How did one finding
open the next?

In 1636 CE Marin Mersenne listened closely to one plucked string and heard quiet extra tones ringing above it. In 1701 CE Joseph Sauveur measured them: a string vibrates whole, in halves, in thirds and onward, all at the same time.

This museum calls each pure component a partial, numbered from 1 at the fundamental. Sauveur named the whole-number multiples harmonics and coined the word acoustics. An older name, overtone, skips the fundamental, so overtone 1 is partial 2.

This explained the ancient pattern. Two notes an octave or a fifth apart share many of these partials, so they fuse. A rough pair sets neighbouring partials beating against each other, a mechanism Hermann von Helmholtz worked out in 1863 CE. A sampled tone carries almost no partials to share, so every control here uses a reedy tone.

Earlier pages showed two waves meeting again after a few cycles. Partials say why that matters. When the cycles of two tones line up, their partials land on the same frequencies, and each tone reinforces part of the other. A pair that almost lines up puts partials a few hertz apart instead, and partials that close together beat. The wave picture and the partial picture describe one event.

Even the major triad was waiting inside the string: partials 4, 5 and 6 of any deep note spell it out. Those whole-number sections of one string are what the Greek ratios were counting, and the monochord marks the same list on a ruler.

This is the answer discovery 2 could not give. Number decides what the ear enjoys because the number counts partials.