This is my answer (but it is not just a kind of Q-A (as I really didn't see this relationship before!)
Starting with the smallest primes:
They are 1,2,3,5,7,11 ...
That's what we have learnt in the primary school: these numbers can only be divided by 1 and by themselves.
the overtones, (harmonics) and the frequencies: note that I've edited wavelength to length of string!
in the 1. column of the chart is Length
of string in the 2. the Frequency factor in the 3. the resulting tone
1 => 1 = e.g. C
1/2 => 2 = c
1/3 => 3 = g
1/4 => 4 = c (8va of c 1/2 x 1/2)
1/5 => 5 = e
1/6 => 6 = g (8va of g 1/2 x 1/3)
1/7 => 7 = b7
1/8 => 8 c (8va of c 1/2 x 1/4)
1/9 => 9 d (5th of g 1/3 x 1/3)
1/10 => 10 e (8va of e 1/2 x 1/5)
as we can see the bold frequencies are the new overtones all others are multiples of already derived overtones which can obviously be divided by an other prime that has already been derived. This tabula shows that each new harmonic tone must obviously be identical with the next prime number that can't be divided in another number than one or itself. That's why some may say this is trivial but it has not been trivial to me until now! There is not a similarity or correlation between the harmonics and the primes. They are identical.
With this last sentence I meant to say:
Prime numbers and overtones are one and the same thing - only in different terms and medias, as all not-primes numbers must be octavas or fifths of overtones of those we have already developed:
But I see now what I have missed: the pythagorean comma: 7 octavas are not equal 12 fifths. I had completely forgotten this point!
Here I have found an aritcle saying the same:
The harmonic numbers are equivalent to the values of the source harmonics in all previous discussions of harmonic evolution. Since we only studied those up to Quintality, most were prime. But as we can see from the chart, numerically, most source harmonics and harmonic multipliers are non-prime, any of which is a product of primes – its own series of harmonics and multipliers. We will call the first in the series its root harmonic, which is thus distinct from the product itself – the source harmonic. It determines the root spiritual or physical nature of the product’s lineage.
It says equivalent what I think this is better than one and the same.
and here's another link that says they primes and harmonics are eyuivalent: