Historically, keyboard instruments could be tuned so that natural harmonic intervals were available in at least one key. This is possible because each note can be tuned separately. Even with these systems however, not all intervals worked in all positions. Keys could be "near" or "distant" and the "wolf note" - a particularly dissonant interval named for its howl - was ever present.
The reason this cannot be done on a guitar is because there are six strings and the range of each string overlaps with its neighbour over at least three quarters of it's compass so that the same note can appear in multiple positions. If, for example, you attempt to tune the low E and the A to a perfect fourth, because the strings are divided by the constant ratio scale the A at the fifth fret on the E would sound sharp relative to the open A because the interval defined by the 5th fret is a wider one than the one being employed between the two open strings. You could, of course, move the fret a little way to accommodate the error but it would then be in the wrong place to define the D on the A string and the g on the D string, et cetera...
Furthermore, how would the open strings sound?
If you tuned each course to a perfect harmonic interval you would end up with a very flat top string because each fourth is represented by a ratio of 4/3 and there are four of these in the two octave span of the guitar, plus one major third at a ratio of 5/4 evaluating to an interval between the low and high E of 320/81 and not the mandatory 4/1 that would be a true double octave.
There are guitars that attempt to resolve these discrepancies - the Fretwave and the so-called True Temperament guitars by Anders Thidell - but it's hard to see how they don't cause more problems than they solve. and even harder to see how they overcome the problem of fitting standard tuning into a two octave interval without stretching the fourths and thirds or resorting to a unique tuning or tuning to a chord, requiring a complete reworking of tradtional guitar technique.
Modern music is irrevocably wedded to equal temperament. All keyboards and other fixed pitch instruments are built to use it and there is little place for a modern guitar built to use earlier systems of intonation except in the vanishingly small special interest groups devoted to the performance of early music. In these arenas a modern instrument would have no place since authenticity is the name of the game and a guitar or lute that actually plays true to a key would be anything but authentic...
