--- 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.