POSSIBLE SOLUTION(S)

1) The middle C scale (all 13 tones.) Each tone is played for 1 second.

Input := 

Do[
 Play[Sin[2 Pi 263 2^(i/12) t], {t, 0, 1}],
  {i, 0, 12}]

2) The middle C Scale: (I simply cut and past the proper tones from the scale above.

3) Checking that octaves increase by powers of 2...

Let's check with octaves based in D.

Input := 

d = 263 2^(2/12);
Do[Play[Sin[2 Pi 2^i d t], {t, 0, 1}],
 {i, -1, 2}]

4) Mary Had a Little Lamb...

Here are the tones that I'll need.

Input := 

c = 263;
d = 263 2^(2/12);
e = 263 2^(4/12);
g = 263 2^(7/12);

Input := 

e1 = Play[Sin[2 Pi e t], {t, 0, 0.4}]

Output =

-Sound-

Input := 

d1 = Play[Sin[2 Pi d t], {t, 0, 0.4}]

Output =

-Sound-

Input := 

c1 = Play[Sin[2 Pi c t], {t, 0, 0.4}]

Output =

-Sound-

Input := 

g1 = Play[Sin[2 Pi g t], {t, 0, 0.4}]

Output =

-Sound-

Input := 

d2 = Play[Sin[2 Pi d t], {t, 0, 0.8}]

Output =

-Sound-

Input := 

e2 = Play[Sin[2 Pi e t], {t, 0, 0.8}]

Output =

-Sound-

Input := 

g2 = Play[Sin[2 Pi g t], {t, 0, 0.8}]

Output =

-Sound-

Input := 

c3 = Play[Sin[2 Pi c t], {t, 0, 1.2}]

Output =

-Sound-

And here's the song. (Before each pasted tone, is the note followed by a number representing the relative duration of the tone.) The tones should be pasted in in the following order:

e1 - d1 - c1 - d1 - e1 - e1 - e2 - d1 - d1 - d2 - e1 - g1 - g2 - e1 - d1 - c1 - d1 - e1 - e1 - e1 - e1 - d1 - d1 - e1 - d1 - c3.