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Introduction to
Harmonic Series' Principles
An harmonic series is
a mathematical series which corresponds to the order of numbers
(1,2,3,4,5,6,etc...), and also contains an orientation. The order of numbers
series can then be applied to a given base number to create an harmonic series
which references the base number. This process can be accomplished in two ways -
multiplying the base number by the order of numbers series or dividing the base
number by the order of numbers series. The orientation of an harmonic series is
one of two types - overtone or undertone. The mathematical process of
multiplying or dividing can be applied to both the overtone and the undertone
orientation but there is a catch. Both the overtone and undertone series always
exist simultaneously but must always be mathematically opposite for any specific
application. This means that for a given application, if the overtone series
process is multiplication, then the undertone series process must be division.
For a different application the mathematical processes might be reversed. The
following two statements are the only two options for these harmonic series.
If the overtone series uses multiplication then the undertone
series uses division.
If the overtone series uses division then the undertone series uses
multiplication.
The table below shows the process of applying an order of
numbers series to a base number of 100 by using both multiplication and
division.
Multiplication
Division
100 x 1 = 100
100 )  1 = 100
100 x 2 = 200
100 ) 2 = 50
100 x 3 = 300
100 ) 3 = 33.33
100 x 4 = 400
100 ) 4 = 25
100 x 5 = 500
100 ) 5 = 20
100 x 6 = 600
100 ) 6 = 16.67
Etc...
Multiplication series of 100 = 100, 200, 300, 400, 500, 600, ...
Division series of 100 =
100, 50, 33.33, 25, 20, 16.67, …
The concept of harmonic series' is
easily demonstrated by applying them to any simple sound generating structure,
such as stretched string or a pipe. For this presentation we are going to use a
monochord which has a single stretched string (Diagram 1).
Diagram 1
The monochord is an instrument or tuning instructional device which consists of
a sound box, a string stretched over the length of its body, and a movable
bridge. Once it is tuned and the tension is set, the property of string length
can be manipulated by adjusting the movable bridge. The movable bridge
effectively acts as a nodal end to shorten the string length. Along the body of
the sound box are length marks to show where to place the movable bridge in
order to provide effective string lengths of 1,
2, etc... The following examples use a
monochord with a string tuned to a frequency of 131 Hz.. Only the
first 6 harmonic frequencies will be shown.
Overtone Frequencies
Overtone frequencies are obtained by multiplying the base frequency 131 Hz. by
the order of numbers 1, 2, 3, 4, 5, 6,... The overtone string lengths which
correlate to these frequencies are obtained by dividing the entire string length
by the order of numbers 1, 2, 3, 4, 5, 6,... For the overtone series the base
frequency and the entire string length provide what is known as the fundamental
which is the starting or reference number for the harmonic series (number 1).
The calculations for an overtone series starting at 131 Hz. are shown in the
following table.
Frequencies
String Lengths
131 Hz. x 1 = 131
Entire String Length ) 1 = 1
131 Hz. x 2 = 262
Entire String Length ) 2 = 1
131 Hz. x 3 = 393
Entire String Length ) 3 = 2
131 Hz. x 4 = 524
Entire String Length ) 4 = 3
131 Hz. x 5 = 655
Entire String Length ) 5 = 4
131 Hz. x 6 = 786
Entire String Length ) 6 = 5
Etc...
The 2nd harmonic is twice
the frequency of the base frequency 131 Hz. To obtain the 2nd
harmonic frequency the string length must be divided in half by adjusting the
movable bridge to the middle of the string so that the length of string from the
bridge to the string end is 1 the entire
string length. By plucking this 1 string
length, a frequency sound of 262 Hz. will be produced which is twice the
frequency of the fundamental 131 Hz. The results of carrying out this process
for subsequent harmonics of the series is shown in the table above and in
Diagram 2.
When the bridge divides the string in
half , playing the string on either side of the bridge will result in a string
length which is double the fundamental frequency. For the 2
division of the string, the bridge can be placed on the right or left 2
of the string. The 3rd harmonic frequency is obtained by plucking the
2 string length and not the B
string length. (If someone wanted to sound the middle 2
of the string (also triple the fundamental frequency) it would require two
bridges to section it off). All subsequent divisions function like the 2
division. For this example the bridge is consistently placed on the divisions to
the right of the string center, and the string length being plucked is to the
right of the bridge. (Note - a mirror reversal of Diagram 2 would still provide
the same frequency results). [To play the strings of Diagram 2 an MP3
player is required. Click here to obtain a free MP3
player]
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The most important aspect of this
process is to understand that the fundamental frequency for this example is the 5
string length and not the entire string length. The entire string length is
actually the 6th harmonic position in the series. Because a string
longer than the entire string length is not obtainable for this example, the
undertone harmonic series is restricted to 6 frequencies or lengths.
