A string and a ruler
Divide a string and the ear agrees.
Stretch a string, pluck it, then press it down at its middle and pluck again. The new tone sounds like the old one moved higher, so alike that we give both the same name. Stop the string at two thirds and a different tone appears, one that sits against the open string without any grinding. The distance between two pitches is an interval, and this museum is a map of them.
Greek experimenters marked the friendly stopping points on an instrument built for measuring, the monochord, and found plain fractions at every one. A shorter length vibrates faster, in the same simple ratio the fraction gives. The monochord in the lobby sounds 110 Hz. This bench holds a string half that length, so its open tone is 220 Hz, the same name higher.
A famous legend says Pythagoras first heard these ratios at a forge, in hammers weighing 12, 9, 8 and 6 pounds. A struck hammer rings at a pitch set by its shape and length, so the weights prove nothing, and the tale fails as physics.
People kept telling the forge story because it promised that number decides what the ear enjoys. Measurements on stretched strings bear that promise out.
Play each stop alone and hear the pitch rise as the string shortens. Then hold two thirds against the open string, and eight ninths after it. Listen for one pair to settle and the other to grind.