A logarithm you can play

Every key is the one before it times the twelfth root of two. Equal steps along the keys are equal multiplications in pitch — the keyboard is a ruler for logarithms.

play the keyboard — each key is labelled with its frequency

Tap or drag across the keys. The numbers are hertz — wiggles per second. Plug a MIDI keyboard in and it plays this one, lighting each key as you hold it.

climb the ladder — one key at a time

261.6 Hz

0 steps above middle C

one step up = × 1.0595 (the 12th root of 2)

261.6 × 1.0595⁰ = 261.6 Hz

Do it twelve times and every little ×1.0595 multiplies up to exactly ×2 — a doubling, one octave. Equal steps on the keys, equal multiplications in pitch.

the straight keys, the curving pitch

The keys are evenly spaced, but the frequency curves upward — it multiplies, not adds. Undo that curve with a log and you get the straight line of the keys back. The keyboard is a logarithm ruler.

the twelve steps of one octave, from middle C

step	note	× from C	frequency
0	C	×1.0000	261.6 Hz
1	C#	×1.0595	277.2 Hz
2	D	×1.1225	293.7 Hz
3	D#	×1.1892	311.1 Hz
4	E	×1.2599	329.6 Hz
5	F	×1.3348	349.2 Hz
6	F#	×1.4142	370.0 Hz
7	G	×1.4983	392.0 Hz
8	G#	×1.5874	415.3 Hz
9	A	×1.6818	440.0 Hz
10	A#	×1.7818	466.2 Hz
11	B	×1.8877	493.9 Hz
12	C	×2.0000	523.3 Hz

Notice the bottom row: ×2.0000, exactly double the top. Twelve equal multiplications, one clean octave.

For the classroom

Learning goals

Each semitone multiplies frequency by the same factor, 2^(1/12) ≈ 1.0595 — a geometric sequence, not an arithmetic one.
Twelve equal steps multiply to exactly ×2: (2^(1/12))¹² = 2, one octave. Repeated multiplication = an exponent.
Equal spacing on the keys but exponential frequency is the definition of a logarithmic scale — the keyboard is a log ruler you can play.

Try this

1.Start at middle C and press + twelve times, reading the ×-from-C column aloud. Ask the class to predict the frequency at step 12 before you arrive (it doubles).
2.Pick any key, note its Hz, then find the key exactly one octave up and check the number doubled. Try it three times across the keyboard.