Double-flats vs. Naturals

In yesterday's post I told the story of the "why?" question to a colleague who I knew would understand it.  I think he truly did.

He quickly countered with a reference to another Bartok piece, this one from the Mikrokosmos, in which there is a fleeting A-double-flat, "G!" he pointed out.  This brings up two fascinating issues:

l.  My student is blind, therefore does not read.  Could he get the difference between G and A-double-flat?  Could I get him to hear the difference?  Is there a difference?

2.  When I was a child I was convinced that learning to read music efficiently was the clue to success as a musician, so I read voraciously.  The things I did not readily understand I changed to make them accessible.  Double-flats, for example.  I transposed them all into naturals.  There was no such thing as a double-flat, in my esteemed 12-year-old opinion, and nobody took the trouble to indicate that such a thing did not only exist, but might make a significant difference.

They are both played on the white key that customarily called G, so what, if anything, is the difference?  The white key G struck at the point of maximum leverage, i.e., at the outward tip of the key, produces a full range of G overtones.  Moving the finger in toward the fallboard reduces the number of audible G overtones until, at the innermost position of poorest leverage it is possible to hear it as something other than a G. 

Suddenly there are no longer merely twelve notes per octave, seven white and five black keys; there are twenty-one white plus five black keys, since each white key takes on the potential of being either a double-flat or a double-sharp.  (C might be construed either  as D-double-flat or as B#.)  I do count notes - Tonal Refraction has provided insight into the importance of the number of tones in a piece.

Despite the fact that some music theorists walk on the ceiling when I point this out, it is not uncommon for Brahms to use twenty-one. 

One sonata I studied has twenty-four.  Who wrote it?  Schubert.