You are right, this is not a discussion worth having when you can do the experience. If you have a digital synthesizer, just try the experiment that I describe and listen whether you can tell the periodic from the non-periodic waveform. You say that they are perfectly distinguishable, but I just did the experiment (even drawing the lissajous figures) and there's no real difference except the perceived pitch.
I didn't say they were perfectly distinguishable. You have reasons simple ratios don't result in physical waveform attributes relevant to music. I posit you should just really think about that for like five seconds. Otherwise I don't know what to tell you. Look up Pythagoras or maybe what a wave is or something.