Harmonic Series

The physical reason certain intervals sound consonant

The harmonic series is the natural pattern of frequencies that vibrating strings (like those on the guitar or in a piano) and air columns (like in your voice or a trumpet) produce. Every sound has a fundamental frequency f, and comes with varying amounts of other higher frequencies that shape its timbre. These frequencies occur at integer multiples of the fundamental f: 2f, 3f, 4f, etc. The frequency ratios between consecutive harmonics correspond approximately to intervals in equal temperament:

2:1 = octave
3:2 = perfect fifth
4:3 = perfect fourth
5:4 = major third
6:5 = minor third

The later an interval appears in the harmonic series the less consonant it sounds. Music theory revolves around describing how we aim to pattern consonance and dissonance intentionally.

One frequency, and no harmonics

Harmonic seriesEach row is one harmonic of the note — n times the fundamental's frequency, about 1/n of its strength — faster and fainter as it climbs. The rows in ink are where the chosen interval first arises. The bottom row is the one your ear actually receives: those rows added, a single wave. The shaded band is one repeat of it — the shorter that pattern, the sooner the ear settles on it as one sound. Harmonic 1 alone — the fundamental, a pure tone at f. Pick an interval above to see where it first appears. The harmonic series, interval by interval →

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