Re: Fibonacci and Stern-Brocot

Pierre,

Thanks for posting this observation. The Stern-Brocot tree is a branch of
Wilson's scale tree, but the tree of Euclid which you describe actually
comes closer to the system of scale-derivation that I was describing in my
last post. It's a very useful tool for enumerating the possible ways of
extending a scale along these lines. The observation that these are
isomorphic is similar to the point that my proposed method captures all the
MOS's derivable by beginning with an arbitrary generator and taking
successive approximations.

jason

Jason

Just a little precision. I don't have a name for the tree mentionned. What
I name Euclid's tree is distinct. It's the Stern-Brocot complete reversing
like that :

1/4 4/3 3/5 5/2 2/5 5/3 3/4 4/1
1/3 3/2 2/3 3/1
1/2 2/1
1/1

If a/b correspond to node (1)-110000100111 in Stern-Brocot,
it correspond to node (1)-111001000011 in Euclid

and vice-versa. It's a simple permutation.

The construction algorithm here is simpler depending only of actual node a/b :

left(a/b) = a/(a+b)
right(a/b) = (a+b)/b.

It's useful for calculating, by example, the node of a/b in Stern-brocot by
Euclid's algorithm. Final operation is simple reversing. Example (53/41
12/41 12/29 12/17 12/5 7/5 2/5 2/3 2/1 1/1) was on

http://www.aei.ca/~plamothe/tangents.htm

There are many ways to easily approach nice properties of rationality that
was often hidden in abstract mathematical theories (networks,
approximations, modules, etc.)

Pierre