My guitar teacher told me that we never use the terms "tenths, twelves," etc when we talk about the next octave. I asked why, he told me he didn't even know, so I assume that it's just for identification purposes or something. So does anyone know the real reason why this is, if there is one?



I think it's because 10th is the same as the third, and it doesn't really affect the chord other than accenting that note a bit more. The same goes for 5th/12th. Things like the 9th, 11th etc actually add something new to the chord which is why we consider them worth naming
When you have an interval larger than an octave it usually has the same harmonic purpose as the interval formed an octave below, therefore when doing a bare bones analysis of a chord, say involving the notes C4, E5, and G6 (Octave ranges), its still a C major chord, regardless of the Major 10ths and stuff that are formed from it.

But, when adding a 7th or 9th to a major/minor chord, you should definitely include it, as its part of the tonality of the chord. Plus, CM7add2, or addX might imply that its a more closed voicing rather than stacks of thirds leading up to the final extension.