An interesting

scale discovered by

psychologist Roger Shepard and mentioned in

Douglas Hofstadter's book

Gödel, Escher, Bach: An Eternal Golden Braid. In it, the

volume of each

pitch in a scale is

weighted according to its distance from

middle C (or whatever pitch it happens to be centered on). As the pitch approaches the middle tone, it gets louder; after the middle tone, it softens into inaudibility. The example in

the book (fig. 148) has the scale extend from C1 to C5.
Thus, a

chromatic scale is played using all

octaves of the note within the

range, so that when you end up at the end of the scale, you're playing the same notes as the beginning of it. The scale can go up

forever and still stay in the same place.

Hofstadter, with his colleague Scott Kim, actually used this scale using a computer to play J.S. Bach's *Endlessly Rising Canon*, so that instead of ending up an octave above its beginning, it ended up right where it started. Weird stuff.

To hear the scale for yourself, type this into Mathematica:

Play[Sum[(1 - 1/24*Abs[12*n + Floor[4*x] - 24])*
Sin[2*Pi*110.1*2^(n+1/12*Floor[4*x])*x], {n, 0, 3}],
{x, 0, 3}]

This particular scale happens to be centered around

A below

middle C. To change the

center, take the

frequency of the center tone you want,

divide it by 4, and

substitute it for "110.1".

Repeat as desired. If you repeat it multiple times, you can sense the continuity between the end and the beginnning.