Why do call jazz extensions 9/11/13 instead of 2/4/6?

A More Practical Way Of Thinking 

The main reason for this is that it’s more practical to think of these notes as the extensions (9/11/13) than the lower tones 2/4/6.

For example if you are reading a lead sheet, you will frequently see symbols asking for extended chords such as C-9, D13#11 or G13b9.

If you memorise the scale degrees as 9/11/13 – you will find it much easier to remember and find these voicings.

Also jazz nomenclature rarely uses 2 and 4. There are some instances such as ‘add2’ chords but again these are rare. There is the sus4 chord too which is more common but 9/11/13 chords are much more common in jazz and so it makes sense to learn, memorise, and refer to these tones as upper extensions.

At the end of the day it is the same note, the 2 is the 9, the 11 is the 4, and the 13 is the 6, but thinking of them in terms of extensions (past the 7th) is simply more effective and practical.
The unusual extension is the 13…

The 13 is slightly different… for example if you have the 9 and 13 in the chord but no 7th it becomes a 6/9 chord. 

So basically if the major 7th or b7th is not in the chord the 13 is referred to as the 6. 

The 6/9 is a common chord and you will see it used in many of my jazz standard lessons. A voicing for C69 would be C-G in the left hand and E-G-A-D in the right hand. Notice there is no ‘B’ in the chord which is the 7th and so the 13 is now referred to as the 6.

Conversely, it’s rare that you would have the #11 in the chord without the major or dominant 7th. There are some examples such as the slash chord Dmaj played over a Cmaj ( IImaj / Imaj ) which can give you a #11 sound but again its usage is rare.

Again it’s rare that you have the 2 in the chord without the 7. If it was it can be called an ‘add2’ for example Cadd2 which would be a C Major triad with the 2 voiced somewhere in the chord.

Complicated I know! It gets much easier with time.