I have a few nuggets of basic music theory as they apply to the guitar. This first one is about the difficulty of tuning guitars – and indeed any musical instrument – accurately, which I will show by looking at the mathematics of note frequencies.
In standard tuning, guitar strings are pitched either a perfect 4th (5 frets) or a major 3rd (4 frets) apart, and the ratios of frequencies are 4:3 (perfect 4th) and 5:4 (major 3rd). Therefore we multiply the frequency of each string’s open note by the appropriate ratio to get the frequency of the next string’s open note.
The bottom and top strings of a guitar are two octaves apart, which means doubling the frequency twice. The frequency of the low E on a guitar is actually 82.4 Hz, if we use A440 tuning, but we will use a notional base note of 100 Hz (it’s not so far out anyway, that’s about 3 or 4 semitones higher). Hence, doubling that twice, the top string should be at 400 Hz.
However, if we go up string by string, we get the following:
E – Pitch of lowest string = 100
A – Up a fourth (4/3) = 133.3
D – Up a fourth (4/3) = 177.8
G – Up a fourth (4/3) = 237.0
B – Up a third (5/4) = 296.3
E – Up a fourth (4/3) = 395.1
So we reach 395 Hz instead of 400 Hz, which is about a fifth of a semitone out. And that’s why you can’t tune your guitar!