Article: Why Johnny can't tune
Are you frustrated that your guitar doesn't really play in tune? Well, the purpose of this little article is to explain why fretted instruments can never play every note in perfect tune, and what you the player can do to overcome the inherent shortcomings of fretted instruments.
The intervals used in music can be expressed as simple frequency ratios. An octave, for example, is a 2 to 1 ratio. The interval of a fifth is a 3 to 2 ratio. Thus, the E above middle A is 660 cycles, which is three halves of 440.
If you keep stacking fifth intervals by multiplying by three halves, after 12 you've used up all the notes of the chromatic scale and you're back at A (just like going once around the Circle of Fifths). Thus, if you multiply 440 by three halves twelve times, you get 57,088.38865. Of course, this frequency is outside the range of hearing, but dividing by two lowers this very high A pitch by exactly an octave. Do this a few more times and you'll arrive at 446 cycles. We went up by fifths and down by octaves, but didn't wind up at the same pitch. So, if you tune an octave out of successive exact (or perfect) fifths, it is too wide.
If we squeeze a fifth interval a bit (two hundredths of a semitone) we can stack them as before and come out with an exact multiple, thereby getting a perfect octave. This squeezed interval is called tempered. All of the intervals except the octave must be tempered (either squeezed or stretched) in order to make the octave come out right.
Here's a practical example: Take a mandolin, which is tuned in fifths; the lowest string is a G. Tune the D strings a pure fifth above this and repeat with the A and E strings. Now play a G note on the third fret of the E strings; it will be sharp to the low open G strings, because each fifth was just a smidgen too wide to produce a perfect octave. This is not the result of a misplaced bridge; it's just the way it is.
The lesson is that you can't tune a fretted instrument accurately with harmonics because you are tuning to pure intervals. Tempered intervals are indeed a little "out of tune" but necessarily so in order to make all keys on the instrument equally pleasing to the ear. Electronic tuners know this and use tempered intervals to measure pitches. If you tune by ear, here are some tips: Don't use harmonics, and don't use chords to tune. Do use unisons and octaves to compare strings, and use one string as a reference for all the rest.