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UNIT-IV

HARMONICS

Fundamental Frequency and Harmonics

Each natural frequency that an object or instrument produces has its own characteristic vibrational mode

or standing wave pattern. These patterns are only created within the object or instrument at specific

frequencies of vibration; these frequencies are known as harmonic frequencies, or merely harmonics. At

any frequency other than a harmonic frequency, the resulting disturbance of the medium is irregular and

non-repeating. For musical instruments and other objects that vibrate in regular and periodic fashion, the

harmonic frequencies are related to each other by simple whole number ratios. This is part of the reason

why such instruments sound pleasant

A harmonic of a wave is a component frequency of the signal that is an integer multiple of the

fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f,

3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental

frequency, therefore the sum of harmonics is also periodic at that frequency. Harmonic

frequencies are equally spaced by the width of the fundamental frequency and can be found by

repeatedly adding that frequency. For example, if the fundamental frequency (first harmonic) is

25 Hz, the frequencies of the next harmonics are: 50 Hz (2nd harmonic), 75 Hz (3rd harmonic),

100 Hz (4th harmonic) etc.

Contents

1 Characteristics 2 Harmonics and overtones 3 Harmonics on stringed instruments

o 3.1 Table 4 Other information 5 See also 6 References 7 External links

Characteristics

Many oscillators, including the human voice, a bowed violin string, or a Cepheid variable star,

are more or less periodic, and so composed of harmonics.

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Most passive oscillators, such as a plucked guitar string or a struck drum head or struck bell,

naturally oscillate at not one, but several frequencies known as partials. When the oscillator is

long and thin, such as a guitar string, or the column of air in a trumpet, many of the partials are

integer multiples of the fundamental frequency; these are called harmonics. Sounds made by

long, thin oscillators are for the most part arranged harmonically, and these sounds are generally

considered to be musically pleasing. Partials whose frequencies are not integer multiples of the

fundamental are referred to as inharmonic. Instruments such as cymbals, pianos, and strings

plucked pizzicato create inharmonic sounds.[1][2]

The untrained human ear typically does not perceive harmonics as separate notes. Rather, a

musical note composed of many harmonically related frequencies is perceived as one sound, the

quality, or timbre of that sound being a result of the relative strengths of the individual harmonic

frequencies. Bells have more clearly perceptible inharmonics than most instruments. Antique

singing bowls are well known for their unique quality of producing multiple harmonic partials or

multiphonics.

Harmonics and overtones

The tight relation between overtones and harmonics in music often leads to their being used

synonymously in a strictly musical context, but they are counted differently leading to some

possible confusion. This chart demonstrates how they are counted:

Frequency Order Name 1 Name 2

1 · f = 440 Hz n = 1 fundamental tone 1st harmonic

2 · f = 880 Hz n = 2 1st overtone 2nd harmonic

3 · f = 1320 Hz n = 3 2nd overtone 3rd harmonic

4 · f = 1760 Hz n = 4 3rd overtone 4th harmonic

Harmonics are not overtones, when it comes to counting. Even numbered harmonics are odd

numbered overtones and vice versa.

In many musical instruments, it is possible to play the upper harmonics without the fundamental

note being present. In a simple case (e.g., recorder) this has the effect of making the note go up

in pitch by an octave; but in more complex cases many other pitch variations are obtained. In

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some cases it also changes the timbre of the note. This is part of the normal method of obtaining

higher notes in wind instruments, where it is called overblowing. The extended technique of

playing multiphonics also produces harmonics. On string instruments it is possible to produce

very pure sounding notes, called harmonics or flageolets by string players, which have an eerie

quality, as well as being high in pitch. Harmonics may be used to check at a unison the tuning of

strings that are not tuned to the unison. For example, lightly fingering the node found halfway

down the highest string of a cello produces the same pitch as lightly fingering the node 1/3 of the

way down the second highest string. For the human voice see Overtone singing, which uses

harmonics.

While it is true that electronically produced periodic tones (e.g. square waves or other non-

sinusoidal waves) have "harmonics" that are whole number multiples of the fundamental

frequency, practical instruments do not all have this characteristic. For example higher

"harmonics"' of piano notes are not true harmonics but are "overtones" and can be very sharp, i.e.

a higher frequency than given by a pure harmonic series. This is especially true of instruments

other than stringed or brass/woodwind ones, e.g., xylophone, drums, bells etc., where not all the

overtones have a simple whole number ratio with the fundamental frequency.

The fundamental frequency is the reciprocal of the period of the periodic phenomenon.

This article incorporates public domain material from the General Services Administration

document "Federal Standard 1037C".

Harmonics on stringed instruments

Playing a harmonic on a string

The following table displays the stop points on a stringed instrument, such as the guitar (guitar

harmonics), at which gentle touching of a string will force it into a harmonic mode when

vibrated. String harmonics (flageolet tones) are described as having a "flutelike, silvery quality

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that can be highly effective as a special color" when used and heard in orchestration.[3] It is

unusual to encounter natural harmonics higher than the fifth partial on any stringed instrument

except the double bass, on account of its much longer strings.[4]

Harmonic Stop note

Sounded note relative to

open string

Cents

above

open

string

Cents

reduced

to one

octave

Audio

2 octave octave (P8) 1,200.0 0.0 Play (help·info)

3 just perfect fifth P8 + just perfect fifth (P5) 1,902.0 702.0 Play (help·info)

4 second octave 2P8 2,400.0 0.0 Play (help·info)

5 just major third 2P8 + just major third (M3) 2,786.3 386.3 Play (help·info)

6 just minor third 2P8 + P5 3,102.0 702.0

7 septimal minor third 2P8 + septimal minor

seventh (m7) 3,368.8 968.8 Play (help·info)

8 septimal major second 3P8 3,600.0 0.0

9 Pythagorean major second 3P8 + Pythagorean major

second (M2) 3,803.9 203.9 Play (help·info)

10 just minor whole tone 3P8 + just M3 3,986.3 386.3

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11 greater unidecimal neutral

second

3P8 + lesser undecimal

tritone 4,151.3 551.3 Play (help·info)

12 lesser unidecimal neutral

second 3P8 + P5 4,302.0 702.0

13 tridecimal 2/3-tone 3P8 + tridecimal neutral sixth

(n6) 4,440.5 840.5 Play (help·info)

14 2/3-tone 3P8 + P5 + septimal minor

third (m3) 4,568.8 968.8

15 septimal (or major) diatonic

semitone

3P8 + just major seventh

(M7) 4,688.3 1,088.3 Play (help·info)

16 just (or minor) diatonic

semitone 4P8 4,800.0 0.0

First, consider a guitar string vibrating at its natural frequency or harmonic frequency. Because

the ends of the string are attached and fixed in place to the guitar's structure (the bridge at one

end and the frets at the other), the ends of the string are unable to move. Subsequently, these

ends become nodes - points of no displacement. In between these two nodes at the end of the

string, there must be at least one antinode. The most fundamental harmonic for a guitar string is

the harmonic associated with a standing wave having only one antinode positioned between the

two nodes on the end of the string. This would be the harmonic with the

longest wavelength and the lowest frequency. The lowest frequency

produced by any particular instrument is known as the fundamental

frequency. The fundamental frequency is also called the first

harmonic of the instrument. The diagram at the right shows the first

harmonic of a guitar string. If you analyze the wave pattern in the guitar

string for this harmonic, you will notice that there is not quite one

complete wave within the pattern. A complete wave starts at the rest

position, rises to a crest, returns to rest, drops to a trough, and finally returns to the rest position

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before starting its next cycle. (Caution: the use of the words crest and trough to describe the

pattern are only used to help identify the length of a repeating wave cycle. A standing wave

pattern is not actually a wave, but rather a pattern of a wave. Thus, it does not consist of crests

and troughs, but rather nodes and antinodes. The pattern is the result of the interference of two

waves to produce these nodes and antinodes.) In this pattern, there is only one-half of a wave

within the length of the string. This is the case for the first harmonic or fundamental frequency of

a guitar string. The diagram below depicts this length-wavelength relationship for the

fundamental frequency of a guitar string.

After a discussion of the first three harmonics, a pattern can be recognized. Each harmonic

results in an additional node and antinode, and an additional half of a wave within the string. If

the number of waves in a string is known, then an equation relating the wavelength of the

standing wave pattern to the length of the string can be algebraically derived.

