Harmony & Scales

Ratios, octaves and twelve semitones

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Octave = 2:1, fifth = 3:2, fourth = 4:3. Simple ratios sound consonant.
• Equal temperament: each semitone multiplies the frequency by $\sqrt[12]{2} \approx 1.0595$.
• Twelve pure fifths overshoot seven octaves (the Pythagorean comma), so tuning is always a compromise.

Steps

1. The octave
2. Seeing consonance
3. A perfect fifth
4. Equal temperament
5. Seven semitones
6. Why not use pure fifths?