Electric Bass · maths & physics
The bass makes waves you could measure with a tape measure. Divide the speed of sound by the pitch and the low E turns out to be 8.33 metres long — then find out why the string that plays it has to be the fat one, and why you feel the note in your chest.
The bass makes waves you could measure with a tape measure. Not a metaphor — an actual length, in actual metres, that you could pace out across the room. Every other instrument here works in ratios and fractions. The bass works in metres, because its notes are so slow that one single wave is longer than the room you are sitting in.
A sound wave has a length. Divide the speed of sound by the frequency and you get it — in metres. Do it for the bass's low E and the answer is bigger than the room.
the formula, and nothing else
λ = v / fSound travels through air at about 343 metres every second. A note at f hertz sends out f whole waves in that same second. So those f waves have to share 343 metres between them — and each one is 343 ÷ f metres long. That is the whole idea. One division.
the headline
343 ÷ 41.2 = 8.33 metres
That is the low E string on a bass. One single wave of it is 8.33 metres long — taller than a classroom ceiling (2.8 times over), longer than 4 people lying end to end. The wave does not fit in the room. It is still a wave; the room is just too small to hold one.
the sum, for this string
343 ÷ 41.2 = 8.33 m
That is 4.9 people long, or 2.8 classroom ceilings.
low = long. high = short. it is an inverse proportion.
You have met inverse proportion before — the reciprocal graph, y = 1/x, the one that swoops down and never touches the axis. Here it is with units on it. Double the frequency and you halve the wavelength. Multiply the frequency by fifty and the wavelength shrinks to a fiftieth. Nothing is added or subtracted; everything divides.
Same bars, same ruler. The bass's low B is 65× longer than that 2000 Hz squeak — because 2000 is 65× the frequency, and the two numbers are reciprocals of each other.
Three things set a string's pitch, and they do not pull their weight equally. Length is a straight inverse. Tension and mass are square roots — and a square root is a much lazier lever.
three knobs, one pitch — Mersenne's law
f = (1 / 2L) · √(T / μ)the string now plays
41.2Hz
×1.00 the open E
three ways to double the pitch — set them and compare
f ∝ 1/L. Halve it, pitch doubles. Cheap, direct — but reaching the bass's low E on a thin guitar-gauge string would need a neck about 1.3 m long, double a guitar's.
f ∝ √T. ×4 the pull for ×2 the pitch; ×9 for ×3. Strings snap well before the maths runs out.
f ∝ 1/√μ. ×4 the mass per metre for half the pitch. The only lever a maker can push hard without breaking anything — so they wind the string thick.
Now look at a real bass. The neck is one length for all four strings. The tensions are near enough equal, or the neck would twist. So the only lever left is μ — and sure enough, the strings get visibly fatter as the notes get lower. The fat string is not decoration. It is the square root, in metal.
The last thing wavelength explains: why a bass is felt as much as it is heard.
A high note is a fast, tight flutter of air — thousands of tiny shoves a second, each one over almost before it starts. The bass's low E does something different. It pushes the air 41.2 times a second in slabs 8.33 metres long: a long, slow, patient shove, then a long slow pull, over and over. Your eardrum reads it as a note. Your ribs, your chest, the floor and the windows — things far too big and heavy to be moved by a flutter — are exactly the right size to be nudged by a shove that long. That is why you feel a bass line in your body and not a piccolo.
one push of low E
4.16 m
half a wavelength — the length of a single squeeze of air
Turn it up, and if you have a real speaker, put a hand on it.
A bass string is tied down at both ends, so it cannot vibrate in just any old shape. It has to fit a whole number of half-waves between the nut and the bridge — and the places in between that never move are the reason harmonics exist.
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 31 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 × 30.9 =
61.7Hz
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.
Six or seven pieces of wood and metal, and every one of them is doing a job you can now name: 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. The two green dots are the fixed ends: whatever note you play, a still point always sits at each of them.
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 pickups, then the amp and speaker. A thin string moves almost no air, so electricity does the carrying and the speaker cone does the shoving. 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