Pythagoras (c. 582 - 497 BC) is reported to have discovered that halving the length of a tensioned musical string will cause its musical pitch to rise an octave. On a guitar, the 12th fret is half way along the speaking length of the open string. Stop the string on the 12th fret, and the length of string remaining to sound, between the finger and the bridge, is 1/2 the original length. The note produced is an octave above that of the open string.

Pythagoras is also said to have discovered that stopping a string so that two thirds of its length is allowed to sound, produces a note a perfect fifth (5 steps up the major diatonic scale) above the note of the open string. On a guitar, stopping the string on the 7th fret allows 2/3 of the open string's speaking length to sound, and produces a note a perfect fifth above it. For example, stop on the 7th fret of the E string and you get the note B, a perfect fifth above.

Finally, Pythagoras is also said to have discovered that stopping a string so that three quarters of its length is allowed to sound, produces a note a perfect fourth (four steps up the major diatonic scale) above the note of the open string.  On a guitar, stopping the string on the 5th fret allows 3/4 of the open string's speaking length to sound, and produces a note a perfect fourth above it. For example, stop on the 5th fret of the E string and you get the note A, a perfect fourth above.

These ratios of the open string length to the stopped string lengths are

For an octave   2:1

For a perfect fifth   3:2

For a perfect fourth   4:3

These are called harmonic ratios.

A rising perfect fourth followed by a rising perfect fifth, rises a total of one octave. On the E string, stop on the 5th fret to get an A, a perfect fifth above the E. The jump from this A to the E on the 12th fret is a perfect fifth.

Harmonic ratios can be manipulated like musical intervals. Musical intervals can be "added together", like the E to A + A to E example above. But harmonic ratios, representing musical intervals, must always be multiplied together rather than added together.

We just said that a perfect fifth and perfect fourth "added together" make an octave. So, writing this as harmonic ratios, we would multiply the ratios, and say

4:3 X 3:2 = 2:1

OR

Putting intervals together

Pythagoras is also credited with discovering what would happen if perfect fifths and fourths were put together to make octaves. If you put an perfect fifth together with a perfect fourth, you get an octave. For example, C up to G is a fifth, G up to C is a fourth, and the interval between the two Cs is an octave.

If you tuned the G slightly sharp, you would make both the fifth and the fourth slightly mistuned. The fifth C to G would be "wider" than a perfect fifth, and the fourth G to C would be slightly "narrower" than a perfect fourth.

Similarly if you tuned the G slightly flat, you would again make both the fifth and the fourth slightly mistuned. The fifth C to G would be "narrower" than a perfect fifth, and the fourth G to C would be slightly "wider" than a perfect fourth.

On a keyboard, or on a guitar, there are a number of sequences of fifths or fourths, either rising or falling, that could be followed, that start and end on the same note, and that pass through each and every other note only once.

For example, we could play C, G, D , A , E, B, F#, C#, Ab, Eb, Bb, F, C.

(A note on the grammar: you might notice we carried out an enharmonic change from C# to A-flat. There would always have to be one of these).

All is not what it seems

All such sequences must contain 12 different notes. Probably the easiest way to try this is on a keyboard, but you can do it on a guitar.

As it happens, if every one of the fifths and fourths in any such sequence is tuned as a perfect interval, by using the ratios 3/2 or 4/3 (or their inversions 2/3 or 3/4), then the final note in the sequence turns will not coincide with the starting note. We will end up with a note at a slightly different pitch to the one we started with. The starting and ending notes will be different by a small micro-interval called the Pythagorean Comma. 

This discrepancy turns up in a number of other ways, too. Instead of attempting to start and finish a sequence on the same note, we can arrange it so that the starting and finishing notes are one or more octaves apart. Try this on the piano or guitar.

You will probably notice there seems to be no such problem on the guitar or piano, assuming it is tuned well. This is in fact because on these instruments the fourths and fifths are not tuned as perfect intervals by using the ratios 3/2 or 4/3 (or their inversions 2/3 or 3/4).

The mathematics

It does not matter how you construct the sequence, or where you use fourths or fifths, rising or falling: The final octave, or multiple octave, will end up mistuned by a Pythagorean Comma. 

The largest distance apart we can make the starting and finishing notes, turns out to be 7 octaves, and we can do this if we use only fifths in the sequence. In other words, the sequence

C, G, D , A , E, B, F#, C#, Ab, Eb, Bb, F, C

could mean there is a rising fifth between any two adjacent notes. The two Cs would then be 7 octaves apart. Try this on a full sized piano with a concert compass, starting on the lowest C. Or on a seven octave piano, try

A , E, B, F#, C#, Ab, Eb, Bb, F, C, G, D, A

Again, on the piano, there may seem to be no problem, and in any case, the two Cs are so far apart it's hard to tell anyway, and other complex factors come into play in deciding whether the two Cs agree.

Mathematically we would say that the twelve fifths make an interval of

3/2 X 3/2 X 3/2.....etc.,  twelve times altogether.

The seven octaves between the starting and ending notes would be

2/1 X 2/1 X 2/1.....etc., seven times altogether.

Multiplying these out gives two different answers, one for the 12 fifths, and another for the seven octaves. The musical difference between the two different ending notes is the ratio between these two answers:

Here, the 1.014 is the approximate harmonic ratio of the Pythagorean Comma.

Equal temperament

Modern tuning gets around the problem of the Pythagorean comma by tuning by the system known as equal temperament. Rather than using the harmonic ratios, as discovered by Pythagoras, the octave is divided into 12 equally sized semitones.

The result of this is that none of the intervals are tuned according to their associated harmonic ratio, except the octave.

Because none of the intervals except the octave are tuned harmonically, all the intervals will contain audible beating in their soundscapes.

Badly tuned intervals are also characterised by the fact that they contain beating in their audible soundscape. The difference between temperament, and plain poor tuning, is that in temperament, the speed of the beating is carefully controlled, in relation to particular intervals. It is not just random or accidental.

Equal temperament must be taken into account in tuning the guitar, because the fret positions are designed for it.

Equal temperament is not the only problem affecting guitar tuning. A natural feature of string behaviour called inharmonicity, and any errors in the fret positions, heights, or neck angle, can demand further tuning techniques to be applied, in addition to equal temperament tempering.

For more on this, see:  

Tuning the guitar