Help me understand why does the first inversion in the key of C goes to the 6th instead of the 5th? I.e. E-G-C instead of E-G-B. etc, etc, etc...
An inversion deals with the notes of 1 chord only. It doesn't go to the 6th of anything, you don't add any notes, you don't take any notes out, you just take the bottom note of the chord, and move it to the top. So, using the C major chord:
C-E-G
When you invert this chord, you take the C and move it behind the G, leaving everything else the same:
C-E-G --> E-G-C
Then, you invert it again by taking the E and moving it behind the C, leaving everything else the same:
C-E-G --> E-G-C --> G-C-E
You keep inverting until you end up back where you started. Does that make better sense?