Have you heard of the golden ratio? Well, what about the gold-colored guitar fret ratio?
I’ve been listening lately to classical guitar musicians on YouTube. Such skill and artistry is utterly astonishing. After a while, though, I got curious about how the guitars were made and how they achieved their frequency outputs. I noticed something. I took a screen shot of a guitar’s strings and fretboard and, by counting pixels, I measured the dimensions seen in Figure 1:
Figure 1 Guitar fret positioning is definitely not random, acoustically speaking.
I then looked at the ratios of a string’s fret-to-base length (my choice of term) to that of the next shorter one (Figure 2):
Figure 2 Length ratios: the ratio of each fret position taken in pixels divided by that of the next fret closer to the base is the ratio of the frequencies of each note of the string. That ratio is nominally 2^(1/12) =1.05946… which is approximated in each and every case. The average of the calculations as shown here comes to 1.058422 which is only 0.1% in nominal error.
I discovered that in spite of my crudeness in using the pixels, the ratios come out very close to the twelfth root of two. That ratio is the ratio of adjacent note frequencies of a tempered musical scale. If the full length of a string is taken as “Do” in its particular key, the fret positions yield the twelve-tone-scale arrangement of notes seen in Figure 3. In hindsight, I guess I should have intuitively known this but I didn’t. I do now.
Figure 3 Tempered scale: taking the full length of each string as the note “Do” as in Do-Re-Mi-Fa-Sol-La-Ti-Do scaling, we see how each fret position corresponds to one of the twelve notes (including sharps and flats) of the Western culture’s twelve-tone scale.
It should be noted that singer Jimmie Rodgers once admitted that he couldn’t really play the guitar as he performed, so he tuned the six strings of his guitar to be in open string harmony. Then when he was performing, he would keep one finger, his thumb, across all six strings at the same time across the fretboard. Since all six strings followed the above length versus note pattern, the six strings were always in harmony. He was using a single, movable chord.
Watch how his left hand does that in the following YouTube videos and enjoy the music.
John Dunn is an electronics consultant and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).
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