heheh. C13 definitely has a 9th in it. i remember cuz its another one of those things that was never really explained to me.
the 6th 'A' is different to the 13th 'A' obviously cuz they are an octave apart. its the same key register but it'll sound different in terms of pitch and frequency which is an important distinction.
the reason we use 'add' is when we want to add 9th or 11th or 13th or whatever single notes to an existing chord, but not the notes in between. only really applies with the larger chords.
if a C13 is C, E, G, Bb, D, A
a C9add13 would a C9 chord (C, E, G, Bb, D) + 13th (A)
by extension
a C7add13 would a C7 chord (C, E, G, Bb) + 13th (A)
no 9th.
so a C7 add 6 = C, E, G, Bb + the 6th note of the scale A (not the 13th A which has a much higher frequency)
because you are counting up that would kind of mean you are playing C, E, G, A, Bb
which i thought would be C6 + 7th. im confused about that example and it would be cool if you could check with your piano teacher to clear this up
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