Just Intonation

The perfect fifth is 3:2 — the third harmonic dropped down an octave. The perfect major third is 5:4 — the fifth harmonic dropped down two octaves. These are not approximations but exact integer ratios, and when two tones in a just-intoned interval sound together, their waveforms align into a single periodic pattern. This is the "ringing" quality of baroque and folk music played in one key.

The major triad uses the ratios 1:5/4:3/2 (root, major third, fifth). The minor triad swaps the major third for a minor third of 6:5 — found by dividing the fifth (3/2) by the major third (5/4): 3/2 × 4/5 = 6/5. The result, 1:6/5:3/2, is also a set of perfect ratios. But other thirds and fifths in the scale will not be true — only the triad built on the tonic rings perfectly.

Just intonation actually has two different whole tones: T₁ = 9/8 = 1.125 and T₂ = 10/9 = 1.111, plus a semitone S = 16/15 = 1.0667. This means the same scale degree can have a slightly different pitch depending on the chord — a richness that equal temperament smooths away.