[Math] Generalizations of the golden and silver ratios, and their significance

Question

$\Phi$, or the golden ratio, is basically $\frac{a+b}{a}=\frac{a}{b}$. The silver ratio corresponds to a similar idea of: $\frac{2a+b}{a}=\frac{a}{b}$. I've read on Wikipedia that both of these ratios are well known and have some appearance in nature (not the mystical hogwash stuff). In addition, there is the relation of the Fibonacci and Pell Numbers respectively.

Funny enough, I stumbled on these ratios originally by myself just messing around with numbers. I also noticed that the silver ratio minus 1 approximates $\sqrt{2}$ (which is how I came to be fooling around with this).

However I don't seem to find anything about further manipulation of this ratio, such as: $\frac{3a+b}{a}=\frac{a}{b}$, $\frac{4a+b}{a}=\frac{a}{b}$ and so forth. Do these ratios have any special properties that are found in nature? Perhaps called the copper ratio or something?

Answer 1

Answer thanks to Foo Barrigno:

They have less and less importance as you increase in the height. If you look at the wikipedia page on the silver ratio (http://en.wikipedia.org/wiki/Silver_ratio) you can find section on "silver means" that generalizes it to "metallic means" and then specifically mentions the "bronze mean".