Five-String Banjo · maths & physics
The banjo has a real drum head on the front — so πd and πr² stop being homework and start being an instrument. Stretch the circle, discover why doubling the width quadruples the skin, then meet the short fifth string: three quarters of a length, four thirds of a pitch, the same two numbers flipped over.
The banjo is a circle and a fraction. Every other stringed instrument in this book is a box with a hole in it; the banjo has a drum bolted to the front — a real circle, with a real diameter you can measure — and a fifth string that is deliberately short, a piece of wire that only starts three-quarters of the way along. Two ideas from two different maths lessons, screwed together into one instrument.
A banjo head is about 11 inches across. That single number gives you the whole thing: how much rim to bolt down, and how much skin has to stretch across it.
stretch the head — both formulas, live
C = πd · A = πr²around the rim
π × 11 =
34.6inches
The hoop that clamps the skin down. Grows in a straight line with d.
skin stretched across
π × 5.50² =
95.0sq in
The vibrating surface. Grows with the square of d — much faster.
Nudge the slider and watch the two numbers race. At 11″ — the standard banjo head — the rim is 34.6″ around and the skin is 95.0 sq in. The circumference creeps up. The area bolts.
double the width → double the rim, but four times the skin
A bass drum head is 22″ — exactly twice the banjo's 11″. So the rim is exactly twice as long: easy, obvious, boring. The skin is four times as big, which is neither obvious nor boring, and it is the reason a bass drum sounds nothing like a banjo. Both circles below are on the same scale. Trust your eyes: the big one is not twice the small one, it is four of them.
the rim — ×2
69.1 ÷ 34.6 = 2.00
Because C = πd and d is the only thing that changed. Twice the diameter, twice the way round. A length doubles like a length.
the skin — ×4
380.1 ÷ 95.0 = 4.00
Because A = πr², and the r got squared. Double r and you double it twice over: π(2r)² = 4πr². Four 11″ heads — 4 × 95.03 = 380.13 sq in — add up to exactly one 22″ head.
And that is what your ear is hearing. A drum head's pitch drops as it gets wider: more skin to shift, more mass swinging, a slower vibration, a lower note. The banjo picked a small head on purpose — 11 inches is a bright, snappy, cutting circle. Stretch that same skin over a 22″ hoop and you get the thud at the bottom of a marching band. The whole difference is one squared number. There is more on struck heads in the drum kit's room.
The banjo's little drone string does not start at the top of the neck. It starts at the 5th fret — so it is about three-quarters of a string, and that fraction turns upside down on its way to your ear.
three-quarters of a string
Press any string at the 5th fret and you shorten the ringing part to 2^(−5/12), which works out at 0.749 — near enough three quarters. The banjo's 5th string is that idea made permanent: its tuning peg is screwed into the side of the neck right at the 5th fret, so the string is only ever about ¾ as long as the others. It is the one string on the instrument that is a fraction by construction.
the length
3 / 4
0.749
shorter string
the fraction flips
↺
Pitch is inversely proportional to length. Turn the fraction upside down and you have the pitch.
the pitch
4 / 3
1 ÷ 0.749 = 1.335
higher note — a perfect fourth
That is the whole of inverse proportion, in one string. 3/4 of the length gives 4/3 of the frequency — the same two numbers, swapped over. Musicians have a name for the 4 : 3 ratio: they call it a perfect fourth, and it is the sound of the first two notes of Auld Lang Syne. Hear it: play the 4th string open, then press the same string at the 5th fret.
now the drone itself
The 5th string is shorter and thinner and tighter, so it ends up higher than the fraction alone would put it — the ¾ length buys a fourth, and the gauge does the rest. Play it against the 3rd string and you will hear how far above the neck strings it sits. That is why banjo players call it a drone: it stays on the same high G while everything underneath moves.
Strum a banjo without touching a fret and it plays G major. Count the note names and you will see why — and the frequencies land on 4 : 5 : 6, the tidiest ratio in music.
g D G B D — five strings, three note names
Open G tuning reads g D G B D. Five strings, but write down only the different names and the list collapses to three: G, B, D. Those three stacked up are a G major chord. So the banjo is tuned to a chord already — which is why bluegrass rolls sound full with barely any left hand at all.
and the numbers underneath: 4 : 5 : 6
| note | frequency | ÷ by G, ×4 | tidy ratio |
|---|---|---|---|
| G | 196.0 Hz | 4.00 | 4 |
| B | 246.9 Hz | 5.04 | 5 |
| D | 293.7 Hz | 5.99 | 6 |
Divide all three frequencies by the lowest one and multiply by four, and the answers land within a whisker of the whole numbers 4, 5, 6. Every major chord ever played is that ratio. The middle note is a hair sharp here — banjos, like pianos, are tuned in the compromise that lets every key work — but the shape is unmistakable: the simplest three whole numbers that are not just octaves of each other.
Concept ii was one fraction frozen into a spare string. This is every fraction at once: a string tied at both ends can only vibrate in whole numbers of half-waves, and the places in between that never move are where a banjo's ghost notes live.
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 147 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 × 146.8 =
293.7Hz
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.
A banjo is a neck, a drum and five strings — one of them deliberately short. Every piece is doing one of three jobs: make the note, hold the ends still, or make it loud.
tap a number — what does that bit actually do?
Drawn to the real fret formula. Note the fifth string starting halfway up the neck — and the two green dots, 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 drum head. A thin string moves almost no air; a big stretched skin 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.
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