The
Science
of Piano
A piano is a machine for turning a finger’s push into vibrating air. Felt hammers strike steel strings under huge tension; a wooden soundboard makes the sound loud; dampers stop it. The physics explains why bass strings are thick, why every piano sounds different, why tuners “stretch” octaves, and how to protect the ears you listen with.
- 00First Principlesp.2
- 01How a piano makes soundp.4
- 02Strings: length & tensionp.5
- 03Overtones & timbrep.6
- 04Inharmonicity & tuningp.7
- 05Digital pianos & your earsp.8
- ✦Self-check · Make it · 5 Questions · Side Questp.9
You'll need: pencil, calculator, any keyboard (real, digital, online or paper), rubber bands, a small box, a ruler, and a free tuner or sound-level app.
Principles
A first principle is a basic truth you can build on. All sound is vibration travelling through air. A piano makes strings vibrate, makes those vibrations bigger and lets you control when they start and stop.
What do I already know?
What is necessary?
Everything in this module is built from four bricks. Rate yourself on each one, then check the example to see if your rating holds up.
What is the objective?
By the end, you’ll be able to explain how a piano makes sound and how to listen safely.
- 01Trace the path from key to hammer, string, soundboard and ear.
- 02Use length and tension to predict a string’s pitch.
- 03Explain overtones and why instruments sound different.
- 04Explain inharmonicity and stretch tuning.
- 05Compare acoustic and digital pianos.
- 06Use decibels to plan safe listening and practice.
The list on the left is ours. Now make it yours. What do you want out of this?
How much do you know about the science of piano right now? Shade the boxes in pencil. You'll come back at the end (p.9) and shade it again in pen.
Why start this way? When you know what you already have (1), what you need (2) and where you're going (3), you learn faster and you notice the gaps. Every module on the Piano Path starts with these same three questions.
How a piano makes sound
Inside every acoustic piano is the same chain: a hammer hits a string, and a soundboard makes the vibration loud.
From finger to air
Lots of strings
- Most notes have 3 strings tuned together; lower notes have 2, the lowest bass notes 1.
- A modern piano has about 230 strings.
- Together they pull with roughly 16–20 tonnes of tension, so the frame is cast iron.
Exact numbers vary by model; concert grands are often said to reach higher totals.
Three feet, three jobs
| Pedal | What it does |
|---|---|
| Right: sustain (damper) | Lifts all dampers, so notes keep ringing and strings resonate together |
| Left: soft (una corda) | On grands, shifts the hammers so they hit fewer strings; on uprights, moves hammers closer |
| Middle: sostenuto (if present) | Sustains only notes held when it’s pressed; on some uprights it’s a practice mute |
Sympathetic strings
On an acoustic piano: silently press and hold middle C so its damper lifts. Then strike the C an octave lower, hard, and let go. Can you still hear middle C ringing? That’s resonance. (On a digital piano, check if it has a “string resonance” setting.)Why must the hammer bounce off the string straight away?
Strings: length & tension
Every piano string is a physics equation you can hear.
What sets a string’s pitch
- Shorter string → higher pitch. Halve the length: one octave up.
- Tighter string → higher pitch. To double the pitch you need 4 × the tension.
- Heavier string → lower pitch. That’s why bass strings are wrapped in copper wire.
The rule: f = (1 ÷ 2L) × √(T ÷ μ), where L is length, T tension and μ mass per metre.
Tension of an A4 string
The bass problem
A0 is 4 octaves below A4. Using the same plain wire and tension, its string would need to be 24 = 16 times longer: 16 × 0.4 m = 6.4 m. No room has space for that! Instead, makers wrap bass strings with copper (more mass), use somewhat longer strings and cross them over the others (overstringing, Lens 4).
Rubber-band lab
Stretch a rubber band around a small open box. Pluck it. Then slide a pencil under it to shorten the vibrating part to half. Did the pitch jump about an octave? Use a free tuner app to check. Then stretch it tighter: what happens?A string plays 220 Hz. What would it play at half the length? At 4 × the tension?
Overtones & timbre
One piano key sounds like one note, but it’s really a blend of many.
One note, many pitches
A vibrating string moves as a whole and in halves, thirds, quarters… at the same time. Each part adds a quieter, higher pitch called an overtone or harmonic. Their frequencies are (nearly) whole-number multiples of the lowest one, the fundamental.
| Harmonic | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Hz (A2 = 110) | 110 | 220 | 330 | 440 | 550 | 660 |
| Note | A2 | A3 | E4 | A4 | C♯5 | E5 |
Hidden chords
Why a piano sounds like a piano
Timbre (tone colour) depends on which overtones are strong, and how the sound starts and fades. A piano note has a sharp attack from the hammer, then fades away; an organ or synth pad can hold steady.
- Harder hammer felt → brighter sound, more high overtones.
- Where the hammer hits matters: makers usually strike near one end, roughly 1/7 to 1/9 along, which affects which overtones ring.
- Players change timbre too, through how fast they press and how they pedal.
