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Michael Lee's Mathematics and Physics

 

Music

 

Why the frets on a guitar get closer together as you move towards the bridge

 

The ancient Greeks, found three intervals particularly pleasing to the ear and it follows common mathematical ratios. If you take any open string and pluck it and then play the string again by shortening its length by a half (it's the 12th fret), you get an octave, the first two notes of "Somewhere Over the Rainbow." or a 2:1 ratio. Whereas if you pluck any open string and then shorten the string by a third (it's the 7th fret) you get a perfect fifth or the first two notes of "Twinkle, Twinkle Little Star." or a 3:2 ratio. Finally, if you shorten any string by a fourth (it's the 5th fret), you get a perfect fourth, the first two notes of "Here Comes the Bride." or a 4:3 ratio. 

 

To see what I mean. Put the guitar down on a something; I use my couch and play it like it were a piano; this is so you can see the proportions by eyeballing it.  Grab a capo and let's make a perfect fourth. Let's use the sixth string, but it doesn't matter which one, pluck it then pluck it with the left finger about a quarter the way down.  Your finger should be on the fifth fret and listen; you can play "Here Comes the Bride."  Clamp the capo on the fifth fret.

 

Now ask yourself, where is the next fourth up from where the capo is located? Well, again shorten the sixth string by a quarter from the capo to the bridge.   Eyeball it and you'll see your finger is on the 10th fret and you can play "Here Comes the Bride."

 

Now notice something. With a tape measure of some sort, measure the distance from the nut to the capo; that was the first perfect fourth we played. Next, measure the distance from the capo to the 10th fret; that is the second perfect fourth. On my guitar, the former was six and a half inches and the latter is four and three quarter inches. So, the interval length of the string for a fourth is diminishing as we move towards the bridge. This is true for all the other intervals  such as the perfect fifth and the octave.  This is why the frets get closer together towards the bridge.