4 min read

What does a piano sound like?

Twelve equal steps, one power law

For students, teachers, and the simply curious. No prior knowledge of logarithms needed — if you can do 3 + 5, you can follow this to the end.

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What does a piano sound like?

Play a C on the piano. That note vibrates 262 times per second (262 Hz).

Now play the C one octave higher. That note vibrates exactly twice as fast: 524 Hz.

C262C♯277D294D♯311E330F349F♯370G392G♯415A440A♯466B494C523Click a bar to hear the note — each is ×1.0595 higher in pitch

Between those two C's there are 12 keys. Here's the clever part: the piano is tuned so that each key is the same percentage higher in pitch than the one before. Not the same amount higher — the same factor.

Let's call that unknown factor r. You start at 262 Hz and multiply by r for each key:

×r×r×r···×r×rC₄262 HzC₅524 Hz12 steps of ×r = ×2 (one octave)

After 12 keys you arrive at 524 Hz, which is exactly 2 × 262. So those twelve multiplications by r together multiply by 2:

×r×r×r···×r×r1212 × multiply by r = ×2

Repeated multiplication by the same number — that's a power:

r ↑ 12 = 2

Look at this formula: it has the shape a ↑ b = c, where r is in the position of a — the left side of ↑. From the rule card: left side unknown → use ↓.

r = 2 ↓ 12
2 ↓ 12(2 down 12)

About 1.0595. So each piano key is roughly 6% higher in pitch than the previous one.

In school, this answer would be written as ¹²√2. Same number — but good luck seeing where it came from. With the notation, the equation r ↑ 12 = 2 directly mirrors what the piano does, and ↓ gets you the answer in one step.

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2 ↑ 3 = 82 = 8 ↓ 3left unknown? use down3 = 8 ⇓ 2right unknown? use double-down