A logarithm you can play
iWalk up the keys and watch every note multiply by 1.06. Twelve steps = one doubling.
for teachers: exponents · logarithms · 12th root of 2Open →
Never touched a keyboard? Start with the big button — you will be playing a tune in about five minutes.
eighty-eight keys · the math & physics of the keyboard
The piano is a logarithm you can play.
Every other instrument hides its math. The piano puts it in a straight line you can touch: equal steps along the keys are equal multiplications in pitch. Learn the keyboard and you've learned exponents, logarithms, and the powers of two — by ear.
tap a marker to learn the part
Lid up, you can see the whole harp: the strings run long on the left and short on the right — that is the pitch, laid out as a length.
play it — a real grand piano, sampled note by note
Tap or drag across the keys. Every white key is marked; the bold letters are the C's — your landmarks.
the same still points, on a piano string
Where the hammer hits — and the note the piano leaves out on purpose
A piano string is tied down at both ends, exactly like a guitar's, so it plays by exactly the same rule: it can only vibrate in whole loops, and every one of those loops has still points at each end. But a piano string is hit, not plucked — and where the hammer lands turns out to be a decision somebody made, on purpose, about which note to leave out.
FractionsPhysics: waves
move the hammer and watch harmonic 7 wake up or go quiet
strength of harmonic n ∝ sin(nπp)where the hammer hits1/7.00 along
p = 0.143 of the way along
draw harmonic
how much of each harmonic this hammer wakes up
Tap a bar to hear that harmonic on its own. A bar goes to nothing when the hammer has landed on one of that harmonic's still points — you cannot start something moving by hitting the one spot that never moves.
the note the piano quietly leaves out
Set the hammer to 1/7 and watch the 7th bar collapse. Harmonic 7 has still points at 1/7, 2/7, 3/7… — so a hammer landing on 1/7 arrives at a place that harmonic never moves, and barely wakes it at all. The same blow also flattens the 14th and the 21st, for the same reason.
And harmonic 7 is exactly the one you would want asleep. Seven times the frequency lands 1200 × log₂7 = 3369 cents above the note — about 31 cents below the nearest key on the keyboard, roughly a third of a semitone out. Against the piano's own tuning it sounds sour and clangy. Hit the string a seventh of the way along and it never really shows up.
Real pianos use somewhere between 1/7 and 1/8, moving nearer the end as the strings get shorter and higher. It is not a rule of nature; it is two centuries of makers listening, and landing on the same fraction.
hear it for yourself
The third button is what a piano would sound like if its hammer hit in the wrong place. Listen for the buzz sitting on top of the note.
the math & physics of this instrument
Four ways the piano is math.
Each one is a short lab you can run in front of a class — with a sound to poke and a “for the classroom” card at the end.
Octaves are doublings
iiEvery octave doubles the number: 55, 110, 220, 440. Powers of two you can hear.
for teachers: powers of 2 · exponential growthOpen →The spiral that never closes
iiiWhy 12 keys? Twelve fifths nearly equal seven octaves — but miss by a whisker.
for teachers: ratios · powers · the Pythagorean commaOpen →Why the piano cheats — and you can hear it
ivPure fifth vs piano fifth: hear them wobble against each other. That wobble is the math.
for teachers: beats |f₁−f₂| · equal temperamentOpen →Twelve notes, then it starts again
vKeep adding fifths and you tour all twelve notes. Add thirds and you only ever see three. That's gcd.
for teachers: mod 12 · gcd & coprime · irrational numbersOpen →
New to it? Learn the basics — notes, chords & first tunes →
Wondering why the hammer hits where it does? See the still points on a piano string →
Bringing this to a classroom? See the school workshops → or explore the cross-instrument Math Lab →
the maths & physics of this instrument
The piano is a logarithm you can play with both hands.
Eighty-eight keys laid out in a straight line, evenly spaced — and the frequencies behind them are not evenly spaced at all. They multiply. Walking up the keyboard one key at a time means multiplying by the same number over and over, which is exactly what a logarithmic scale is; the piano just happens to be one you can sit at.
i
Every key multiplies
Indices · exponential growth
f = 440 × 2^((n − 69) ÷ 12). One key up is × 2^(1/12) = 1.0595. Twelve of those compound to exactly 2 — an octave.
Equal steps to your hand, equal ratios to your ear: the definition of a logarithmic ruler.
open it →
ii
Twelve keys, a clock, and a common factor
Modular arithmetic · highest common factor
Step by s semitones round the twelve and you visit 12 ÷ gcd(s, 12) notes before returning. Fifths (s = 7) visit all twelve; major thirds (s = 4) visit only three.
The circle of fifths isn't a music-theory diagram to memorise — it's what gcd does, made audible.
open it →
iii
A fraction the piano cannot play
Rational versus irrational numbers
A pure fifth is 3/2 = 1.5 exactly. The piano's fifth is 2^(7/12) = 1.4983 — irrational, and deliberately slightly wrong.
Here is a real instrument choosing an irrational number on purpose, so that all twelve keys work.
open it →
iv
Two notes, and a throb you can count
Subtraction as a rate
Play f₁ and f₂ together and the sound pulses |f₁ − f₂| times a second. 440 and 444 throb four times a second.
A subtraction you count out loud — and exactly how a tuner works.
open it →