This information is summarized in the table below.

Harm.

#

# of

Waves

in String

# of

Nodes

# of

Anti-

nodes

Length-

Wavelength

Relationship

1 1/2 2 1 Wavelength = (2/1)*L

2 1 or 2/2 3 2 Wavelength = (2/2)*L

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3 3/2 4 3 Wavelength = (2/3)*L

4 2 or 4/2 5 4 Wavelength = (2/4)*L

5 5/2 6 5 Wavelength = (2/5)*L

The above discussion develops the mathematical relationship between the length of a guitar

string and the wavelength of the standing wave patterns for the various harmonics that could be

established within the string. Now these length-wavelength relationships will be used to develop

relationships for the ratio of the wavelengths and the ratio of the frequencies for the various

harmonics played by a string instrument (such as a guitar string).

Determining the Harmonic Frequencies

Consider an 80-cm long guitar string that has a fundamental frequency (1st harmonic) of 400 Hz.

For the first harmonic, the wavelength of the wave pattern would be two times the length of the

string (see table above); thus, the wavelength is 160 cm or 1.60 m. The speed of the standing

wave can now be determined from the wavelength and the frequency. The speed of the standing

wave is

speed = frequency • wavelength

speed = 400 Hz • 1.6 m

speed = 640 m/s

This speed of 640 m/s corresponds to the speed of any wave within the guitar string. Since the

speed of a wave is dependent upon the properties of the medium (and not upon the properties of

the wave), every wave will have the same speed in this string regardless of its frequency and its

wavelength. So the standing wave pattern associated with the second harmonic, third harmonic,

fourth harmonic, etc. will also have this speed of 640 m/s. A change in frequency or wavelength

will NOT cause a change in speed.

Using the table above, the wavelength of the second harmonic (denoted by the symbol 2) would

be 0.8 m (the same as the length of the string). The speed of the standing wave pattern (denoted

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by the symbol v) is still 640 m/s. Now the wave equation can be used to determine the frequency

of the second harmonic (denoted by the symbol f2).

speed = frequency • wavelength

frequency = speed/wavelength

f2 = v / 2

f2 = (640 m/s)/(0.8 m)

f2 = 800 Hz

This same process can be repeated for the third harmonic. Using the table above, the wavelength

of the third harmonic (denoted by the symbol 3) would be 0.533 m (two-thirds of the length of

the string). The speed of the standing wave pattern (denoted by the symbol v) is still 640 m/s.

Now the wave equation can be used to determine the frequency of the third harmonic (denoted

by the symbol f3).

speed = frequency • wavelength

frequency = speed/wavelength

f3 = v / 3

f3 = (640 m/s)/(0.533 m)

f3 = 1200 Hz

Now if you have been following along, you will have recognized a pattern. The frequency of the

second harmonic is two times the frequency of the first harmonic. The frequency of the third

harmonic is three times the frequency of the first harmonic. The frequency of the nth harmonic

(where n represents the harmonic # of any of the harmonics) is n times the frequency of the first

harmonic. In equation form, this can be written as

fn = n • f1

The inverse of this pattern exists for the wavelength values of the various harmonics. The

wavelength of the second harmonic is one-half (1/2) the wavelength of the first harmonic. The

wavelength of the third harmonic is one-third (1/3) the wavelength of the first harmonic. And the

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wavelength of the nth harmonic is one-nth (1/n) the wavelength of the first harmonic. In equation

form, this can be written as

n = (1/n) • 1

These relationships between wavelengths and frequencies of the various harmonics for a guitar

string are summarized in the table below.

Harm.

#

Freq.

(Hz)

Wavelength

(m)

Speed

(m/s) fn / f1 n / 1

1 400 1.60 640 1 1/1

2 800 0.800 640 2 1/2

3 1200 0.533 640 3 1/3

4 1600 0.400 640 4 1/4

5 2000 0.320 640 5 1/5

n n * 400 (2/n)*(0.800) 640 n 1/n

The table above demonstrates that the individual frequencies in the set of natural frequencies

produced by a guitar string are related to each other by whole number ratios. For instance, the

first and second harmonics have a 2:1 frequency ratio; the second and the third harmonics have a

3:2 frequency ratio; the third and the fourth harmonics have a 4:3 frequency ratio; and the fifth

and the fourth harmonic have a 5:4 frequency ratio. When the guitar is played, the string, sound

box and surrounding air vibrate at a set of frequencies to produce a wave with a mixture of

harmonics. The exact composition of that mixture determines the timbre or quality of sound that

is heard. If there is only a single harmonic sounding out in the mixture (in which case, it wouldn't

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be a mixture), then the sound is rather pure-sounding. On the other hand, if there are a variety of

frequencies sounding out in the mixture, then the timbre of the sound is rather rich in quality.

