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.
Greek experimenters marked the friendly stopping points on an instrument built for measuring, the monochord, and found plain fractions at every one.
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 stopping point, then walk all four stops. Listen for the pitch to rise as the vibrating length shortens, and for the half and two-thirds stops to sound closely related to the open string. Shorter lengths vibrate faster in simple frequency ratios. Those audible relationships turn a ruler into a map of musical intervals.