🎹 Equal Temperament Explained

About this simulation

12-tone equal temperament (12-TET) divides the octave into twelve identical steps, each a frequency ratio of 2^(1/12) ≈ 1.05946. That uniformity lets instruments play in every key with the same interval quality, but it means almost every interval is very slightly "out of tune" compared to the pure, whole-number frequency ratios used in just intonation — the tuning system built from the harmonics that naturally ring inside a vibrating string or column of air.

How it works

• The frequency wheel shows all 12 chromatic notes as paired bars around a circle: purple bars are equal temperament, teal bars are just intonation, both scaled to height by their frequency ratio to the root note.
• The 3D piano in front plays real tones through the Web Audio API — click a white or black key to hear its equal-tempered pitch, or click a bar on the wheel to hear that note in the currently emphasised tuning.
• Deviation (¢) is measured in cents (1/100 of a semitone) — the difference between the just-intonation ratio and the equal-tempered pitch for the same scale degree.
• The "Play scale" buttons step through the octave so you can compare how the two systems drift apart and snap back into unison at the root and octave.

Controls

• Root note — sets the tonic that just-intonation ratios are built from.
• Concert pitch (A4) — shifts the whole tuning reference, like a tuning fork (415–466 Hz).
• Octave — chooses which octave's absolute frequencies are played and displayed.
• Show just-intonation bars — toggles the teal comparison bars on the wheel.

Did you know?

The major third in equal temperament is about 14 cents sharper than the pure 5:4 just ratio — enough that trained ears and a cappella singers routinely "bend" toward just intonation, even though nearly every piano and fretted instrument on Earth is tuned equal-tempered.