Rak greppbräda

The distances between the frets are calculated mathematically and would be perfectly in tune on an ideal instrument with no string hight (action) and strings without stiffness.

That the action on an ideal instrument is zero, tells us indirctly that the ideal fretboard is perfectly straight. The tensioned, straight string touches the top of each fret.

Probably all modern guitar fretboards have something called "relief". The fretboard is slightly curved along its length, towards the strings. This makes it easier to play up the neck as the action gets lower, but intonation will suffer.

On a straight fretboard the string height increases constantly. On a straight fretboard the effect of the bridge compensation also increases constantly as the distances between the frets constantly decreases. This is why the bridge compensation works perfectly. With a bent fretboard the constantly increasing string height is lost. Only two notes on a string will get the correct pitch - the open string wich is tuned to pitch, and the note used as reference when the string was intonated (commonly the 12:th fret).

Relief is checked by simultaneously fretting the first and last fret and measure the largest distance between the string and the frets. On guitars the relief is usually largest at the 7 - 8 fret. Normal relief is 0.3 - 0.5 mm.

0.3 mm may sound little, but if the angle at the nut is maintained, the last fret will be about 1.2 mm closer to the string. That's a lot when it comes to intonation!

Almost all fretboards are transverselly arched (radiused). The arch improves playability. Barred chords are more comfortable for the left hand and fingerpicking with the right hand is easier. An arched fretboard can be cylindrical, conical or compound.

The cylindrical arch is most common. It is easier for the luthier to make, but it has a flaw. The strings are not parallell along the fretboard. The closer to the bridge you get, the more the strings are spred. This causes a phenomen best described with an experiment. Hold a stright edge along the cylindrical side of a bottle. When the straight edge is parallell to the bottle the whole leght of the edge touches bottle. But, if you slightly change the angle of the straight edge you will notice how the side of the bottle is arching along the edge. Due to the spreading of the strings a cylindric fretboard is arching towards the string. Exactly the oposite to relief.

To keep the fretboard stright along the strings on an arched fretboard, the fretboard has to be conical. If the conicity is proportional to the string spread, the fretboard can be kept perfectly straight along the strings, wich is optimal for good intonation.

Compound is relief added to a conical fretboard. A compound radiused neck is curved in two dimensions, across the the strings and along the strings. Exaggerated the surface of such a fretboard would look like a horses saddle. As said before, relief is not good for intonation.