THE LIVING LIBRARYMonday, October 21, 2002 - 05:03 pm

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 thenbe applied to a given base number to create an harmonic series which references the basenumber. This process can be accomplished in two ways - multiplying the base number bythe 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. Themathematical process of multiplying or dividing can be applied to both the overtone andthe undertone orientation but there is a catch. Both the overtone and undertone seriesalways exist simultaneously but must always be mathematically opposite for any specificapplication. This means that for a given application, if the overtone series process ismultiplication, then the undertone series process must be division. For a differentapplication the mathematical processes might be reversed. The following two statementsare 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 basenumber 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 simplesound generating structure, such as stretched string or a pipe. For this presentation we aregoing 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 soundbox, a string stretched over the length of its body, and a movable bridge. Once it is tunedand the tension is set, the property of string length can be manipulated by adjusting themovable bridge. The movable bridge effectively acts as a nodal end to shorten the stringlength. Along the body of the sound box are length marks to show where to place themovable bridge in order to provide effective string lengths of 1, 2, etc... The followingexamples use a monochord with a string tuned to a frequency of 131 Hz.. Only the first 6harmonic frequencies will be shown.

Overtone Frequencies

Overtone frequencies are obtained by multiplying the base frequency 131 Hz. by the orderof numbers 1, 2, 3, 4, 5, 6,... The overtone string lengths which correlate to thesefrequencies 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 theharmonic series (number 1). The calculations for an overtone series starting at 131 Hz. areshown 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 2ndharmonic frequency the string length must be divided in half by adjusting the movablebridge to the middle of the string so that the length of string from the bridge to the stringend is 1 the entire string length. By plucking this 1 string length, a frequency sound of 262Hz. will be produced which is twice the frequency of the fundamental 131 Hz. The results ofcarrying out this process for subsequent harmonics of the series is shown in the tableabove and in Diagram 2.

When the bridge divides the string in half , playing the string on either side of the bridgewill result in a string length which is double the fundamental frequency. For the 2 divisionof the string, the bridge can be placed on the right or left 2 of the string. The 3rd harmonicfrequency is obtained by plucking the 2 string length and not the B string length. (Ifsomeone wanted to sound the middle 2 of the string (also triple the fundamentalfrequency) it would require two bridges to section it off). All subsequent divisions functionlike the 2 division. For this example the bridge is consistently placed on the divisions to theright 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). [Toplay the strings of Diagram 2 an MP3 player is required.]

Diagram 2

Once the appropriate string lengths of the overtone series have been developed, andmarkings have been made to correlate with these lengths, the monochord can befunctionally used to generate a playable overtone series. Diagram 3 shows a monochordwith the overtone positions marked.

Diagram 3

Undertone frequencies can be produced by generating longer string lengths then the firstor reference string length. Since this demonstration is limited to the monochord which is asingle string, lengths of string longer than the fundamental string length are not physicallyviable because a string cannot magically expand to greater lengths. So how can anundertone series be obtained from a single string? The answer lies in two principles: theequal division marking of a string and a fundamental frequency which correlates to thesmallest division length of an equal division marked string. Diagram 4 shows how this

process works. In this example a monochord is marked with 6 equal length divisions.Obviously an harmonic series is not obtained by all lengths being the same. In order toprovide different lengths which equate to an undertone harmonic series, one simply startsby placing the movable bridge of the monochord to the far division on one side of thestring and moves consecutively to each division until the entire string length is obtained.For Diagram 4 this process begins by using the far right string length which is 5 the lengthof the entire string. The undertone 2nd harmonic is obtained by moving the bridge to thenext position which is 2 the length of the string. Since the 2 length is twice the 5 length itgenerates the undertone 2nd harmonic frequency. Each successive undertone harmonic isobtained by simply moving to the next position on the string. [To play the strings ofDiagram 4 an MP3 player is required.]

Diagram 4

The most important aspect of this process is to understand that the fundamental frequencyfor this example is the 5 string length and not the entire string length. The entire stringlength is actually the 6th harmonic position in the series. Because a string longer than theentire string length is not obtainable for this example, the undertone harmonic series isrestricted to 6 frequencies or lengths. For an undertone orientation, successive string lengths are obtained by multiplying thefundamental string length by the order of numbers. Undertone frequencies are obtained bydividing 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 5fundamental length, etc. The table below shows the mathematical derivation of theundertone string lengths and their respective frequencies. Keep in mind that theundertone 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 stringdivision series is not a mirror image, reverse direction, or directmathematical inversion of the overtone string division series. For example, it isobvious that the 2nd harmonic length of the undertone series is 2 which is not the same asthe 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, anddirection mathematical inversions of each other. These cases occur when the overtone andundertone series share the same fundamental.

When do the two series share the same fundamental? For the previous explanations of theovertone and undertone series the string was physically divided by a manual process butthis is not necessary to generate the harmonic series on a single string. Objects which arecapable of producing harmonics will naturally generate both harmonic series when theobject is set into motion. For example, when a string is plucked the mechanics of the stringwill naturally divide the string according to the overtone divisions previously shown. Theindividual 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 whichwas 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 andphysicists, that a string cannot expand itself into larger and larger lengths. They areabsolutely right about the inability of a string to expand to larger lengths, but expanding tolarger lengths physically is not how the undertone series is exhibited by the mechanics of astring. The three properties of a string which can be altered are the length, the mass, andthe tension. Since the length and the mass are constant the only property capable ofchanging is the tension. Physicists have also assumed that the tension remained constantbecause no one is increasing or decreasing the string tension during the time the string isin vibration. The failure of recognition is in the principle that when a string is pulled backand released an applied tension is added to the string which has harmonic properties. Theundertone tension is the applied tension imposed when the string is pulled out to start thestring vibration. We know that the fundamental wave moves through a constantly reducingamplitude based on harmonic reduction which means that the applied tension of the stringwill also move through an harmonic energy reduction which means that the resulting wavewill move through the undertone harmonic series.

The only text in the world which explains in detail, and shows the proof, for the naturaloccurrence of the undertone series on a string is the science essay in this site). The readermust understand that this knowledge is not just important for music and color theory, orprinciples of physics, but that it has far reaching effects in the field of electronics. Theknowledge of how undertones occur in sound emitting or translating devices will cause acomplete redesign of every device which incorporates harmonic series. For example, amicrophone is designed to faithfully translate a sound wave into an electrical signal whichcan be recorded. Because all microphones have been developed with engineers only havingknowledge of the overtone harmonic series, no microphone in existence has evertranslated a sound wave accurately. This means that every single recorded sound is aninaccurate reproduction. The infusion of the knowledge of undertones into the electronicsfield will generate an unprecedented expansion of new and better devices.