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Harmonics

Introduction

- Harmonic is obtained by a method of playing

stringed instruments - playing the instrument

while holding a finger against the string, but

very lightly so that it doesn't touch the

fretboard. Another meaning used has to do with

the relationship between frequencies. Although

these meanings are related, try to be sensitive

to these different definitions.

Physics of the Plucked String

- Pitch
- The pitch of a plucked string is dependent on

three factors the length of the string, the

tension in the string, and the mass density of

the string (i.e. the mass divided by the string

length). If you increase the tension (like when

bending a string) or make the string shorter, the

pitch increases. The frequency also goes up as

the mass density decreases. This last property is

why guitar strings get thinner as you move

towards the higher strings, and why lighter gauge

strings are 'slinky' and easier to bend.

Wave Theory

- There is an inverse relationship between

frequency and wavelength. As frequencies

increase, the wavelength becomes shorter. But if

you double the wavelength, the frequency would be

halved. - The simplest wave is a pure tone
- Any waveform can be broken up in to a series of

sinusoids of different frequencies - If we view the shape of a string just before it

is released as a waveform, we can then consider a

single sinusoidal component at a time. In a

vibrating string, each sinusoidal waveform

defines the boundaries that the string moves

between. Places where the string does not vibrate

are called nodes or zeros.

- The reflected wave is again inverted, so, after

the reflection, we have two waves (with the same

frequency and amplitude) travelling in opposite

directions. At the fixed end they add to give no

motion - zero displacement after all it is this

condition of immobility which causes the inverted

reflection. But if you look at the solid line on

the diagram (representing the sum of the two

waves) you'll see that there are other points

where the string never moves. They occur half a

wavelength apart. These motionless points are

called nodes of the vibration, and they play an

important role in all of the instrument families.

Halfway between the nodes are antinodes points

of maximum motion. But note that these peaks are

not travelling along the string the combination

of two waves travelling in opposite directions

produces a standing wave - This is shown in the figure, which represents a

time sequence (time increases from top to

bottom). Think of it as representing a series of

snap shots of the waves. The blue wave is

travelling to the right, the green to the left.

The red line is their sum the red wave is what

happens when the two travelling waves add

together (superpose is the technical term). Note

the positions (nodes) where the two travelling

waves always cancel out, and the others

(anitnodes) where they add to give an oscillation

with maximum amplitude.

Natural Frequencies

- A frequency is called a natural frequency if it

can exist without any driving source. With our

string, there is a physical interpretation of

natural frequencies. A frequency is a natural

frequency if its waveform has nodes that match up

with the ends of the string. When you pluck a

string, you are causing it vibrate at these

natural frequencies. The lowest frequency at

which this occurs is the fundamental - the pitch

you hear.

The Harmonic Series

- The harmonic series is a mathematical definition,

generally used when talking about frequencies.

The harmonic series is important in musical

applications because most instruments (including

guitar) produce sounds that contain harmonic

frequencies. The natural frequencies of the

string mentioned above form a harmonic series.

Harmonics and modes of vibration

- The string on a musical instrument is (almost)

fixed at both ends, so any vibration of the

string must have nodes at each end. Now that

limits the possible vibrations. For instance the

string with length L could have a standing wave

with wavelength twice as long as the string

(wavelength 2L) as shown in the first sketch in

the next series. This gives a node at either end

and an antinode in the middle.

- Let's work out the relationships among the

frequencies of these modes. For a wave, the

frequency is the ratio of the speed of the wave

to the length of the wave f v/wavelength.

Compared to the string length L, you can see that

these waves have lengths 2L, L, 2L/3, L/2. We

could write this as 2L/n, where n is the number

of the harmonic. - The fundamental or first mode has frequency f1

v/wavelength v/2L,The second harmonic has

frequency f2 v/l2 2v/2L 2f1The third

harmonic has frequency f3 v/l3 3v/2L

3f1,The fourth harmonic has frequency f4 v/l4

4v/2L 4f1, and, to generalise, - The nth harmonic has frequency fn v/ln nv/2L

nf1.

- All waves in a string travel with the same speed,

so these waves with different wavelengths have

different frequencies as shown. The mode with the

lowest frequency (f1) is called the fundamental.

Note that the nth mode has frequency n times that

of the fundamental. All of the modes (and the

sounds they produce) are called the harmonics of

the string. The frequencies f, 2f, 3f, 4f etc are

called the harmonic series. This series will be

familiar to most musicians, particularly to

buglers and players of natural horns. If for

example the fundamental is the note C3 or viola C

(a frequency of 131 Hz), then the harmonics would

have the pitches shown in the next figure. These

pitches have been approximated to the nearest

quarter tone. The octaves are exactly octaves,

but all other intervals are slightly different

from the intervals in the equal tempered scale. - The figure shows the musical notation for the

first twelve harmonics on a C string. When you

play the sound file, listen carefully to the

pitch. The seventh and eleventh harmonics fall

almost halfway between notes on the equal

tempered scale, and so have been notated with

half sharps.

- On a guitar tuned in the usual way, the B string

and high E string are approximately tuned to the

3rd and 4th harmonics of the low E string. If you

pluck the low E string anywhere except one third

of the way along, the B string should start to

vibrate, driven by the vibrations in the bridge

from the harmonic of the first string. If you

pluck the low E string anywhere except one

quarter of the way along, the top E string should

be driven similarly. Guitarists often begin to

tune-up in the following way first tune the 4th

harmonic of the low E string, the 3rd of the A

string and the top E all to the same note then

tune theB string (B3) to the 3rd hamonic of the

first (E2) then tune the 4th harmonic of the A

string to the 3rd of the D string. This method

cannot be extended succesfully to the G string

because it is usually too thick and stiff, so it

is better tuned by octaves, using the frets. For

several reasons (see the notes at the end of this

page), this method of tuning is only approximate,

and one needs to retune the octaves afterwards.

The best tuning is usually a compromise that must

be made after considering what chords you will be

playing and where you are playing on the

fingerboard.

Calculating Pitches

- Perhaps the way to figure out what pitches the

harmonics are is to use a tuner. Begin with the

pitch of the open string, and consider it as the

base of the harmonic series (the fundamental).

Playing a harmonic at the twelfth fret generates

a pitch equivalent to the second harmonic in the

open string, which is an octave above. The next

harmonic, at the seventh fret, produces a pitch

equal to the third harmonic in the open string -

an octave and a fifth above the fundamental in

the open string. A harmonic at the 5th fret is

the fourth harmonic, two octaves above the open

string. Using this process to select one of the

harmonics in the open string may be why the

playing technique is called a harmonic.

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