The monochord, made playable

Every fret sits where the string is shortened by the twelfth root of two. That's why the frets crowd together as you climb — and why the twelfth fret lands exactly halfway.

a real fretboard — tap any fret to pluck it

fret spacing:

See how the frets bunch up as you go right? Each one shortens what's LEFT by the same fraction — so the gaps shrink.

pick a fret — where does it sit?

string shortened to2^(−5/12) = 0.7492

fret sits from the nut25.1% = 162.5 mm

string still ringing485.5 mm

note on the low stringA

the shrinking gaps (25.5″ scale)

fret	from nut	mm	note
1	5.6%	36.4	F
2	10.9%	70.7	F#
3	15.9%	103.1	G
4	20.6%	133.7	G#
5	25.1%	162.5	A
7	33.3%	215.5	B
12	50.0%	324.0	E

Luthiers call it the rule of 18: each new fret takes about 1/17.82of the string that's left. The exact number is the twelfth root of two — mark the frets by multiplying, never by measuring equal steps.

For the classroom

Learning goals

A fret at position n shortens the string to 2^(−n/12) of its length — a geometric, not arithmetic, sequence.
Because each fret multiplies what remains, the distances between frets shrink toward the bridge (the 'rule of 18', exactly 17.817).
The 12th fret sits at exactly ½ the scale length — halving a string raises it one octave, tying the guitar straight back to the monochord.

Try this

1.Toggle 'evenly (wrong)' vs 'real' and pluck the 12th fret in each. Ask why equal spacing can't work — where would the octave land?
2.From the table, subtract consecutive 'mm' values (fret 1→2, 2→3 …). Students see the gaps shrink, and can predict the next gap before reading it.