For an undertone orientation,
successive string lengths are obtained by multiplying the fundamental string
length by the order of numbers. Undertone frequencies are obtained by dividing
the fundamental frequency by the order of numbers. If 5
length is the fundamental than 2 length is
twice the 5 fundamental length, 1
is three times the 5 fundamental length, etc.
The table below shows the mathematical derivation of the undertone string
lengths and their respective frequencies. Keep in mind that the undertone
process used the same entire string length frequency of 131 Hz.
String Length
Frequency
Fundamental = Entire String Length )
6 x 1 = 5
= 131 Hz. x 6 ) 1 = 786 Hz.
2nd harmonic = Entire String Length )
6 x 2 = 2
= 131 Hz. x 6 ) 2 = 393 Hz.
3rd harmonic = Entire String Length )
6 x 3 = 1
= 131 Hz. x 6 ) 3 = 262 Hz.
4th harmonic = Entire String Length )
6 x 4 = B
= 131 Hz. x 6 ) 4 = 197 Hz.
5th harmonic = Entire String Length )
6 x 5 = M
= 131 Hz. x 6 ) 5 = 157 Hz.
6th harmonic = Entire String Length )
6 x 6 = 1 = 131 Hz. x 6 )
6 = 131 Hz.
The principle point
to understand is that, on a single string, the undertone string division
series is not a mirror image, reverse direction, or direct mathematical
inversion of the overtone string division series. For example, it is obvious
that the 2nd harmonic length of the undertone series is 2
which is not the same as the 4 length which
is the 2nd position of a reversed overtone series. There are,
however, other cases in which these two harmonic series are mirror images,
reverse directions, and direction mathematical inversions of each other. These
cases occur when the overtone and undertone series share the same fundamental.
When do the two series share the same
fundamental? For the previous explanations of the overtone and undertone series
the string was physically divided by a manual process but this is not necessary
to generate the harmonic series on a single string. Objects which are capable of
producing harmonics will naturally generate both harmonic series when the object
is set into motion. For example, when a string is plucked the mechanics of the
string will naturally divide the string according to the overtone divisions
previously shown. The individual can visibly see the wave on the string move in
smaller and smaller divisions. However, the mechanics of the string do not
generate equal divisions of the string which was the manual process used to show
an undertone harmonic series.
So how does the mechanics of the
string naturally generate the undertone harmonic series? This goes back to the
fundamental problem posed by music theorists and physicists, that a string
cannot expand itself into larger and larger lengths. They are absolutely right
about the inability of a string to expand to larger lengths, but expanding to
larger lengths physically is not how the undertone series is exhibited by the
mechanics of a string. The three properties of a string which can be altered are
the length, the mass, and the tension. Since the length and the mass are
constant the only property capable of changing is the tension. Physicists have
also assumed that the tension remained constant because no one is increasing or
decreasing the string tension during the time the string is in vibration. The
failure of recognition is in the principle that when a string is pulled back and
released an applied tension is added to the string which has harmonic
properties. The undertone tension is the applied tension imposed when the string
is pulled out to start the string vibration. We know that the fundamental wave
moves through a constantly reducing amplitude based on harmonic reduction which
means that the applied tension of the string will also move through an harmonic
energy reduction which means that the resulting wave will move through the
undertone harmonic series.
The only text in the world which explains in detail, and shows the proof, for the
natural occurrence of the undertone series on a string is the
science essay in this site). The reader must understand that this knowledge is
not just important for music and color theory, or principles of physics, but
that it has far reaching effects in the field of electronics. The knowledge of
how undertones occur in sound emitting or translating devices will cause a
complete redesign of every device which incorporates harmonic series. For
example, a microphone is designed to faithfully translate a sound wave into an
electrical signal which can be recorded. Because all microphones have been
developed with engineers only having knowledge of the overtone harmonic series,
no microphone in existence has ever translated a sound wave accurately. This
means that every single recorded sound is an inaccurate reproduction. The
infusion of the knowledge of undertones into the electronics field will generate
an unprecedented expansion of new and better devices.
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