Fret placement is the most mathematical part of guitar building, and it is also the part where a small error has the largest consequence. A fret in the wrong place does not just sound out of tune — it sounds out of tune in a way the player cannot correct by bending or by tuning differently.
The good news is that the maths is settled. You are not inventing anything; you are placing frets where the ratios say they go, and then checking that the instrument agrees.
The twelfth fret is half the scale length
This is the anchor for everything else. A string stopped at the twelfth fret vibrates at half the length of the open string, which produces an octave. So the twelfth fret sits exactly at the midpoint of the scale length, and that is the one position you can verify with a tape measure.
Every other fret follows from the twelfth root of two. To go up one semitone you shorten the string by a factor of the twelfth root of two, roughly 0.9439. Each successive fret is that factor applied again to the remaining length.
Before you trust any fret layout, measure the twelfth fret. If it is not at the midpoint of the scale length, nothing else on the board can be right, and you have found the problem in ten seconds.
Equal temperament versus just intonation
Equal temperament divides the octave into twelve equal steps. Just intonation uses small whole-number ratios, which produce purer intervals in one key but are badly out of tune in others. Pythagorean tuning is a third system, built on stacked perfect fifths.
Almost every guitar is fretted to equal temperament, because equal temperament is the only system where all twelve keys are usable. The other systems matter mostly as a way of understanding how much error equal temperament accepts.
| Interval | Equal temperament | Just intonation | Error (cents) |
|---|---|---|---|
| Major third | 400 cents | 386.3 cents | +13.7 |
| Perfect fourth | 500 cents | 498.0 cents | +2.0 |
| Perfect fifth | 700 cents | 702.0 cents | −2.0 |
| Major sixth | 900 cents | 884.4 cents | +15.6 |
| Octave | 1200 cents | 1200 cents | 0 |
Compensation is why the saddle is not straight
The maths assumes an ideal string that is perfectly flexible and has no stiffness. A real string has stiffness, and a real string has to be pushed down to the fret, which stretches it slightly and raises the pitch. Both effects make the fretted note sharper than the maths predicts.
Compensation corrects for this by making the vibrating length slightly longer than the theoretical scale length. That is why the saddle is angled and why the bass strings are usually set back further than the treble strings — they are stiffer and need more compensation.
- Set the bridge position from the scale length, then add compensation at the saddle.
- Compensate the bass strings more than the treble strings.
- Check intonation at the twelfth fret against the harmonic, not the open string alone.
- A guitar can only be intonated for one string gauge at a time. Changing gauge means re-intonating.
Diagnosing an instrument that will not play in tune
Check the nut height first
If the nut slots are too high, every fretted note near the nut is pulled sharp. This is the most common cause and the easiest to fix.
Check the relief
Too much relief raises the action in the middle of the neck, which pushes those notes sharp when you fret them.
Check the fret positions
Measure the twelfth fret against the midpoint of the scale length. If it is wrong, the whole board is wrong.
Check the fret level
A high fret shortens the vibrating length. Level and crown before you adjust anything else.
Only then adjust the saddle
Saddle compensation is the last correction, not the first. Adjusting it while the nut is wrong is chasing your own tail.
Fret positions are usually given in millimetres and relief in thousandths of an inch. Pick one system and convert everything into it before you start cutting, so you never mix a 0.010 in shim with a 0.25 mm target.
A guitar is in tune when the maths, the frets and the saddle all agree. Two out of three is an out-of-tune guitar.