A Comparison of Passive Filters and Active Filters

The Advantages of Each Filter Type:

PASSIVE ACTIVE

no power supply

required

can handle large

currents and high

voltages

very reliable

least number of

components for given

filter

noise arises from

resistances only

no bandwidth limitation

no inductors

easier to design

high Zin, low Zout for minimal loading

can produce high gains

generally easier to tune

small in size and weight

The Disadvantages of Each Filter Type:

PASSIVE ACTIVE

inductors large for

lower frequencies

some inductors (non-

toroidal) may require

shielding

limited standard sizes,

often requiring variable

power supply required

susceptible to intermodulation,

oscillations

susceptible to parasitics from DC

output offset voltage and input bias

currents

op amp gain bandwidth constrained

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inductors and therefore

tuning

low tolerance inductors

(1-2%) very expensive

must be designed with

consideration to input

and output loading

generally not amenable

to miniaturization

no power gain possible

no voltage gain

op amp slew rate constrained

can require many components

Active and Passive filters – A Comparison:

The simplest approach to building a filter is with passive components (resistors, capacitors,

and inductors). In the R-F range it works quite well but with the lower frequencies, inductors

create problems. AF inductors are physically larger and heavier, and therefore expensive. For

lower frequencies the inductance is to be increased which needs more turns of wire. It adds to the

series resistance which degrades the inductor’s performance.

Input and output impedances of passive filters are both a problem, especially below RF. The

input impedance is low, that loads the source, and it varies with the frequency. The output

impedance is usually relatively high, which restricts the load impedance that the passive filter

can drive. There is no isolation between the load impedance and the passive filter. Thus the load

will have to be considered as a component of the filter and will have to be taken into

consideration while determining filter response or design. Any change in load impedance may

significantly alter one or more of the filter response characteristics.

An active filter uses an amplifier with R-C networks to overcome these problems of passive

filters. Originally built with vacuum tubes and then transistors, active filters now normally are

centered around op-amps. By enclosing a capacitor in a feedback loop, the inductor (with all its

low frequency problems) can be eliminated. By proper configuration input impedance can be

increased. The load is driven from the output of the op-amp, giving a very low output

impedance. Not only does this improve load drive capability, but the load is now isolated from

the frequency determining network. Thus variation in load will have no effect on the

characteristics of the active filter.

The amplifier allows us to specify and easily adjust passband gain, passband ripple, cutoff

frequency, and initial roll-off. Because of high input impedance of the op-amp, large value

resistors can be used and therefore size and cost of the capacitors used are reduced. By selecting

a quad op-amp IC, steep roll-off can be built in very little space and at very little cost.

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Active filters also have their limitations. High frequency response is limited by the gain

bandwidth (GBW) and slew-rate of the op-amps. High frequency op-amps are expensive,

making passive filters a more economical choice for RF applications. Active filter needs a power

supply. For op-amps this may be two supplies. Variations in the power supplies output voltage

may affect, to some extent, the signal output from the active filter. In multi-stage applications,

the common power supply provides a bus for high frequency signals. Feedback along the power

supply lines may cause oscillations unless decoupling techniques are rigorously applied. Active

devices, and therefore active filters, are much more susceptible to RF interference and ionization

than are passive R-L-C filters. Practical considerations limit the Q of the bandpass and notch

filters to less than 50. For circuits requiring very selective (narrow) filtering, a crystal filter,

because of its high Q value, will prove to be the best.

Although active filters are most widely employed in the field of communications and signal

processing, they are used in one form or another in almost all sophisticated electronic systems.

Radio, TV, telephone, RADAR, space-satellites, and biomedical equipment are but a few

systems that make use of active filters.