.org/physics/applets/overtones
• The Guitar A Applet: http://www.bsharp.org/physics/applets/guitar
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.
© 2010 Samuel Hokin. All rights reserved. Copies may be downloaded from www.EngageEngineering.org. This material
may be reproduced for educational purposes.