Harmonics are the stacked ratios of primes. There's no 9th harmonic, there's the overtonal stacking of the two 5th harmonics. Or more literally, when you play a maj 9th on the piano, your ear is filling in the missing perfect 5th between them.
Harmonics means all the multiples of the fundamental. See https://en.wikipedia.org/wiki/Harmonic_series_(music)
People talk about primes because of their significance in mapping harmonic relationships.
Please don't talk about "overtonal stacking" that's not a thing. "Overtones" means any of the actual tones in a sound that are higher than the lowest one, whether the sound is harmonic or not.
Your piano claim is wrong also. What's happening is that the 4th harmonic of the major ninth matches the 9th harmonic of the lower note. And you do not fill in the "missing 5th" unless you happen to have trained yourself to imagine it.
How does 5+5=9? (Wrong I know but what is right?)
Isn't the 9th harmonic a distinct frequency in a musical note? I thought it was side effect of how physical vibrations ripple out or something...
Where can I learn more about what you're talking about? It's very interesting. Thanks.
But the 5+5=9 part is correct. It's a weird artifact of note-counting. C D E F G is 5 letters. C to G is called "a fifth" (which is a horrendous use of language, but it's a perversion of saying "we got to the fifth letter"). G to D is also "a fifth". Stacking them means C to G and G to D, and you don't count the G twice, that's why 5+5=9 instead of 5+5=10. But if we had actually counted the steps through the letters instead of counting the starting note, it would be 4+4=8, and that's logical and correct, but you can't say that to musicians because that's not how music jargon works.
A perfect 5th is a 3/2 ratioof the base frequency (3rd harmonic). If you square that fraction you'll get 9/4, which is a 9/major 2nd.