First published in the spring 2009 edition of Open Position, my “String Theory” column was created and designed to enlighten and inspire guitarists to study music theory. Playing an instrument is a multifaceted journey of discovery, and I believe understanding the language of music through theoretical principles enhances one’s playing and overall understanding of music. I find it most important for guitarists in particular to learn these theoretical concepts because of the inherent difficulty of seeing theory on the instrument itself. The guitar is not a theory-friendly instrument.
Enjoy this reprint of the inaugural edition of my “String Theory” column with more to come!
Welcome to the first installment of my new column entitled “String Theory.”
String Theory is a column intended for guitarists who are interested in all aspects of music theory, specifically as it pertains to the guitar.
Chapter One: The Circle of Fifths
“There is geometry in the humming of the strings” —Pythagoras
Ever wonder who came up with the Circle of Fifths, that simple diagram which has come to symbolize the heart of the entire Western Music system?
Or why the distance portrayed in the circle between each key is based upon the interval of the perfect fourth and perfect fifth? I find these topics to be really interesting so I began investigating and the more I researched the subject the more fascinating it became.
As the story goes a man walked by a blacksmith’s shop one day and heard some interesting sounds emanating from the shop. Each time the blacksmith struck a piece of metal with one of his hammers a secondary sound was heard, a harmonic. That man was Pythagoras of Samos. Pythagoras (570-500 B.C.) the well known mathematician and scholar is credited with discovering the Harmonic Series.
“If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music… I get most joy in life out of music.” ~ Albert Einstein
The Harmonic Series as pictured above is a series of pitches (partials) that occur naturally and sound concurrently with a string being plucked or a note being sounded, in this case the note C2. This series is based upon certain frequency ratios and these ratios and their relative strength in conjunction with the fundamental (plucked string) determine an instruments timbre.
What Pythagoras discovered was that the most prominent harmonic ratios were 2:1 perfect octave, 4:3 perfect fourth, and 3:2 perfect fifth. You’ll notice that in the Western Music system each of these intervals was given the name “perfect”, the idea being that if they exist in nature, God must have created them therefore they were described as “perfect.” Pythagoras discovered the Harmonic Series after experimenting with a monochord. (see diagram below)
A monochord has a wooden body with a string attached to both ends. A movable bridge placed under the string determines the pitch. Pythagoras determined that stopping the string at its mid point produced a note an octave higher than the fundamental. On the guitar this would be achieved by fretting any of the strings at the twelfth fret. An octave is represented by the ratio of 2:1. Further experimentation led Pythagoras to discover that the perfect fourth (4:3) and fifth (3:2) intervals were also very prominent. Pythagoras and his followers believed that the underlying truth to the universe could be found in the language of numbers and that the language of numbers represented the language of God. He also believed that the number 10 had great significance and considered it the perfect number. The Pythagoreans discovered that if you put the numbers associated with the three main musical ratios together you would create an equilateral triangle. The Pythagoreans gave this figure the name Tetractus. A Tetractus is created by layering four tiers such as in the diagram below. If you look at the diagram more closely you can see all of the prominent ratios involved in the Harmonic Series. This simple pyramid contains tiers comprised of the numbers 1, 2, 3 and 4. Thus the ratios 2:1, 3:2 and 4:3 are all contained in this image. The final caveat is that 1, 2, 3 and 4 add up to the number 10, the perfect number. So for the Pythagoreans the Tetractus represented the “musical numerical order of the cosmos”*
You can imagine how excited Pythagoras must have been at his discovery. It led to further inquiry and discoveries concerning among other things the evolution of astronomy and geometry, ultimately leading to the concept of the “Music of the Spheres.”
*excerpt taken from “The Music of Pythagoras” by Kitty Ferguson
Based on Pythagoras’ discoveries it seems logical that the mathematics surrounding the intervals of a perfect octave, fifth and fourth would become a means of representing the underlying principles and spawn the evolution of the Western Music system. The basic organizing principle behind the circle of fifth’s is easy enough to understand. It takes twelve keys a perfect fifth apart to return to the key you started from, see below:
The note B#/C that is arrived at is actually slightly sharp from the original note C that we started on and this discrepancy is known as the “Pythagorean Comma.” Two of the earliest representations of the diagram of the Circle of Fifth’s are shown above. The circle above on the left was drawn by Nikolay Diletsky in 1679. The other circle was drawn by Johann David Heinichen in 1711. After Pythagoras’ experiments and discoveries it was quite clear that the intervals/ratios of a perfect octave, fourth and a fifth, would play a significant role in the development of the four disciplines of study of Pythagoras’ time termed the Quadrivium, those being Arithmetic, Astronomy, Geometry and Music. His work also played a significant role in the development of the construction of modern stringed musical instruments such as the violin, cello, bass and guitar and the way in which they were tuned. The cello and violin are tuned in perfect fifth’s C G D A and G D A E respectively and the guitar of course is primarily tuned to perfect fourths, E A D G B E. Pentatonic scales, specifically the Major Pentatonic scale, is a series of five notes all a perfect fifth apart. (C G D A E). The Pentatonic scale predates the twelve tone western system and in fact every culture on earth seems to have incorporated some form of pentatonic scale in their traditional music affirming the notion of an awareness on some level of the Harmonic Series. In terms of the electric guitar, is it any wonder that a power chord comprised of a root, perfect fifth and octave being played at high volume with distortion, resulting in enhanced harmonic activity, would make such a strong sonic statement and be so universally accepted? Isn’t it also interesting that the number 10, the perfect number, can be seen as the numbers one and zero, the binary system upon which all computer programming language and digital code is based. The Harmonic Series has consistently played an important role throughout history in every facet of human culture. From Music to Modern Cosmology every element in our universe is comprised of vibrations, waves and frequency’s. From the macrocosmos down to the quantum level, everything hidden or perceivable is vibrating and being aware of this fact only makes being a musician even more gratifying. We are tapping into the language which permeates all existence by playing an instrument. Through our guitars we are speaking the mysterious code of the heavens. My brief discourse on this subject is only scratching the surface and there are many great websites and books written about these and similar topics. The fascinating thing for me is that the language of music, comprised of vibration at varying frequencies, exists independently of us as human beings and that through Pythagoras’ experiments and others like him we are able to manipulate and transform these vibrations into beautiful cognizant music. Every civilization throughout history has discovered and created music using these harmonies and ratios. Perhaps it can be said that the language of music is not only the language of number but the language of humanity and the cosmos as well.
Pythagoras of Samos
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