Re: Questions whose answers I don't know

--- In crazy_music@y..., xed@e... wrote:
> FROM: mclaren
> So here are some of the things I don't
> know in music:
<snip>
> To turn it around, I also don't know
> any general method of generating transcendental
> numbers -- this is vital, since NJ NET
> tunigns are typically produced by transcendental
> irrational numbers. (To recap: we have
> the counting numbers [positive integers],
> then the integers [positive and negatve],
> then rational fractions [ratios of integers],
> then algebraic irrational numbers [produced
> by the solution of an algebraic equation with
> a finite number of terms], then transcendental
> irrational numbers [produced by the solution of
> a trasncendental equation, viz., an equation
> involving the number e], then real numbers,
> then transfinite numbers, then hypperreal
> numbers [in non-standard analysis], and finally
> hyperinfinite numbers [1 divided by an infinitesimal].

Don't forget our pals the surreal numbers. John Conway invented them
while working on game theory. They include all the usual suspects,
plus the hyperreals, and odd things like (aleph null)/pi.

As far a generating transcendental numbers, you can generate a subset
by taking x=a^b where a is algebraic (but not 0 or 1) and b is a non
rational algebraic number. For instance, 2^(sqrt(2)) is transcendental
.

<snip>
> We now return you to the regularly scheduled lies
> and misinformation and cult fanaticism of the internet.
> ---------
> --mclaren

Awwww gee, Brian. You were soing so well.

John Starrett

<snip>
> As far a generating transcendental numbers, you can generate a
> subset
> by taking x=a^b where a is algebraic (but not 0 or 1) and b is a non
> rational algebraic number. For instance, 2^(sqrt(2)) is
> transcendental

For further reference, the theorem is called the Gelfond-Schneider
theorem. I think this set has measure 1, and in any case these are
dense, so you can generate all the non-just, non equal tempered scales
you could ever use with just these.

p.s. In the spirit of good fellowship, my little tweak "... you were
doing so well" should have had a smiley emoticon.