Q:

Is there a relationship between prime numbers and music?

Is there a relationship between prime numbers and the series of overtones and their frequencies?
I've found links to Euler's research about math and about music, and I got goose bumps when I found that here is a whole universe of correlation. 
Can someone breakdown this theory for dummies like me?
http://www.tonalsoft.com/monzo/euler/euler-en.aspx
(http://eulerarchive.maa.org//pages/E033.html)
I mean especially the correlation of string-length, overtones, and prime numbers.
After thinking about it and trying to explain it to me, I've found that this is very simple, but I never realized this relationship. 

A:

There is one observation with respect to primes. No prime power (except 0) is a power of any other prime. Thus no number of stacked fifths will be equal to any number of stacked octaves. (Taking a fifth to be a ratio of 3:2). Thus, any useful music over more than a few notes will need tempering. 
"Pythagorean" tuning uses only ratios using 2 or 3. "Just" tuning uses ratios using 2, 3, and 5.  The Pythagorean third becomes 81/64 and the Just third is 5/3; these don't match. 
Other than this, there isn't much except for figuring out how to temper the difference between (for example) 7 octaves and 12 fifths in a practical manner.

A:

I would say "trivially, yes".
Yes, because music can be analyzed and is often created in ways that involve numbers and fundamentals of algebra (such as addition and multiplication), and once numbers become involved, and particularly when multiplication (and division) is involved, prime numbers become significant.
Trivially, because every branch of human thought that can be analyzed and/or developed using numbers and basic algebra, and particularly multiplication/division, has a meaningful interaction with prime numbers.
That is because the very nature of prime numbers is they create patterns in how numbers in general are multiplied and divided.
Here is a list of only some of the areas of music where prime numbers have important interaction with the concept because of the usefulness of multiplication and/or division of whole numbers:

frequency ratios and intervals
time signatures and rhythms
tuning, intonation, and scales
resonance, damping, and instrument construction
acoustics
etc.

There are overlaps between some of those areas, as many will surely notice.
Specifically regarding Euler's ideas, they don't seem particularly helpful to me, at least from a musical point of view. They might be interesting in their own way, but I think there's a reason that people don't frequently refer to these ideas when talking about music. Also, Euler's mathematical innovations are much more exciting.