Hear the harmonics
On an acoustic piano, silently hold down the C an octave and a fifth above a low C (G3 above C2). Strike the low C hard and release it. The G keeps sounding: it’s the 3rd harmonic of the low C!What is the 8th harmonic of A2 (110 Hz)? What note is it?
Inharmonicity & stretch tuning
Real strings are imperfect, and good tuning works with that.
Inharmonicity
Real piano strings are stiff steel, not perfectly floppy. Stiffness makes the higher overtones come out slightly sharp: harmonic 2 is a bit more than 2 × the fundamental, harmonic 3 a bit more than 3 ×. This is called inharmonicity. Short, thick strings (in small pianos and the very top and bottom) have the most.
This is one reason big concert grands often sound “cleaner” than small uprights.
Stretch an octave
The Railsback curve
In 1938, physicist O. L. Railsback measured well-tuned pianos and found that, compared with exact equal temperament, the high notes were tuned sharp and the low notes flat: a gentle S-shaped curve. Amounts differ for every piano, often a few cents in the middle and up to about 30 cents at the extremes. Many technicians now use apps that measure each string’s inharmonicity, but tuning by ear is still widely taught.
Octave check
Play A3 and A4 together, then A6 and A7 (if you have them). Do the high octaves sound wider or narrower than the middle ones? On a digital piano, find out if it has a “stretch tuning” setting.Why would a small upright piano need more stretch than a concert grand?
Digital pianos & your ears
Not every piano has strings, and every pianist needs healthy ears.
Strings vs speakers
| Acoustic | Digital | |
|---|---|---|
| Sound | Real strings and soundboard | Recorded samples or a computer model |
| Tuning | Needs a technician, about 1–2 times a year | Never needs tuning |
| Volume | Can’t turn it down (some have mutes) | Volume knob and headphones |
| Touch | Real hammer action | Weighted keys imitate it, some very well |
| Cost & size | Larger, usually more expensive | Smaller, often cheaper |
Safe listening time
A common workplace guideline: 85 dBA for 8 hours, and every +3 dB halves the safe time.
| Level (dBA) | 85 | 88 | 91 | 94 | 97 |
|---|---|---|---|---|---|
| Safe hours | 8 | 4 | 2 | 1 | ½ |
Guidelines differ by country and agency; these are rules of thumb, not medical advice.
Decibels add up fast
Decibels are logarithmic: +10 dB means 10 times the sound energy, and +20 dB means 100 times. A piano played loudly can reach the 80s or 90s up close, and practice rooms are small. Hearing damage builds slowly and doesn’t heal.
- Keep headphones at about 60% volume or less, and take breaks.
- Musicians’ earplugs lower volume evenly, so music still sounds clear.
- Ringing ears or muffled hearing after playing are warning signs: rest them, and see a professional if it continues.
Measure your practice
Use a free sound-level app (they’re approximate). Measure soft playing, loud playing and your headphones at your usual volume. How long could you listen each day, using the table?A rehearsal room measures 94 dBA. Using the rule, how long is a safe daily exposure? What if you lower it by 6 dB?
Quick self-check
Six questions, covering every stop. Show your thinking. Answers are upside down at the bottom; no peeking till you're done.
1 · What does the damper do?
2 · What happens to pitch if you halve a string’s length?
3 · Why are bass strings wrapped in copper?
4 · What is the 3rd harmonic of a 110 Hz note?
5 · What is stretch tuning, and why is it needed?
6 · Using the +3 dB rule, how long is safe at 91 dBA?
Re-shade your knowledge meter
How much do you know about the science of piano now? Flip back to p.3 and shade the meter again in pen (or shade this one). Did any of the four bricks on p.2 move from "Not yet" to "Got it"?
The string lab
Test the string rule yourself: change the vibrating length of a string and measure the pitch with a tuner app. Does physics match your ears?
1 · My setup
- Rubber band over a box (a simple “monochord”)
- Guitar, ukulele or other string instrument
- A piano with the lid open (with permission)
- A digital piano or online keyboard (for comparison)
2 · Prediction
If I halve the length, the pitch will…
3 · Results
| Vibrating length (cm) | Tuner reading | Hz (if shown) |
|---|---|---|
4 · Must-haves
- At least 5 lengths measured
- Half length tested against full length
- A graph of my results
- One source of error I noticed
5 · What I found
Did halving the length double the frequency? What surprised you?
Safety first: rubber bands and wires can snap: keep your face back and wear glasses if you have them. Never put tools inside an acoustic piano without the owner’s permission. Keep the volume comfortable.
5 Questions
I Still Have
This module didn't answer everything. Good. Write 5 new questions it didn't answer, one of each type below. There are no wrong questions, only ones nobody's asked yet.
★ Star the one question you'd most like answered. You'll need it on the next page.
1What I'll find out
Break your big question into 2 or 3 smaller ones you could actually answer.
2Where I'll look
3How I'll share it
Who I'll share it with:
Done by: