Hear the Ratios

Sit at a piano or open a keyboard app. Play middle C and G together — a perfect fifth, ratio 3:2. The sound fuses smoothly because the frequencies share a common harmonic. Now play C and F-sharp — a tritone, ratio 45:32. It sounds tense and unstable because the ratio involves larger numbers.

Try the octave test. Play middle C, then C one octave up. They sound like "the same note" because the frequency ratio is 2:1 — every overtone of the higher note is already present in the lower.

Spend ten minutes experimenting with chord pairs. You will hear what the Pythagoreans are traditionally credited with noticing 2,500 years ago — that harmony lives in simple integer fractions.