Why some notes love each other

Play two notes at once. When their speeds make a small, tidy ratio — like 3 : 2 — they sound smooth. Messy ratios sound crunchy.

two notes at once — the low one stays at 220 Hz (A3)

pick a ratio for the high note

or slide anywhere in between× 1.500 = 330.0 Hz

the ratio

3 : 2

Perfect fifth

musician's ruler

702 ¢

cents = 1200 × log₂(ratio) · 100 ¢ = one piano step

how do they get along?smooth

crunchysmooth

the two waves, and their sum

Tidy ratios make the bottom line repeat in a short, calm pattern. Messy ratios make a pattern that almost never repeats — your ear hears that as roughness.

Small numbers sound friendly

When the high note wiggles exactly 3 times for the low note's 2, their waves line up again and again — your ear loves that. 45 wiggles against 32? They almost never line up. That's the tritone, the “spooky” sound in film music.

The musician's ruler (a gentle log)

Ears hear multiplying, not adding: every doubling sounds like the same jump (an octave). So musicians measure pitch with a ruler built on log₂ — called cents. 1200 ¢ makes an octave, 100 ¢ makes one piano step. It turns × into +, which is exactly what a logarithm is for.

For the classroom

Learning goals

Consonance follows ratio simplicity: 2:1, 3:2 and 4:3 sound smooth; complex ratios like 45:32 sound rough.
A frequency ratio — not a frequency difference — is what defines a musical interval.
Cents introduce logarithms concretely: 1200·log₂(r) turns multiplied frequencies into added steps.

Try this

1.Play 3:2, then slide slowly away from it while holding Both. Have students raise a hand the instant it turns sour, then check how many cents they drifted.
2.Ratio detective: one student picks a preset in secret; the class hears Both and votes silky / sweet / spicy / crunchy — then guesses which numbers made it.