the thing (any thing) that vibrates creates both fundamentals and simultaneously creates a new set of harmonics, again, each time
Not if it vibrates as a pure sine wave, when there are no harmonics. You would be better off looking at Fourier analysis which shows that any repeating waveform can be represented by a set of sine waves - the fundamental and the odd harmonics.
Strangely, audiophiles tend to inhabit the frequency domain and talk about bass, midrange and treble, whereas in real life sound exists in the time domain which gives rise to discussions about Prat.
You can mathematically convert that repeating waveform into a set of frequencies using a Fourier transform. You can also take the frequency set and convert it back to (almost) the original waveform.
There is an unfortunate glitch as the original waveform approaches a square wave in form. The reconstructed waveform has a sharp spike surrounded by ripples. You can see these as fringes if you look at sharp edges in a magnified digital photograph from a compressed file.