The harmonics exist in a series of whole number multiples of the fundamental in the ratios 1/2/3/4/5/6 and it's the relationship between these harmonics that define our perception of what sounds "right".
Remember that what we think of as a linear "step" in music is actually a ratio of two frequencies; the octave is defined by a doubling of the frequency so the first step ratio in the above series is 1/2. Following on from this is the perfect fifth defined by the ratio 2/3, the perfect fourth is 3/4, and the major and minor thirds by 4/5 and 5/6 respectively.
Interestingly these successive intervals exactly spell out a major chord...
The harmonic series unfortunately gives us no value for the tone or semitone so these need to be derived from the larger intervals of the harmonic series. There is a gap of a tone between the fourth and fifth so dividing one by the other gives us a value for the tone of 8/9. There is also a step of a semitone between the major and minor third so this gives us a value for the semitone of 24/25. Two of these multiplied together then will obviously be 8/9; except it is not, neither is the step between the major third and the perfect fourth the same value as the step between the major and minor third; it's actually 15/16...
We have now encountered the first problem associated with our devotion to the harmonic series; it doesn't add up.
It is, in mathematical terms, irrational.
We perceive it as being "correct" because neighbouring tones whose fundamental frequencies are the same as - or multiples of - a harmonic in another tone will share a range of harmonics and difference tones ( link will open new window) which mesh together to produce few interference effects. However there is no fixed value for any of the smaller intervals nor do the larger harmonic intervals add up to an octave; four minor thirds in sequence evaluate to a ratio of 2.07 while three major thirds evaluate to 1.95. the only intervals which actually combine to create a perfect octave are the fourth and fifth and that's only because a fourth is an inverted fifth.
You may have noticed that we have got this far without even mentioning the guitar; that's because - and take good note of this - it's not a guitar problem! The problem is with music itself...
It is a problem for the guitar because it is a fixed pitch instrument and in common with all other fixed pitch instruments the maker needs to be able to define a series of values for the pitch of each note which, in the harmonic system, don't actually exist.
