Lab ii · the guitar
Rest a finger lightly over the 12th, 7th or 5th fret and a bell-like ghost note rings. Each one is the string splitting into whole halves, thirds or quarters — the harmonic series, on a string you can touch.
the three node frets are lit — the ghost notes live there
Touch here and the string can only vibrate in 2 equal loops. It rings at 2× the open pitch — one octave higher.
110.0 × 2 = 220.0 Hz
Touch here and the string can only vibrate in 3 equal loops. It rings at 3× the open pitch — an octave + a fifth higher.
110.0 × 3 = 330.0 Hz
Touch here and the string can only vibrate in 4 equal loops. It rings at 4× the open pitch — two octaves higher.
110.0 × 4 = 440.0 Hz
A string held at both ends can only vibrate in whole loops: 2, 3, 4 of them, never two-and-a-half. Touching a node kills the fundamental and lets one of those whole-number splits ring alone. The fractions you touch — 1/2, 1/3, 1/4 — are the same tidy fractions from the magic-string lab, and the pitches you get are the harmonic series, note for note.
Here is the picture behind the sound. The string is tied down at the nut and at the bridge, so it can only fit whole numbers of half-waves between them — and between the loops sit points that never move at all.
pick a harmonic and watch the string
y = A · sin(nπx / L) · cos(2πft)Tap anywhere along the string to rest a finger on it. The picture runs at a few wiggles a second so you can watch it — the real string does this 82 times a second, which is far too fast to see.
loops of string
2
2 places of biggest movement — the antinodes.
still points
3
Always one more than the loops, because the two ends are tied down and count too.
the frequency
2 × 82.4 =
164.8Hz
what that sounds like
one octave up
1200 cents above the open string
where the still points are — in fractions, and on a real neck
Harmonic 2 stands still at every k/2 of the way along the string. Two of those are the fixed ends — the nut at 0 and the bridge at 1 — which is why there are always 3 of them and only 2 loops.
| still point | along the string | on the neck | touch it |
|---|---|---|---|
| 1/2 | 50.0% | the 12th fret (12.00) |
Those three numbers — 12, 7 and 5 — are why guitarists, bassists and banjo players all know the same three frets by heart. The 12th fret is the halfway point (1/2, harmonic 2), the 7th sits a third of the way along (1/3, harmonic 3) and the 5th a quarter (1/4, harmonic 4). Nobody chose them. The fractions did.
why a light touch is enough
You are not pressing the string down. You are just resting a finger on it — saying “this spot is not allowed to move.” Every harmonic that needed to swing there gets stopped instantly. Every harmonic that was already standing still there never notices you. So one light touch does not add a note — it deletes all the others, and what is left is the harmonic (and its multiples) that had a still point under your finger. Musicians call it a natural harmonic; it is the sound of subtraction.
Now that the nut and the bridge have a job title — the two fixed ends where a still point always sits — the rest of the instrument falls into place too.
tap a number — what does that bit actually do?
Drawn to the real fret formula — every fret line is where it would be on an actual neck. The two green dots are the fixed ends, where a still point always sits.
These are the only parts that actually make the pitch. Pluck one and it swings between its two tied-down ends — and because both ends are held still, it can only ever swing in whole loops: one, two, three, never two-and-a-half. Fat, heavy strings swing slowly and sound low; thin ones swing fast and sound high. That is the whole reason the strings on any instrument get visibly fatter as the notes get lower.
three jobs, and only three
The string, and nothing else. Its length, its tightness and its weight decide the pitch — and the pegs and the frets are just handles for two of those three.
The nut and the bridge. They are why a still point always sits at each end, why the string can only swing in whole loops, and therefore why harmonics exist at all.
The wooden top. A thin string moves almost no air; a big flat soundboard moves a lot of it. None of this changes the note — only how much of it reaches you.
Every string instrument ever built is those same three jobs, shared out differently. Once you can name the job a part is doing, you can look at an instrument you have never seen before — a sitar, a harp, a double bass — and work out most of it from first principles.
For the classroom