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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.

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