Tenney height times error

One factiod which may or may not have anything to do with what Carl
asked is that taking H*e, where p/q approximates to x, H is the Tenney
height = pq, and e = |x-p/q|, leads to something you can use as a bound.
If I ask for the list of p/q such that H*e is less than 1 for
x=log2(3/2), I get 1/2, 1/3, 2/3, 3/5, 4/7, 7/12, 10/17, 17/29, 24/41,
31/53, 38/65, 55/94... If there's an advantage to using the numerator
rather than squaring the denominator, I don't know it.

At 06:51 PM 5/30/2006, you wrote:
>One factiod which may or may not have anything to do with what Carl
>asked is that taking H*e, where p/q approximates to x, H is the Tenney
>height = pq, and e = |x-p/q|, leads to something you can use as a bound.
>If I ask for the list of p/q such that H*e is less than 1 for
>x=log2(3/2), I get 1/2, 1/3, 2/3, 3/5, 4/7, 7/12, 10/17, 17/29, 24/41,
>31/53, 38/65, 55/94... If there's an advantage to using the numerator
>rather than squaring the denominator, I don't know it.

I think H*e puts more ordering than I'm looking for. I just want
the simplest ratio within an interval.

-Carl

Carl Lumma wrote:

> I think H*e puts more ordering than I'm looking for. I just want
> the simplest ratio within an interval.

If you want the simplest ratio then the semiconvergents should do it. Check a number theory site (like cut the knot) to see if that's what you're doing. If you want the simplest ratios, then check everything in the scale tree within simple bounds that cover the range, until they get too complex.

Graham