Borrowed these from another site. These articles explain why your guitar (ANY guitar, actually, no matter the maker or cost) can sound out of tune, even when tuned with the most accurate tuner. There have been many attempts to sort this issue, from compensated nuts and saddles to staggered or fanned frets (Dingwall bass frets are fanned for a different reason), but none really sort the issue. Go ahead, get your tuner and check your €15,000 signature Gibson LP, but don’t blame the maker…… it’s all down to mathematics.
Before a guitar can stay in tune, you have to be able to get it in tune. This requires a guitar that is properly intonated. A guitar is considered to be in proper intonation when notes and chords do not get more and more out of tune as you play up the neck. Proper intonation does not mean that it is perfectly in tune at every fret, just that it is equally out of tune at any given point. A properly intonated guitar should sound acceptably in tune for all your chords from the first fret to the highest fret.
Guitars don’t play perfectly in tune! They are only relatively in tune. They are in what is called a “tempered” tuning. Even on a perfectly maintained instrument, if you tune it so that the G# in the first position E chord is sweetly in tune, the open G in your first position C chord will be flat. If you tune the C# in a first position A chord to be sweet, the open B will be flat. Guitars are tuned in a compromised tuning that makes that C# and G# a little sharp so that the open strings are only a little flat. Fortunately for most of us, our ears have become accustomed to these compromises so that it doesn’t sound unpleasant. Still, you will often see classical guitarists re-tuning for different pieces that feature specific intervals – these paragraphs are taken from Dean Markley’s website.
Tuning has always been a bugbear for guitarists. Every guitar player – and every guitar builder and repairer – is familiar with the problem. No matter how good the instrument, and how well tuned and adjusted, it never sounds perfectly in tune in all positions and keys.
This is not the fault of the guitar. It is not designed to play perfect intervals (except for octaves and unisons) in any position, or any key. It is designed to play the equal-tempered scale, and it is perfectly possible to adjust and intonate almost any well-made guitar so that it plays this scale pretty accurately. The problem with equal temperament, though, is that it is artificial, a mathematical construct, and it conflicts with the physical properties of real-world strings.
Real-world strings produce harmonics which are pure fractions of the speaking length of the string. The ancient Greeks and Chinese knew about the pure intervals, and constructed their musical scales around them. But Nature throws a spanner in the works by making the natural tone row irregular, so instruments tuned in this way cannot modulate to different key signatures without adding more intervals to the octave.
There is another problem in that 7 pure octaves and 12 pure fifths do not add upp the same:
7 octaves = (2/1) ^7 = 128
12 fifths = (3/2) ^12 = 129.74
The discrepancy works out to 24 cents (almost exactly a quarter-tone), and is known as the “Pythagorean Comma”. Finding a way around these problems has been the cause of much controversy and many bitter arguments among music theorists for two and a half millenia.
To make a fixed-interval instrument with 12 notes in the octave useable in all the key signatures, the purity of the intervals has to be compromised. This is called “tempering”. A temperament is a specific way of dividing the Pythagorean comma among the intervals of the octave. There many alternative ways to do this on keyboard instruments, and it is only in the last 150 years that equal temperament has taken over as the accepted standard.
As far as the guitar and other fretted instruments having 12 straight, unbroken frets to the octave are concerned, equal temperament is the only choice. Back in 1581, Vincenzo Galilei (Galileo’s father), explained the need for equal semitones logically and correctly – “since the frets are placed straight across the six strings, the order of diatonic and chromatic semitones is the same on all strings. In chords, therefore, a C# might be sounded on one string, and a Db on another – this will be a very false octave unless the instrument is in equal temperament.”
Equal temperament divides the octave into twelve exactly equal semitones. The resulting equal divisions are a logarithmic function of the speaking length of the string, rather than pure fractions, and thus are not a true analog of the natural harmonic series.
Equal temperament is the ultimate compromise. Tonal purity is sacrificed for ease of modulation. Depending on your viewpoint, equal temperament either a) makes every key equally in tune, or b) makes every key equally out of tune… The idea is to make it possible to play all intervals and chords, in all keys, with the same relative accuracy. Although every key is very slightly out of tune, every key is also useable. No key sounds worse than any other key. The same applies to all chords. Theoretically, that is. In practise certain intervals and chords can still sound dissonant. Thirds are especially troublesome, as the even-tempered minor third is 16 cents flat to the “pure” minor third and the even-tempered major third is 14 cents sharp of pure. The equal-tempered major sixth is 16 cents sharp of just, and the equal tempered major seventh is 12 cents sharp of just. The only interval which is identical in the two scales is the octave.