Hear the ratios
Turn your sound on. Each button plays the base note, then a note whose frequency is the base multiplied by a ratio, then both together. Simple ratios sound smooth. Listen for the slow wobble in the last button, Twelve fifths vs seven octaves.
What's going on
A note is a vibration, and its frequency is how many vibrations happen each second. Double the frequency and you get the same note an octave higher. Multiply by 3/2 and you get a perfect fifth, the interval at the start of "Twinkle, Twinkle". Two notes sound smooth together when their frequencies are in a simple ratio, because their waves line up often.
Go up twelve fifths from a note and you should land back on the same note seven octaves up. But (3/2)¹² = 129.75 while 2⁷ = 128. They miss by about 1.4 percent, the Pythagorean comma. The last button plays 440 Hz against 440 × 129.75 / 128 ≈ 446 Hz and you hear the mismatch as a beat, a wobble about six times per second.
Pianos solve this by making every semitone the same ratio, the twelfth root of 2, so twelve of them give exactly 2. Every interval except the octave is then very slightly out of tune, and almost nobody notices.
Sheet: Why there are twelve notes.