si16,
Sorry. I may not have been clear about that.
A9=A C# E G B
Amaj9=A C# E G# B
Aadd2=A C# E B
The 7 or b7 must be present in the chord for the 2 to "become" a 9. Otherwise it's just a 2 regardless of octave.
If the 2 replaces the 3 it would be called sus2. If the 3 is present it's an add2.
In Classical Theory the add2 chord was called a 9/8 suspension and had to be resolved to the triad. In Modern theory the add2 is a variant that stands on its own.
Examples of Classical 9/8 suspension:
--x--x----0---0
--3--1----3---1
--0--0----0---0
--2--2----x---x
--3--3----3---3
--x--x----x---x
Note that in these examples the "9" does reside in the second octave. But then something like this comes along:
---0---
---1---
---0---
---0---
---3---
---3---
Regards,
Monk
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