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The name implies that you are adding a note, but we already notate that explicitly in the chord name and quality.
We say Cm7, not Cm add 7. If we want a 9th, we can go Cm9, but is Cm add 9 correct too? Is there a difference?
So, what exactly is an add chord? When is the notation used?
The 'add' modifier is used if a note above the 7th is added to a triad, and if the lower tensions are not part of the chord. That's why there's a difference between a C9 and a C(add9) chord. The first has a (flat) 7th, the other one doesn't:
C9 = C E G Bb D
C(add9) = C E G D
Another usage is to add notes that would otherwise replace another note, as is the case with an added 4th. In a sus4 chord, the 4 replaces the third, but in an add4 chord both the third and the fourth are present.
Note that when higher tensions are added on top of the 7th, then "add" is not used, even if some of the lower tensions are not present. E.g., a C13 chord must have a b7 and a 13, but it could be played without the 9, and it would still be perceived as (and called) a C13, and definitely not C(add 13).
The difference between C9 and C add9 is that the latter chord doesn't contain the 7th.
True, most chords are clear in their make-up from the name. However, sometimes, there needs to be an extra note added and it's more clear to write that at the end of a chord's name.
Csus2, for example, needs C D G, as the sus knocks out the 3rd of the chord, E. But what about if we wanted to have a D note as well? C E G and D. That's where the 'add' part comes in handy. We can either call it Cadd2, or more commonly, Cadd9. Cadd9 sounds better as the D is not played next to the C and E anyway.
I suppose C6 could be called Cadd6, except generally speaking, any add notes are above the octave, so there's no need. Any chord that contains a number higher than 7 will need that 7 in it anyway, so Cmaj9 will be C E G B D. The only way to name that chord, but not have the 7th (B) played, would be to call it Cadd9.
“Add” is used in chord symbols in certain cases where the usual assumptions don’t apply.
Strictly speaking, a chord with an extension (9, 11, or 13) contains the triad indicated by the note letter (root, third, and fifth), the seventh, the interval of the extension, and all lower extensions. (See here for the complete picture.) A ninth chord has no lower extensions, so a Cm9 is C E♭ G B♭ D. The seventh (B♭) and the ninth (D) are both important to the quality of this chord. But a composer might not want the B♭ to be played. How is this solved? The symbol becomes Cm(add9).
The same applies to higher extensions. Strictly speaking, a Cm13 chord is C E♭ G B♭ D F A, though it’s rarely played that way in practice. Musicians may choose to eliminate the fifth (G), the ninth (D), and/or the eleventh (F). That leaves the most important qualities of the chord in place; the root, the minor third, the seventh, and the thirteenth. Again, if a composer wanted an A in the chord without a B♭, the chord could be written as C6 or C(add13). By convention, a six chord is a triad plus a major sixth interval, so “add” would be redundant, but “add” is necessary in C(add13) to indicate that no B♭ is played. (The “add13” suggests a voicing where the A is an octave higher than 6, thus C6 is a far more common chord symbol.)
“Add” is also used for minor and major seconds, and perfect and augmented fourths when these aren’t meant to be suspensions. For example, Cadd4 is C E F G whereas C F G is Csus4. Confusingly, some publishers and composers use C4 to mean Cadd4 while other use it for Csus4. Obviously, it’s better to be explicit when you can.
When chord symbols get too complicated, they lose their utility. They’re meant to make things easier, not more complicated. In cases where specific voicings are desired, it’s better to write in standard notation.
First, C9 denotes traditionally a dominant chord: that's C7 with the major 9th. This must resolve somewhere (most standardly to F).
Now Cadd9 is a C major triad with the major 9th. This is a major chord which functions as a tonic.
You can pretty much always substitute Cadd9 for CMaj9, which is CMaj7 with the major 9th. The former is more "pop", the latter more "jazz".
Likewise, Cm9 is Cm7 with the major 9th, and those two chords are roughly interchangeable (and also with Cm11 in a jazz context) and function as II chords in a II/V movement (Cm7/F7).
