The circle refuses to close
Twelve fifths overshoot seven octaves.
Keep stacking the chain from discovery 4. Twelve pure fifths should land on the same pitch as seven octaves. They miss.
The miss is 23.46 cents, called the Pythagorean comma. Powers of 3 cannot equal powers of 2.
A keyboard with fixed pipes or strings must place that gap somewhere. Lay eleven of the fifths down pure and the twelfth fifth in the chain comes out sour, at 678 cents. Tuners called it a wolf, and its throb is the whole 23.46-cent gap made audible. The howl lives in the upper partials an organ pipe supplies, so the pure fifth and the wolf both use a reedy tone.
In 1523 CE Pietro Aron described a meantone method: narrow most of the fifths by the same small amount, and the common thirds come out close to 5:4. Temperament carries the problem through seven playable steps, including a meantone wolf.
Play the pure fifth, the wolf, then the comma alone. Listen for the 3:2 fifth to hold still, the 678-cent wolf to throb about nine times a second, and the close comma tones to beat.