How can you construct the proper diatonic scales from each note on the circle?

There's a number of ways. One way is to use this pattern: major 3rd, minor 3rd, major 3rd, minor 3rd, minor 3rd, major 3rd, minor 3rd.

For example, let's use C. Starting at C we go up a major 3rd (or 4 half-steps) to E. Then go up a minor 3rd (or 3 half-steps) to G. Then go up another major 3rd to B. Then another minor 3rd to D. Then another minor 3rd to F. Then a major 3rd to A. Then a minor 3rd to C'. So the order is:

C E G B D F A C'

Now arrange these notes starting at C in a circle in alphabetical order: C (C') D E F G A B and draw lines starting from C to each note in the order obtained as shown above. You get a 7-pointed star! And this works with any key and whether major or minor.

Pick another key--say Ab. The order we obtain is:

Ab C Eb G Bb Db F Ab'

Now arrange them in a circle in alphabetical order and then draw a line from one note to the next in the sequence above--a 7-pointed star.

Another demonstration of how music and geometry are related.