The capo
#music/capo · Note 6 of 9 · part of Guitar
A capo is a movable nut.
That is the whole device. Everything below is a consequence of it.
The physics
A stretched string's fundamental frequency is
f = (1 / 2L) · √(T / μ)
- L — the vibrating length
- T — the tension
- μ — mass per unit length
A capo changes only L. The string is the same string at the same tension, so tension and mass drop out and what is left is
f ∝ 1 / L
Halve the length, double the frequency. That is why the 12th fret — exactly halfway along the string — is an octave. Not by convention: by measurement.
Where the frets are
Guitars use equal temperament: the octave is split into twelve equal ratios, not twelve equal distances. One semitone is
2^(1/12) ≈ 1.059463
To raise the pitch one semitone, the length must shrink by the inverse:
L′ = L / 1.059463 ≈ 0.943874 · L
So fret n sits at
distance from nut = L · (1 − 2^(−n/12))
Fret 1 lands at L × 0.056126, or L / 17.817 — the old luthier's "rule of
18". Each fret is that same fraction of the remaining length, which is why
frets crowd together as you climb. The spacing is geometric, and it has to be,
because pitch is.
A worked example
The open 5th string is A = 110 Hz.
| Capo | Length | Frequency | Note |
|---|---|---|---|
| none | L | 110 Hz | A |
| fret 2 | 0.891 L | 110 × 2^(2/12) = 123.5 Hz | B |
| fret 5 | 0.749 L | 110 × 2^(5/12) = 146.8 Hz | D |
| fret 12 | 0.500 L | 110 × 2 = 220 Hz | A, one octave up |
What it does musically
A capo at fret n raises every string by n semitones. Standard tuning becomes:
| Capo | 6th | 5th | 4th | 3rd | 2nd | 1st |
|---|---|---|---|---|---|---|
| 0 | E | A | D | G | B | E |
| 2 | F♯ | B | E | A | C♯ | F♯ |
| 3 | G | C | F | B♭ | D | G |
| 5 | A | D | G | C | E | A |
| 7 | B | E | A | D | F♯ | B |
Your shapes do not change. The chord they sound moves up n semitones:
| Capo | C shape | G shape | D shape | A shape | E shape | Am | Em |
|---|---|---|---|---|---|---|---|
| 0 | C | G | D | A | E | Am | Em |
| 1 | C♯ | G♯ | D♯ | A♯ | F | A♯m | Fm |
| 2 | D | A | E | B | F♯ | Bm | F♯m |
| 3 | D♯ | A♯ | F | C | G | Cm | Gm |
| 4 | E | B | F♯ | C♯ | G♯ | C♯m | G♯m |
| 5 | F | C | G | D | A | Dm | Am |
| 7 | G | D | A | E | B | Em | Bm |
Read it as arithmetic: sounding chord = shape + capo frets, counted in semitones. Nothing else to remember.
In sargam: Sa moves, and nothing else does
Sargam and the scale makes the point that Sa is not a pitch, it is home — and that every fret pattern is a set of offsets from wherever you put it. A capo is that idea made physical.
Play open C shapes and Sa = C. Clamp a capo at fret 2 and Sa = D. The
offsets 0 2 4 5 7 9 11 12 are untouched. Re is still two frets above Sa; Ga is
still four; the Ga→Ma and Ni→Sa′ half steps land exactly where they did.
The capo has not changed the music's relationships at all. It has only chosen a different Sa. A singer who cannot reach the top of a song in C is not asking for different music — they are asking for a different Sa, and this is the two-second way to give them one.
Why bother, when barre chords exist
A barre chord transposes too, so the capo is not about avoiding work. It is about open strings.
When you barre, every string is stopped by your finger. With a capo, every string is stopped by a bar of metal or rubber and the strings above it ring open. Those open strings:
- sustain far longer than a fretted note
- vibrate sympathetically when other notes are struck
- carry a different overtone balance — brighter, with more upper partials
That is a timbre you cannot get from a barre chord, in any key. Playing in E♭ with a capo at 3 and G shapes sounds like a G-shaped guitar; playing E♭ with barre chords sounds like a barred guitar. Both are E♭. They are not the same sound.
Three good reasons, then:
- Match a voice without relearning anything.
- Keep the open-string ring in keys that would otherwise demand barres.
- Two guitars, one song — capo one of them and the same chords become two different voicings, which is most of what makes two acoustic guitars sound large.
What it does not do
- It does not change intervals, or the key's internal relationships.
- It does not transpose you. If you are reading a chart in F and put a capo at 1 playing E shapes, you are still responsible for knowing it sounds in F.
- It does not fix a tuning problem. It usually reveals one.
Practical, and one more piece of physics
Place it just behind the fret — as close as you can without sitting on top of the fret wire. On the fret itself it buzzes; too far back and it pulls the string sharp.
Use the least pressure that stops the buzz. Squeezing does not just clamp the string, it stretches it — which raises T in the equation above, which raises the pitch. An over-tightened capo sharpens the strings, and it sharpens the thin ones most. This is the single most common reason a capo'd guitar sounds out of tune when the open guitar was fine.
Retune after fitting it, especially above the 5th fret, where small errors in placement are a larger fraction of the remaining string length.
Partial capos clamp only some strings — five, or three — which is a way of producing an open tuning without retuning anything. A different note.
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