The monochord, made playable
iWhy do the frets get closer together? Because each one multiplies, it doesn't add.
for teachers: 12th root of 2 · the rule of 18 · geometryOpen →
Never held a guitar? Start with the big button — first string, first fret, first tune.
six strings · the math & physics of the fretboard
Two thousand years ago it was one string and a movable bridge. The guitar just fixes the bridges in place and calls them frets — each one dropped exactly where the twelfth root of two says it belongs.
plate vi · the acoustic guitar
play it — tuned E A D G B E, with true fret spacing
Tap any fret to pluck it. Notice the frets crowd together toward the right — that shrinking is the math.
the math & physics of this instrument
Each is a short lab with a sound to poke and a “for the classroom” card — the frets, the ghost-note harmonics, and the physics of a plucked string.
Why do the frets get closer together? Because each one multiplies, it doesn't add.
Touch the string over a fret — don't press — and a ghostly bell rings. That's fractions.
Long, loose and fat = low. Short, tight and thin = high. Slide the three knobs and hear it.
New to it? Learn the basics — open strings, chords & a first tune →
Start where it began — the one-string monochord → or bring the guitar to a class with the school workshops →
the maths & physics of this instrument
Two thousand years ago the Greeks stretched a string over a box and moved a bridge along it to find which fractions sounded good together. The guitar is that experiment, finished and mass-produced: someone worked out where the divisions go, and then hammered metal into the neck at exactly those points. Every fret is a fraction that was decided in advance.
Exponential decay · geometric sequences
Fret n leaves 2^(−n/12) of the string. Fret 12 leaves exactly one half, which is why it sounds an octave. The frets crowd together as they climb because each is a fixed multiple of the last.
The maths is physically inlaid into the instrument, and you can measure it with a ruler.
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Division · iterative construction
Each fret sits one 17.817th of the remaining length further along — because 1 ÷ (1 − 2^(−1/12)) = 17.817. Luthiers rounded it to 18 for centuries.
A real trade using a real approximation, with the error it costs you.
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Unit fractions · whole-number multiples
Rest a finger at 1/2, 1/3 or 1/4 of the string and it rings at 2×, 3× or 4× the open pitch — the harmonic series, selected by hand.
The reciprocal relationship between where you touch and what you hear, done in one gesture.
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Waves · fractions of a length
In harmonic n the still points sit at k ÷ n along the string, and the widest swings at (2k+1) ÷ 2n. Harmonic 3 stands still at 1/3 and 2/3.
A guitar string is the standing wave in the textbook, at a size you can put a finger on.
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